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Sound transmission through acoustic metamaterials and prepreg processing science

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Sound transmission through acoustic metamaterials and prepreg processing science

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SOUND TRANSMISSION THROUGH ACOUSTIC METAMATERIALS AND
PREPREG PROCESSING SCIENCE

by

William Thomas Edwards

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
MECHANICAL ENGINEERING

December 2020

Copyright 2020 William Thomas Edwards
ii

Acknowledgements
To my parents, Tom Edwards and Dr. Ruth Yopp-Edwards: thank you for blessing me
with the opportunity to pursue a higher education and supporting me every step of the way. I
will always remember the hundreds of calls from the car on the way back to my apartment to
celebrate, vent, think out loud, or ask for help. You came through every time.
To my old friend, Dr. Mark Anders: thank you for recommending me for a position at
USC. Thank you for the countless hours spent indulging technical discussions. Thank you for
your fraternity in the lab, in street, and at the weather mark.
To my advisor, Prof. Steve Nutt: thank you for the opportunity to work on interesting and
meaningful projects. Your mentorship has helped me to grow as a researcher, communicator,
and citizen. Thank you pushing me to focus, but also allowing me to take time off for adventures
in Germany, France, Portugal, Nicaragua, Benin, the middle of the Pacific Ocean, Puerto Rico,
and dozens of places in the Pacific Northwest and Southwest.
To Dr. Tim Centea: thank you for setting an example. I strive to emulate not only your
technical excellence, but—even more so—your kindness, integrity, generosity, and infectious
joy.
To my Defense Committee Members: thank you for your time, insight, and guidance.
To my brother, Dan Edwards, and the Slowik family: thank you for always supporting
and encouraging me, even when I declined or cancelled family functions to work on my degree.
To our Lab Manager, Yunpeng Zhang: thank you for making sure all the instruments and
facilities I used were always in working order. Your assistance saved me many long hours and
headaches.
iii

To the student research assistants—Jacob Iuele, Ashton Meginnis, Patricio Martinez,
Rebecca Beiter, Brice Tanner, Elliott Hoppe, and Trisha Palit: thank you for your assistance
collecting much of the data presented herein.
To John “Johnson” Schmidt, Ryan “All the Dark Chocolate You’ll Ever Need” Jones,
Jeong “Jeongson” Choi, Mark “Murk/Sanders” Anders, and Victoria “Oh, That’s Just Fetal Brain
Tissue” Wolsley: y’all were pretty good roommates. Except Cat. He wasn’t great.
To my shipmates, Scott “Floyd/Lloyd/Boyde” Barber, Steve Calhoun, Jim Barber, Billy
Wright, and especially Duncan Cameron, Roland Vollmann, and Mark Anders: thank you for
keeping my back strong, my sails trimmed, and my rum chilled. Regular escapes to Santa
Monica Bay (or Hawaii via Long Beach, CA) kept the barnacles off my rudder, and being
involved in your lives has enriched mine.
To Mr. and Mrs. J, Joe, and Marcos of JJ’s Sandwich Shop, and to the women who
worked USC Habit Burger: thank you for fueling my degree. Regularly seeing each of you
brightened my day, and helped me to find community in LA.

iv

Table of Contents
Acknowledgements ......................................................................................................................... ii
Table of Contents ........................................................................................................................... iv
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
CHAPTER 1. Introduction.............................................................................................................. 1
1.1. Motivation ................................................................................................................ 1
1.2. Acoustic Metamaterials ........................................................................................... 1
1.3. Prepreg Processing ................................................................................................... 2
1.4. Scope of Dissertation ............................................................................................... 5
CHAPTER 2. Analytical Model for Low-Frequency Transmission Loss Calculation of Zero-
Prestressed Plates with Arbitrary Mass Loading ............................................................................ 8
2.1. Introduction .............................................................................................................. 8
2.2. Theory .................................................................................................................... 13
2.2.1. Boundary Conditions and Eigenfunctions ...................................................... 15
2.2.2. Eigenvalue Problem ........................................................................................ 16
2.2.3. Transmission Loss .......................................................................................... 20
2.3. Results and Discussion .......................................................................................... 21
2.3.1. Validation ........................................................................................................ 21
2.3.2. Effect of Boundary Conditions on Vibroacoustic Properties ......................... 26
2.4. Conclusions ............................................................................................................ 35
CHAPTER 3. Transmission Loss and Dynamic Response of Hierarchical Membrane-Type
Acoustic Metamaterials ................................................................................................................ 37
3.1. Introduction ............................................................................................................ 37
3.2. Methods.................................................................................................................. 42
3.2.1. Fabrication of Membrane Array ..................................................................... 42
3.2.2. Transmission Loss Testing and Modal Analysis ............................................ 43
3.2.3. Analytical Model ............................................................................................ 45
3.3. Results and Discussion .......................................................................................... 57
3.3.1. Hierarchical Acoustic Metamaterial: Characteristic Performance ................. 58
3.3.2. Parametric Effect of Inertial Inclusions .......................................................... 64
3.3.3. Accuracy and Extensions to the Modeling Approach..................................... 66
3.4. Conclusions ............................................................................................................ 69
CHAPTER 4. Process Robustness and Defect Formation Mechanisms in Unidirectional
Semipreg ....................................................................................................................................... 71
4.1. Introduction ............................................................................................................ 71
4.2. Methods.................................................................................................................. 74
4.2.1. Semipreg Production ....................................................................................... 74
v

4.2.2. Uncured Prepreg Characterization .................................................................. 76
4.2.3. Laminate Fabrication ...................................................................................... 78
4.2.4. Laminate Characterization .............................................................................. 79
4.3. Results and Discussion .......................................................................................... 80
4.3.1. Uncured Prepreg Characterization .................................................................. 80
4.3.2. Process Reliability Study ................................................................................ 83
4.3.3. Void Formation Study..................................................................................... 89
4.4. Conclusions ............................................................................................................ 95
CHAPTER 5. Processability of Overaged Prepreg ....................................................................... 97
5.1. Introduction ............................................................................................................ 97
5.2. Methods................................................................................................................ 102
5.2.1. Prepreg Characterization ............................................................................... 103
5.2.2. Laminate Manufacturing and Characterization ............................................ 103
5.2.3. Cure Cycle Tailoring..................................................................................... 104
5.2.4. Demonstration of Prepreg Life Extension via Cure Cycle Modification ..... 105
5.3. Results and Discussion ........................................................................................ 106
5.3.1. Room Temperature Aging ............................................................................ 106
5.3.2. Freezer Aging................................................................................................ 111
5.3.3. Cure Cycle Tailoring..................................................................................... 115
5.3.4. Demonstration of Prepreg Life Extension via Cure Cycle Modification ..... 117
5.4. Conclusions .......................................................................................................... 121
CHAPTER 6. Conclusions and Suggested Future Work ............................................................ 124
6.1. Acoustic Metamaterials ....................................................................................... 124
6.1.1. Conclusions and Outcomes ........................................................................... 124
6.1.2. Suggested Future work ................................................................................. 125
6.2. Prepreg Processing ............................................................................................... 126
6.2.1. Conclusions and Outcomes ........................................................................... 126
6.2.2. Suggested Future Work................................................................................. 129
References ................................................................................................................................... 132

vi

List of Tables
Table 2.1: Coefficients of entries in 𝐂 and 𝐌 matrices................................................................. 17
Table 2.2: Comparison of analytic and finite element eigenfrequency predictions (centrally
located mass, 𝜉𝑀 ,𝐶𝑀 = 𝜂𝑀 ,𝐶𝑀 = 0.5) ..................................................................................... 23
Table 2.3:Comparison of analytic and finite element eigenfrequency predictions (eccentric mass
placement, 𝜉𝑀 ,𝐶𝑀 = 0.75,𝜂𝑀 ,𝐶𝑀 = 0.625) ............................................................................ 25
Table 2.4: Comparison of eigenfrequencies for identical plate-mass system with all combinations
of clamped and simply supported boundary conditions ............................................................... 27

vii

List of Figures
Figure 2.1: Definition of geometric and mathematical variables for modeling plate behavior. ... 13
Figure 2.2 Finite element model implementation for CCSS plate with eccentric mass highlighting
element types and boundary conditions. ....................................................................................... 22
Figure 2.3: Analytical prediction for the first six unique mode shapes of CCCC plates. Modes 3
and 8 are omitted, because they occur at the same frequency and are simple 90° rotations of
modes 2 and 7, respectively. ......................................................................................................... 24
Figure 2.4: Comparison of mode shapes 1, 4, and 7 (left to right) for simply supported
boundaries as predicted using (a) analytic and (b) finite element techniques. ............................. 25
Figure 2.5: Comparison of normalized shapes of mode 4 for various boundary conditions. ....... 28
Figure 2.6: Comparison of mode 6 for Class A plates. CCCC and SSSS plates are characterized
by an anti-symmetric response that is not observed in CSCS plates. ........................................... 30
Figure 2.7: Analytical prediction for the first six mode shapes of CCSS plates. ......................... 31
Figure 2.8: Transmission loss through plates belonging to Class A as a function of frequency. . 32
Figure 2.9: Transmission loss through plate-mass systems belonging to Class B as a function of
frequency....................................................................................................................................... 33
Figure 2.10: Transmission loss through plates with and without mass loading, demonstrating the
relative importance of boundary conditions ................................................................................. 34
Figure 3.1: Photograph of completed hierarchical membrane-type metamaterial acoustic barrier.
....................................................................................................................................................... 43
Figure 3.2: Schematic representing two tiers of design hierarchy: array level and individual cell
level. .............................................................................................................................................. 43
Figure 3.3: Metamaterial sample mounted in transmission loss chamber with clamped
boundaries. .................................................................................................................................... 44
Figure 3.4: Definition of geometric and mathematical variables for modeling array behavior. .. 46
Figure 3.5: Definition of geometric and mathematical variables for modeling membrane
behavior......................................................................................................................................... 52
Figure 3.6: Characteristic transmission loss performance of hierarchical acoustic metamaterial. 59
Figure 3.7: First modal response of homogenized array as predicted analytically (left) and
measured experimentally (right). .................................................................................................. 60
viii

Figure 3.8: Second quasi-symmetric modal response of array as predicted analytically (left) and
measured experimentally (right). .................................................................................................. 62
Figure 3.9: Analytical prediction of second symmetric mode shape in individual membrane cells.
....................................................................................................................................................... 63
Figure 3.10: Parametric effects of 𝑀𝐴 on transmission properties for 𝑀𝑀 = 0 g. ...................... 65
Figure 3.11: Parametric effects of 𝑀𝐴 on transmission properties for 𝑀𝑀 = 1.6x10 − 4 kg. ... 66
Figure 4.1: Key steps in the mask-and-press process used to produce semipreg. ........................ 75
Figure 4.2: Micrographs of cross sections from cold-cured semipreg and control prepreg at
equivalent scale. ............................................................................................................................ 81
Figure 4.3: Morphology of modified and unmodified semipreg resin strips with diagrams
showing resin strip cross section for each..................................................................................... 82
Figure 4.4: Resin viscosity for semipreg and control prepreg resins. ........................................... 83
Figure 4.5: Bulk defect content for semipreg and control prepreg laminates produced under
baseline and a variety of adverse conditions. ................................................................................ 84
Figure 4.6: Microtomography data from the Baseline semipreg laminate highlighting (A) all
pores, (B) resin rich volumes between plies 2 and 3, and (C) pores between plies 2 and 3. ........ 85
Figure 4.7: Surface defect content for semipreg and control prepreg laminates produced under
baseline and a variety of sub-optimal conditions. ......................................................................... 86
Figure 4.8: Image of the surface of the Ply Drop semipreg laminate indicating each of the two
types of surface defects found in semipreg laminates. ................................................................. 87
Figure 4.9: Images of semipreg at the tool-prepreg interface for select times during layup and
processing. .................................................................................................................................... 89
Figure 4.10: Surface defect content and resin strip cross section schematic for unmodified and
modified semipreg laminates produced under baseline conditions. ............................................. 93
Figure 4.11: Comparison of resin strips on semipreg before and after conditioning at 35 °C and
90% relative humidity for 24 hours. ............................................................................................. 95
Figure 5.1: Sub-ambient glass transition temperature of prepreg aged at room temperature. .... 106
Figure 5.2: Micrographs of laminates produced from room temperature aged prepreg using the
MRCC. ........................................................................................................................................ 108
Figure 5.3: Average short beam strength of laminates aged at room temperature (labels indicate
strength normalized against the F12R0 case). .............................................................................. 109
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Figure 5.4: : Micrographs of laminates produced from freezer aged prepreg using the MRCC. 112
Figure 5.5: Average short beam strength of laminates aged in the freezer (labels indicate strength
normalized against the F12R0 baseline case). .............................................................................. 113
Figure 5.6: Viscosity profile of resin squeezed from prepregs with different aging conditions. 114
Figure 5.7: Relationship between effective flow number and (A) ramp rate for given gelation
temperatures and (B) gelation temperature for given ramp rates. .............................................. 116
Figure 5.8: Relationship between effective flow number and super-ambient hold time for given
deviations in ramp rate and gelation temperature from the MRCC. ........................................... 116
Figure 5.9: Viscosity profiles and corresponding effective flow number of overaged prepreg (T g
= 15.8° C) cured using the MRCC and a modified cure cycle. .................................................. 118
Figure 5.10: Comparison of microstructure in laminates produced from (left) prepreg aged for 42
days at room temperature, cured using the MRCC and (right) prepreg aged for 44 days at room
temperature, but cured using the modified cure cycle show in Figure 5.9. ................................ 119
Figure 5.11: Short beam strength for laminates produced from prepreg with various aging
conditions processed using the MRCC (F12R0, F12R42, F19.5R0) and a modified cure cycle (F18R44
Mod). ........................................................................................................................................... 120

1

Introduction

1.1. Motivation
Mankind has a long history of leveraging composites as engineering materials; one of
the oldest examples is wattle and daub construction, which has been used since at least 4000
B.C. [1]. Composites, defined as a combination of two or more materials (e.g., metals,
ceramics, or polymers), are of interest because they are often characterized by more favorable
properties than observed in each constituent material alone. The last century has witnessed a
revolution in engineering materials. The invention and widespread adoption of synthetic
polymers in particular has enabled myriad new possibilities, including a wide variety of
composite materials.
In this dissertation, two classes of composite materials are studied. First, composite
metamaterials designed for use as acoustic barriers are discussed. Second, structural composites
comprised of carbon fiber reinforced polymers (CFRPs) produced from prepreg are examined.
Each of these topics is briefly introduced below.
1.2. Acoustic Metamaterials
Metamaterials are defined as materials that exhibit properties not typically observed in
traditional materials [2]. Metamaterials are produced from multiple materials arranged
inteintionally, often in a periodic structure. Metamaterials derive their properties from the
arrangement and periodicity of structures rather than from the intrinsic properties of their
constituent materials. Unique properties observed in metamaterials have generated interest in
exploring their application in guiding and controlling the transmission of electromagnetic and
2

acoustic waves in ways not possible for conventional materials [3,4]. Typically, periodic unit
structures in metamaterials are smaller than the wavelength (electromagnetic or acoustic) of
interest [5].
Acoustic metamaterials are of interest for use as acoustic barriers in weigh-critical
applications. The acoustic mass law provides a first order approximation of the transmission
efficiency of sound waves through a monolithic material with zero stiffness. The acoustic mass
law indicates that low-frequency sound is particularly difficult to isolate, and suggests that the
most simplest method for reducing sound transmission is to increase the mass of the acoustic
barrier. This is, of course, at odds with the need for weight efficient structures in applications
where parasitic weight must be kept to a minimum. Furthermore, most engineering materials
block the transmission of less sound than estimated by the acoustic mass law.
As the importance of sound attenuation through weight-critical structures has grown and
mass law based strategies have proven impractical, engineers have pursued alternative
approaches for sound attenuation. Membrane-type acoustic metamaterials have demonstrated
sound attenuation significantly higher than mass law predictions for narrow, tunable
bandwidths. Similar phenomena can be achieved with plate-like structures. Both types of
structures are explored in this work.
1.3. Prepreg Processing
Structural composite materials have gained increasing traction as the primary materials
used in many applications where performance is paramount. The aerospace industry, for
example, has adopted CFRPs as the predominant structural material in many commercial and
military aircraft because CRFPs are characterized by higher specific strength and stiffness than
the materials they replaced (typically metals). By replacing metallic structures with composites,
3

performance improvements have been unlocked, including increased flight range and fuel
efficiency.
While there are many methods for processing fiber reinforcement and thermoset resin
into composite structures, most composite aerospace structures are produced from prepreg
processed at elevated temperature and (often) elevated pressure. Prepreg is comprised of a fiber
bed (unidirectional or woven) that has been combined with a partially cured (B-staged) resin,
typically applied as continuous films to one or both sides of the fiber bed. Prepreg is widely
used because it offers several advantages over other processing routes. Specifically, prepreg is
easy to handle, enables precise control of fiber volume fraction, does not require expensive two-
sided tooling or resin injection infrastructure, and accommodates use of higher viscosity resins.
Because the resin in prepreg is pre-catalyzed, however, prepreg storage time must be carefully
tracked and expired prepreg has been traditionally discarded.
Prepreg is processed into composite structures by first cutting and stacking individual
prepreg plies into a layup. Engineers provide layup technicians with guidance for the fiber
orientation, ply shape, and ply location for each layup. Ply cutting is often automated, and ply
stacking can be done by hand using automated systems to improve precision in critical parts. A
Layup is done on the tooling that prepreg will subsequently be cured on, and the tooling largely
determines the shape of the resulting composite structure. Once laid up on the tooling, a variety
of consumable materials are placed over and adjacent to the uncured layup. Edge-breathing
dams are used to enable gas transport out of ply edges and prevent resin from flowing out of the
part during cure. A perforated release film is placed on top of the laminate to prevent it from
bonding with subsequent consumables (but not prevent through-thickness gas transport). The
edge breathing dams and release film are then covered with a nonwoven breather cloth that
4

provides a continuous network of gas evacuation pathways. Finally, the layup and consumables
are sealed under a vacuum bag.
To cure the laid up prepreg into a composite structure, a vacuum is applied to the
vacuum bag and the assembly is placed into an autoclave or oven. A multi-step temperature
cycle tailored to the system is typically recommended by each prepreg manufacturer, but
typically four primary phenomenon occur during processing. First, gas is removed at room
temperature during a “debulk” at which time vacuum applied. Next, temperature is increased
such that resin viscosity drops, and resin flow saturates dry regions of fiber bed. Temperature is
typically then held constant until the resin crosslinking reaction has advanced enough that resin
has gelled and will no longer flow. Finally, peak temperature stability and mechanical
properties are assured when the resin crosslinking reaction is completed during post cure
(typically at a higher temperature than at which gelation occurred).
The first prepregs were produced using a solution dipping process that yielded prepregs
that were fully saturated with uncured resin. In this process, dry fibers are run through a bath
containing a combination of uncured resin and a solvent. The solvent lowers the viscosity of
the resin enough that fiber bed saturation is assured. After saturation with resin, prepreg is
heated to evaporate excess solvent. Solvent, unfortunately, cannot be entirely removed without
significantly advancing the resin degree of cure, and solution dipped prepreg often suffers from
voids caused by residual solvent vaporizing during cure. To eliminate the introduction of
solvent during prepregging, hot-melt prepregging was developed. In this process, resin is first
filmed to a controlled thickness. Resin films are then laminated onto one or both sides of a fiber
bed, and a series of temperature-controlled compaction rollers encourage partial fiber bed
saturation.
5

In risk averse applications, prepreg is cured in an autoclave. The compaction pressure
applied during laminate processing suppresses void nucleation and growth, making autoclave
processing the most reliable method for curing prepreg into composite structures. There are
economic and environmental incentives, however, to develop cure procedures that provided the
same level of quality assurance without requiring an autoclave. The relationship between
prepreg format and the reliability of vacuum-bag-only processing is explored in this work.
Economic and environmental incentives also motivate a study of the mechanisms by which
prepreg ages and ultimately spoils.
1.4. Scope of Dissertation
CHAPTER 2: This chapter presents an analytical model for prediction of transmission
loss through rectangular plates arbitrarily loaded with rigid masses, accommodating any
combination of clamped and simply supported boundary conditions. Equations of motion are
solved using a modal expansion approach, incorporating admissible eigenfunctions given by the
natural mode shapes of single-span beams. The effective surface mass density is calculated and
used to predict the transmission loss of low-frequency sound through the plate-mass structure.
To validate the model, finite element results are compared against analytical predictions of
modal behavior and shown to achieve agreement. The model is then used to explore the
influence of various combinations of boundary conditions on the transmission loss properties of
the structure, revealing that the symmetry of plate mounting conditions strongly affects
transmission loss behavior and is a critical design parameter.
CHAPTER 3: A deployment-scale array of locally resonant membrane-type acoustic
metamaterials (MAMs) is fabricated. The acoustic performance of the array is measured in a
transmission loss chamber, and a complex interaction between the individual cell and the array
6

length scales is shown to exist. Transmission behavior of both the membrane and the array are
independently studied using analytical models, and a method for estimating transmission loss
through the structure that combines vibroacoustic predictions from both length scales is
presented and shown to agree with measurements. Degradation of transmission loss
performance often associated with scaling individual MAM cells into arrays is explained using
analytical tools and verified using laser vibrometry. A novel design for hierarchical locally
resonant acoustic metamaterials is introduced, and experimental and analytical data confirm this
approach offers an effective strategy for minimizing or eliminating the efficiency losses
associated with scaling MAM structures.
CHAPTER 4: This work explores how modifications to prepreg format can increase
process robustness. A unidirectional (UD) prepreg is produced with a customized,
discontinuous resin distribution, henceforth referred to as semipreg. The semipreg is shown to
exhibit through-thickness permeability orders of magnitude greater than conventional hot-melt
VBO prepregs. The semipreg also is found to be less sensitive to variations in process
conditions than conventional VBO prepreg. In situ process monitoring allows observation and
identification of two defect formation mechanisms arising during cure of the custom prepreg.
Resin feature topography is shown to play a critical role in these mechanisms, indicating its
importance to the design of next generation VBO semipregs.
CHAPTER 5: This study investigates the processing behavior and mechanical
performance of prepreg as it accumulates storage time and out time. During aging at room
temperature, prepreg was observed to advance in degree of cure, and sub-ambient glass
transition temperature was confirmed to be a good indicator of prepreg resin exposure time.
Microstructural analysis and mechanical testing of laminates produced from room temperature
7

aged prepreg showed the onset age-induced porosity is correlated with reduced short beam
strength. The primary mechanism by which porosity remains within laminates produced from
prepreg exceeding room temperature storage limits was observed to be insufficient resin
infiltration into dry regions of prepreg fiber bed. Prepreg stored in the freezer, however, did not
undergo significant crosslinking; resin glass transition temperature and rheological behavior
were found to be unchanged after 20 months of freezer storage (167% of the manufacturer’s
recommended storage life). Resin achieved uniform distribution in all freezer aged laminates,
indicating that insufficient resin flow was not the reason for age-induced complications.
CHAPTER 6: This section summarizes the primary contributions yielded from each of
the four studies described in the preceding chapters. Suggestions are made for possible future
work.

8

Analytical Model for Low-Frequency Transmission Loss
Calculation of Zero-Prestressed Plates with Arbitrary Mass Loading
1

2.1. Introduction
The attenuation of low-frequency sound through weight-critical structures has
historically been a challenging task. Noise reduction approaches using traditional engineering
materials typically rely on increasing the mass of acoustic barriers (the acoustic mass law), but
this is an especially inefficient mechanism for attenuation of low-frequency sound waves [6].
Recently developed membrane-type acoustic metamaterials (MAMs), however, have been
shown, both theoretically and experimentally, to attenuate substantially more energy than mass
law predictions for the low-frequency regime (100-1000 Hz) [7]. These structures are quasi-
planar and comprised of a membrane under tension with fixed boundaries, loaded with one or
more masses. As with other locally resonant sonic materials, MAMs exhibit negative dynamic
mass at acoustic excitation wavelength regimes larger than the characteristic length of the
membrane structure itself, and this behavior manifests unique transmission minima and maxima
within the low frequency domain [7,8]. The frequency response of MAMs is known to be
governed by the geometry, density, and tension of the membrane, and by the weight, location,
and geometry of attached masses [9–11]. The effects of stacking multiple layers of MAMs and
creating two dimensional arrays with multiple membrane-mass cells reportedly alter the
bandwidth, amplitude, and number of the transmission loss minima and maxima [12].
As efforts to scale up MAMs are undertaken, the need for efficient predictive tools
increases. Finite element models of MAM vibroacoustic behavior achieve acceptable agreement

1
This study was published in the Journal of Vibration and Acoustics in August 2019.
9

with experimental measurements. However, implementation of such models is complex and
generally does not complement automated analysis and optimization. This is especially true
when geometric parameters are to be varied, as this requires successively recreating the mesh on
which finite element are defined. Analytical techniques for predicting transmission loss through
acoustic metamaterials, in contrast, offer a more manageable implementation that meaningfully
improves optimization efficiency.
Analytical modeling of sound transmission through membranes was initially explored by
Ingard [13], who described transmission through a pretensioned, uniform, circular membrane
and recognized the potential for using such structures as weight-efficient acoustic insulation.
Around the same time, the influence of rigidly attached masses on the eigenfrequencies of
vibrating membranes was also being explored [14,15]. Interest in analytical modeling of sound
transmission through mass loaded membranes, however, was not pursued until after extensive
experimental and finite element data had demonstrated that such structures hold promise for
weight-efficient, low-frequency sound insulation.
Analytical modeling efforts have culminated in a model proposed by Chen et. al. [16]
that uses a point matching approach, considers the coupling of the membrane with the
surrounding acoustic fluid, and offers accuracy at arbitrarily high frequency. The model
accommodates both circular and rectangular membranes with an arbitrary number of masses.
This method, however, (1) results in a nonlinear eigenvalue problem, and (2) requires repeated
numeric integration of Green’s function across the membrane surface to solve the acoustic wave
equation. Executing each of these operations is computationally cumbersome and slow.
To avoid these inefficient operations, Langfeldt et. al. [17] presented a model in which
membrane displacement is expanded in the eigenmodes of the unloaded membrane, creating a
10

linear eigenvalue problem that is efficiently solved. Normal incidence transmission loss is
approximated at low frequency by determining the effective surface mass density, which does
not require numerical integration. In the low-frequency regime (wavelengths smaller than the
characteristic dimension of the membrane), this model remains the most accurate and efficient
method for optimizing transmission loss through MAMs.
As engineers continue to explore optimization of MAM structures, stiffer materials have
been considered for use as the “membrane” component of MAMs. Several analytic models have
been developed to describe sound transmission through single- and double-panel barriers
[18,19]. These models accurately describe transmission through plates where all boundaries are
clamped or all boundaries are simply supported; however, they do not accommodate plates with
a combination of clamped and simply supported boundaries. Further, as presented, these
models do not incorporate the influence of one or more masses bonded to either panel. Such
capability is desirable, because the presence, size, and location of bonded masses give designers
the ability to tune the frequencies of transmission loss maxima and minima.
To accommodate the addition of a mass bonded to a stiff acoustic barrier, Chen et al.
[20] presented an analytical model describing transmission through a rectangular, pretensioned
Kirchhoff-Love plate. The model achieves reasonable agreement with finite element
predictions, but suffers the same numerical inefficiencies present in their membrane model [16].
Namely, it requires solving a nonlinear eigenvalue problem and repeated numerical integration.
Furthermore, the model (1) applies only to rectangular membranes clamped along all four
edges, (2) requires doubly symmetric mass placement, and (3) captures only doubly symmetric
plate motion.
11

The work presented herein aims to overcome the numerical inefficiencies and
limitations of the Chen plate model by adapting the numerical framework presented by
Langfeldt et. al. [17] to describe transmission through mass loaded plates. Specifically, although
the influence of prestress is not considered, the model presented herein avoids requiring the
solution of a nonlinear eigenvalue problem. Instead, the model takes advantage of a long
history of optimized standard eigenvalue solvers to find solutions to a linear eigenvalue
problem. By avoiding the need to solve a nonlinear eigenvalue problem, the model offers
substantially faster computational times than the plate model presented by Chen et. al. Further,
the model presented herein avoids the repeated numerical integration of Green’s function
required in the model presented by Chen et. al., streamlining an otherwise time-consuming
operation.
In addition to improvements in numerical efficiency, the model presented herein
accurately captures the behavior of a wider variety of mass loaded plates. While previous work
has accommodated only clamped boundary conditions, the model presented in this work
accommodates any combination of clamped and simply supported edges. This is especially
important to designers who may not be able to achieve perfect clamping on all plate edges in
realistic deployment situations. Indeed, we show that when a clamped edge is opposite a simply
supported edge, plate metamaterial behavior can exhibit additional transmission loss minima as
compared to plates with uniform fastening. An additional advantage over Chen’s plate model is
that the symmetry requirements imposed on mass placement location are relaxed. Instead of
requiring that masses be placed such that there is symmetry across both the plate midlines, the
model presented here accommodates an arbitrary number of masses placed at any location on
the plate with no symmetry requirements. Further, while the Chen model only predicted modal
12

behavior that was symmetric across both plate midplanes, the model presented herein can
accurately predict asymmetric modal behavior, which is critical for accurate prediction of
transmission loss for plates that do not have identical boundary conditions on all edges and for
plates with eccentric mass placement.
The value of this work lies in three primary achievements. First, the model presented
herein is the only analytical model to describe low-frequency sound transmission loss through
plate-like acoustic metamaterials where mass placement is not limited by symmetry
requirements. Second, the model accommodates various boundary conditions and combinations
of boundary conditions that have hitherto been absent from the literature, significantly widening
the scope of acoustic transmission problems that can be addressed using analytic techniques.
Finally, new insights into the influence of boundary conditions on the transmission loss through
plate-like acoustic metamaterials are presented. Although the relationship between boundary
conditions and frequency response of vibrating plates is well known, this work presents the first
explicit demonstration of how boundary conditions influence transmission loss performance of
mass loaded plates, and reveals the critical importance of mounting conditions in determining
plate-like metamaterial performance.
In the section below, we present the analytical theory describing vibration of the coupled
plate-mass system and the corresponding equations of motion, the eigenfunctions used for each
set of boundary conditions, the method for solving the resulting linear homogeneous eigenvalue
problem, and the method for transmission loss calculation. In the Results and Discussion
section, we validate the model by comparison to results obtained using established analytical
and finite element techniques, and use the model to investigate the influence of boundary
conditions on the transmission loss profile of plate-mass structures. We demonstrate that
13

asymmetric mounting gives rise to additional transmission loss maxima and minima when
compared to symmetric boundary conditions.
2.2. Theory

Figure 2.1: Definition of geometric and mathematical variables for modeling plate behavior.

Consider a homogeneous isotropic plate of dimensions 𝐿 𝑥 and 𝐿 𝑦 , surface mass
density 𝑚 ′ , and bending stiffness 𝑇 . Bonded to the plate is a rigid inertial inclusion of mass 𝑀
of arbitrary shape, the center of mass of which is located according to [𝑥 𝑀 ,𝑦 𝑀 ] in the { 𝐱⃗ ,𝐲 ,𝐳 }
coordinate frame. As shown in Figure 2.1, the origin is located at one corner of the plate, the
positive x- and y-axes along the plate edges. The out-of-plane displacement of the plate is a
function of position and time given by 𝑤 (𝑥 ,𝑦 ,𝑡 ), and the vibration of the plate is described
according to Kirchhoff-Love plate theory written as

𝑚 ′
∂
2
∂𝑡 2
𝑤 (𝑥 ,𝑦 ,𝑡 )+ 𝑇 ∇
2
∇
2
𝑤 (𝑥 ,𝑦 ,𝑡 )= 𝑃 (𝑥 ,𝑦 ,𝑡 )+ 𝑓 ′(𝑥 ,𝑦 ,𝑡 )
(2.1)
14

where ∇
2
= ∂
2
∂𝑥 2
⁄ + ∂
2
∂𝑦 2
⁄ is the Laplace operator in Cartesian coordinates,
𝑃 (𝑥 ,𝑦 ,𝑡 ) is the acoustic pressure acting on the plate, and 𝑓 ′(𝑥 ,𝑦 ,𝑡 ) is the coupling force
resulting from inertial inclusions mounted to the plate. Assuming harmonic time dependence
and normally incident acoustic excitation, we can write
𝑤 (𝑥 ,𝑦 ,𝑡 )= 𝑤̂ (𝑥 ,𝑦 )e
i𝜔𝑡
(2.2)
𝑃 (𝑥 ,𝑦 ,𝑡 )= 𝑃 ̂
(𝑥 ,𝑦 )e
i𝜔𝑡
(2.3)

𝑓 ′(𝑥 ,𝑦 ,𝑡 )= 𝑓 ̂
′(𝑥 ,𝑦 )e
i𝜔𝑡
(2.4)
For the sake of brevity, the time dependence of these terms will be omitted from
mathematical expressions below. We can also introduce the dimensionless parameters given by
𝜉 = 𝑥 /𝐿 𝑥 𝜂 = 𝑦 /𝐿 𝑦 𝜁 = 𝑧 /𝐿 𝑥 𝑢 = 𝑤̂ /𝐿 𝑥 Λ = 𝐿 𝑥 /𝐿 𝑦 𝛽 = 𝑃 ̂
𝐿 𝑥 /𝑇 𝑘 2
= 𝑚 ′𝜔 2
𝐿 𝑥 2
/𝑇 𝛾 = 𝑓 ̂
′ /(𝑇 𝐿 𝑥 )
(2.5)
to simplify Eq. (2.1) into the form

−𝑘 2
𝑢 +
∂
4
𝑢 ∂𝜉 4
+ 2𝛬 2
∂
4
𝑢 ∂𝜉 2
∂𝜂 2
+ 𝛬 4
∂
4
𝑢 ∂𝜂 4
= 𝛽 + 𝛾 (2.6)
We next approximate the influence of the inertial inclusions. To do so, we employ a
point matching approach where the coupling force that is applied over a continuous domain is
approximated as a set of 𝐼 forces that act at discrete points within and on the boundary of the
domain of the inclusion. With this in mind, the coupling force can be expressed

𝛾 = ∑𝛾 𝑖 𝛿 (𝜉 − 𝜉 𝑖 )𝛿 (𝜂 − 𝜂 𝑖 )
𝐼 𝑖 =1
(2.7)
where 𝛾 𝑖 is the dimensionless coupling force contributed by the 𝑖 𝑡 ℎ
collocation point,
and 𝛿 is the Dirac delta function. We choose to solve the resulting equation of motion using a
15

modal expansion approach, approximating the response of the system as a finite linear
combination of 𝑁 eigenfunctions 𝛷 𝑖 (𝜉 ,𝜂 ),𝑖 = { 1,2,…,𝑁 } , which are determined by the
boundary conditions and Cartesian geometry of the system.
2.2.1. Boundary Conditions and Eigenfunctions
The eigenfunction 𝛷 𝑛 (𝜉 ,𝜂 ) can be separated into the product of two dimensionally
independent functions: 𝛷 𝑛 (𝜉 ,𝜂 )= 𝜙 𝑛 𝑥 (𝜉 )𝜙 𝑛 𝑦 (𝜂 ) . The index 𝑛 then runs from 1 to 𝑁 = 𝑁 𝑥 𝑁 𝑦 ,
where 𝑁 𝑥 and 𝑁 𝑦 are the number of eigenfunction expansion terms used in each of the 𝑥 and 𝑦
dimensions. The indices 𝑛 𝑥 and 𝑛 𝑦 are determined such that 𝑛 = 𝑁 𝑦 (𝑛 𝑥 − 1)+ 𝑛 𝑦 . The
functions 𝜙 𝑛 𝑥 and 𝜙 𝑛 𝑦 are determined by the boundary conditions in each corresponding
dimension (e.g., 𝜙 𝑛 𝑥 is determined by the boundary conditions at 𝑥 = 0 and 𝑥 = 𝐿 𝑥 ). Eq. (2.8)
gives these expressions, where SS and CC indicate opposing boundaries are both simply
supported and clamped, respectively, and CS indicates one boundary is clamped and the other
simply supported.
𝜙 𝑘 (𝜉 )= {
sin (𝑎 𝑘 𝜉 ), SS
cosh(𝑎 𝑘 𝜉 )− cos(𝑎 𝑘 𝜉 )− 𝑏 𝑘 (sinh(𝑎 𝑘 𝜉 )− sin(𝑎 𝑘 𝜉 )), CC or CS
(2.8)
The coefficients 𝑎 𝑘 are the nontrival solutions to the characteristic equation associated
with each set of boundary conditions given by Eq. (2.9) [21].
sin(𝑎 𝑘 )= 0 for SS
(2.9) cos(𝑎 𝑘 )cosh(𝑎 𝑘 )= 1 for CC
tan(𝑎 𝑘 )= tanh (𝑎 𝑘 ) for CS
The corresponding coefficients 𝑏 𝑘 are found by solving Eq. (2.10) to ensure CC and CS
eigenfunctions satisfy the appropriate boundary conditions:
16

𝜙 𝑘 (𝜉 )|
𝜉 =0,1
=
𝑑 𝜙 𝑘 (𝜉 )
𝑑𝜉 |
𝜉 =0,1
= 0
for CC
(2.10)

𝜙 𝑘 (𝜉 )|
𝜉 =0,1
=
𝑑 𝜙 𝑘 (𝜉 )
𝑑𝜉 |
𝜉 =0
= 0
for CS
Using the Rayleigh-Ritz method, we can approximate the unitless modal displacement
as a linear combination of appropriately chosen eigenfunctions according to Eq. (2.11).

𝑢 ≈ ∑ 𝑞 𝑛 𝛷 𝑛 𝑁 𝑛 =1
= ∑ ∑ 𝑞 𝑛 𝜙 𝑛 𝑥 (𝜉 )𝜙 𝑛 𝑦 (𝜂 )
𝑁 𝑦 𝑛 𝑦 =1
𝑁 𝑥 𝑛 𝑥 =1
(2.11)
Note that the set of eigenfunctions in the 𝑥 -direction, 𝜙 (𝜉 ), does not have to be identical
to the set of eigenfunctions in the 𝑦 -direction, 𝜙 (𝜂 ) : that is, any combination of clamped and
simply supported boundaries can be studied.
2.2.2. Eigenvalue Problem
After substituting Eq. (2.7) and Eq. (2.11) into Eq. (2.6), we rearrange the summations
into matrix operations of the form
(𝐂 − 𝑘 2
𝐌 )𝐪 = 𝛽 𝐛 + 𝐋𝛄 (2.12)
where the dimensionless coupling-force vector 𝛄 contains entries [𝛾 1
,𝛾 2
,…,𝛾 𝑁 ]
T
, and the
stiffness matrix 𝐂 = (𝑐 𝑚𝑛
) 𝜖 ℝ
𝑁 ×𝑁 and the mass matrix 𝐌 = (𝑚 𝑚𝑛
) 𝜖 ℝ
𝑁 ×𝑁 have entries given
by

𝑐 𝑚𝑛
∗
= {
𝑓 𝑥 ,𝑚 4
+ 2(𝑔 𝑥 ,𝑚 )(𝛬 2
𝑔 𝑦 ,𝑚 )+ 𝛬 4
𝑓 𝑦 ,𝑚 4
for 𝑚 = 𝑛 0 else
(2.13)

𝑚 𝑚𝑛
= {
ℎ
𝑥 ,𝑚 ℎ
𝑦 ,𝑚 for 𝑚 = 𝑛 0 else
(2.14)
17

and the coupling matrix 𝐿 = (𝑙 𝑚𝑛
) 𝜖 ℝ
𝑁 ×𝐼 and forcing vector 𝐛 = (𝑏 𝑛 ) 𝜖 ℝ
𝑁 have entries given
by
𝑙 𝑚𝑛
= 𝜙 𝑛 𝑥 (𝜉 𝑛 )𝜙 𝑛 𝑦 (𝜂 𝑛 )
(2.15)
𝑏 𝑛 = 𝑑 𝑛 𝑥 𝑑 𝑛 𝑦
(2.16)
The values of the coefficients 𝑓 𝑛 , 𝑔 𝑛 , ℎ
𝑛 , and 𝑑 𝑛 are listed in Table 2.1 as functions of
the boundary conditions imposed on the plate.
Table 2.1: Coefficients of entries in 𝐂 and 𝐌 matrices
Coefficients for Various Boundary Conditions
𝑓 𝑛 ℎ
𝑛 𝑔 𝑛 𝑑 𝑛
Clamped-Clamped 𝑎 𝑛 1 𝑏 𝑛 𝑎 𝑛 (2 − 𝑏 𝑛 𝑎 𝑛 ) 2𝑏 𝑛 𝑎 𝑛 −1
[1 − (−1)
−𝑛 ]
Clamped-Pinned 𝑎 𝑛 1 𝑏 𝑛 𝑎 𝑛 (1 − 𝑏 𝑛 𝑎 𝑛 )
𝑎 𝑛 −1
[(−1)
𝑛 +1
√𝑏 𝑛 2
+ 1 − √𝑏 𝑛 2
− 1 + 2𝑏 𝑛 ]
Pinned-Pinned
𝑎 𝑛 √2

1
2

𝑎 𝑛 2
4

1
𝑎 𝑛 [1 − (−1)
𝑛 ]

To further constrain the system of equations (unknown term 𝜸 remains in Eq. (2.12)), we
write a separate set of equations describing the motion of the attached rigid mass in the
coordinate frame { 𝒙⃗ ⃗ ′,𝒚⃗ ⃗
′
,𝒛⃗ }, centered at the center of mass of the inertial inclusion (Figure 2.1).
As shown in Eq. (2.17), the displacement anywhere within the domain of the inclusion can be
described using the position of the center of mass, 𝑢 𝐴 ,𝐶𝑀
, and two terms, 𝛼 𝜉 ′ and 𝛼 𝜂 ′, describing
rotation about the 𝒙⃗ ⃗
′
- and 𝒚⃗ ⃗
′
-axes, respectively.

𝑢 𝑀 (𝜉 ′
,𝜂 ′)= 𝑢 𝑀 ,𝐶𝑀
− 𝛼 𝜂 𝜉 ′
+
1
𝛬 𝛼 𝜉 𝜂 ′
(2.17)
Eq. (2.17) is rearranged by expressing 𝑢 𝑀 ,𝐶𝑀
as a function of dimensionless frequency
parameter 𝑘 , dimensionless mass parameter 𝜇 = 𝑀 /(𝑚 ′𝐿 𝑥 2
) , and the point-force coupling terms
𝛾 𝑖 according to
18

𝑢 𝑀 ,𝐶𝑀
=
1
𝜇 𝑘 2
∑𝛾 𝑖 𝐼 𝑖 =1
(2.18)
where 𝑀 is the total mass of the inertial inclusion mounted to the plate. We can similarly
express the rotational terms 𝛼 𝜉 and 𝛼 𝜂 as shown below, where dimensionless rotational inertia
parameters are normalized according to 𝜗 𝜉 = 𝐽 𝑥 ′/(𝑀 𝐿 𝑥 2
) and 𝜗 𝜂 = 𝐽 𝑦 /(𝑀 𝐿 𝑥 2
) , and 𝐽 𝑥 ′ and 𝐽 𝑦 ′
are moments of inertia about the 𝒙⃗ ⃗ ′ - and 𝒚⃗ ⃗ ′ -axes.

𝛼 𝜉 =
1
𝜇 𝑘 2
𝛬 𝜗 𝜂 ∑𝜉 𝑖 ′
𝛾 𝑖 𝐼 𝑖 =1
(2.19)

𝛼 𝜂 =
1
𝜇 𝑘 2
𝜗 𝜂 ∑𝜂 𝑖 ′
𝛾 𝑖 𝐼 ∗
𝑖 =1
(2.20)
Using these relations, we can rewrite Eq. (2.17) as

𝑢 𝐴 (𝜉 ′
,𝜂 ′)=
1
𝜇 𝑘 2
∑(1 +
𝜉 ′
𝜉 𝑖 ′
𝜗 𝜂 +
𝜂 ′
𝜂 𝑖 ′
𝛬 2
𝜗 𝜉 )
𝐼 𝑖 =1
𝛾 𝑖 (2.21)
Because the inertial inclusion is perfectly bonded to the plate, we have mathematically
identical motion in the plate and inclusion at each of the collocation points, implying
𝑢 𝐴 (𝜉 𝑚 ′
,𝜂 𝑚 ′
)= 𝑢 (𝜉 𝑚 ,𝜂 𝑚 ) for 1 ≤ 𝑚 ≤ 𝐼 (2.22)
which can be written in matrix form according to

−𝐋 T
𝐪 +
1
𝑘 2
𝐆𝛄 = 0 (2.23)
where the matrix 𝐆 = (𝑔 𝑚𝑛
) 𝜖 ℝ
𝐼 ×𝐼 has entries given by

𝑔 𝑚𝑛
=
1
𝜇 (1 +
𝜉 ′
𝜉 𝑖 ′
𝜗 𝜂 +
𝜂 ′
𝜂 𝑖 ′
𝛬 2
𝜗 𝜉 ) (2.24)
19

We can now combine equations Eq. (2.12) and Eq. (2.23) into a single block matrix
system of the form

[
𝐂 − 𝑘 2
𝐌 −𝐋 −𝐋 T 𝐆 𝑘 2
⁄
][
𝒒 𝜸 ] = 𝛽 [
𝐛 0
] (2.25)
the homogeneous form for which can be expressed as the generalized eigenvalue problem given
by
𝐀𝐱 = 𝑘 2
𝐁𝐱 (2.26)
where

𝐀 = [
𝐂 −𝐋 0 𝐆 ], 𝑩 = [
𝐌 0
𝐋 T
0
], and 𝐱 = [
𝐱 𝑞 𝐱 𝛾 ] (2.27)
This equation can now be solved using standard solvers to identify the first 𝐾
eigenvalues (indicating modal frequencies) and eigenvectors (indicating coupling forces at
collocation points and modal coefficients for eigenfunction weighting coefficients). If we
assume that the deformation of the membrane under steady-state vibration can be approximated
by a linear combination of the first 𝐾 eigenmodes, we can write [𝐪 T
𝛄 T
]
T
≈ 𝐗𝐜 (𝑘 ) , arranging
eigenvectors into a matrix 𝐗 𝜖 ℝ
(𝑁 +𝐼 )×𝐾 , where the 𝑖 𝑡 ℎ
column of the matrix is the 𝑖 𝑡 ℎ

eigenvector of Eq. (2.26), and 𝐜 = [𝑐 1
,𝑐 2
,…,𝑐 𝐾 ]
T
is a vector containing the modal contribution
factors. We can solve for these modal contribution factors with inhomogeneous Eq. (2.25),
using the identities in Eq. (2.27) and pre-multiplying by the matrix 𝐗 T

𝐗 T
𝐀𝐗𝐜 − 𝑘 2
𝐗 T
𝐁𝐗𝐜 = 𝛽 𝐗 T
[
𝐛 0
] (2.28)
20

Using the identity 𝐀𝐗 = 𝐁𝐗𝚲 , where 𝚲 is a diagonal matrix with entries corresponding
to the first 𝐾 dimensionless eigenfrequencies extracted from Eq. (2.26), we can calculate the
coefficients 𝐜 as a function of dimensionless frequency 𝑘 according to

𝐜 (𝑘 )= 𝛽 (𝚲 − 𝑘 2
𝐈 )
−1
(𝐗 T
𝐁𝐗 )
−1
𝐗 T
[
𝐛 0
] (2.29)
2.2.3. Transmission Loss
We calculate transmission through the plate under the assumption that sound radiation
behavior is governed primarily by the surface-averaged vibration amplitude. This assumption is
appropriate for frequencies where the acoustic wavelength 𝜆 is greater than the characteristic
length of the plate √𝐿 𝑥 2
+ 𝐿 𝑦 2
[22]. Because we are interested primarily in low-frequency
performance, this assumption is not particularly restrictive and will be discussed more
thoroughly in the next section. From [17], the effective mass density of the plate can be
calculated as a function of excitation frequency according to

𝑚 ̃′= −
𝑚 ′
𝑘 2
[
𝐛 T
0
]𝐗𝐜
(2.30)
The effective mass density 𝑚 ̃
′
is used in accordance with the acoustic mass law to
finally calculate the transmission coefficient 𝑡 according to
1
𝑡 = 1 +
i𝜔 𝑚 ̃
′
2𝜌 0
𝑐 0
(2.31)
where 𝜌 0
and 𝑐 0
are the density and speed of sound of the acoustic fluid through which sound is
being transmitted. Using the transmission coefficient, transmission loss in dB can be calculated
per 𝑇𝐿
0
= −20 log
10
|𝑡 |.

21

2.3. Results and Discussion
To facilitate the discussion of various boundary conditions, the following convention
will be adopted for the remainder of this document. Each edge is assigned C or S to indicate
whether it is clamped or simply supported. Each plate is named according to the four letters
corresponding to the conditions at each boundary. The first letter in the series corresponds to
the edge at 𝑥 = 0, and each subsequent letter describes the adjacent edge in a counterclockwise
direction. For example, a plate with boundary conditions CCSC would be clamped at the three
edges defined by 𝑥 = 0, 𝑦 = 𝐿 𝑦 , and 𝑦 = 0 and simply supported along the edge 𝑥 = 𝐿 𝑥 .
2.3.1. Validation
To validate the efficacy of the analytic model developed above, we compare it against
two established and generally accepted approaches. In the first approach, the eigenfrequencies
of plates without inertial inclusions (𝑀 = 𝐼 = 0) were compared to analytical results collected
by Belvins [23]. All combinations of clamped and simply supported boundary conditions were
considered, as were both square and rectangular plates (Λ = 0.5). For 𝑁 = 3600, analytical
model predictions of dimensionless eigenfrequencies achieved agreement with published
values, having less than 0.7% difference in all cases.
To confirm that the influence of inertial inclusions is appropriately captured, a finite
element model was implemented. The plate was modeled as a thin shell comprised of parabolic
triangular elements. The inertial inclusion was modeled as a solid body with parabolic
tetrahedral elements. An element size of 0.001 m with a 5e-5 m tolerance was specified,
resulting in approximately 113,000 nodes and 58,000 elements. High mesh quality was verified
by examining element aspect ratios and Jacobian. The average element aspect ratio was 1.73
with a maximum aspect ratio of 3.08 and 98.3% of elements characterized by aspect ratios of
22

less than 2. The average Jacobian was 1.0000 with a minimum of 1.0000 and a maximum of
1.0002. Boundary conditions were implemented assigning immovable or fixed restraints to
simply supported and clamped plate edges, respectively. The immovable condition
accommodated rotation of the edge nodes, but restricted all translation, while the fixed
condition did not allow rotation or translation of edge nodes. Figure 2.2 shows the
implementation of this finite element model for the case of CCSS boundaries with an eccentric
mass located at 𝑥 𝑀 = 0.12 m, 𝑦 𝑀 = 0.10 m. In this figure, the simply supported boundaries are
symbolized with arrows along each primary axis, indicating that edge nodes translation
constrained. Clamped edges are symbolized with flanged arrows pointing along each primary
axis, indicating that edge nodes are translation and rotation constrained.

Figure 2.2 Finite element model implementation for CCSS plate with eccentric mass highlighting element types and boundary
conditions.

Material properties were assigned to correspond to aluminum (𝐸 = 69 GPa, 𝜌 = 2700
kg/m
3
, 𝜈 = 0.33), and geometric parameters were set to 𝐿 𝑥 = 𝐿 𝑦 = 0.16 m and ℎ = 0.001 m.
23

Only one inertial inclusion was considered: a cubic body with side lengths of 0.01 m and total
mass 0.011 kg, located at the center of the plate (𝑥 𝑀 = 𝑦 𝑀 = 0.08 m). The inertial inclusion
was assigned material properties corresponding to pure lead (Pb) and modeled as a linear elastic
body perfectly bonded to the plate. A frequency analysis was conducted and solved using the
Intel Direct Sparse solver, yielding the first twenty eigenfrequencies and corresponding modal
responses. Finite element data were compared to analytical predictions for a plate-mass system
with corresponding material and geometric properties. Numerical parameters were given by
𝑁 = 3600 (𝑁 𝑥 = 𝑁 𝑦 = 60) , 𝐼 = 16, and 𝐾 = 100.
Table 2.2: Comparison of analytic and finite element eigenfrequency predictions (centrally located mass, 𝜉 𝑀 ,𝐶𝑀
= 𝜂 𝑀 ,𝐶𝑀
= 0.5)

Clamped Boundaries
(CCCC)
Simply Supported
Boundaries
(SSSS)
Mode Index Analytic FEA Error Analytic FEA Error
1 251.1 Hz 247.4 Hz 1.47% 149.8 Hz 149.0 Hz 0.53%
2 681.4 Hz 676.8 Hz 0.68% 465.1 Hz 464.4 Hz 0.15%
3 681.4 Hz 676.8 Hz 0.68% 465.1 Hz 464.4 Hz 0.15%
4 1016 Hz 974.7 Hz 4.06% 726.1 Hz 713.9 Hz 1.68%
5 1087 Hz 1054 Hz 3.04% 773.1 Hz 767.6 Hz 0.71%
6 1301 Hz 1279 Hz 1.69% 961.1 Hz 958.0 Hz 0.32%
7 1526 Hz 1481 Hz 2.95% 1199 Hz 1190 Hz 0.75%
8 1526 Hz 1481 Hz 2.95% 1199 Hz 1190 Hz 0.75%

Table 2.2 compares the analytic and finite element predictions for the first eight
eigenfrequencies of plates with CCCC and SSSS mounting conditions. Analytical predictions
achieve accurate agreement with finite element results. The average difference between the
analytic and finite element results for the first eight modes is 1.41%, with a maximum error of
4.06%. Analytic predictions for these mode shapes corresponding to these eigenfrequencies are
pictured in Figure 2.3 for CCCC plates. Mode 3 and mode 8 are omitted from this figure as
these modes are simple 90° rotations of modes 2 and 7, respectively. In all figures depicting
24

modal response, solid contours indicate positive plate deflection in the 𝒛⃗ direction, and dashed
contours indicate negative deflection. The color of contour lines correlates with deflection
amplitude, where brighter lines indicate higher magnitude than darker lines. Clamped
boundaries are indicated by a hatched region along the corresponding edge, otherwise edges are
simply supported. The mode shapes captured in Figure 2.3 are distinct yet analogous to the
corresponding shapes exhibited by plates with SSSS boundary conditions.

Figure 2.3: Analytical prediction for the first six unique mode shapes of CCCC plates. Modes 3 and 8 are omitted, because they
occur at the same frequency and are simple 90° rotations of modes 2 and 7, respectively.

The modal response shapes predicted by each method similarly achieve agreement:
Figure 2.4 shows a comparison of normalized displacement maps of modes 1, 4, and 7
generated by finite element and analytic techniques. This figure highlights consistency between
the two methods for doubly symmetric plate motion—modes 1 and 4—and antisymmetric-
symmetric plate motion—mode 7—about midplanes normal to 𝒙⃗ ⃗ and 𝒚⃗ ⃗ . Because the mass was
25

placed centrally for model verification, modal responses were exclusively symmetric and
antisymmetric (for the boundary conditions considered in Table 2.2). However, the ability to
predict both types of response confirms the absence of symmetry limitations present in previous
work.

Figure 2.4: Comparison of mode shapes 1, 4, and 7 (left to right) for simply supported boundaries as predicted using (a) analytic
and (b) finite element techniques.

Table 2.3:Comparison of analytic and finite element eigenfrequency predictions (eccentric mass placement, 𝜉 𝑀 ,𝐶𝑀
=
0.75,𝜂 𝑀 ,𝐶𝑀
= 0.625)
CCSS CSCS
Mode Index Analytic FEA Error Analytic FEA Error
1 214.2 Hz 211.0 Hz 1.49% 251.9 Hz 247.8 Hz 1.63%
2 499.8 Hz 480.0 Hz 3.96% 485.7 Hz 470.9 Hz 3.05%
3 583.6 Hz 575.7 Hz 1.35% 585.5 Hz 574.7 Hz 1.84%
4 865.9 Hz 841.7 Hz 2.79% 848.9 Hz 838.0 Hz 1.28%
5 1016 Hz 996.2 Hz 1.95% 971.3 Hz 950.7 Hz 2.12%
6 1076 Hz 1058.0 Hz 1.67% 1115 Hz 1093 Hz 1.97%
7 1347 Hz 1310 Hz 2.75% 1295 Hz 1271 Hz 1.85%
8 1405 Hz 1361 Hz 3.13% 1421 Hz 1406 Hz 1.06%
26

For confidence in the ability to accurately accommodate eccentric mass placement and
combinations of simply supported and clamped boundary conditions, additional data were
generated for plates with eccentric mass placement (𝑥 𝑀 = 0.12 m, 𝑦 𝑀 = 0.10 m) on plates with
CCSS and CSCS boundaries. Table 2.3 compares analytic and finite element predictions for the
first eight eigenfrequencies of such plates. Analytical predictions again achieve reasonable
agreement with finite element results. The average difference between the analytic and finite
element results for the first eight modes is 2.12%, with a maximum error of 3.96%.
The results verify that the analytical model presented herein accurately captures modal
behavior of the mas loaded plate. Furthermore, the model accurately captures symmetric (about
both midplanes) and asymmetric plate motion, significantly relaxing symmetry requirements
present in previous work. This aspect is particularly critical for investigating plates with
dissimilar opposing boundaries (CCCS, CCSS, CSSS), for plates with multiple masses that are
not characterized by symmetric loading about both midplanes (parallel to 𝒛⃗ 𝒙⃗ ⃗ and 𝒚⃗ ⃗ 𝒛⃗ ), and for
plates with a single eccentric mass.
2.3.2. Effect of Boundary Conditions on Vibroacoustic Properties
Modal Response
To investigate the effect of boundary conditions on the transmission loss profile of plate
structures, all combinations of clamped and simply supported edges were considered. The same
geometric and material parameters used to describe the plate and inertial inclusion are given in
the validation section above. Table 2.4 compares the eigenfrequencies of analogous modes for
all combinations of boundary conditions considered. Plates with fewer clamped edges exhibit
lower natural frequencies (with frequencies for CCSS generally being lower than for CSCS
plates). For corresponding eigenmodes, eigenfrequencies were 20-40% lower in SSSS plates
27

than CCCC plates. Generally, lower-order modes were more strongly influenced by mounting
conditions.
Table 2.4: Comparison of eigenfrequencies for identical plate-mass system with all combinations of clamped and simply
supported boundary conditions
Mode
Index
CCCC CCCS CSCS CCSS CSSS SSSS
1 251.1 228.0 210.8 201.2 176.8 149.8
2 681.4 572.8 516.4 539.8 485.6 465.1
3 681.4 658.1 645.5 566.7 529.5 465.1
4 1016 936.7 818.6 891.7 793.1 726.2
5 1087 995.0 935.8 917.1 840.3 773.2
6 1301 1212 1186 1108 1047 961.1

As the natural frequencies of this system are directly related to its stiffness, the boundary
conditions studied can be ranked from stiffest to least stiff according to: CCCC, CCCS, CSCS,
CCSS, CSSS, SSSS. One notable exception to this trend is mode 4 in CCSS plates. The
eigenfrequency of this mode is greater than the corresponding eigenfrequency in CSCS
mounted plates. This anomalous increase in frequency is accompanied by a corresponding
change in mode shape: Figure 2.5 shows that, although occurring at different frequencies, nearly
all plates exhibit analogous response shapes except for the CCSS plate. The unique response of
the CCSS plate is attributed to the doubly mismatched nature of its opposing boundaries, giving
rise to a mode shape that differs substantially from its analog in other plates.
28

Figure 2.5: Comparison of normalized shapes of mode 4 for various boundary conditions.

Indeed, when discussing the influence of boundary conditions on mode shape, it is
helpful to divide the boundary conditions studied into two classes. The first class we identify as
Class A, defined by plates in which each pair of opposing edges is identically supported—this
class includes CCCC, CSCS, and SSSS conditions. The second class we identify as Class B, in
which plates have at least one pair of opposing boundaries where a clamped edge is opposite as
simply supported edge—this class includes CCCS, CCSS, and CSSS conditions. For improved
clarity, in the remainder of this paper, each boundary condition will include a subscript
indicating to which class it belongs (e.g., CCCCA belongs to Class A and CSSSB belongs to
Class B).
Plates belonging to Class A produced modal responses that were all either symmetric or
anti-symmetric about each midplane parallel to 𝒚⃗ ⃗ 𝒛⃗ and 𝒙⃗ ⃗ 𝒛⃗ . A representative example of Class A
mode shapes can be seen in Figure 2.3 which shows the first eight modes of the CCCCA. The
29

mode shapes of CSCSA and SSSSA plates are unique but analogous to those pictured in Figure
2.3. Modes 2, 3, 7, and 8 can be seen to exhibit anti-symmetric responses across only one
midplane of the plate, while mode 5 was characterized by anti-symmetric response across each.
The volume displaced during vibration on either side of these planes of anti-symmetric response
is equal and opposite, implying that these modes will not contribute to the transmission of
acoustic energy through such plates.
For all plates belonging to Class A, only modes 1, 4, and 6 were characterized by
symmetric motion across both the 𝒚⃗ ⃗ 𝒛⃗ -midplane and the 𝒙⃗ ⃗ 𝒛⃗ -midplane. Of these modes, only
modes 1 and 4 displace a non-zero volume of acoustic fluid during vibration for all three
boundary conditions. These modes for CCCCA plates can be seen in Figure 2.3 and mode 4 for
all plates is shown in Figure 2.5. Figure 2.6 shows that mode 6 only displaces a non-zero
volume for CSCSA plates. In this figure, two midplanes of anti-symmetry in CCCCA and SSSSA
plates can be identified for mode 6: each plane defined by the 𝒛⃗ a line from one corner of the
plate to the opposite corner. The existence of these midplanes of anti-symmetric motion
indicates that mode 6 will not contribute to the transmission of acoustic energy the plates with
CCCCA and SSSSA boundary conditions. For CSCSA plates, however, the motion about these
planes is asymmetric—but not antisymmetric—and the volume displaced during by mode 6
during vibration is non-zero. Only activation of modes with non-zero volume displacement will
propagate acoustic energy through the plate.
30

Figure 2.6: Comparison of mode 6 for Class A plates. CCCC and SSSS plates are characterized by an anti-symmetric response
that is not observed in CSCS plates.
Plates belonging to Class B demonstrated modal responses that exhibit symmetric/anti-
symmetric motion about only one plane. For CCCSB and CSSSB cases, eigenmodes exhibit this
behavior about the midplane parallel to the only pair of matching boundaries (i.e., 𝒙⃗ ⃗ 𝒛⃗ and 𝒚⃗ ⃗ 𝒛⃗ ,
respectively). Strictly symmetric or anti-symmetric modal responses, however, are not observed
about the midplane parallel to the pair of mismatched boundaries. In all cases, deflection was
larger on the side of the midplane corresponding to the simply supported boundary than on the
clamped side. In plates with CCSSB boundaries, modal responses were asymmetric about both
the 𝒙⃗ ⃗ 𝒛⃗ - and 𝒚⃗ ⃗ 𝒛⃗ -midplanes, but symmetry (or anti-symmetry) was observed about the diagonal
midplane containing 𝒛⃗ , the origin, and the point (𝐿 𝑥 ,𝐿 𝑦 ,0) . This can be clearly seen in Figure
2.7 where the first six modes for CCSSB plates are pictured. Asymmetric deformation about
these midplanes is explained by higher plate compliance on the simply supported side than on
the clamped side, resulting in larger deformation on the simply supported side. An important
implication of this asymmetry is that fewer modes oscillate with zero volume velocity.
Specifically, for CCCSB boundaries, the only modes to exhibit zero net volume displacement
are modes 3, 5, and 7; for CSSSB boundaries, only modes 2, 5, and 8 exhibit zero net volume
displacement. This result indicates that plates belonging to Class B have a more complex
31

transmission loss profile than plates belonging to Class A because there are more modes that
propagate acoustic energy.

Figure 2.7: Analytical prediction for the first six mode shapes of CCSS plates.

Transmission Loss
Figure 2.8 compares the transmission loss profile of a single plate-mass system with
each type of boundary condition in Class A. The frequency range considered spans 10 Hz to
2000 Hz, although the assumptions presented in the previous sections are only satisfied for
frequencies less than 1515 Hz. Fortunately, frequencies of primary interest for targeted acoustic
treatments (between modes 1 and 4) are captured accurately for the plate sizes studied. In all
cases, there is a transmission loss minimum at the frequency of the first mode. First, as the
number of clamped boundary conditions decreases, the maxima and minima of the transmission
profile are shifted to lower frequency ranges, a result that is consistent with the trend evident in
32

Table 2.4. Further, the transmission loss through the structure at frequencies below the first
mode where acoustic transmission is stiffness dominated is indeed strongly influenced by the
boundary conditions of the plate: more compliant mounting conditions result in significantly
larger displacements and hence higher acoustic transmission, leading to differences of up to 12
dB.

Figure 2.8: Transmission loss through plates belonging to Class A as a function of frequency.

Note that the first two minima in Figure 2.8 for all cases correspond to the frequencies
of mode 1 and 4 from Table 2.4, and no features indicate any influence of modes 2, 3, 5, 7, or 8
on any curve. In the cases of CCCCA and SSSSA plates, mode 6 also appears to have no
influence on the transmission properties; however, for CSCSA plates, this is not true. Figure 2.8
shows that for CSCSA plates, an additional transmission loss minimum and maximum exist in
the vicinity of mode 6 (at 1186 Hz). These additional features arise from the mismatched
boundary conditions, which promote a modal response that has nonzero volumetric
33

displacement across the 𝑥𝑦 -plane and increases transmission efficiency. In the cases with
CCCCA and SSSSA boundaries, the symmetry of boundary conditions ensures the
corresponding mode has zero net displacement.
A similar effect can be seen in the class of asymmetric boundary conditions. Figure 2.9
shows the transmission loss performance of the remaining plate-mass systems studied, from
which it is apparent that the vibroacoustic behavior of systems with asymmetric mounting
conditions is considerably more complex than systems with symmetric boundaries. In the case
of symmetric boundaries, many eigenmodes have zero net displacement across the 𝑥𝑦 -plane.
However, the corresponding modes in plates with asymmetric boundaries result in small, but
non-zero, displacements, yielding efficient transmission at these frequencies.

Figure 2.9: Transmission loss through plate-mass systems belonging to Class B as a function of frequency.

34

When Figure 2.8 is compared against Figure 2.9, the importance of ensuring that the
prescribed boundary conditions are enforced is evident. Even one insufficiently clamped
boundary results in a sharp transmission loss dip with 100 Hz bandwidth at a frequency near the
peak transmission loss of the fully clamped case. This presents a problem for MAM designers,
as the improper installation of these structures can markedly reduce efficiency in the frequency
range they may be designed to attenuate. Precisely how important the boundary conditions are
relative to other design parameters is captured in Figure 2.10, which shows that the effect of
different mounting conditions can more strongly influence the transmission loss curve than the
effects of an added mass. In this figure, the first and second transmission loss minima for an
SSSSA plate with bonded 0.011 𝑘𝑔 mass (150 Hz) relocates to a higher frequency when the
edges are clamped (250 Hz) than when the mass is removed (187 Hz).

Figure 2.10: Transmission loss through plates with and without mass loading, demonstrating the relative importance of
boundary conditions

35

2.4. Conclusions
An analytical model was developed and validated to describe acoustic transmission loss
through plate structures with attached rigid masses of arbitrary number, shape, weight, and
location under various boundary conditions. A point matching approach was used to
approximate the coupling force between a Kirchhoff-Love plate and attached rigid mass as a
finite set of discrete point forces. Admissible functions given by natural mode shapes of single-
span beams with appropriate boundary conditions were used to solve the resulting equations of
motion using a modal expansion approach. The representative linear eigenvalue problem was
presented and its components were defined for rectangular plates with any combination of
clamped and pinned boundaries. The effective surface mass density of the plate was calculated
as a function of excitation frequency for normally incident acoustic waves and used to
determine the transmission loss through the metamaterial. Edge mounting conditions of the
plate-like metamaterial were shown to strongly influence the acoustic performance of the
structure, with asymmetric mounting conditions giving rise to additional transmission loss
maxima and minima when compared to symmetric boundaries. Boundary conditions also
strongly influenced the off-modal transmission properties.
The model presented offers improved utility over previous work by allowing for
prediction of antisymmetric plate motion, while inheriting the efficiency and accuracy of the
numerical scheme presented by Langfeldt et. al [17]. The model can be implemented using
standard linear algebra methods and solved efficiently using existing techniques and packages,
but its accuracy is limited to acoustic frequencies below the frequency corresponding a
wavelength equal to the characteristic length of the plate. The model can accommodate any
number of masses placed arbitrarily on the plate, however, masses are assumed to be ridged.
36

This assumption is not limiting for masses that are sufficiently small or stiff, however, the
model is not well suited to capturing the behavior of large, thin, compliant mass loading.
The practical significance of this work is threefold. First, we provide a theoretical
demonstration that the same phenomena giving rise to favorable transmission loss properties
exhibited by MAMs can also be inspired in stiffer materials systems. This result opens the
design space of locally resonant, two-dimensional acoustic metamaterials. Second, we
demonstrate the critical importance of boundary conditions and explore their influence on plate-
like acoustic metamaterial performance. Finally, the efficient analytical tools presented here
give engineers and acousticians a toolbox for streamlining and optimizing the design and
frequency response of plate-type acoustic metamaterials.

37

Transmission Loss and Dynamic Response of Hierarchical
Membrane-Type Acoustic Metamaterials
2

3.1. Introduction
Designing acoustic barriers that reject low-frequency noise transmission has been a
particularly challenging task for materials scientists and acousticians. Typical approaches for
reducing sound transmission through an acoustic barrier generally rely on increasing the
thickness or density of the barrier in question. Transmission efficiency through these structures
can be predicted using the acoustic mass law, but experimental data indicate that many
structures underperform this benchmark [24]. Further, the acoustic mass law indicates that for
such strategies to be effective in the low-frequency regime (20-2000 Hz), a significant addition
of mass is required. For weight-critical applications, this additional mass may be untenable, and
alternative solutions are demanded.
To address the need for a slim, weight-efficient acoustic barrier effective at low
frequencies, membrane-type acoustic metamaterials (MAMs) were conceived [7]. Experimental
and finite element modeling demonstrated that membrane-type acoustic metamaterials
significantly outperform the acoustic mass law in certain frequency ranges [10]. Comprised of
one or more masses bonded to a pre-tensioned membrane, MAMs exhibit a characteristic
transmission profile that can be tuned with respect to frequency by adjusting the size of the
masses or changing the tension in the membrane [9]. The influences of membrane geometry and
mass location have been measured using a transmission loss tube and predicted using finite
element models [11]. While MAMs can be entirely passive structures, work has been done to

2
This study was published in the Journal of Vibration and Acoustics in April 2020.
38

explore the active frequency response tuning and energy harvesting using a variety of
approaches [25–27].
Structures similar to MAMs have been in investigated for use in duct silencing where a
tensioned membrane (with no mass bonded to it) replaced a portion of the duct wall to reflect
gracing incident noise [28]. In such structures, acoustic propagation is directed primarily
parallel to the surface of the membrane, whereas MAMs are typically tuned for normally-
incident acoustic propagation. Demonstration of duct-membrane systems both with and without
sealed backing cavities provide acousticians with a better understanding how membrane
geometry and tension influence acoustic performance of such structures [29,30].
While finite element tools have been shown to accurately predict the acoustic
performance of MAMs, their implementation is not well suited for some design and
optimization challenges. Parametric analysis of one or more geometric variables (e.g., size,
shape, location, or number of masses) would be particularly inefficient using finite element
methods, as it would require successive remeshing of the bodies involved. Alternatively,
analytical methods can be easily implemented, efficiently used, and seamlessly integrated with
automated design optimization tools. This has driven the development of several analytical tools
for prediction of sound transmission through MAMs with a variety of geometries. Chen et al.
developed coupled vibroacoustic analytical models for both circular and Cartesian membranes
under tension with fixed boundaries, using a point matching approach to capture the influence
of the attached mass [31]. Langfeldt et al. simplified the numerical implementation of this
model by using dimensionless parameters to compose a linear eigenvalue problem and
decoupling the membrane from the surrounding air to avoid costly numerical integration [17].
39

Both models assume that the stiffness of the membrane contributes no restorative force during
MAM excitation and deformation.
Other analytical models were developed to capture the influence of bending stiffness on
MAM performance. An acoustically-coupled analytical model describing membrane motion
according to pre-tensioned plate-like dynamic equations was developed, but was limited to
MAMs with clamped boundaries and required any masses to be placed symmetrically about
both the midplanes of the surface of the membrane [20]. Efforts to develop a similar model that
considers bending effects, relaxes symmetry and boundary condition limitations, and integrates
the numerical efficiency of the model presented by Langfeldt culminated in analytical tools that
accommodate any combination of simply supported and clamped boundaries and allow for an
arbitrary number of masses with no symmetry requirements [32].
Efforts to scale MAMs beyond a single cell have investigated arranging several
membranes in series and in parallel. Arranging MAMs in series—such that acoustic energy is
transmitted sequentially through each membrane structure—has been shown to increase
transmission loss, and independent tuning of each MAM allows designers more control over the
spectral response of the system [33]. Initial investigation into scaling MAMs in the in-plane
direction has been conducted by the same authors by creating an array of several membranes,
creating parallel transmission paths for acoustic energy. Small samples have been fabricated and
tested in a transmission loss tube with results indicating that even incremental scaling (from one
membrane to four) results in the decay of favorable transmission properties [12]. Modeling
efforts describing multi-celled arrays of MAMs in a single panel have resulted in an analytical
model that produces more realistic predictions for the transmission loss through arrays of
40

membranes that can each be independently tuned [34]. This model, however, assumes that each
edge of each membrane cell is fixed (i.e., the membrane support grid is rigid).
Efforts to deploy increasingly larger arrays of membrane cells have been thwarted by
similar decay of the theoretical transmission loss performance. It has been primarily thought
that this performance decay is linked to motion of the array frame (the substrate on which the
membranes are bonded), where this motion violates the assumption that individual membrane
resonators have fixed boundaries. To reduce the magnitude of frame motion—and thereby
mitigate this issue, stiffer materials have been investigated for use as a substrate by the authors.
Preliminary results have suggested this is not a comprehensive solution.
Some efforts have been made to capture the influence of array compliance in multi-
celled MAM arrays using numerical and analytical techniques. Langfeldt et al. use numerical
techniques to begin exploring the importance of considering grid compliance when estimating
transmission loss for very low frequencies [35]. Subsequently, the relationship between MAM
boundary compliance and transmission loss were further explored when an analytical model
was developed that accommodates elastic MAM cell edges [36].
An alternative strategy for addressing performance knockdowns associated with
membrane array scaling is presented here. We suggest that a hierarchical approach can be taken:
the array of MAMs can be considered analogous to an individual membrane cell where the
bending stiffness of the array plays the same role as the tension in the membrane cell and a mass
of appropriate size is bonded to the array. In hierarchical structures, material is organized at
different length scales, and structural elements are comprised of one or multiple sub-levels of
structural organization. Such hierarchically organized materials occur naturally in tooth enamel
and spider silk for example, and it is their hierarchical organization that results in emergent
41

mechanical properties [37,38]. Prior to this work, the principles of hierarchical design have been
successfully applied to a variety of acoustic metamaterials to broaden the frequency ranges in
which desirable behavior occurs. For example, Zhang and To presented a hierarchical phononic
crystal that achieved bandgaps an order of magnitude larger than those exhibited in
conventional structure [39]. A variety of other hierarchical metamaterials have been studied
[40–42], but to the authors’ knowledge, this work represents the first time such principles have
been applied to membrane-type acoustic metamaterials.
To demonstrate the effectiveness of a hierarchical design approach in controlling the
dynamic response of an array of MAMs, we detail the first fabrication and experimental
characterization of a deployment-scale, hierarchical MAM. The primary contributions of this
work are (1) to explain the decay of transmission loss performance associated with scaling from
one MAM to an array of MAMs, (2) to demonstrate a design solution in the form of a novel
hierarchical acoustic metamaterial structure, and (3) to explore and explain the behavior of such
a hierarchical membrane structure. These contributions are enabled by the application and
verification of analytical modeling tools that enable uncovering of the physical mechanism by
which the transmission loss profile of an array of membranes differs from that of an individual,
identically tuned membrane. The modeling tools developed for this purpose also provide
predictive tools to the acoustician for use when designing hierarchical MAMs.
The section below contains (a) details regarding the fabrication of a hierarchical MAM
array, (b) the methods used to measure transmission loss and modal response, and (c) details
about the implementation of analytical models describing each length scale of the structure. In
Results and Discussion, experimental and analytical data explaining the transmission loss
performance characteristics of hierarchical MAMs are presented. Further, the influences of
42

changing the size of the mass bonded to the array and to each membrane are demonstrated and
discussed. Finally, a summary of findings is shared, and the implications thereof are explored.

3.2. Methods
3.2.1. Fabrication of Membrane Array
An array of thirty-six MAM cells was fabricated by first milling a six-by-six grid of
4x10
-2
m by 4x10
-2
m square holes through a 0.251 m by 0.251 m by 7.5x10
-3
m thick plate of
aluminum. The holes were located such that each of the four exterior edges of the plate were
3x10
-3
m wide, and the material remaining between each hole was 1x10
-3
m across. After
machining, the surface of the plate was covered with an epoxy adhesive over which a 7.62x10
-5

m thick film of polyethylene terephthalate (PET) was draped. The assembly was placed into an
oven and warmed to 120°C during which time the PET was bonded to the aluminum plate. After
the epoxy adhesive cured, the assembly was removed from the oven and allowed to cool, during
which time dissimilar thermal expansion properties introduced a residual tensile stress in each
of the membrane cells.
In the center of each prestressed membrane cell, an annular inertial inclusion of interior
diameter 9.53x10
-3
m

and exterior diameter 1.59x10
-2
m was bonded. Investigations using
inclusions of thickness 5.1x10
-4
m (mass 1.6x10
-4
kg) as well as thickness 1.1x10
-3
m (mass
3.20x10
-4
kg) were conducted; in each case, a spray adhesive was used to bond the inclusions to
the membrane cells. Figure 3.1 shows a photograph of the completed MAM array. The system’s
two levels of hierarchy are depicted schematically in Figure 3.2, where the entire array structure
and large centrally bonded mass (not pictured in Figure 3.1) constitute one level of the
hierarchy, and an individual membrane cell with corresponding inclusion constitute another.
43

Figure 3.1: Photograph of completed hierarchical membrane-type metamaterial acoustic barrier.

Figure 3.2: Schematic representing two tiers of design hierarchy: array level and individual cell level.

3.2.2. Transmission Loss Testing and Modal Analysis
A small-scale, two-chamber transmission loss test facility, constructed in accordance
with ASTM 2249-02 [43], was used to measure random-incidence transmission loss through the
membrane metamaterial array with various and mass loading conditions (on both the individual
membrane cells and the array structure). The facility was comprised of a reverberant chamber
(15 𝑚 3
) and an anechoic chamber (12 m
3
), separated by a square (0.241 m x 0.241 m) orifice
used to hold the test panel. To ensure clamped boundary conditions on the membrane array, the
44

test panels were mounted covering the orifice using an aluminum frame and tightly bolted at
sixteen equally spaced locations around the perimeter of the sample as shown in Figure 3.3.
Fasteners were tightened with a torque-controlled hand drill to achieve maximally consistent
mounting conditions.

Figure 3.3: Metamaterial sample mounted in transmission loss chamber with clamped boundaries.
An acoustic signal was generated in the reverberant chamber, where nine nonparallel,
reflective walls produced a diffuse sound field measured using a rotating boom sound pressure
level microphone to determine a spatially averaged sound pressure level intensity. In the
anechoic chamber, the transmitted acoustic energy was measured using an intensity probe on a
motorized gantry located 0.17 m away from the test panel. Measurements were collected at
equally spaced locations to determine an average transmission loss through whole membrane
array. The frequency range of the noise source was 100 Hz-6.4 kHz and source-side pressure
levels were maintained at between 90 and 95 dB for all tests. Previous work determined the
45

cutoff frequency of the diffuse field—due to the geometry of the reverberant chamber—to be
315 Hz [44].
For several samples, the out-of-plane motion of the array during vibration was
determined using a laser vibrometer (Ometron VH300+ Laser Doppler Vibrometer Type 8329)
to measure the motion of 85 discrete points within the domain of the aluminum array.
Vibrometry measurements were collected at each midpoint along each edge and each corner of
each membrane cell, excluding cell edges and corners on the boundary of the array. For each
measurement, sound pressure levels in the reverberant room were also recorded. The frequency
range of the noise source was 100 Hz-6.4 kHz. For each of the 85 points measured, auto- and
cross-correlations were used calculate the H2 transfer function relating the motion of the array
(vibrometer signal) to the acoustic excitation (sound pressure signal) as a function of frequency.
At frequencies where coherence between these signals was high for many of the measured
points, the data were used to determine natural frequencies and recover an estimate of their
corresponding modal responses.
3.2.3. Analytical Model
To further confirm and study the transmission properties of hierarchical MAMs,
analytical models were implemented, capturing the behavior of individual membrane cells, the
entire array, and the two length scales in concert (referred to going forward as the compound
system). Below, we describe in detail the mathematics used to model each length-scale of the
hierarchical MAM array structure. In the first subsection, we present relevant details of the
fourth order model used to approximate the vibration of the array, described as a Kirchhoff-
Love plate with homogenized stiffness and mass properties and clamped boundaries. In the next
subsection, we highlight the parallel mathematical framework used to describe the motion of
46

individual membrane cells, described according to second order dynamics, and define
analogous terms and parameters. Finally, we present an approach for predicting the transmission
loss through the compound structure using both models in conjunction.
Modeling Array Behavior: Plate-like Dynamics

Figure 3.4: Definition of geometric and mathematical variables for modeling array behavior.

Using the model presented in [32], we approximate the array structure as a monolithic
isotropic plate of dimensions 𝐿 𝑥 ∗
and 𝐿 𝑦 ∗
, homogenized surface mass density 𝑚 ′
𝐴 , and
homogenized effective bending stiffness 𝑇 ∗
. A rigid inertial inclusion of mass 𝑀 𝐴 is bonded to
the array, its center of mass is located at coordinates [𝑥 𝑐𝑚
∗
,𝑦 𝑐𝑚
∗
], and its rotational moments of
inertia about the 𝒚⃗ ⃗
∗
′ and 𝒙⃗ ⃗
∗
′ axes are given 𝐽 𝑥 ∗ and 𝐽 𝑦 ∗ respectively, where the coordinate frame
{ 𝒙⃗ ⃗
∗
′ , 𝒚⃗ ⃗
∗
′,𝒛⃗
∗
} is defined such that its origin is located at the center of mass of the inertial
inclusion. As shown in Figure 3.4, the origin of the coordinate frame { 𝒙⃗ ⃗
∗
, 𝒚⃗ ⃗
∗
,𝒛⃗
∗
} is located at
one corner of the plate with the positive 𝒙⃗ ⃗
∗
-axis and 𝒚⃗ ⃗
∗
-axis oriented along edges of the plate
and the 𝒛⃗
∗
-axis orthogonal to the plate surface with the positive direction defined in accordance
with the right-hand rule. The out-of-plane displacement of the plate is given by 𝑤 ∗
(𝑥 ∗
,𝑦 ∗
,𝑡 )
and its motion is governed by Kirchhoff-Love plate dynamics
47

𝑚 𝐴 ′
𝜕 2
𝜕 𝑡 2
𝑤 ∗
(𝑥 ∗
,𝑦 ∗
,𝑡 )+ 𝑇 ∗
∇
2
∇
2
𝑤 ∗
(𝑥 ∗
,𝑦 ∗
,𝑡 )
= 𝑃 (𝑥 ∗
,𝑦 ∗
,𝑡 )+ 𝑓 𝐴 ′
(𝑥 ∗
,𝑦 ∗
,𝑡 )
(3.1)

where ∇
2
= 𝜕 2
𝜕 𝑥 2
⁄ + 𝜕 2
𝜕 𝑦 2
⁄ is the Laplace operator in Cartesian coordinates, 𝑃 (𝑥 ∗
,𝑦 ∗
,𝑡 ) is
the acoustic pressure acting on the array, and 𝑓 𝐴 ′
is the coupling force resulting from inertial
inclusions mounted to the array. Assuming harmonic time dependence and normally incident
incoming acoustic excitation, we can write
𝑤 ∗
(𝑥 ∗
,𝑦 ∗
,𝑡 )= 𝑤̂
∗
(𝑥 ∗
,𝑦 ∗
)𝑒 𝑖𝜔𝑡 (3.2)
𝑃 ∗
(𝑥 ∗
,𝑦 ∗
,𝑡 )= 𝑃 ̂
∗
(𝑥 ∗
,𝑦 ∗
)𝑒 𝑖𝜔𝑡 (3.3)

𝑓 𝐴 ′
(𝑥 ∗
,𝑦 ∗
,𝑡 )= 𝑓 𝐴 ′
̂
(𝑥 ∗
,𝑦 ∗
)𝑒 𝑖𝜔𝑡 (3.4)
For the sake of brevity and clarity, the time dependence of these terms is omitted from
mathematical expressions below.
𝜉 ∗
= 𝑥 ∗
/𝐿 𝑥 ∗
𝜂 ∗
= 𝑦 ∗
/𝐿 𝑦 ∗
𝜁 ∗
= 𝑧 ∗
/𝐿 𝑥 ∗
𝑢 ∗
= 𝑤̂
∗
/𝐿 𝑥 ∗
Λ
∗
= 𝐿 𝑥 ∗
/𝐿 𝑦 ∗
𝛽 ∗
= 𝑃 ̂
∗
𝐿 𝑥 ∗
/𝑇 ∗
𝑘 ∗
2
= 𝑚 𝐴 ′
𝜔 2
𝐿 𝑥 ∗
4
/𝑇 ∗
𝛾 ∗
= 𝑓 𝐴 ∗
′
̂
𝐿 𝑥 2
/𝑇 ∗
(3.5)
The dimensionless parameters given by Eq. (3.5) simplify Eq. (3.1) to

−𝑘 ∗
2
𝑢 ∗
+
𝜕 4
𝑢 ∗
𝜕 𝜉 ∗
4
+ 2Λ
∗
2
𝜕 4
𝑢 ∗
𝜕 𝜉 ∗
2
𝜕 𝜂 ∗
2
+ Λ
∗
4
𝜕 4
𝑢 ∗
𝜕 𝜂 ∗
4
= 𝛽 ∗
+ 𝛾 ∗
(3.6)

The influence of the rigid mass is captured using a point-matching approach to
approximate the coupling force that is applied over a continuous domain as a finite set of 𝐼 ∗

forces that act at discrete points within and on the boundary of the domain of the inclusion. The
coupling force is expressed
48

𝛾 ∗
= ∑𝛾 𝑖 ∗
𝛿 (𝜉 ∗
− 𝜉 𝑖 ∗
)𝛿 (𝜂 ∗
− 𝜂 𝑖 ∗
)
𝐼 ∗
𝑖 =1
(3.7)
where 𝛾 𝑖 ∗
is the dimensionless coupling force contributed by the 𝑖 𝑡 ℎ
collocation point and 𝛿 is
the Dirac delta function. Substituting Eq. (3.7) into Eq. (3.6), the resulting equation is solved
using a modal expansion approach, approximating the response of the system as a finite linear
combination of 𝑁 ∗
eigenfunctions Φ
∗
(𝜉 ∗
,𝜂 ∗
) determined by the boundary conditions and
geometry of the system. In this implementation, the boundaries of the array are assumed to be
rigidly clamped to a support structure, resulting in boundary conditions given by

𝑢 ∗
(0,𝜂 ∗
)= 𝑢 ∗
(𝜉 ∗
,0)= 𝑢 ∗
(𝜉 ∗
,𝐿 𝑦 ∗
)= 𝑢 ∗
(𝐿 𝑥 ∗
,𝜂 ∗
)= 0
𝜕 𝑢 ∗
(0,𝜂 ∗
)
𝜕𝜉
=
𝜕 𝑢 ∗
(𝜉 ∗
,0)
𝜕𝜂
=
𝜕 𝑢 ∗
(𝜉 ∗
,𝐿 𝑦 ∗
)
𝜕𝜉
=
𝜕 𝑢 ∗
(𝐿 𝑥 ∗
,𝜂 ∗
)
𝜕𝜂
= 0
(3.8)
To satisfy these boundary conditions, the eigenfunction Φ
𝑛 𝑥 𝑛 𝑦 ∗
(𝜉 ∗
,𝜂 ∗
)=
Φ
𝑛 𝑥 ∗
(𝜉 ∗
)Φ
𝑛 𝑦 ∗
(𝜂 ∗
) is chosen such that Φ
𝑛 ∗
(𝜛 )= cosh(𝑎 𝑛 𝜛 )− cos(𝑎 𝑛 𝜛 )− 𝑏 𝑛 (𝑠𝑖𝑛 ℎ(𝑎 𝑛 𝜛 )−
𝑠𝑖𝑛 (𝑎 𝑛 𝜛 )) is the 𝑛 𝑡 ℎ
mode shapes of a clamped-clamped single span beam. The coefficients 𝑎 𝑛
are the solutions to
cos(𝑎 𝑛 )cosh(𝑎 𝑛 )= 1 (3.9)
and the corresponding coefficients 𝑏 𝑛 are determined according to

𝑏 𝑛 =
sinh(𝑎 𝑛 )+ sin(𝑎 𝑛 )
cosh(𝑎 𝑛 )− cos(𝑎 𝑛 )
(3.10)
The unitless modal displacement is then approximated according to

𝑢 ∗
≈ ∑ 𝑞 𝑛 ∗
Φ
𝑛 ∗
𝑁 ∗
𝑛 =1
= ∑ ∑ 𝑞 𝑛 𝑥 𝑛 𝑦 ∗
Φ
𝑛 𝑥 ∗
(𝜉 ∗
)Φ
𝑛 𝑦 ∗
(𝜂 ∗
)
𝑁 𝑦 ∗
𝑛 𝑦 =1
𝑁 𝑥 ∗
𝑛 𝑥 =1

(3.11)
where 𝑁 ∗
= 𝑁 𝑥 ∗
𝑁 𝑦 ∗
and 𝑛 = 𝑁 𝑦 ∗
(𝑛 𝑥 − 1)+ 𝑛 𝑦 .
49

After substituting Eq. (3.7) and Eq. (3.11) into Eq. (3.6), the equations of motion can be
arranged into matrix form according to

(𝑪 ∗
− 𝑘 ∗
2
𝑴 ∗
)𝒒 ∗
= 𝛽 ∗
𝒃 ∗
+ 𝑳 ∗
𝜸 ∗
(3.12)
where the dimensionless coupling-force vector 𝜸 ∗
contains entries [𝛾 1
∗
, 𝛾 2
∗
,…,𝛾 𝑁 ∗
∗
]
𝑇 , the
stiffness matrix 𝑪 ∗
= (𝑐 𝑚𝑛
∗
) 𝜖 ℝ
𝑁 ∗
×𝑁 ∗
, the mass matrix 𝑴 ∗
= (𝑚 𝑚𝑛
∗
) 𝜖 ℝ
𝑁 ∗
×𝑁 ∗
have entries
given by

𝑐 𝑚𝑛
∗
= {
𝑎 𝑛 ∗
4
+ 2Λ
∗
2
(𝑏 𝑛 2
𝑎 𝑛 2
(2 − 𝑏 𝑛 𝑎 𝑛 )
2
)+ Λ
∗
4
𝑎 𝑛 4
for 𝑚 = 𝑛 0 else
(3.13)

𝑚 𝑚𝑛
∗
= {
1 for 𝑚 = 𝑛 0 else
(3.14)
and the coupling matrix 𝑳 ∗
= (𝑙 𝑚𝑛
∗
) 𝜖 ℝ
𝑁 ∗
×𝐼 ∗
and forcing vector 𝒃 ∗
= (𝑏 𝑛 ∗
) 𝜖 ℝ
𝑁 ∗
have entries
given by
𝑙 𝑚𝑛
∗
= Φ
𝑛 𝑥 ∗
(𝜉 𝑛 ∗
)Φ
n
y
∗
(𝜂 𝑛 ∗
)
(3.15)

𝑏 𝑛 ∗
= {
16𝑏 𝑛 𝑥 𝑏 𝑛 𝑦 𝑎 𝑛 𝑥 𝑎 𝑛 𝑦 for 𝑛 𝑥 𝑛 𝑦 odd
0 else
(3.16)
A second set of equations is developed to describe the motion of the inertial inclusion.
This set of equations is written in the coordinate frame { 𝒙⃗ ⃗
∗
′,𝒚⃗ ⃗
∗
′,𝒛⃗
∗
} whose origin is located at
the center of mass of the inertial inclusion mounted to the array (see Figure 3.4). The
displacement anywhere within the domain of the rigid inclusion can be related to the position of
its center of mass 𝑢 𝐴 ,𝐶𝑀
and two terms 𝛼 𝜉 ∗ and 𝛼 𝜂 ∗ describing its rotation about the 𝒙⃗ ⃗
∗
′ and 𝒚⃗ ⃗
∗
′
axes, respectively.

𝑢 𝐴 (𝜉 ∗
′
,𝜂 ∗
′)= 𝑢 𝐴 ,𝐶𝑀
− 𝛼 𝜂 ∗𝜉 ∗
′
+
1
Λ
∗
𝛼 𝜉 ∗𝜂 ∗
′
(3.17)
50

The additional unknown terms in Eq. (3.17)—𝑢 𝐴 ,𝐶𝑀
,𝛼 𝜂 ∗, and 𝛼 𝜉 ∗—can be expressed as
a function of dimensionless frequency parameter 𝑘 ∗
, dimensionless mass parameter 𝜇 ∗
=
𝑀 𝐴 /(𝑚 𝐴 ′
𝐿 𝑥 ∗
2
), dimensionless rotational inertia parameters 𝜗 𝜉 ∗ = 𝐽 𝑥 ∗/(𝑀 𝐴 𝐿 𝑥 ∗
2
) and 𝜗 𝜂 ∗ =
𝐽 𝑦 ∗/(𝑀 𝐴 𝐿 𝑥 ∗
2
) , and the point-force coupling terms 𝛾 𝑖 ∗
according to

𝑢 𝐴 ,𝐶𝑀
=
1
𝜇 ∗
𝑘 ∗
2
∑𝛾 𝑖 ∗
𝐼 ∗
𝑖 =1
(3.18)

𝛼 𝜉 ∗ =
1
𝜇 ∗
𝑘 ∗
2
Λ
∗
𝜗 𝜉 ∗
∑𝜉 𝑖 ∗
′𝛾 𝑖 ∗
𝐼 ∗
𝑖 =1
(3.19)

𝛼 𝜂 ∗ =
1
𝜇 ∗
𝑘 ∗
2
𝜗 𝜂 ∗
∑𝜂 𝑖 ∗′
𝛾 𝑖 ∗
𝐼 ∗
𝑖 =1
(3.20)
Using these relations, Eq. (3.17) becomes

𝑢 𝐴 (𝜉 ∗
′
,𝜂 ∗
′)=
1
𝜇 ∗
𝑘 ∗
2
∑(1 +
𝜉 ∗
′𝜉 𝑖 ∗
′
𝜗 𝜂 ∗
+
𝜂 ∗
′𝜂 𝑖 ∗
′
Λ
∗
2
𝜗 𝜉 ∗
)
𝐼 ∗
𝑖 =1
𝛾 𝑖 ∗
(3.21)
Since the inertial inclusion is perfectly bonded to the array, its motion will match
identically the motion of the plate at each of the 𝐼 ∗
colocation points, implying
𝑢 𝐴 (𝜉 𝑚 ∗′
,𝜂 𝑚 ∗′
)= 𝑢 ∗
(𝜉 𝑚 ∗
,𝜂 𝑚 ∗
) for 1 ≤ 𝑚 ≤ 𝐼 ∗
(3.22)
which can be written in matrix form according

−𝑳 ∗
𝑇 𝒒 ∗
+
1
𝑘 ∗
2
𝑮 ∗
𝜸 ∗
= 0 (3.23)
where the matrix 𝑮 ∗
= (𝑔 𝑚𝑛
∗
) 𝜖 ℝ
𝐼 ∗
×𝐼 ∗
has entries given by

𝑔 𝑚𝑛
∗
=
1
𝜇 ∗
(1 +
𝜉 ∗
′𝜉 𝑖 ∗
′
𝜗 𝜂 ∗
+
𝜂 ∗
′𝜂 𝑖 ∗
′
Λ
∗
2
𝜗 𝜉 ∗
) (3.24)
Combining equations Eq. (3.12) and Eq. (3.23) into a single block matrix, the resulting
equations of motion can be expressed
51

[
𝑪 ∗
− 𝑘 ∗
2
𝑴 ∗
−𝑳 ∗
−𝑳 ∗
𝑇 𝑮 ∗
𝑘 ∗
2
⁄
][
𝒒 ∗
𝜸 ∗
] = 𝛽 ∗
[
𝒃 ∗
𝟎 ] (3.25)
whose homogeneous form can be expressed as the generalized eigenvalue problem given by

𝑨 ∗
𝒙 ∗
= 𝑘 ∗
2
𝑩 ∗
𝒙 ∗
(3.26)
where

𝑨 ∗
= [
𝑪 ∗
−𝑳 ∗
𝟎 𝑮 ∗
], 𝑩 ∗
= [
𝑴 ∗
𝟎 𝑳 ∗
𝑻 𝟎 ], and 𝒙 ∗
= [
𝒙 𝒒 ∗
𝒙 𝜸 ∗
] (3.27)
The solution to this eigenvalue problem is found using standard solvers to identify the
first 𝐾 ∗
eigenvalues (indicating dimensionless modal frequencies) and eigenvectors (indicating
coupling forces at collocation points and modal coefficients for eigenfunction weighting
coefficients).
The steady state behavior of the structure under forced vibration can be determined by
approximating the solution to Eq. (3.25) as a finite linear combination of the first 𝐾 ∗

eigenmodes. This approximation is captured in Eq. (3.28)
[
𝒒 ∗
𝑇 𝜸 ∗
𝑇 ]
𝑇 ≈ 𝑿 ∗
𝒄 ∗
(𝑘 ∗
) (3.28)
where 𝑿 ∗
𝜖 ℝ
(𝑁 ∗
+𝐼 ∗
)×𝐾 ∗
is a matrix of eigenvectors such that the 𝑖 𝑡 ℎ
column corresponds to the
𝑖 𝑡 ℎ
eigenvector of Eq. (3.26) and 𝒄 ∗
= [𝑐 1
∗
,𝑐 2
∗
,…,𝑐 𝐾 ∗
∗
]
𝑇 is a vector containing modal contribution
coefficients. The modal contribution coefficients can be determined by substituting Eq. (3.28)
into Eq. (3.25), pre-multiplying by the matrix 𝑿 ∗
𝑇 , taking advantage of the identity 𝑨 ∗
𝑿 ∗
=
𝑩 ∗
𝑿 ∗
𝚲 ∗
(where 𝚲 ∗
𝜖 ℝ
𝐾 ∗
×𝐾 ∗
is a diagonal matrix with entries corresponding to the first 𝐾 ∗

dimensionless eigenfrequencies extracted from Eq. (3.26)), and rearranging to express these
coefficients 𝒄 ∗
as a function of dimensionless frequency 𝑘 ∗
and according to
52

𝒄 ∗
(𝑘 ∗
)= 𝛽 ∗
(𝚲 ∗
− 𝑘 ∗
2
𝑰 )
−1
(𝑿 ∗
𝑇 𝑩 ∗
𝑿 ∗
)
−1
𝑿 ∗
𝑇 [
𝒃 ∗
𝟎 ] (3.29)
Modeling Cell Behavior: Membrane-like Dynamics

Figure 3.5: Definition of geometric and mathematical variables for modeling membrane behavior.

Using an analogous mathematical framework, the individual membrane cells are
described according to a model presented by Langfeldt et al. [17]. The rectangular membrane is
given dimensions 𝐿 𝑥 and 𝐿 𝑦 , surface mass density 𝑚 𝑀 ′
, is subject to a uniform tension force per
unit length 𝑇 , and is taken to have perfectly fixed edges. Bonded in the center of the membrane
is an annular inertial inclusion of mass 𝑀 𝑀 . Geometric parameters and variables are defined in
Figure 3.5, where the origin of the { 𝒙⃗ ⃗ ,𝒚⃗ ⃗ ,𝒛⃗ } coordinate frame is located at one corner of the
membrane with the positive 𝒙⃗ ⃗ - and 𝒚⃗ ⃗ -axes oriented along edges of the membrane. The out of
plane motion of the membrane is given by 𝑤 (𝑥 ,𝑦 ,𝑡 ) which evolves according to the
inhomogeneous wave equation given by Eq. (3.30).

𝑚 𝑀 ′
𝜕 2
𝜕 𝑡 2
𝑤 (𝑥 ,𝑦 ,𝑡 )− 𝑇 ∇
2
𝑤 (𝑥 ,𝑦 ,𝑡 )= 𝑃 (𝑥 ,𝑦 ,𝑡 )+ 𝑓 𝑀 ′
(𝑥 ,𝑦 ,𝑡 )
(3.30)
Equation (3.30) is normalized using dimensionless parameters analogous to those used
in modeling the array. These parameters are given by Eq. (3.5) when the “*” symbol is dropped
from each term and the subscript “𝑀 ” (for membrane) is substituted for the subscript “𝐴 ” (for
53

array) where appropriate. Modal responses of the membrane cells can then be approximated as a
linear combination of 𝑁 eigenfunctions. The eigenfunctions used to approximate the
membrane’s modal response are given by Φ
𝑛 (𝜉 ,𝜂 )= Φ
𝑛 𝑥 𝑛 𝑦 (𝜉 ,𝜂 )= sin(𝑛 𝑥 𝜋𝜉 )sin(𝑛 𝑦 𝜋 𝜂 )
where 𝑛 𝜖 { 1,2,…,𝑁 }, 𝑛 𝑥 𝜖 { 1,2,…,𝑁 𝑥 }, 𝑛 𝑦 𝜖 { 1,2,…,𝑁 𝑦 }, 𝑁 = 𝑁 𝑥 𝑁 𝑦 , and 𝑛 = 𝑁 𝑦 (𝑛 𝑥 −
1)+ 𝑛 𝑦 . The matrix equation of motion this produces is analogous to Eq. (3.12).

(𝑪 − 𝑘 2
𝑴 )𝒒 = 𝛽 𝒃 + 𝑳𝜸 (3.31)
The entries of the stiffness matrix 𝑪 = (𝑐 𝑚𝑛
) 𝜖 ℝ
𝑁 ×𝑁 , mass matrix 𝑴 = (𝑚 𝑚𝑛
) 𝜖 ℝ
𝑁 ×𝑁 ,
forcing vector 𝒃 = (𝑏 𝑛 ) 𝜖 ℝ
𝑁 , and coupling matrix 𝑳 = (𝑙 𝑚𝑖
) 𝜖 ℝ
𝑁 ×I
have entries given by

𝑐 𝑚𝑛
= {
𝜋 2
4
(𝑛 𝑥 2
+ Λ
2
𝑛 𝑦 2
) for 𝑚 = 𝑛 0 else
(3.32)

𝑚 𝑚𝑛
= {
1
4
for 𝑚 = 𝑛 0 else
(3.33)

𝑏 𝑛 = {
4
𝜋 2
𝑛 𝑥 𝑛 𝑦 for 𝑛 𝑥 𝑛 𝑦 odd
0 else
(3.34)

𝑙 𝑚𝑖
= sin(𝑚 𝑥 𝜋 𝜉 𝑖 )sin(𝑚 𝑦 𝜋 𝜂 𝑖 ) (3.35)
The coupling effect between the continuously vibrating membrane and the rigid, bonded
mass is approximated by the same point-matching approach as described in the previous
section. A selection of 𝐼 discrete colocation points, located within (and on the boundary of) the
domain of the annular inclusion, was used to approximate continuous-domain coupling.
Equations describing the motion of the inertial inclusion on the membrane are given according
to Eq. (3.21) when the “*” symbol is dropped from each of the terms and the subscript “𝑀 ” is
substituted for the subscript “𝐴 ” in 𝑢 𝐴 . Further, the equivalence relation between equations of
54

motion for each body at each of the 𝐼 colocation points is given in matrix format by Eq. (3.23),
and entries to the matrix 𝑮 are given by Eq. (3.24)—again, the “*” symbol is dropped from each
of the terms in both equations.
The resulting matrix equation of motion describing vibration of the membrane cell is
given by

[
𝑪 − 𝑘 2
𝑴 −𝑳 −𝑳 𝑇 𝑮 𝑘 ∗
2
⁄
][
𝒒 𝜸 ] = 𝛽 [
𝒃 𝟎 ] (3.36)
The homogeneous part of the resulting equation is solved in the same fashion as above:
by finding the first 𝐾 eigenvalues and eigenvectors of
𝑨𝒙 = 𝑘 2
𝑩𝒙 (3.37)
where

𝑨 = [
𝑪 −𝑳 𝟎 𝑮 ], 𝑩 = [
𝑴 𝟎 𝑳 𝑻 𝟎 ], and 𝒙 = [
𝒙 𝒒 𝒙 𝜸 ] (3.38)
These solutions are then used to approximate the inhomogeneous solution, calculating
the mode participation factors 𝒄 in the same method as outlined above according to

𝒄 (𝑘 )= 𝛽 (𝚲 − 𝑘 2
𝑰 )
−1
(𝑿 𝑇 𝑩𝑿 )
−1
𝑿 𝑇 [
𝒃 𝟎 ] (3.39)

Transmission Loss: Compound Structure
The dynamic response of each of the two length scales of the array of MAMs is used to
estimate the acoustic transmission properties of the system. Transmission efficiency through
both the membrane and the array are each determined under the assumption that the surface-
averaged vibration amplitude dominates the sound radiation behavior of the system. This
55

assumption is appropriate for frequencies with acoustic wavelength 𝜆 > √𝐿 𝑥 ∗
2
+ 𝐿 𝑦 ∗
2
larger than
the characteristic length of the array. Because low-frequency performance is of primary interest,
this assumption is not particularly restrictive and is discussed more thoroughly below. The
surface-averaged vibration amplitude is used to determine the effective mass density of the
structure, from which transmission loss can be predicted in accordance with the acoustic mass
law. In accordance with Newton’s Second law, the effective mass density of the compound
structure 𝑚 ̃
𝐶𝐸
′
is expressed

𝑚 ̃
𝐶𝐸
′
=
〈𝑃 〉
〈𝑤 ̈ 𝐶𝐸
〉
=
〈𝑃 〉
𝜔 2
〈𝑤 𝐶𝐸
〉
(3.40)
where 〈 〉 indicates the surface average of the quantity it encloses. Because the total
summed area of the membranes is roughly equivalent to the area of the array itself, the surface
average motion of the compound structure 〈𝑤 𝐶𝐸
〉 can be estimated as the sum of the surface
average motion of the array and the individual membrane cells: 〈𝑤 𝐶𝐸
〉 = 〈𝑤 ∗
〉 + 〈𝑤 〉. Using the
relation in Eq. (3.40), rearranging terms, and making the appropriate substitutions for
dimensionless parameters, the effective mass density of the compound structure is given by

𝑚 ̃
𝐶𝐸
′
= (𝑚 ̃
′
−1
+ 𝑚 ̃
∗
′
−1
)
−1

(3.41)
where

𝑚 ̃
∗
′
= −
𝑚 𝐴 ′
𝑘 ∗
2
[
𝑏 ∗
𝑇 0
]𝑋 ∗
𝑐 ∗
(3.42)

𝑚 ̃
′
= −
𝑚 𝑀 ′
𝑘 2
[
𝑏 𝑇 0
]𝑋𝑐
(3.43)
The acoustic mass law is used to calculate the transmission coefficient 𝑡 according to
1
𝑡 = 1 +
𝑖𝜔 𝑚 ̃
𝐶𝐸
′
2𝜌 0
𝑐 0

(3.44)
56

where 𝜌 0
and 𝑐 0
are the density and speed of sound of the acoustic fluid through which
sound is being transmitted (typically air). In this study, the transmission coefficient of the
individual membrane cell, the array, and the compound structure were each determined and
compared. Finally, transmission through the compound structure can be calculated according to
𝑇𝐿
0
= −20 log
10
|𝑡 |.
In the work presented below, the following geometric parameters and derived quantities
were used and held constant when modeling the array, membrane, and compound structures:
𝐿 𝑥 ∗
= 𝐿 𝑦 ∗
= 0.241 m, ℎ = 7.5 × 10
−3
m, [𝑥 𝑀 ∗
, 𝑦 𝑀 ∗
] = [0.1205 m, 0.1085 m], 𝐿 𝑥 = 𝐿 𝑦 =
0.040 m, 𝑥 𝑀 = 𝑦 𝑀 = 0.020 m, 𝑇 = 750 Nm
-1
, and 𝑚 𝑀 ′
= 0.0971 kg m
-2
. The numerical
parameters used were given by 𝑁 𝑥 ∗
= 𝑁 𝑦 ∗
= 𝑁 𝑥 = 𝑁 𝑦 = 40, 𝐼 ∗
= 44 and 𝐼 = 16. Simulations
were conducted for combinations of 𝑀 𝐴 = { 0,2.5,10.3,20.5} × 10
−3
kg and 𝑀 𝑀 =
{ 0,1.6,3.2} × 10
−4
kg using the appropriate corresponding values of 𝐽 𝑥 ∗, 𝐽 𝑦 ∗, 𝐽 𝑥 and 𝐽 𝑦 given
inclusion geometry.
The areal mass and flexural rigidity of the homogenized array (with no masses bonded
to individual membrane cells) were estimated according to 𝑚 𝐴 ′
= 1.16 kgm
-2
and 𝑇 ∗
= 30.77
Nm
-2
. The method for estimating the areal mass of the array was measuring the mass of the
array, adhesive, and membrane after assembly (but prior to the placement of individual masses
on each membrane cell), subtracting the mass of the edge regions of the array (which are
clamped in the test fixture during transmission loss measurement), and dividing the remaining
mass by the area of the test window (0.05801 m
2
). The mass of the array, adhesive, and bonded
membrane was measured to be 0.128 kg, the mass of the array edges clamped in the test fixture
were theoretically calculated to be 0.0608 kg, and the resultant areal mass of the homogenized
array was determined to be 𝑚 𝐴 ′
=1.16 kgm
-2
. When individual membrane cells had masses
57

bonded to them, the areal mass of the homogenized array was increased appropriately. For
example, when 0.16 g masses are added to each of the 36 individual membrane cells, this
mass is assumed to be distributed uniformly, and the areal mass of the homogenized array is
increased to 𝑚 𝐴 ′
= 1.26 kgm
-2
to account for this additional inertia. The flexural rigidity of the
homogenized array was estimated fixing the areal weight of the homogenized array in the
manner previously described, then fitting a value to 𝑇 ∗
such that the analytical model
predictions match experimental data for the fourth eigenfrequency of homogenized array
vibration. This frequency was measured to be approximately 1.8 kHz for the array when 0.16 g
masses were attached to individual membrane cells and no mass was attached to the array
(𝑚 𝐴 ′
= 1.26 kgm
-2
, 𝑀 𝐴 = 0). The resultant homogenized flexural rigidity of the array used for
modeling was estimated to be 𝑇 ∗
= 30.77 Nm
-2
.

3.3. Results and Discussion
The transmission loss was measured and analytically predicted through twelve different
configurations of the MAM array structure. The results demonstrated a complex interaction
between the two length scales, showed that new transmission properties—present for neither
length scale independently—can be achieved in the compound structure, and established the
importance of considering this interaction when scaling MAM structures. In this section, we
first present transmission loss results characteristic of the hierarchical metamaterial structure
fabricated for this study and explain the vibroacoustic behavior responsible for features of
interest. Subsequently, we demonstrate the effect of varying the mass of the inertial inclusions
bonded to both the array and the individual membrane cells. Finally, we discuss the limitations
58

of the modeling approach presented in the previous section and indicate possible future
extensions or improvements the model.
3.3.1. Hierarchical Acoustic Metamaterial: Characteristic Performance
The experimentally measured transmission loss through the array of thirty six MAM
cells is plotted in Figure 3.6 for the case of 𝑀 𝐴 = 2.5 × 10
−3
kg and 𝑀 𝑀 = 1.6 × 10
−4
kg.
Experimental data in this figure are compared against analytical predictions for transmission
loss through the individual membrane cells, the homogenized array, and the compound
structure. It is apparent that the transmission loss properties of the compound structure arise
directly from the behavior of, and interaction between, each length scale of the structure.
Previous work has demonstrated that the low-frequency propagation of acoustic energy through
membrane- and plate-type acoustic metamaterials is determined by the modal responses of such
structures. Similarly, it is apparent that the modal responses of each length scale of the
hierarchical metamaterial contribute to the transmission properties of the structure. Below the
modal contributions responsible for each local minima and maxima are explained. When
interpreting Figure 3.6, it is worth noting that estimates for the performance of neither length
scale individually (the membrane cell scale or the homogenized array scale) is anticipated to
match the estimates produced by the compound model or experimental data across the whole
frequency range. Features such as minima in estimates of the performance of each length scale
are apparent in the compound model over narrow frequency ranges, but (appropriately) when
transmission loss is non-zero for both length scales, a more complex interaction between the
two levels of hierarchy occurs. Rather than to achieve good matching with the compound model
or experimental data, Figure 3.6 includes estimates of the performance of each length scale to
59

explain the origin of transmission loss maxima and minima apparent in the compound model
prediction and experimental measurements.

Figure 3.6: Characteristic transmission loss performance of hierarchical acoustic metamaterial.

Four experimentally measured transmission loss local minima can be seen in Figure 3.6
at frequencies of 315 Hz, 765 Hz, 1455 Hz, and 3200 Hz. At each of these frequencies, the
transmission loss of the compound structure is dominated by the behavior of one of its two
constituent length scales. That is, the behavior of the homogenized array dictates the
transmission loss behavior at 315 Hz and 1455 Hz, while the behavior of the individual MAM
cells dictates the behavior at 765 Hz and 3200 Hz. At each of these frequencies, the compound
structure is nearly transparent to acoustic propagation, and the admittance of nearly all acoustic
energy through the structure is explained by the excitation of a mode (either modes of the
homogenized array or modes of the membrane cells) characterized by with nonzero average
surface motion during oscillation.
The first transmission loss minimum in Figure 3.6, near 315 Hz, corresponds to
activation of the first fundamental mode of the homogenized array. While analytic techniques
60

predicted this mode to occur at approximately 360 Hz, experimental data indicated the first
resonance of the array occurs near the transmission loss minimum at 315 Hz. Acoustic energy is
efficiently transmitted through the hierarchical acoustic metamaterial at this frequency—
resulting in a transmission loss of nearly zero—because the surface averaged motion of the
array oscillating in this mode is nonzero. Further, because the effective mass density of the
homogenized array (𝑚 ̃
∗
′ ) is nearly zero at the first fundamental mode of array vibration, the
term contributed by the array to Eq. (3.41) far outweighs the term contributed by the membrane
cell level behavior. This explains why array level behavior dominates the performance of the
compound structure near the first transmission loss minimum.

Figure 3.7: First modal response of homogenized array as predicted analytically (left) and measured experimentally (right).

The analytically predicted (left) and experimentally measured (right) modal responses of
the array oscillating in its first mode of vibration are pictured in Figure 3.7. Experimental mode
shape plots were produced by fitting a surface to the 85 vibrometer measurements using a
locally weighted scatterplot smoothing algorithm, normalizing out of plane deformation, and
generating contour lines and shading to indicate the magnitude and direction of out of plane
array deformation. In Figure 3.7, as for all figures showing modal response of the membrane or
array, regions of deformation below the neutral plane appear darker than regions of deformation
61

above the neutral plane. A normalized deformation value of negative one corresponds to black
coloration, and normalized deformation of positive one corresponding to white coloration. The
similarity of the analytical and experimental modal responses pictured in Figure 3.7 indicates
the efficacy of the modeling approach presented herein. Discrepancies between the two modal
response shapes can likely be attributed to noise in the laser vibrometry data and imperfectly
clamped array edges.
The second transmission loss minimum corresponds to activation of the first mode of the
individual MAM cells. Analytic and experiment data agree that this mode is located at
approximately 765 Hz. The shape of membrane deformation under excitation at this frequency
is analogous to first mode of the vibrating array (see Figure 3.7), and similarly produces a
nonzero surface averaged displacement during oscillation, resulting in efficient transmission
through the structure, and dominating performance over array scale behavior at this frequency.
From Figure 3.6, it is apparent that the third analytically predicted and experimentally
measured transmission loss minimum can be attributed to the dynamics of the array.
Experimental and theoretical results agree that the fourth fundamental mode of array vibration is
located at approximately 1455 Hz. Activation of this mode dominates transmission loss
characteristics of the compound structure at this frequency. Because the surface averaged
deformation of the array significant at this frequency, acoustic energy is efficiently transmitted,
resulting in a transmission loss minimum. The analytically predicted and experimentally
measured mode shapes for this frequency achieve excellent agreement and are shown in Figure
3.8. Unlike with the first transmission loss minimum of the compound structure, the
experimental and analytical data indicate nonzero transmission loss at this frequency despite the
analytic model of the homogenized array indicating that zero transmission loss should be
62

achieved. Indeed, experimental measurements indicate that transmission loss of no less than 12
dB in the vicinity of this frequency. This discrepancy can likely be attributed to the frequency
band averaging performed during experimental data collection and compound model result
processing. The effect of averaging transmission loss data over a 1/8
th
octave band is
exacerbated by the relatively narrow frequency range at which this mode of the array achieves
large amplitude vibration.

Figure 3.8: Second quasi-symmetric modal response of array as predicted analytically (left) and measured experimentally
(right).

The fourth experimental transmission loss minimum in Figure 3.6 is correlated with the
fourth modal response (second symmetric response) of the individual membrane cells.
Experimental and analytical results locate this mode at approximately 3200 Hz. An analytical
prediction of the mode shape at this frequency is shown in Figure 3.9. The apparent nonzero
surface averaged displacement seen in this figure explains the efficiency of acoustic
transmission through the structure at this frequency.
63

Figure 3.9: Analytical prediction of second symmetric mode shape in individual membrane cells.

At each of the minima discussed, the transmission properties of the structure are
dominated by the vibratory behavior of a single length scale, where resonance results in large
volumetric displacement across the neutral plane of the structure. Transmission loss maxima,
however, are not dominated by the behavior of either length scale individually, and instead
require the sum of volumetric displacements for each length scale to be zero. For example,
modeling data describing the compound structure indicate the first transmission loss peak in
Figure 3.6 occurs at approximately 430 Hz where the structure achieves a transmission loss of
35 dB. There is no local maximum for either the array or membrane length scales at this
frequency, however, and the transmission loss predicted at each length scale is found to be 14
dB and 13 dB for the array and membrane, respectively. The increased transmission loss when
compared against either individual length scale can be attributed to the phase difference
between the motion of the array and the membrane cells. At this frequency, the motion of the
array lags the acoustic excitation signal by about pi radians, while the motion of the individual
64

membrane cells is approximately in phase with the excitation signal, resulting in near zero net
displacement during oscillation, and efficient rejection of acoustic energy.
Like the first transmission loss maximum, the second and third maxima in the analytical
compound structure predictions are located at frequencies for which the surface averaged
displacement of the homogenized array is predicted to be equal and opposite to the surface
averaged displacement of the individual membrane cells. Such frequencies are located at or
immediately adjacent to frequencies at which the transmission loss curves of the two lengths
scales cross. The second transmission loss maximum of the compound structure is predicted to
be at 1275 Hz, near the frequency where the transmission loss curves of the homogenized array
and membrane cell cross over at 20.3 dB. The third transmission loss maximum is similarly
located at such a crossover point at 1775 Hz. These analytically predicted transmission loss
maxima correspond reasonably well with experimental data, indicating the effectiveness of the
modeling approach presented in this paper.
3.3.2. Parametric Effect of Inertial Inclusions
By varying the size of the inertial inclusions incorporated into each length scale of the
hierarchical acoustic metamaterial, the frequencies at which characteristic transmission loss
peaks and dips can be shifted. Figure 3.10 shows measured and predicted transmission
properties through the structure studied for 𝑀 𝑀 = 0 kg and various values of 𝑀 𝐴 . As 𝑀 𝐴
increases, the frequency at which the first transmission loss dip occurs, corresponding to the
first symmetric eigenmode of the array, decreases. Similarly, the frequency of the first
transmission loss peak is also shifted to a lower frequency range. Note, however, that features
on the curve above approximately 800 Hz are unaffected by changes in 𝑀 𝐴 . This is because the
dynamics of the membrane dominate the transmission performance of the structure above this
65

frequency, resonating with large amplitude over a wide frequency band. Further, the lack of
inertial inclusions on the individual membrane cells results in a second symmetric mode that is
unfavorable for sound rejection.

Figure 3.10: Parametric effects of 𝑀 𝐴 on transmission properties for 𝑀 𝑀 = 0 g.

The parametric influence of 𝑀 𝐴 can be further characterized as shown in Figure 3.11,
where the size of the mass on the membrane cells is given 𝑀 𝑀 = 1.6x10
−4
kg. In this figure,
the influence of the second symmetric mode of the array becomes clearer. The eigenfrequencies
associated with this mode for each case of increasing 𝑀 𝐴 are given as 1791 Hz, 1600 Hz, 1377
Hz, and 1297 Hz, respectively. For each case, the transmission efficiency of the structure is
enhanced in the neighborhood of this frequency, and transmission loss is reduced. This
phenomenon is largely responsible for the decay of transmission loss performance that is
associated with scaling of MAM structures. The figure demonstrates the importance of
66

considering the dynamics of the array, as the modal responses can provide efficient parallel
transmission paths that are highly efficient in some frequency ranges.

Figure 3.11: Parametric effects of 𝑀 𝐴 on transmission properties for 𝑀 𝑀 = 1.6x10
−4
kg.

3.3.3. Accuracy and Extensions to the Modeling Approach
Readers may notice several differences between the analytical predictions and
experimental data presented in Figure 3.10 and Figure 3.11. Some of these differences can be
attributed to the averaging necessary for collection of experimental transmission loss data that is
not present in the analytical predictions. Because experimental data are averaged over a 1/8
th

octave band around each sampling frequency, the magnitude of the transmission loss maxima
and minima measured are significantly less extreme than is estimated by the analytical tool.
Averaging of the experimental data is largely responsible for the difference between model
predictions and measurements in Figure 3.11 in the range of 1-2 kHz. When analytical data are
67

subjected to 1/8
th
octave band averaging over this range and sampled at the same frequencies as
experimental data, the minimum forecast transmission loss increases from 0 dB to 8-14 dB
(depending on array mass loading condition), and the maximum forecast transmission loss data
decreases from infinite to 30-45 dB. Averaging of the analytical data also entirely obfuscates
the transmission loss maxima and minima analytically predicted to occur just below 1 kHz in
Figure 3.11. Indeed, such local extrema are not seen in the measured transmission loss data.
Overall, averaging the estimated transmission loss results in a much closer match with
experimental measurements, and the effect of such averaging is not always intuitive due to the
logarithmic nature of transmission loss data and frequency sampling.
Beyond the difference between averaged and unaveraged transmission loss data, the
experimental and analytical results likely deviate from one another due to variability in
membrane tension and mass size and location within each membrane cell. While each MAM
cell in the array was intended to be identical, imperfections in the machining of the array,
thickness of the membrane, adhesion of the bond between the two, and location and size of the
mass placed on each membrane cell no doubt resulted in a distribution of similarly tuned
membrane cells. The effect of this on the transmission loss through the structure would serve to
further reduce the extremity of the maxima and minima predicted analytically throughout
frequency range of interest. Such a difference in the tuning of individual membrane cells is
likely responsible for differences in the shape of the experimental and analytical transmission
loss profiles near the first maximum in Figure 3.10 and Figure 3.11. At this transmission loss
maximum, the volume velocity of the sum of membrane cells is approximately equal to and
opposite of the volume velocity of the array structure. This results in inefficient transmission
through the metamaterial, yielding high transmission loss. Analytical tools assume that all
68

membrane the cells are oscillating with identical shape and that they are perfectly phase
matched. The inherent variability in the manufacturing of the array of MAMs, however,
prevents such synchronistic motion from being physically realized. The effect of this
distribution of frequency responses in individual membrane cells is such that the measured
transmission loss maximum is broader but of reduced amplitude around this frequency when
compared against analytical estimates.
Some of the differences between the estimated and measured transmission loss data are
due to the limitations of approximating the membrane array as a homogenized plate rather than
a true array. Data indicate that this assumption is a useful first-order approximation, but authors
acknowledge that the eigenfrequencies and mode shapes of a grid do not match identically with
those of an equivalent isotropic, homogeneous plate. Further, the nature of the homogenization
scheme used, whereby the flexural rigidity was determined using the measured response of the
array near 1.6 kHz, is such that the analytical estimate of behavior is anticipated to be accurate
near this frequency, but accumulate error when moving toward significantly higher or lower
regimes. Indeed, this is demonstrated in Figure 3.10 and Figure 3.11 where analytical and
experimental data deviate most significantly at the lowest frequencies considered. Authors
anticipate that the accuracy of the model could be improved if the array length scale of the
hierarchical structure was modeled as a grid rather than a homogenized plate.
One additional way that the accuracy of the model could be improved would be to
include the effect of motion coupling between the array and membrane cell scales.
Incorporation of such a coupling term would require simultaneous solving of the membrane and
array length scale behaviors. The impact of considering this coupling would be most significant
in the low-frequency range where array deformation is the largest. In high-frequency ranges
69

where the amplitude of array deformation is smaller, the assumption of zero-displacement
membrane cell edges is better satisfied, and the consideration of motion coupling would have a
smaller impact.
It is worth noting here that while this modeling approach was uniquely developed to
describe hierarchical MAMs, and it cannot be used to study different types of hierarchical
metamaterials, the model can be extended to capture the effect of additional degrees of
hierarchy in these structures. If, for example, each membrane cell was comprised of an array of
MAMs, then modeling this additional length scale could be done in a manner similar to how the
membrane and array length-scale behavior was estimated, and combined into the compound
model in the same manner as described in Section 3.2.3.3 above.

3.4. Conclusions
The acoustic behavior of deployment-scale arrays of locally resonant MAM structures
was investigated. Vibroacoustic behavior responsible for the decay of transmission loss
properties typically associated transitioning from a single MAM cell to an array of cells was
explained using experimental and analytical results. A novel hierarchical design approach was
proposed, considering both the individual membrane cells and the array of membranes as
independent, locally resonant acoustic metamaterials that can each be tuned by varying stiffness
and mass parameters. A theoretical approach for predicting transmission loss through such
structures was presented and validated against experimental data. Results indicated the
importance of considering dynamic properties of the frame used to mount individual MAM
cells and demonstrated that hierarchical design can be an effective tool to maximize
transmission loss performance in regimes of interest.
70

The practical significance of this work is that it provides acousticians and materials
scientists with an effective strategy for maximizing the performance of MAM structures. The
analytical tools presented herein grant designers a toolbox that can be used prior to costly
sample fabrication and testing. Engineers can confirm and ensure favorable interaction between
membrane- and array-scale dynamics during the design process. The primary limitation of the
technique presented herein, however, is that the required addition of mass to the membrane
array compromises the weight-efficiency of the structure. Further, as the size of MAM arrays
increases, assumptions about the phenomena dominating acoustic transmission through the
structure begin to break down. As a consequence, larger MAM array structures may face
inherent limitations to the frequencies in which they can operate or the efficiency they can
achieve.

71

Process Robustness and Defect Formation Mechanisms in
Unidirectional Semipreg
3

4.1. Introduction
To address the growing need for rapid, flexible, robust, and cost-effective production
routes for high-performance composite structures, this work aims to demonstrate that UD
prepreg formats with through-thickness gas permeability (semipregs) are less sensitive to
challenging VBO processing conditions and part features (such as poor vacuum, ply drops, etc.)
than commercial hot-melt prepreg. Additional objectives of this work are to identify primary
mechanisms leading to defect formation during cure of semipregs and to elucidate the
relationship between prepreg format and these mechanisms. The objectives are motivated by a
need to inform the design of next-generation optimized semipreg formats. The study
contributes to a broader goal of accelerating the adoption of VBO prepregs to reduce the
environmental and economic costs of composites manufacturing associated with autoclave
processing.
For use in risk-averse applications such as commercial aviation, carbon fiber reinforced
polymers (CFRP) are normally processed by curing prepreg in an autoclave; however,
autoclaves (1) are expensive to purchase, install, operate, and maintain, (2) require a large
footprint, (3) limit production flexibility, and (4) may not accommodate large or irregular parts.
VBO prepreg processing offers an alternative that retains the precise control of fiber volume

3
This study was published in the Advanced Manufacturing: Polymer & Composites Science in October
2020.
72

fraction, fiber alignment, and availability of higher performance resins associated with prepreg
processing, while replacing the autoclave with a standard oven [45].
Most VBO prepregs are produced via a solventless hot-melt process that yields prepreg
with continuous resin film(s) partially or fully impregnated into one (or both) side(s) of the ply
[46,47]. A comparison of fully and partially saturated prepreg formats was first conducted in
1987 by Thorfinnson and Biermann, who showed that partially impregnated prepregs yielded
nearly void-free laminates, whereas laminates made from fully saturated prepreg contained
extensive porosity [48]. The difference in part quality was attributed to prepreg format:
unimpregnated fibers create a network of pathways through which gases can be evacuated
during cure. Evacuation through these channels, however, occurs in the plane of the ply
(mainly in the fiber direction) and requires that the laminate perimeter remain permeable to
gases (e.g., through use of edge breathing dams that ensure pathways are not sealed by
consumables) [49].
Today, nearly all commercial OoA prepregs rely on engineered vacuum channels
(EVaCs) to achieve low porosity. Such prepregs can achieve autoclave-equivalent consolidation
and quality, yet require only simple ovens for cure [50]. However, oven-cure of OoA prepregs
lacks the process robustness of autoclave cure, and ideal layup, bagging, and processing
conditions are required [51,52]. Part quality can also be compromised by improper handling
and storage of prepregs (aging, exposure to moisture, etc.) [50,53]. In fact, even under ideal
conditions, OoA manufacturing of large parts and complex geometries remains challenging.
For large parts, residual and evolved gasses must travel long distances, and longer breathe-out
distances generally correlate with higher void content [54]. This problem can be mitigated to a
certain degree by employing longer vacuum holds prior to cure, but at the cost of additional out-
73

time and slower production rates [55]. Fabrication of void-free components with complex
geometries or internal ply drops can also be challenging, since evacuation paths can become
occluded, resulting in higher defect levels, particularly at corners and ramps (ply drops) [56,57].
Recently, the design of though-thickness evacuation paths into VBO prepreg has shown
promise for overcoming limitations in process reliability apparent in conventional hot-melt
prepregs. Semipregs feature discontinuous resin distributions and through-thickness
permeability that exceeds that of conventional OoA prepregs by orders of magnitude [58].
Laminates produced with semipregs exhibited near-zero surface or bulk porosity, even under
challenging process conditions [59]. The resin distribution in the semipreg was determined as a
function of the fiber bed weave architecture and the roll-coating process by which it was
produced. As such, the resin distribution could not be easily tailored, nor could the prepregging
process be used to produce discontinuities in resin deposited on UD fiber beds. Inability to
customize resin distribution irrespective of fiber bed architecture has heretofore limited the
ability to conduct controlled studies to determine the relationship between semipreg format and
process robustness. This limitation applies especially to UD prepregs, which comprise a
significant fraction of prepreg used in aerostructures.
This work describes a lab-scale process for producing UD semipregs with controlled
resin distributions, measurements of through-thickness permeability, and demonstrations that
the high through-thickness permeability imparts process robustness. A custom UD semipreg
was designed, fabricated, characterized, and compared against a commercial VBO prepreg. The
through-thickness permeability of semipreg was measured and compared against that of the
commercial prepreg. Laminates were cured using both materials, and the influence of ideal and
adverse conditions on part quality was investigated using microscopy and high-resolution x-ray
74

tomography to measure and map sample porosity. Two types of surface defects were identified,
and in situ monitoring of the tool-ply interface provided direct observations of the mechanisms
by which each type of defect formed. The importance of minor changes in resin feature
topography and the effect on void formation was demonstrated by curing laminates with
modified resin features and comparing the quality of resulting laminates.
4.2. Methods
4.2.1. Semipreg Production
All through-thickness permeable prepreg was produced using a mask-and-press process
to selectively transfer resin film onto a UD fiber bed. A toughened epoxy resin (PMT-F4A,
Patz Materials & Technologies) was used to produce the semipreg and was procured as a
continuous 152 g/m
2
(+/- 6 g/m
2
) film on a backing paper. A 305-315 g/m
2
UD non-crimp fiber
system (#2583, FibreGlast) was selected consisting of 12K tows supported by (1) polyester
threads hot-bonded (orthogonal to the fiber direction) every 8.5 mm to one side of the fabric and
(2) minimal binder powder (0.7 wt. %).
Key steps of the mask-and-press prepregging process and the resulting prepreg are
pictured in Figure 4.1. Resin film, supported by backing paper, was first covered by a release
paper, then passed through an automated cutter (R19 Desktop Vinyl Cutter, Vinyl Express),
which scored the release paper in a prescribed pattern. Scored regions of the release paper were
removed, partly exposing the resin film beneath and creating a mask of release paper. Masked
resin films were arranged on each side of a ply of UD carbon fiber such that the release paper
mask was in contact with the dry fibers. The assembly was placed in a hydraulic press (G30H-
18-BCX, Wabash MPI) and pressed at 400 kPa for 50 minutes at room temperature
(approximately 22 °C) to bond exposed resin film to each side of the fiber bed. Masks,
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untransferred resin film, and backing paper were then removed, resulting in a semipreg ply on
which resin was distributed according to the areas removed from the scored release paper mask.

Figure 4.1: Key steps in the mask-and-press process used to produce semipreg.

All semipreg was produced using release film masks designed such that 5.0 mm strips of
resin were exposed with a periodicity of 10.0 mm, resulting in 50% coverage of the resin film
by the mask. The ratio of exposed area to masked resin film area was determined in
conjunction with the areal weight (or thickness) of the resin film to produce prepreg with a resin
content comparable to aerospace prepreg (33 wt. %). All semipreg featured symmetric resin
distribution across the midplane of each ply. Symmetry was achieved by aligning opposing
masks across the fiber bed during pressing. While only one resin distribution pattern was
investigated, the process described is not limited to parallel strip-type geometries and can be
used to produce semipregs with various distributions (squares, dots, grids, etc.), without a
requirement for periodicity. The process was tedious but was suited to lab-sale production of
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experimental formats, affording flexibility, affordability, and compactness, while requiring
minimal resin and fabric to operate.
Semipreg resin topography modification. Samples of prepreg were produced via the
method described in Section 4.2.1 and immediately modified. All modifications were intended
to effect changes in the topography of semipreg resin features. To modify resin feature
topography, prepreg plies were covered on both sides with a release paper and returned to the
hydraulic press, where they were subjected to 400 kPa for 50 minutes.
Three types of release paper were used to modify the resin feature topography of
semipregs: a smooth release paper, a diamond-textured release paper, and a crosshatch-textured
release paper. The texture of each release paper produced changes in the morphology of resin
strips on semipreg. Henceforth, unmodified prepreg produced via the mask-and-press method
will be referred to as simply as semipreg while prepregs modified by subsequent pressing with
smooth, diamond-textured, and crosshatch-textured release papers will be referred to as SPA,
SPB, and SPC, respectively.
4.2.2. Uncured Prepreg Characterization
Uncured plies of semipreg were characterized to determine resin feature size and
distribution, speed of through-thickness gas transport, and resin flow properties during
processing. For comparison, similar characterization was conducted for a commercial UD tape
prepreg (Cycom 5320-1 IM7 12K 145gsm 33% RW UD, Solvay). The commercial prepreg
characterized will be referred to as “control” prepreg.
Resin distribution. A digital light microscope (VHX-5000, Keyence) was used to
measure resin distribution on the surface of uncured plies of semipreg. Resin feature size was
determined by measuring the width of each resin strip at twenty randomly selected locations
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(across four different plies) and averaging. To measure resin penetration into the fiber bed,
samples were “cold-cured” at room temperature in an ammonia vapor bath for 10 days [60].
After cold-curing, samples were sectioned, polished, and imaged.
Through-thickness permeability. Through-thickness permeability was measured for
semipreg, dry UD carbon tape (#2583, FibreGlast), and control prepreg using a custom fixture
[61]. For each test, a single ply was laid over a cavity (supported by a honeycomb insert) and
held in place by vacuum sealant tape. Vacuum sealant tape overlapped all sides of each ply to
minimize (in-plane) gas transport through ply edges during testing. Perforated release film,
breather cloth, and a vacuum bag—consumables typically used in VBO prepreg processing—
were then laid up over test samples. The test fixture featured two pressure sensors (PX32B1,
Omega): one connected to the cavity (on the “cavity side” of the sample) and one connected to
the volume between the vacuum bag and the sample (“bag side” of the sample).
The permeability coefficient for each sample was determined using falling pressure
tests, using Darcy’s Law to describe gas flow through a porous fiber bed [62,63]. A vacuum
port in the text fixture was used to apply vacuum to the bag side of the test article, while the
pressure was recorded on each side of the sample. Pressure recordings were terminated when
pressure on the cavity side of the sample stabilized within 1% of the pressure measured on the
bag side of the sample or after 16 hours.
Rheology. A parallel plate rheometer (AR 200ex, TA Instruments) was used to measure
resin film viscosity. Measurements were performed over a cure cycle beginning with a 1.5°
C/min ramp from room temperature to 121° C, followed by a dwell at 121° C until resin
gelation occurred. This cure cycle corresponds to the cycle used to cure semipreg laminates and
control material (Section 4.2.3). Equivalent viscosity data for the resin comprising the control
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material could not be measured (neat resin was not available). However, the material has been
studied previously, and a model published by Kim et. al. was used to estimate the viscosity
profile for the control resin for comparison of resin flow properties between semipreg and
control prepreg [64].
4.2.3. Laminate Fabrication
Two studies were conducted. A Process Reliability Study was designed to determine the
relationship between through-thickness gas permeability and process reliability by comparing
(unmodified) semipreg against the control prepreg under challenging but commonly
encountered processing conditions. A Void Formation Study was intended to determine the
relationship between resin topography and void formation by comparing unmodified semipreg
against semipreg that was modified as described in Section 4.2.1. The following subsections
detail material and process conditions for the laminates produced.
Process Reliability. Ten laminates were produced for this study: five from semipreg and
five from control prepreg. Except where otherwise noted, all laminates: (1) were cured using
standard consumables (edge breathing dams, perforated release film, breather cloth, vacuum
bag) on a polished aluminum caul plate coated with release agent (Frekote 770-NC, Henkel);
(2) measured 140 x 140 mm; (3) followed a [0°,90°]2S (for semipreg) or [0°2,90°2]2S (for control
prepreg) stacking sequence; and (4) were cured using the baseline cure cycle. Laminates from
the control prepreg were laid up with twice the number of plies as semipreg laminates of similar
thickness (~3.2 mm) from the two materials. The baseline cure cycle was defined according the
recommended cure cycle (for the control prepreg) and consisted of a four-hour room
temperature vacuum hold (RT-VH), followed by a ramp at 1.5 °C/min to 121 °C, a two-hour
dwell at 121 °C, and a ramp at -1.5 °C/min to room temperature.
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One “Baseline” laminate was produced under baseline conditions for each prepreg.
Laminates were also cured from each material under modified baseline conditions, each
simulating a challenging scenario commonly encountered in industrial practice. For “No RT
Hold” laminates, baseline conditions were modified to remove the four-hour RT-VH. For
“Sealed Edges” laminates, baseline conditions were modified by replacing edge breathing dams
with vacuum sealant tape placed against and over the edges of the layup. For “Humidity
Exposed” laminates, prepreg was conditioned at 90% relative humidity at 35 °C for 24 hours
prior to layup and cure. For “Ply Drop” laminates, larger plies were used (229 x 229 mm) and
additional plies (90 x 90 mm) were added at the midplane. Embedded plies were laid up
according to [0°,90°]S and [0°2,90°2]S for semipreg and control laminates, respectively.
Void Formation. Four laminates were produced, one from each of the four formats of
unmodified and modified semipreg. Each sample was processed identically: using the baseline
cure cycle, and a 100 x 100 mm, two-ply, [0°, 90°] layup was cured from each of semipreg,
SPA, SPB, and SPC. Laminates were cured on a glass tool plate coated with release agent using
standard consumables. The interface between the tool plate and the prepreg was monitored and
recorded during cure using a digital camera.
4.2.4. Laminate Characterization
Bulk porosity. The bulk porosity was measured for each laminate as part of the Process
Reliability Study. Two measurements were performed on polished sections (200 x 3.2 mm)
from the center of each laminate and averaged to produce an indicator of laminate quality.
Porosity was determined for each section by binarizing the images to distinguish pores, then
dividing the pore area by the total cross section area of the laminate.
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Micro-CT. A 24 x 62 x 3.2 mm volume of cured semipreg was imaged with high-
resolution x-ray microtomography (XT H 225ST, Nikon) using Mo-Kα incident radiation
(𝜆 =0.071 nm, 50 kV/400 𝜇 A). The scan yielded material density data at a resolution of 1.086
µm per voxel. Software (Visual Studio Max) was used to analyze microtomography data (1) to
corroborate bulk defect measurements, (2) to visualize the size, shape, and distribution of voids,
and (3) to identify the presence and distribution of resin-rich regions.
Surface quality. Surface quality was measured for all laminates. The percentage of the
tool-side surface covered in defects was determined by imaging the laminate, binarizing the
images to distinguish between defective and defect-free areas, and calculating the percentage of
the surface with defects. A digital microscope (Edge AM7815MZTL, Dino-Lite Digital
Microscope, USA) was used to capture images of the entire surface of each laminate (20x
magnification) with each image corresponding to a 25.4 x 25.4 mm region. The representative
surface defect content for each laminate was determined by averaging the defect levels across
all images of the same laminate.
4.3. Results and Discussion
4.3.1. Uncured Prepreg Characterization
Resin distribution. Measurements of resin strips on semipreg indicated that the mask-
and-press process produced an average resin strip width of 5.3 mm (standard deviation of 1.2
mm). The 5% increase in strip width over the mask geometry was attributed to resin film
stretching and tearing when excess film was removed after pressing. Each resin strip exhibited
distinct edges, and strip surfaces were smooth and roughly planar. Control prepreg was
characterized by continuous resin films on each ply surface. No dry fibers were exposed on
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control prepreg plies and resin film exhibited greater surface roughness than observed in
semipregs.
As shown in Figure 4.2, cold-cured samples indicated no resin penetration into the fiber
bed of uncured semipreg (zero initial degree of impregnation). By contrast, the uncured control
material was 5-100% impregnated, depending on location.

Figure 4.2: Micrographs of cross sections from cold-cured semipreg and control prepreg at equivalent scale.

Modified Resin Topography. Samples of semipreg pressed with textured release paper
retained periodic resin distribution but changes in strip cross section or surface features were
observed. Figure 4.3 shows surface height maps of uncured and modified semipreg samples,
where bluer hues indicate lower regions of the surface (thinner regions of the ply), and yellower
hues indicate higher regions (thicker). Figure 4.3 also shows diagrams approximating each resin
strip cross section. Plies of SPA, created by pressing samples of semipreg between smooth
sheets of release paper, exhibited smoother transitions in ply thickness from dry fiber to resin
strip compared to the step-like change in ply thickness in semipreg. Resin strip morphology
changes also resulted in average resin strip widths ~10% wider in SP A than semipreg. Resin
strips on SPB, created by pressing samples of semipreg between diamond-textured release paper,
exhibited periodic diamond-shaped depressions on the resin strips. Resin strips on SPC, created
by pressing samples of semipreg between crosshatch-textured release paper, exhibited
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crosshatched depressions on all resin strips. These depressions, however, generally exhibited
more rounded edges than on diamond-shaped features on SPB. No change in average strip
width was observed after modification of semipreg to SPB. or SPC.

Figure 4.3: Morphology of modified and unmodified semipreg resin strips with diagrams showing resin strip cross section for
each.

Z permeability. Gas transport in the through-thickness (z) direction was most rapid
through dry fibers, reaching a pressure difference across the ply of < 1 kPa in ~ 30 s. Gas
transport was similarly rapid in the through-thickness direction of semipreg. The presence of
resin strips slowed gas transport, but a pressure difference of < 1 kPa across the ply was
observed after ~ 200 s. The continuous resin films covering both sides of the fiber bed in
control prepreg slowed the transport of gas in the z-direction. Falling pressure tests were
terminated after 16 hours, at which time the pressure difference across the ply was ~ 25 kPa.
Test data yielded permeabilities of 4.2E-9 m
2
, 7.1E-10 m
2
, and 1.5E-16 m
2
for dry
fibers, semipreg, and control prepreg, respectively. Control prepreg permeability measurements
corroborate prior work characterizing the permeability of commercial VBO prepregs
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[59,63,65,66]. Differences in through-thickness permeability between control and semipreg
demonstrate the effectiveness of interrupting the continuity of resin films common to VBO
prepregs for increasing the efficiency of gas transport in the through-thickness direction of a
prepreg ply.
Rheology. Figure 4.4 shows resin viscosity profiles as a function of time for the relevant
period of the baseline cure cycle. The control prepreg resin reached a lower minimum viscosity
(2.84 Pa s) during processing than resin comprising semipreg (5.90 Pa s) and remained at lower
viscosity for a longer period than the resin comprising the semipreg.

Figure 4.4: Resin viscosity for semipreg and control prepreg resins.

4.3.2. Process Reliability Study
The robustness of semipreg against commonly encountered adverse processing
conditions was compared against that of the control prepreg by evaluating the quality of the
laminates produced from each material.
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Bulk porosity. Average bulk porosity measurements for semipreg and control prepreg
are shown in Figure 4.5, where error bars indicate standard deviations. Under all processing
conditions, semipreg laminates exhibited bulk porosity of < 2%, a common pass/fail threshold
for aerospace parts [67]. The laminate produced from humidity-conditioned semipreg had the
lowest bulk porosity at 0.30%, closely followed by the Baseline laminate at 0.46% and the Ply
Drop laminate at 0.51%. Void content was greatest in No RT Hold and Sealed Edges samples.

Figure 4.5: Bulk defect content for semipreg and control prepreg laminates produced under baseline and a variety of adverse
conditions.

Control prepreg cured under baseline conditions exhibited the lowest void content of any
sample at 0.05%; however, the void content of laminates produced from control prepreg was
more sensitive to processing conditions than laminates produced from semipreg. Eliminating
the four-hour RT-VH from the baseline cure cycle increased bulk porosity to 0.81%. Void
content in control prepreg laminates measured > 2% for Humidity Exposed (2.61%) and Ply
Drop (3.36%) samples. Void content in control laminates was most sensitive to occlusion of the
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in-plane evacuation pathways: the Sealed Edge control laminate exhibited nearly 8% porosity
(7.8%).
The location of bulk defects in the Baseline semipreg laminate correlated with the
locations of resin strips on constituent plies. Microtomography data were used to generate a
map of the projected locations of all voids in the baseline semipreg laminate (Figure 4.6A).
Images in Figure 4.6 are oriented such that the structure is viewed in the through-thickness (z)
direction. Voids in semipreg laminates were distributed along a square grid (Figure 4.6A) with
a period roughly equal to the resin strip spacing.

Figure 4.6: Microtomography data from the Baseline semipreg laminate highlighting (A) all pores, (B) resin rich volumes
between plies 2 and 3, and (C) pores between plies 2 and 3.

The relationship between resin strip location/orientation and void location was
determined using the microtomography data to isolate the 0.3 mm thick interface region
between plies 2 and 3 in the layup. Figure 4.6B highlights a projection of the resin-rich regions
in this volume, and a projection of the voids in the same volume is pictured in Figure 4.6C.
Voids are distributed along horizontal lines that correspond roughly to the location of the
horizontal resin strips.
Surface porosity. Average surface porosity measurements for semipreg and control
prepreg are shown in Figure 4.7, where error bars indicate standard deviations. Under all
processing conditions, semipreg laminates exhibited defects on no more than 0.95% of the
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surface. Like bulk porosity results, the laminate produced from humidity-conditioned semipreg
exhibited fewer surface defects than any other laminate produced from semipreg (0.20%).
Large error bars on the data in Figure 4.7 indicate that surface quality was locally variable for
all semipreg laminates, and the relative difference in quality between Baseline, No RT Hold,
Sealed Edges, and Internal Ply Drop semipreg samples was negligible. The surface porosity of
semipreg samples was insensitive to all processing conditions studied.

Figure 4.7: Surface defect content for semipreg and control prepreg laminates produced under baseline and a variety of sub-
optimal conditions.

Despite producing the laminate with the fewest surface defects (0.05%), the surface
quality of parts fabricated from control prepreg was strongly influenced by variations in process
conditions. Omission of the four-hour RT-VH increased average surface porosity by 0.36% (to
0.41%) over the Baseline control laminate. Surface defect levels in laminates produced from
control prepreg were strongly sensitive to occlusion of gas evaluation pathways (Sealed Edges),
introduction of moisture (Humidity Exposed), and the addition of an internal ply drop (Ply
Drop). Average surface porosity for these processing conditions was between 9.9% and 18.5%.
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Inspection of semipreg laminate surfaces revealed two distinct categories of surface
defects. Figure 4.8 shows an image of the surface of the Ply Drop semipreg laminate with each
type of defect indicated. Type 1 defects were each located near the midline of a resin strip, and
their distribution was approximately periodic (10 mm period) along and across resin strips.
Type 2 defects were distributed along lines between adjacent strips. Individual Type 2 defects
were typically much smaller than individual Type 1 defects, but collectively accounted for 45-
75% of total surface defect content. On all samples except for the Humidity Exposed laminate,
Type 2 defects covered 0.3-0.5% of the laminate surface. For the Humidity Exposed laminate,
however, Type 2 defect content was only one tenth that of other samples (0.08%). The
difference in size, location, and distribution between Type 1 and Type 2 defects indicated that
different mechanisms govern the formation of each void type. Such mechanisms are explored
in Section 4.3.3.

Figure 4.8: Image of the surface of the Ply Drop semipreg laminate indicating each of the two types of surface defects found in
semipreg laminates.

88

Discussion. Bulk and surface porosity measurements demonstrate that semipreg is less
sensitive to deviations from baseline process conditions than control prepreg. The quality
difference between semipreg and control laminates was attributed to the presence of through-
thickness gas evacuation pathways present in semipreg, which increased the speed of gas
removal. Compared to control prepreg, semipreg layups had (1) more paths and (2) shorter
paths (≤3.2 mm along z-axis vs. ≤70 mm along x- or y-axis) for removal of entrapped and
evolved gases.
Differences in flow properties of semipreg and control resins emphasize the importance
of prepreg format. Semipreg resin viscosity remained higher and gelled more rapidly than
control resin, leaving less time (vs. control) for gas evacuation, resin flow, and fiber bed
saturation. Resin rich regions remaining on the surface and interior of each sample indicate that
resin flow did not reach equilibrium in semipreg laminates prior to gelation. The difference
between flow properties make the processing window narrower for semipreg than the control
resin, yet semipreg laminates were less sensitive to adverse processing conditions than control
laminates. In principle, a semipreg produced with the control resin would be expected achieve
more uniform resin distribution and fewer flow-related defects (see Type 2 void discussion
below).
The results corroborate previous work indicating the importance of prepreg format in
process robustness and expand on the prior work to provide evidence that the process robustness
associated with through-thickness permeability is not limited to woven prepregs that have fiber
tow crossings.

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4.3.3. Void Formation Study
The primary void formation mechanisms for semipreg were visualized and identified,
and their relationship with resin feature topography was investigated by characterizing the
surface quality of unmodified and modified formats of semipreg.
Defect formation. Figure 4.9 shows a series of images of the glass tool plate-prepreg
interface. Images capture the formation and movement of both types of voids observed in
semipreg laminates.

Figure 4.9: Images of semipreg at the tool-prepreg interface for select times during layup and processing.

Type 1 voids. Type 1 voids were formed by entrapment of gas between resin strips and
the tool plate. Figure 4.9 shows thin (in the z-direction) pockets of gas present at the interface of
resin strips and the tool plate immediately after layup. During the four-hour RT-VH, the shape
and location of these gas pockets evolved: voids tended to move toward the center of resin
strips, decrease in area, and increase in depth. Upon heating, further distortion of voids
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occurred according to the same trend until gelation. Once formed during the initial layup and
vacuum hold, entrapped gases at the tool-resin strip interface remained throughout cure and
appeared in the laminate as Type 1 voids.
Type 1 defects were distributed quasi-periodically: along the length of each resin strip
and transversely aligned across strips. Periodicity and alignment indicate that movement of
entrapped gases during cure was determined by resin distribution (e.g., strip width, orientation,
and spacing). Defect formation is attributed to local variations in compaction that arise from
step-like variability in local ply thickness. Compaction forces are greatest along the strip edges
(due to fiber bridging), and such forces are additive at the intersection of resin strips on adjacent
plies. Intersections of orthogonal resin strips on adjacent plies create areas of greater
compaction force that drive entrapped gases to the respective midline of each strip. This
mechanism explains why Type 1 voids occurred primarily in locations corresponding to the
intersection of strips on the tool side ply and its nearest neighbor.
The nature of gas entrapment and migration that produces Type 1 voids in semipreg
laminates indicates that two mitigation strategies may be effective: (1) minimizing the volume
of gas initially entrapped between resin features and (2) modifying resin features such that stress
concentrations drive voids away from resin strip midlines.
Type 2 voids. Type 2 voids were located at interfaces of resin flow fronts between each
adjacent resin strip. Figure 4.9 shows the degree of fiber bed saturation between adjacent resin
strips at various times during processing. In these images, Type 2 voids are observed to result
from insufficient resin flow into areas of the fiber bed between strips.
Multiple factors can contribute to the formation of Type 2 defects. First, semipreg resin
may not have had an appropriate rheological profile to enable reliable saturation of the region
91

between resin strips. Type 2 defects were observed in regions of the laminate furthest away
from the initial position of resin. Such regions require the most time for resin infiltration and
may remain unsaturated if gelation occurs too rapidly. Surface defect measurements show that
a prepreg with shorter resin flow distances produces laminates with lower Type 2 defect
content: Type 2 defect content was roughly an order of magnitude less for the Humidity
Exposed laminate than for any other semipreg sample. During conditioning of the Humidity
Exposed semipreg, resin flow widened each resin strip, reducing the distance between strips by
approximately 50% (Figure 4.11, discussed more thoroughly below) prior to layup. The
correlation between resin flow distance and Type 2 defect content supports the hypothesis that
Type 2 defects are caused by insufficient flow prior to gelation.
An alternative or additional explanation for the formation of Type 2 defects is that
evacuation pathways may have been sealed prior to full removal of gas from the layup. Once
evacuation paths are saturated with resin, any remaining gas in a sealed (but unsaturated)
volume of the fiber bed would remain entrapped. When unsaturated regions shrink during
impregnation, gas pressure increases until an equilibrium is reached, halting impregnation prior
to full saturation and resulting in voids. Finally, the formation of Type 2 defects may have also
been influenced by the local variations in compaction pressure in the layup created by step-like
discontinuities in ply thickness. Ply thickness variations are anticipated to create regions of
relatively lower compaction pressure in areas of prepreg plies between resin strips. The
occurrence of Type 2 voids may be explained by insufficient local compaction forces between
resin strips at the ply-tool interface. More work is needed to confirm the operation and relative
importance of these mechanisms.
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The nature and evolution of the unsaturated regions resulting in Type 2 defects in
semipreg laminates indicates that defect levels can be reduced by (1) selecting a resin system
with lower viscosity and/or delayed gelation or (2) reducing the distance between resin strips.
Resin topography and surface porosity. Figure 4.10 presents surface porosity data for
semipreg, SPA, SPB, and SPC laminates. Modifying the cross-sectional shape of resin strips to
resemble a bell curve vs. a rectangle (SPA) and modifying the resin surface topography to
contain gas evacuation channels (SPC) both reduced the number of defects relative to the
laminate produced from unmodified semipreg. Process monitoring verified that void reduction
was caused by less air entrapment (vs. semipreg) at the tool-strip interface during layup of SPA.
Further, stress concentrations created by locally variable ply thickness were not as extreme in
SPA (vs. semipreg) because of the rounding of strip edges, which reduced the tendency to
immobilize gases in the center of strips. In the case of SPC, entrapped gases were near
evacuation channels created by the interconnected network of depressions embossed on resin
strip surfaces. These depressions acted as gas removal channels, resulting in fewer, smaller
surface defects in SPC laminates than observed for semipreg laminates. Pressing semipreg with
smooth (SPA) and crosshatch (SPC) release paper resulted in laminates with equivalent surface
quality of 0.08% and 0.12%, respectively.
93

Figure 4.10: Surface defect content and resin strip cross section schematic for unmodified and modified semipreg laminates
produced under baseline conditions.

Semipreg modified by pressing with diamond-textured release paper (SPB) produced the
laminate with the highest void content (1.54%). The roughly threefold increase in surface
porosity compared to semipreg was attributed to the resin strip topography: diamond shaped
impressions left by release paper in resin strips on SPB (1) increased the volume of air entrapped
during layup and (2) immobilized entrapped gases by sealing the resin strip-tool interface,
creating distinct, unconnected voids. Entrapped gases were never effectively removed during
processing and resulted in Type 1 voids in the cured laminate.
Discussion. Observations of resin flow during processing provided insight into the
mechanisms involved in formation of surface defects. Type 1 voids were produced from gases
entrapped at the resin strip-tool plate interface during layup and never removed. Type 2 voids
were related to the flow properties and spacing of resin features on semipreg. Resin strip
topography played a critical role in the formation of Type 1 voids, and changes in resin strip
94

cross section and the embossment of interconnected evacuation channels onto resin strip
surfaces were both effective methods to reduce surface porosity.
The mechanisms by which surface defects form, as identified here, may also cause bulk
defects. The correlation between bulk defect location and resin strip distribution, as revealed by
microtomography, supports this hypothesis. Both internal and surface defects had periodicity
associated with the spacing of resin strips. Bulk defects can be produced via a mechanism
analogous to that which produce Type 1 surface defects: instead of gas entrapment occurring at
the tool-resin strip interface, it occurs at the interface of resin strips on adjacent plies.
The assertion that similar mechanisms produce surface and bulk defects is supported by
bulk porosity data from Humidity Exposed semipreg. The reduction in surface defect content
between Humidity Exposed and Baseline semipreg was initially attributed to an assumed
reduction in resin tack, which has been previously correlated with surface porosity [68]. Such
phenomena, however, do not fully explain the observed reduction in bulk porosity (relative to
Baseline semipreg), where moisture absorbed during humidity conditioning was anticipated to
volatilize during cure, increasing void content over Baseline semipreg.
Rather than tack reduction, changes in resin strip topography were responsible for
minimal defect content in Humidity Exposed semipreg. Figure 4.11 shows micrographs and
height maps highlighting changes in semipreg format resulting from exposure to elevated
temperature and humidity. Softened strip edges and 50% increased strip width indicates resin
flow occurred during conditioning. Furthermore, smooth strip surfaces were replaced with
rough, uneven surfaces. Such features were identified in the Void Formation Study as
associated with a reduction in defect content. Strip morphology changes during conditioning
provide a compelling explanation for the unexpected quality of Humidity Exposed semipreg
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laminates. These results indicate that resin feature topography can be a more significant factor
in determining void content than the presence of excess moisture (due to improper handling or
storage).

Figure 4.11: Comparison of resin strips on semipreg before and after conditioning at 35 °C and 90% relative humidity for 24
hours.

4.4. Conclusions
This work demonstrated a method for producing semipreg with periodic distributions of
resin. UD prepreg with resin strips orthogonal to the fiber direction enhanced through-thickness
gas transport. Compared to a commercial hot-melt OoA prepreg, defect content in laminates
cured from in-house produced prepreg was less sensitive to deviations from baseline cure
conditions. The results indicate the efficacy of through-thickness permeability (and short egress
pathways) for removal of entrapped or evolved gases from UD prepreg during cure. The
evidence indicates that benefits imparted by high through-thickness permeability previously
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observed in woven fabrics do not require tow overlaps or underlaps, and such formats are
similarly effective in UD fiber beds.
Process robustness was particularly notable, particularly because no effort was made to
optimize resin rheology or resin feature geometry (e.g., strips vs. grid), including spacing (i.e.,
periodicity of strips or grid) and topography (i.e., resin feature cross section and surface
texture). Such opportunities, however, were identified in the Void Formation Study.
In situ process monitoring revealed the nature and origin of surface porosity in
semipregs, and two distinct types of surface voids were identified. The mechanisms by which
each void type formed was explained, attributing voids to either gas entrapment by resin
features or insufficient resin flow during processing. Morphology of resin features played an
important role in these mechanisms. Changes to resin strip cross section and the addition of
evacuation pathways on resin feature surfaces each reduced defect content compared to
unmodified semipreg. Microtomography data supported the hypothesis that similar defect
formation mechanisms create both surface and bulk porosity, but further work is required to
prove this.
Broadly, this work represents a step toward addressing the need for rapid, flexible,
robust, and cost-effective high-performance composite manufacturing techniques. Growing
demand for composite structures, limited availability of autoclaves, and the inherently
unsuitable nature of autoclave processing in some scenarios (e.g., in-field repair, large
structures) continue to drive the need for manufacturing methods with improved technical, cost,
and environmental efficiency. The approach presented here effectively transfers the robustness
of autoclave cure into the material itself, and further development will accelerate the adoption of
VBO prepregs.
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Processability of Overaged Prepreg

5.1. Introduction
Many high-performance composite structures are manufactured from thermoset prepregs
cured in autoclaves or ovens. Thermoset prepregs are comprised of a fiber reinforcement (e.g.,
carbon fiber, fiberglass, Kevlar) and a partially-cured (B-staged) resin (e.g., epoxies,
bismaleimides, polyimides) [69]. Prepreg is largely produced via a hot melt process. In this
process, resin is first mixed (catalyzed) and processed into a film of uniform thickness (for
controlled areal weight). Resin films are then laminated on one or both sides of a woven or
unidirectional fiber bed at which time they are heat treated under compaction to control the
degree of resin impregnation into the fiber bed. For prepregs intended for use in out-of-
autoclave/vacuum-bag-only (OoA/VBO) processing scenarios, the result is a fiber bed covered
on both sides by resin film with a region of dry fibers near the midplane of the ply.
The irreversible crosslinking reaction that ultimately gives thermoset composite
structures their mechanical strength is initiated in prepreg resin as soon as it is mixed, even prior
to resin being deposited on a fiber bed [70]. The rate of the crosslinking reaction is determined
by the temperature of the resin. During processing, elevated temperature is used to control and
advance the cure reaction such that crosslinking is completed and prepreg is cured into the
desired composite structure. Prior to processing, however, the resin is continuously
crosslinking at a rate determined by the storage temperature of the prepreg.
As the resin cure reaction progresses, it results in a continuous change in
physicochemical properties of the resin. These properties can produce deviations from intended
handling and processing behavior that complicate prepreg use and result in higher part rejection
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rates [71–73]. Handling issues associated with overaged prepreg include difficulty precisely
cutting ply geometries, insufficient resin tack for layup, and reduced ply compliance (which
makes layup on complex surfaces difficult). Some of these issues can be overcome. For
example, automated cutting and layup systems can achieve high precision ply geometries and
alignments. Furthermore, layup technicians have demonstrated that local heating (e.g., with a
heat gun) during layup can increase tack and ply compliance. These approaches may minimize
the challenges in handling overaged prepreg, but they are nonetheless burdensome and may
require additional equipment.
Physicochemical changes to processing behavior present further challenges during cure
of overaged prepreg. Room temperature aging has been correlated with changes in the
rheological properties of resin during cure [64,74]. Specifically, overaged resin has been
observed to remain at higher viscosity than less aged resin throughout cure. Increased resin
viscosity has been identified as a primary cause of microstructural defects created by
insufficient resin flow into dry regions of the fiber bed [71]. Once deviations from processing
behavior due to resin aging are too severe, the laminates produced from such prepreg will not be
fully consolidated and must be rejected.
Because the aging state of prepreg resin is a critical parameter for assuring the
production of defect-free laminates, prepreg manufacturers provide recommendations for the
allowable out time (time out of the freezer at room temperature) and storage time (time stored in
the freezer). Prior to the expiration of these times, composite manufacturers have reasonable
assurance that handling and processing behavior will complement production of defect-free
laminates using the manufacturer’s cure cycle (MRCC). Beyond the expiration of these times,
there is no assurance that laminates manufactured will achieve mechanical performance
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properties advertised by the prepreg producer. Prepreg producers provide different out life
(allowable time out of the freezer at room temperature) and storage life limits to account for the
temperature dependence of the crosslinking reaction occurring in prepreg resin during storage.
Typically, aerospace prepreg out life and storage life (allowable time stored in the freezer) range
from 14 to 30 days and 6 to 12 months, respectively.
Use of overaged material can lead to increased part rejection, and manufacturers often
choose to simply discard time-expired materials rather than dedicate additional resources to
processing and inspection. Unfortunately, there is presently no effective or economical
approach for reusing, recycling, or otherwise disposing of expired prepreg, and the expiration is
directly responsible for significant loss in economic value and an unnecessary environmental
impact. The initial creation of carbon fibers and uncured resin constituents represents most of
the environmental cost of producing composite structures, and commercial methods are not
presently available for recapturing or recycling this investment [75–77]. Furthermore, time-
expired prepreg that must be disposed of in landfills creates an additional economic and
environmental hazard, especially if prepreg resins are not fully cured prior to disposal,
necessitating special treatment due to the reactive nature of the material.
Industrial practices for monitoring prepreg age accumulation typically rely on tedious,
manual, roll by roll tracking of exposure time at various temperatures; manual tracking is
burdensome and scientifically naïve because it does not rely on direct measurement of prepreg
state. Not only must each roll of prepreg be independently tracked, but tracking must persist as
prepreg is cut from the roll, kitted, laid up, and ultimately cured to assure that out life is not
exceeded. Prepreg is considered usable when resin has not yet exceeded out life or storage life,
and unusable if it has, but the discrete nature of this metric does not map well onto the
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continuous changes in resin chemistry that are occurring as a function of aging. This issue is
complicated by the variability of “room temperature” in various manufacturing environments,
which can result in prepregs with different degrees of resin crosslinking from facility to facility
for the same amount of out time (or seasonally for the same facility).
Some effort has been made to leverage the internet of things to minimize the time and
cost by automating prepreg tracking. One solution uses RFID chips to monitor and log
environmental conditions. Tags that respond to environmental stimuli and indicate when
prepreg has gone bad have also been explored. Neither of these approaches, however, directly
measure the state of the prepreg itself, nor do they provide a continuous framework assessing
the processability of a continuously aging material.
Recent work has demonstrated that prepreg age can be monitored via direct
measurement of prepreg resin [74]. Several properties of partially cured prepreg resin can be
directly measured and have been shown to change as a function of prepreg out time. Expiration
of prepreg out life has been shown not to correspond with a precipitous change in such resin
properties—instead, resin properties change gradually as out life accumulates. In work by Kim
et al., cure kinetics, rheologic behavior, and sub-ambient glass transition temperature (Tg), were
shown to evolve progressively as a function of accumulated time at room temperature
[64,74,78,79].
Other work provides insight as to how defect formation mechanisms operate in overaged
prepreg may be overcome. Centea et al. showed that cure cycles that correspond to lower resin
viscosity resulted in fewer flow-related defects (the predominant type of defect observed in
overaged prepreg) [80]. Kim et al. demonstrated that increasing cure cycle ramp rates and hold
temperatures can reduce viscosity in overaged prepregs, and showed that laminates with a
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defect-free microstructure can be produced from prepregs with up to 49 days of out time
(>160% of manufacturer’s recommended out life) [81].
Despite all this work, questions yet remain. The preponderance of previous work has
examined prepreg state or processability strictly as a function of out time, but almost all prepreg
will have a complex aging state that is the result of aging at various temperatures. In
manufacturing environments, a roll of prepreg may be removed and returned to the freezer
several times over the course of months. It is important to understand the effect of prepreg
storage at (1) room temperature, (2) freezer temperature, and (3) arbitrary combinations of
relevant temperatures. Furthermore, defect formation mechanisms operational in prepreg with
each of these conditions should be identified such that mitigation strategies can be explored. It
is also important to characterize mechanical performance of prepregs as a function of age and
confirm whether the mechanical performance of overaged prepregs can be recovered when
prepreg is processed using modified cure cycles.
In this study, the aging behavior of a commonly used aerospace prepreg is studied.
Laminates were produced from prepregs aged under in and out of the freezer to identify the
onset of age induced processing and performance issues. Where possible, the mechanism by
which laminate performance is compromised for each aging condition is identified and
discussed. Insight into defect formation mechanisms is used to develop a method for cure cycle
modification, which is subsequently demonstrated. A laminate is produced from prepreg with
18 months of freezer time and 44 days of out time, tested, and compared against laminates
produced from prepreg with different aging history. The primary contributions of this work are
(1) showing that the primary mechanism by which prepreg expires when stored at room
temperature is not the same mechanism by which it expires when stored in the freezer, (2)
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developing and demonstrating a method for cure cycle modification on the basis of a direct
measurement of prepreg state, and (3) using mechanical performance measurements in addition
to microstructure images to quantify performance of laminates made from over aged prepreg.
5.2. Methods
All work for this study was conducted using a common DoD-qualified prepreg (Cycom
5320-1 IM7 12K 145gsm 33% RW UD, Solvay). At the time of this study, the MRCC for this
material recommended a range of ramp rates and hold temperatures. A cure cycle from the
middle of these ranges will be referred to nominally as the MRCC throughout this study. The
nominal MRCC used in this study was defined according to the following steps: (1) four hour
room-temperature vacuum hold, (2) ramp at 1.5° C/min to a super-ambient dwell temperature of
70° C, (3) hold at the super-ambient dwell for 2 hours, (4) ramp at 1.5° C/min to the gelation
temperature of 121° C, (5) hold at the gelation temperature for 2 hours, (6) ramp at 1.5° C/min
to the post-cure temperature of 177° C, (7) hold for 2 hours at the post-cure temperature, and (8)
ramp at -1.5° C/min to room temperature.
Due to limited material availability, the prepreg used for this study had already been
stored in the freezer for 12 months at the time that it was received by the researchers. As such,
all of the aging states discussed in this study are complex, in that they are the outcome of
combined freezer and room temperature storage conditions (typically 12-24 months of freezer
storage followed by 0-77 days of room temperature out time). In subsequent sections, prepreg
and laminate samples will be referred to using the nomenclature FxRy, where x is the number of
months stored in the freezer, and y is the number of days of out time accrued prior to layup.

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5.2.1. Prepreg Characterization
Samples of prepreg were stored at room temperature and in the freezer to monitor
differences in aging as a function of storage condition. Prepreg stored at room temperature was
characterized via differential scanning calorimetry (Q2000, TA Instruments) once the material
was received (F12R0) and every seven days for 11 weeks. Prepreg stored in the freezer was
similarly characterized every 30 days for 12 months. Sub-ambient glass transition temperature
was extracted from DSC data and used as a metric for quantifying prepreg age.
5.2.2. Laminate Manufacturing and Characterization
Laminates were produced from room temperature and freezer aged prepreg on the same
schedule as prepreg characterization work was conducted (weekly for room temperature aged,
monthly for freezer stored). All laminates were cured in an oven (DC-1406C, Thermal Product
Solutions) using the manufacturer’s recommended cure cycle and standard VBO consumables.
A layup sequence of [02, 452, 902, -452]s was used in all cases, and resulting laminates measured
approximately 165 x 165 x 2.3 mm.
After cure, laminates were sectioned, mounted, polished, and imaged at 150x
magnification using a light microscope (VHX-5000, Keyence). Two 20 x 2.3 mm cross
sections were prepared from each laminate. Porosity was measured in each micrograph by
binarizing the images to distinguish pores, then dividing the pore area by the total cross section
area of the laminate. Void content was averaged between the two samples from each laminate
to produce an indicator of laminate microstructure quality.
Samples were also taken from each laminate for mechanical testing. Short beam shear
(SBS) testing was conducted (in accordance with ASTM D2344/D2344M – 16) using a tabletop
load frame (5567 Load Frame, Instron). Six SBS samples were tested from each laminate. A
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precision waterjet cutter (ProtoMAX Compact Waterjet, OMAX) was used to remove samples
from each laminate. Samples measured approximately 13.9 x 4.49 x 2.26 mm and were
oriented such that the fiber direction on the top and bottom plies was parallel to the longest edge
of the sample. The span of the test fixture was fixed at 9.2 mm for all samples. Short beam
strength was averaged across all six samples to determine a representative metric of mechanical
performance for each laminate.
Laminate microstructure and mechanical performance as a function of prepreg storage
condition were examined to determining the onset of age-related processing or performance
issues. Where possible, the mechanisms resulting in these issues were identified such that
appropriate steps to address them can be developed and demonstrated.
5.2.3. Cure Cycle Tailoring
An approach was developed to modify the MRCC to extend prepreg storage life. The
basis of this approach relies on controlling the Effective Flow Number of the prepreg resin.
Effective flow number is defined according to

𝑁 𝐹𝑙 ,𝑒𝑓𝑓 = ∫ 𝜂 (𝑡 )
−1
𝑑𝑡 𝑡 𝑔𝑒𝑙 0
((5.1)

where 𝜂 is resin viscosity. This relationship implies that viscosity profiles that have lower
minimum viscosities and remain at low viscosity for longer periods are characterized by
effective higher flow numbers. Because both parameters influence the ability of resin to flow
into dry regions of the fiber bed, higher effective flow numbers are associated with more
efficient fiber bed saturation. As prepreg ages at room temperature, its flow number increases
(because the initial degree of cure advances prior to cure and this is associated with increased
viscosity at all processing temperatures). In the approach described below, increasing the
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effective flow number by modifying the MRCC can address the primary phenomenon
responsible for void formation in room temperature aged prepregs (discussed in detail in section
5.3.1).
To mitigate the decrease in flow number observed for room temperature aged prepreg,
three processing parameters were considered for adjustment. A cure kinetics model developed
in [64] was used to estimate the relationship between these parameters and effective flow
number. The ramp rates of MRCC steps 4 and 6 were varied from 1.5° C/min to 40° C/min,
step 5 gelation temperature was varied from 121° C to 177° C, and super-ambient hold time was
varied from 0 to 2 hours. These ranges were fixed by the operational envelope of the processing
equipment available. Ramp rates in excess of 40° C/min were not feasible, and researchers
determined it was unsafe to allow gelation temperature to exceed the post cure temperature
recommended by the manufacturer.
A script was written to sequentially adjust these parameters until a modified cure cycle
was produced that achieved some effective flow number of interest. This script first increased
the ramp rate, then increased the gelation temperature, and then reduced the super-ambient hold
time. The script proposes cure cycle adjustments to each step only until the target flow number
was achieved or the limit of the allowable parameter range was reached.
The cure cycle modification tool accepts prepreg sub-ambient Tg as an input and
recommends a modified cure cycle that will achieve an effective flow number associated with
full fiber bed saturation. Prepreg age may exceed the manufacturer’s recommended out life so
long as the degree of cure has not advanced to the point that sufficient resin flow is impossible
within the available processing window.
5.2.4. Demonstration of Prepreg Life Extension via Cure Cycle Modification
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The cure cycle tailoring tool described in the previous section was used to generate a
modified cure cycle for production of a laminate from prepreg aged for 44 days at room
temperature. The laminate was laid up in the same stacking sequence as described in Section
5.2.2 with standard VBO consumables and cured using a heat blanket. After cure, the laminate
was analyzed for porosity and mechanical performance in the same manner as described in
Section 5.2.2.
5.3. Results and Discussion
5.3.1. Room Temperature Aging
Sub-ambient Tg. The sub-ambient Tg values extracted from DSC measurements are
shown in Figure 5.1 for all room temperature aged prepreg samples available. Glass transition
temperature for prepreg with zero out time was -2.35° C and increased to 54.1° C as room
temperature storage time accrued up to 28 weeks (196 days). Figure 5.1 shows that sub-
ambient Tg is a strong indicator of the out time accumulated by the prepreg studied in this
project.

Figure 5.1: Sub-ambient glass transition temperature of prepreg aged at room temperature.
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Laminate microstructure. Micrographs of the laminates produced from room
temperature aged prepreg are pictured in Figure 5.2. Laminates aged up to 28 days exhibited
nearly flawless microstructure (< 0.1% porosity) and uniform resin distribution. Beyond 28
days, laminate porosity increased as a function of out time and was predominantly located
within prepreg plies. The onset of porosity is correlated with the presence of resin rich regions
between each prepreg ply and between some adjacent fiber tows in the same ply. Resin rich
interply and intertow regions are visible in the micrographs of sample F12R42 and those with
additional room temperature aging in Figure 5.2. The timing of processing difficulty (porosity
onset) with respect to room temperature aging is well predicted by the manufacturer’s
recommended out life of 30 days.
Short beam strength. The average mechanical performance of laminates cured from
prepreg aged at room temperature is reported in Figure 5.3. In this figure, error bars indicate
standard deviation of short beam strength for the six samples tested, and labels indicate strength
normalized against the performance of the sample with the least accumulated age (F12R0,
referred to as the baseline case). The mechanical performance of samples F12R56 - F12R77 were
omitted because (1) mechanical integrity was insufficient to be reliably measured or because (2)
laminates did not survive SBS sample preparation (e.g., catastrophic delamination during water
jet cutting). It is reasonable to infer that such samples retained little useful mechanical
performance.

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Figure 5.2: Micrographs of laminates produced from room temperature aged prepreg using the MRCC.
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Figure 5.3: Average short beam strength of laminates aged at room temperature (labels indicate strength normalized against the
F12R0 case).
Figure 5.3 shows that for the first three weeks that prepreg accumulates room
temperature out time, short beam strength increases by 10-14%. Once prepreg has accumulated
28 day of out time, mechanical performance has declined back to the level of the F12R0 case.
When the manufacturer’s recommended out life was exceeded, mechanical performance
decreased: the laminate aged for five days beyond the manufacturer’s recommended out life
(F12R35) is characterized by 7% reduction in short beam strength compared to the baseline case.
Furthermore, samples F12R35 and F12R35 were characterized by 47% and 69% knockdowns in
short beam strength compared to baseline. Decreased short beam strength in overaged
laminates was associated with, and can be explained by, the presence and severity of porosity in
laminates.
Aging mechanism during room temperature storage. Sub-ambient Tg data and
micrographs shown in Figure 5.2 provide evidence of the mechanism by which room
temperature aging compromises the processability of the prepreg studied. As prepreg
accumulates out time at room temperature, the resin degree of cure slowly advances. As degree
of cure advances, resin viscosity increases. Increased resin viscosity has a two-fold effect on
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the flow of resin during subsequent cure. First, the viscosity minimum reached during
processing is higher than the minimum viscosity that would have been reached without room
temperature aging. Assuming compaction pressure is constant, increased viscosity slows the
resin flow rate into unsaturated fiber tows. Second, room temperature aged resin remains at a
low viscosity for a shorter period during processing than unaged resin (for the same cure cycle).
A shorter time at low viscosity, means that resin has a narrower processing window during
which fiber bed saturation must occur. The impact of both of these phenomena can be captured
in the resin effective flow number, which deceases for prepregs with increasing age due to these
viscosity changes. Ultimately, the consequence of increased resin viscosity (and corresponding
decrease in effective flow number) is that resin flow fails to partially or completely saturate dry
regions of fiber bed, leading to porosity and mechanical performance issues.
The understanding that resin degree of cure is advancing during room temperature
storage is supported by increases in sub-ambient Tg observed in Figure 5.1. The increase in
sub-ambient glass transition temperature indicates that a chemical change is occurring in the
resin during room temperature aging. Furthermore, previous studies have correlated sub-
ambient Tg with degree of cure for the same type of prepreg as used in this study [74]. That
insufficient resin flow is responsible for porosity in laminates is supported by Figure 5.2, where
it can be observed that as room temperature aging time increases, the uniformity of resin
distribution decreases. In this figure, it is not apparent that any resin flow occurred at all for
samples F12R70 and F12R77. In these samples, resin is observed in high concentrations between
plies, where it would have been originally deposited during prepregging.
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Inadequate gas was considered as an explanation for porosity in samples F12R35 and
beyond, but absence of voids in the interply regions of laminates and evidence outlined above
better support that insufficient resin flow is the primary void formation mechanism.
5.3.2. Freezer Aging
Sub-ambient Tg. No significant change in sub-ambient Tg was observed for prepreg
stored in the freezer for up to 20 months. Sub-ambient glass transition temperature for prepreg
with the baseline aging condition (F12R0) was -2.35° C, and samples stored in the freezer for up
to an additional 8 months (F13R0 - F21R0) ranged between -3.52° C and -1.16° C.
Laminate microstructure. Micrographs of the laminates produced from freezer aged
prepreg are pictured in Figure 5.4. None of the laminates produced from freezer aged prepreg
were observed to have significant porosity content (< 0.1%). Resin distribution uniform, and no
resin rich regions can be observed in laminates produced from freezer aged prepreg. Several
samples could not be produced due to disruption of lab access due to COVID-19. As material
continued to age while labs were unavailable, this data could not be generated once lab access
was restored.
Short beam strength. The average mechanical performance of laminates cured from
prepreg aged in the freezer is reported in Figure 5.5. In this figure, error bars indicate standard
deviation of short beam strength for the six samples tested, and labels indicate strength
normalized against the performance of the sample with the least accumulated age (F12R0,
referred to as the baseline case). Several samples could not be produced due to disruption of lab
access due to COVID-19, and blank spaces remain in their place to better visualize storage time
on the x-axis.

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Figure 5.4: : Micrographs of laminates produced from freezer aged prepreg using the MRCC.
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Figure 5.5: Average short beam strength of laminates aged in the freezer (labels indicate strength normalized against the F12R0
baseline case).
Figure 5.5 shows that for the first three additional months that prepreg accrued storage
time in the freezer, short beam strength did not change significantly. Once prepreg has
accumulated 19 months of total freezer storage time, mechanical performance began to decline,
with the F19R0 sample exhibiting a 6% reduction in strength. This trend continued for samples
F20R0 and F21R0, which were characterized with 86% and 77% of baseline strength,
respectively. The manufacturer’s recommended storage life for this material was 12 months,
but prepreg retained more than 95% of its baseline mechanical performance even when this
period was exceeded by 58% (7 months).
Decreased short beam strength in laminates produced from freezer aged prepreg was not
associated with porosity (no freezer aged laminates exhibited any porosity) or other identifiable
microstructure defect.
Aging mechanism during freezer storage. Prepreg that has been stored in the freezer
does not become unusable via the same aging mechanism as prepreg stored at room
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temperature. Invariant sub-ambient Tg data indicate that freezer storage over the period studied
did not significantly advance resin degree of cure. As seen in Figure 5.6, rheology data confirm
that extended freezer aging had little effect on resin viscosity. Because resin viscosity is
determined by resin temperature and degree of cure, Figure 5.6 also indicates that extended
freezer storage did not significantly alter prepreg degree of cure. Furthermore, the absence of
porosity in any of the micrographs in Figure 5.4 indicates that resin achieved sufficient flow to
saturate fiber tows, and the presence of porosity or other defects is not responsible for the
decline in mechanical performance evident in Figure 5.3.

Figure 5.6: Viscosity profile of resin squeezed from prepregs with different aging conditions.
Additional testing is required to identify precisely how freezer aging impacts the
chemical properties of various resin constituent components. More work is required before any
informed attempt can be made to mitigate this mechanism and address extending prepreg
storage life in the freezer. This result, nonetheless, is valuable because it has been thought by
experts and practitioners in the field that no significant chemical changes occur in the resin
during extended storage below its sub-ambient glass transition temperature. Results presented
above, however, indicate that a chemical change (that is not associated with the degree of cure
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advancing) is likely responsible for mechanical performance knockdowns associated with
extended freezer storage.
5.3.3. Cure Cycle Tailoring
The relationship between resin flow properties and processing parameters was explored
to assess when and how cure cycle tailoring could mitigate the type of defects formed in room
temperature stored prepreg. For a prepreg aged for 44 days at room temperature (sub-ambient
Tg = 15.8° C), Figure 5.7 shows the effective flow number when ramp rate and gelation
temperature deviate from MRCC (baseline) conditions. Figure 5.7A shows effective flow
number vs. ramp rate for various gelation temperatures, and Figure 5.7B shows effective flow
number vs. gelation temperature for various ramp rates.
From Figure 5.7A, as cure cycle ramp rate increases, the effective flow number of
prepreg resin during cure also increases. The relationship between effective flow number and
ramp rate appears to be asymptotic, where the asymptote and rate of convergence are
determined by the gelation temperature. For a baseline gelation temperature of 121° C, there is
little increase in effective flow number beyond a ramp rate of 10° C/min, but for a gelation
temperature of 177° C, effective flow number can be increased by approximately 50% by
increasing ramp rate from 10° C/min to 40° C/min. Figure 5.7B shows that effective flow
number as a function of gelation temperature follows a similar asymptotic trend when ramp
rates are slow (< 5° C/min), but when ramp rates are faster, effective flow number does not
appear to converge in the temperature ranges studied.
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Figure 5.7: Relationship between effective flow number and (A) ramp rate for given gelation temperatures and (B) gelation
temperature for given ramp rates.
Figure 5.8 shows how varying the super-ambient dwell time of the MRCC influences
effective flow number. The figure shows several traces that represent how deviations away
from the MRCC (ramp rate and gelation temperature were independently varied) would impact
effective flow number. For all ramp rates and gelation temperatures considered, there was a
quasi-linear relationship between super-ambient dwell time and effective flow number.
Effective flow number was found to monotonically decrease as a function of super-ambient
dwell time.

Figure 5.8: Relationship between effective flow number and super-ambient hold time for given deviations in ramp rate and
gelation temperature from the MRCC.
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Together, Figure 5.7 and Figure 5.8 suggest that flow number can be increased by (1)
increasing temperature ramp rates, (2) increasing the gelation temperature, and (3) reducing the
length of the super-ambient hold. Furthermore, Figure 5.7 suggests that increasing ramp rate
and gelation temperature simultaneously will have a larger impact on effective flow number
than extreme changes to only one parameter. Such a relationship is important for developing
cure cycle tailoring protocols that minimize deviation from the MRCC while resulting in
sufficient resin flow (high enough effective flow number).
5.3.4. Demonstration of Prepreg Life Extension via Cure Cycle Modification
The prepreg used for demonstration that cure cycle modification can extend storage life
was aged for 44 days at room temperature. At the time of layup, the prepreg was characterized
by a sub-ambient Tg of 15.8° C, corresponding to a degree of cure of approximately 16%. The
viscosity profile associated with curing such prepreg using the MRCC is shown in Figure 5.9.
When cured under the MRCC, effective flow number is only 10.7 Pa
-1
, and resin viscosity never
moves below 150 Pa s. Results described in Section 5.3.1 indicate that such prepreg is cured
using the MRCC, unacceptable levels of porosity will be observed in the resulting laminate due
to insufficient resin flow into dry regions of the fiber bed (see sample F12R42 in Figure 5.2).
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Figure 5.9: Viscosity profiles and corresponding effective flow number of overaged prepreg (Tg = 15.8° C) cured using the
MRCC and a modified cure cycle.
To achieve full fiber bed saturation for overaged prepreg, a modified cure cycle was
proposed. The modified cure cycle was generated such that the resulting cycle would
correspond to a target effective flow Number (Neff, T). The target was fixed at the same effective
flow number achieved by the MRCC for prepreg aged at room temperature for approximately
one week (Tg = 0.5° C). An aging condition of seven days was selected to determine N eff, T
because it provides reasonable assure that the cure cycle modification approach presented herein
would result in sufficient flow enable efficient fiber bed saturation. Using this approach, the
target effective flow number was fixed at 108 Pa
-1
for this study. This effective flow number
was achieved by (1) increasing ramp rates to 40° C/min, (2) increasing the gelation temperature
to 177° C, and (3) reducing the super-ambient hold time to just 50 seconds. The resulting
modified cure cycle, corresponding viscosity profile, and associated flow number are pictured in
Figure 5.9.
The microstructure of the laminate produced from the cure cycle tailored to its aging
condition is pictured on the right in Figure 5.10, where it is compared against the microstructure
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of a laminate produced from prepreg of similar aging condition (42 days of out time accrued, T g
= 15.1° C). It is apparent that cure cycle modification resulted in more uniform resin
distribution and fewer defects (voids). This micrograph provides further evidence that
insufficient resin flow was the primary mechanism for void formation in prepregs overaged at
room temperature. Furthermore, this micrograph suggests that making modifications to
processing parameters is an effective method for controlling laminate microstructure.

Figure 5.10: Comparison of microstructure in laminates produced from (left) prepreg aged for 42 days at room temperature,
cured using the MRCC and (right) prepreg aged for 44 days at room temperature, but cured using the modified cure cycle show
in Figure 5.9.
Figure 5.11 shows the short beam strength of the laminate produced from overaged
prepreg using the cure cycle describe above. In the figure, the average short beam strength is
compared against prepreg with three different aging conditions, each cured using the MRCC.
Error bars correspond to the standard deviation of strength measurements, and labels indicate
short beam strength normalized against the baseline aging condition cured using the MRCC
(F12R0). The average short beam strength of the laminate produced using the modified cure
cycle (F18R44 Mod) was 74 MPa, while the short beam strength of the laminate with similar
room temperature aging—but produced via the MRCC (F12R42)—was found to be only 44
MPa. Processing via the modified cure cycle resulted in a 68% increase in short beam strength
over processing via the MRCC. This increase did not restore the full mechanical performance
of the over-aged prepreg, however. Data in Figure 5.11 also indicate that the laminate produced
120

via the modified cure cycle is characterized by a 12% knockdown in mechanical performance
relative to the laminate produced from fresh prepreg using the MRCC (F12R0). The difference
between these two samples can largely be explained by the additional freezer aging that the
F18R44 Mod sample was exposed to. Prior to room temperature aging and subsequent cure, the
prepreg sample was stored in the freezer for approximately 18 months. Mechanical
performance of a laminate produced using the MRCC from prepreg stored in the freezer for
19.5 months (but with no accumulated room temperature out time) is also shown in Figure 5.11
as sample F19.5R0. This laminate was characterized by nearly a nearly flawless microstructure,
but a 14% reduction in strength compared to the baseline condition was nonetheless observed.
These data support the notion that the mechanical performance knockdown (vs. fresh prepreg)
observed in the F18R44 Mod sample is associated with extended freezer storage, rather than
process-addressable phenomenon. Researchers anticipate that the full mechanical performance
of fresh prepreg could be preserved in overaged prepreg that has accumulated less freezer
storage, but additional testing is required to explore this.

Figure 5.11: Short beam strength for laminates produced from prepreg with various aging conditions processed using the MRCC
(F12R0, F12R42, F19.5R0) and a modified cure cycle (F18R44 Mod).
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Together, Figure 5.10 and Figure 5.11 indicate the efficacy of cure cycle modification as
a method for extending the useful life of room temperature stored prepreg. The example
presented in this demonstration represents the extreme limit of aging condition (and associated
sub-ambient Tg values) for which the approach described herein can produce a cure cycle that
achieves the target effective flow number defined above. Prepregs characterized by sub-
ambient Tg values in excess of 15.8° C would require cure cycles outside the available
processing envelope to achieve the desired flow number. Prepregs characterized by sub-
ambient Tg values between -2.5° C and 15.8° C, however, would require less extreme changes
to the MRCC.
5.4. Conclusions
In this work we characterized processing behavior and mechanical performance of
prepreg as it accumulated storage time and out time. During aging at room temperature, prepreg
was observed to advance in degree of cure, and sub-ambient glass transition temperature was
confirmed to be a good indicator of prepreg resin exposure time. Microstructural analysis and
mechanical testing of laminates produced from room temperature aged prepreg showed the
onset age-induced porosity is correlated with reduced short beam strength. The primary
mechanism by which porosity remains within laminates produced from prepreg exceeding room
temperature storage limits was observed to be insufficient resin infiltration into dry regions of
prepreg fiber bed.
Prepreg stored in the freezer, however, did not undergo significant crosslinking; resin
glass transition temperature and rheological behavior were found to be unchanged after 20
months of freezer storage (167% of the manufacturer’s recommended storage life). All
laminates produced from freezer aged prepreg were observed to be nearly flawless; however,
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mechanical performance of such laminates was observed to decline after between 16 and 19
months of freezer storage. Resin achieved uniform distribution in all freezer aged laminates,
indicating that insufficient resin flow was not the reason for age-induced complications.
Instead, this work suggests different mechanisms are responsible for compromised mechanical
performance in (1) prepreg aged in the freezer and (2) prepreg aged at room temperature. This
finding is important because it has been widely believed that resin crosslinking is the
operational mechanism responsible for spoilage of room temperature and freezer stored prepreg
(with this reaction taking a different amount of time depending on temperature). While this
finding does not suggest that freezer storage entirely suspends resin crosslinking, the results of
Figure 5.4 and Figure 5.5 provide evidence that it is not the primary reason for declining
mechanical performance. More work is necessary to precisely identify the what is occurring as
prepreg accumulates storage time.
The relationship between processing cycles and resin flow properties was leveraged to
develop a method for increasing resin flow (beyond that which would have occurred under the
MRCC) in over aged prepreg. Resin cure kinetics and rheology models were used to estimate
effective flow number, which was shown to be most sensitive to simultaneous changes in ramp
rate and hold temperatures (rather than changes in one parameter at a time). A method for
generating a modified cure cycle based on prepreg sub-ambient glass transition temperature was
developed that relies on changing ramp rate, gelation temperature, and super-ambient hold time
until a target effective flow number is achieved. This method was demonstrated on a prepreg
aged for 44 days and shown to produce a laminate with improved mechanical performance
(compared to a laminate with similar aging but cured using the MRCC). More work exploring
this approach is necessary, including to identify an optimal effective flow number target.
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Because the mechanism by which freezer aged prepreg spoils is not the same as that of
room temperature aged prepreg, the cure cycle modification approach described in this work is
not anticipated to extend prepreg storage life. Whether there is any appropriate method for
storage life extension remains to be seen. Even if such a method is available, adoption may be
slow due to difficultly confirming laminate performance. In many aerospace manufacturing
environments, part quality is inspected using ultrasound or similar non-destructive techniques to
visualize laminate microstructure. This work has demonstrated that laminate microstructure
may not be a strong indicator of mechanical performance in freezer aged prepreg.
Overall, this work shows that simply by changing the cure cycle, materials can be
directly re-used without the need to transfer to different facility or chemical changes/treatments
to the prepreg. Once better understood, deployment of the cure cycle modification approach
requires minimal investment in infrastructure (e.g., a DSC) and technical training, while
providing considerable environmental and economic benefit.

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Conclusions and Suggested Future Work

6.1. Acoustic Metamaterials
6.1.1. Conclusions and Outcomes
1. Analytical model: An analytical model was developed to describe low-frequency
sound transmission loss through plate-like acoustic metamaterials. The model
significantly broadens the scope of acoustic transmission problems that can be
addressed using analytical techniques by relaxing limitations found in previous
literature. Specifically, the model presented in CHAPTER 2 relaxes symmetry
requirements on the placement of rigid masses and accommodates arbitrary
combinations of clamped and simply supported boundary conditions.
2. Insight for plate-type metamaterial designers: The influence of boundary conditions
on the transmission loss through plate-like acoustic metamaterials is better
understood as the result of this work. CHAPTER 2 presented the first explicit
demonstration of how boundary conditions influence transmission loss performance
of mass loaded plates. It was shown that frequencies associated with transmission
loss maxima and minima are, in some cases, more sensitive to the plate boundary
conditions than to the presence or absence of an attached mass. Furthermore, it was
shown plates with asymmetric boundary conditions exhibit additional transmission
loss maxima and minima. An understanding of these behaviors is critical for the
deployment of effective sound reduction systems.
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3. Motion of MAM cell support structure is responsible for performance deviation
when scaling from MAMs: A mechanism by which transmission loss performance of
arrays of identically tuned MAMs deviates from that of an individual MAM was
identified. The motion of the array itself was shown to play a critical role in the
transmission loss through an array of MAMs, and resonance frequencies of the array
were correlated with transmission loss minima not observed for an identically-tuned
single MAM.
4. Conception and demonstration of hierarchical MAM design: A method to control the
interaction between cell and array level length scales in MAM arrays was introduced
and demonstrated. A six-by-six array of MAMs was manufactured and tested in a
transmission loss chamber and the addition of a mass near the center of the array was
shown to shift the frequency response of the array of MAMs. Hierarchical design at
the scale of MAM cells and at the scale of arrays of MAMs was shown to provide
engineers with a method for controlling the frequency response of arrays, thoughtful
use of which would enable minimizing the performance deviation associated with
scale up of MAMs.
5. Tool for estimation of transmission loss through hierarchical MAMs: A tool for
estimating the performance of arrays of MAMs was presented and demonstrated.
Such a tool enables engineers to confirm favorable interaction between membrane-
and array-scale dynamics prior to costly sample fabrication.
6.1.2. Suggested Future work
To address a broader range of scenarios, the analytical model describing transmission
loss through plate-type acoustic metamaterials could be extended to accommodate compliance
126

in attached masses. Such an extension would add a considerable number of degrees of freedom
to the equations of motion but the additional numerical complexity would improve the accuracy
of the transmission loss predictions. This extension would enable more accurate understanding
of the behavior of plate-type acoustic metamaterials that wherein the attached mass or masses
are large, thin, compliant, or some combination of the three. Further generalization of the
analytical model describing plate-type acoustic metamaterials could be achieved by considering
additional boundary conditions. For example, there may be scenarios in which the plate is only
supported on three sides and the fourth side is free.
There remains extensive work to do in improving the modeling approach used to explain
hierarchical MAM performance. First, estimated eigenfrequencies and mode shapes of the
array would be more accurately predicted if the array was modeled as a grid of constrained
single-span beams rather than a homogeneous plate. More critically, however, the model could
be improved by considering motion coupling between the array and membrane cell scales.
Incorporation of such a coupling term would require simultaneous solving of the membrane and
array length scale behaviors. The behavior of each membrane cell would have to be described
independently, significantly increasing the model’s number of degrees of freedom. Also,
appropriate admissible eigenfunctions would have to be identified to describe membrane cell
vibration, allowing for motion of the array. The Expressions for effective surface density and
transmission loss would require appropriate updating. Such changes would critically improve
the rigor of the approach presented in CHAPTER 3.
6.2. Prepreg Processing
6.2.1. Conclusions and Outcomes
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1. The presence of fiber tow overlaps and underlaps is not necessary for through-
thickness permeability: Prior to the work presented in CHAPTER 4, available
literature on semipreg permeability and processing behavior was limited to
semipregs with woven fiber beds. In woven fiber architectures, pinholes are present
intersections of fiber tows and enable though-thickness gas transport. In
unidirectional fiber architectures, however, no such pinholes are present. This work
demonstrated that even without fiber tow overlap or underlap features, unidirectional
semipreg can exhibit through-thickness permeability is increased by several orders
of magnitude when resin continuity is interrupted.
2. Through-thickness permeability improves manufacturing robustness in
unidirectional prepreg: Compared to a commercial hot-melt OoA prepreg, defect
content in laminates cured from in-house produced semipreg was less sensitive to
deviations from baseline cure conditions. The results indicate the efficacy of
through-thickness permeability (and short egress pathways) for removal of entrapped
or evolved gases from UD prepreg during cure. Process robustness was particularly
notable, particularly because no effort was made to optimize resin rheology or resin
feature geometry (e.g., strips vs. grid), including spacing (i.e., periodicity of strips or
grid) and topography (i.e., resin feature cross section and surface texture).
3. Two void formation mechanisms operate during semipreg processing: In situ process
monitoring revealed the nature and origin of surface porosity in semipregs, and two
distinct types of surface voids were identified. Type 1 voids were formed by
entrapment of gas between resin strips and the tool plate, and Type 2 were the result
of insufficient resin flow into areas of the fiber bed between strips.
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4. Resin feature surface topography is a critical design parameter for semipregs:
Morphology of resin features played an important role in the mechanisms
responsible for Type 1 and Type 2 voids. Changes to resin strip cross section and
the addition of evacuation pathways on resin feature surfaces each reduced defect
content compared to unmodified semipreg. Microtomography data supported the
hypothesis that similar defect formation mechanisms create both surface and bulk
porosity, but further work is required to prove this.
5. Mechanical testing of laminates produced from over aged prepreg: The relationship
between prepreg aging condition, processing parameters, and mechanical
performance of resulting laminates is better understood as the result of this work.
Mechanical testing data for prepregs aged at room temperature indicated that
laminate microstructure was a good indicator of mechanical performance; however,
the same was not true for laminates produced from prepreg with extensive freezer
storage time. Laminates produced from prepreg with excess freezer storage time
were uniformly found to be defect free, but mechanical performance was
compromised when storage time exceeded 16-18 months.
6. Two different mechanisms are responsible for aged-induced performance or
processing deviations associated with expired prepreg: The primary mechanism by
which prepreg expires when stored at room temperature is resin crosslinking. The
cure reaction advances slowly during storage, resulting in higher viscosity during
processing and leading to flow-induced porosity. Resin crosslinking, however, was
not the primary mechanism by which prepreg expired during freezer storage.
Prepreg stored in the freezer for up to 22 months (183% of the recommended storage
129

life) was observed to have no measurable change in degree of cure (as indicated by
sub-ambient Tg) and no change in rheological behavior. Despite producing defect-
free laminates for the duration of this study, freezer storage beyond prepreg 16-19
months was characterized by a 6-13% reduction in short beam strength.
7. Prepreg out life extension via cure cycle modification: A method for modifying the
manufacturer’s recommended cure cycle on the basis of a direct measurement of
prepreg state was developed and demonstrated. Users measure and input the sub-
ambient glass transition temperature of over an aged prepreg, and the method
produces a cure cycle that will achieve favorable resin flow parameters (when
possible). This work shows that simply by changing the cure cycle, materials can be
directly re-used, providing considerable environmental and economic benefit.
6.2.2. Suggested Future Work
More work is required to identify optimal formats for robust out-of-autoclave prepreg
manufacturing. Questions remain about the ideal size, shape, and distribution of resin features,
and it is anticipated that optimal resin distribution may be a function of fiber bed architecture.
Work should be undertaken to identify resin distribution patterns and morphologies that
minimize air entrapment during layup and maximize permeability and evacuation path
connectivity for commonly used fiber weave patterns and unidirectional tapes of various. The
influence of prepreg areal weight (or ply thickness) should also be considered when identifying
optimal formats.
Semipreg work should also be followed up with a study that explores the importance of
resin flow behavior. The assumption that changes to resin rheological behavior (specifically,
lower minimum viscosity and less rapid gelation during processing) would improve
130

microstructure uniformity should be tested. This could be done via producing semipreg with an
identical format, but with a different resin (e.g., a resin with a viscosity profile more similar to
the control resin), or it could be achieved via cure cycle modification. The relationship between
void formation mechanism and resin rheology should also be carefully monitored. In particular,
the presence or absence of Type 2 voids should be noted when adjusting resin flow properties.
The aging behavior of prepreg stored in the freezer should be explored in more detail.
Additional work is required to identify precisely the mechanism by which mechanical
performance suffers when prepreg is stored in the freezer for extended periods of time.
Chemical characterization techniques such as high-performance liquid chromatography or
Fourier transform infrared spectroscopy may identify chemical changes in over aged prepreg
resin that are not associated with crosslinking. If possible, once the mechanism (or
mechanisms) responsible for freezer stored prepreg expiration is identified, mitigation strategies
should be explored.
Further work also may be valuable in identifying the appropriateness of effective flow
number as an indicator for resin flow. A study examining laminate quality as a function of
different cure cycles that are each characterized by the same effective flow number would
provide an indication of whether effective flow number is an appropriate metric. If cure cycles
associated with identical effective flow numbers produce laminates of nonuniform quality, then
it would be valuable to propose a modified effective flow number that more accurately
correlates with laminate quality.
Other valuable work would be refining the method proposed for cure cycle modification
to extend prepreg out life. First, the approach would benefit from a more rigorous identification
of appropriate target effective flow number (or target modified effective flow number).
131

Furthermore, the accuracy of the cure kinetics model necessary for cure cycle modification
work should be validated and possible improved for over aged conditions. The cure kinetics
model was developed for unaged prepreg, and it is not known how accurately the cure evolution
or viscosity predictions match reality for prepregs with extensive aging.

132

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Abstract (if available)

Abstract Composites, defined as a combination of two or more materials (e.g., metals, ceramics, or polymers), are often characterized by more favorable properties than observed in each constituent material alone. The last century has witnessed a revolution in engineering materials, and the invention and widespread adoption of synthetic polymers in particular has enabled myriad new possibilities, including a wide variety of composite materials. In this dissertation, two classes of composite materials are studied. First, composite metamaterials designed for use as acoustic barriers are discussed. An analytical model is used to explore the sound transmission properties of a plate-type acoustic metamaterial. Scaling of acoustic metamaterials is also explored with special attention paid to hierarchical design of membrane-type acoustic metamaterials. Second, structural composites comprised of carbon fiber reinforced polymers (CFRPs) produced from prepreg are examined. The relationship between prepreg format and process robustness is examined, and prepreg aging mechanisms are identified for the purpose of developing strategies to extend prepreg storage life.

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Asset Metadata

Creator Edwards, William Thomas (author)

Core Title Sound transmission through acoustic metamaterials and prepreg processing science

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 11/13/2020

Defense Date 10/20/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag acoustic metamaterials,composites,hierarchical design,OAI-PMH Harvest,out-of-autoclave,prepreg,robust processing,semipreg,sound transmission,transmission loss

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Nutt, Steven R. (committee chair), Gupta, Satyandra K. (committee member), Kanso, Eva (committee member), Povinelli, Michelle (committee member), Udwadia, Firdaus (committee member)

Creator Email BoomStacks@gmail.com,wtedward@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-394230

Unique identifier UC11666208

Identifier etd-EdwardsWil-9115.pdf (filename),usctheses-c89-394230 (legacy record id)

Legacy Identifier etd-EdwardsWil-9115.pdf

Dmrecord 394230

Document Type Dissertation

Rights Edwards, William Thomas

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA