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Mechanical characterization of acrylic bone cement

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Mechanical characterization of acrylic bone cement

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Content M E C H A N IC A L C H A R A C T E R IZ A T IO N O F A C R Y L IC B O N E C E M E N T by Sridhar Shankar Narayan A Thesis Presented to the FACULTY OF THE SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN CHEMICAL ENGINEERING December 1988 UMI Number: EP41828 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. D issertation Publishing UMI EP41828 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 This thesis, written by S r id h a r Shankar N arayan under the guidance of h i s Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of M aster o f S c ie n c e in C h em ical E n g in e e r in g D ate I 2 / 8 ../.8 8 . . . . . . . . . . . . . . . . . . . . . . . . . . Faculty Committee ' C Acknowledgements I would like to thank Dr. Ronald Salovey and Dr. Harry McKellop for their advice and guidance throughout the course of this work. Thanks are also due to Dr. Katherine Shing for agreeing to be a member of my committee. Thanks are due to all the people at the Biomechanics laboratory: Eddie, Cheryl, Bin, Alan, Nick, Eric, Robert, Wayne, Cho and Derek. Thanks to Anand Rangarajan for “latexing” this manuscript. Thanks to all the people who made the passage through graduate school a little easier. A special thanks to Khushroo Gandhi for his patience in initiating me into Polymer Science. Thanks to my family and friends who were always there when needed. Finally, thanks to that omnipotent, omnipresent, omniscient entity I choose to call God. Contents L ist o f F igu res v List o f T ables v i i i 1 In tro d u ctio n 1 1.1 Historical Background ............................................................................ 1 1.2 Chemistry and Mechanical Properties of Acrylic Bone Cement . . 4 1.3 Low Modulus Cement ....................................................... 11 1.4 Modeling H is to ry ...................................................................................... 12 2 M a th em a tica l M o d els o f V isc o e la stic ity 18 2.1 Introduction.................................................... 18 2.2 Equation of State Models . . . ........................................................... 20 2.3 Linear Viscoelastic M o d e ls..................................................................... 21 2.3.1 The Maxwell M odel..................... 22 2.3.2 The Kelvin-Voigt M o d e l............................................................. 23 2.3.3 Three and Five Param eter S o lid s ............................................ 24 2.3.4 Stress R e la x a tio n ........................................ 25 2-3-5 Ramp Loading ............................................................................. 26 i i i 2.3.6 Multiple Stress H isto rie s............................................................ 27 3 E x p erim en ta l P ro ced u re 31 4 R e su lts 38 5 D iscu ssio n 51 6 C on clu sion s 72 A F ab rication o f S ilicon e M old s 74 B Load C ell C alib ration 76 C C o n stru ctio n and C alib ration o f th e S train T ran sd u cer 78 /J R eferen ces 80 List of Figures 1.1 Composite Structure of Total Hip R e p la c e m e n t............................... 13 1.2 Two-dimensional representation of bone cement structure after curing 14 1.3 Molecular Weight Distribution of commercial bone cement powder and cured p r o d u c t .............................................................. 15 1.4 Schematic graph showing the exothermic tem perature changes oc curring in acrylic cements during the curing p ro cess...............................16 1.5 Dependence of indentation resistance on monomer concentration of the cement ....................................................................................................... 17 2.1 The Maxwell M odel................................................................................... 28 2.2 The Kelvin M odel....................................................................................... 29 2.3 Three-parameter m o d el............................................................................. 30 3.1 Zimmer vacuum-mixing s y s t e m ............................................................. 35 3.2 Experimental s e tu p 36 i j 3.3 Schematic of the measurement of experimental o u t p u t s ................. 37 4.1 Creep curves for vacuum-mixed Z R C .................................................. 41 4.2 Isochronous Stress-Strain curves for vacuum-mixed ZRC .............. 42 •v 4.3 Test for repeatability of experiment on same specimen at 5.59 M Pa (vacuum-mixed ZRC) ................................................................................... 43 4.4 Test for repeatability of experiment on same specimen at 21.93 M Pa (vacuum-mixed ZRC) ......................................................................... 44 4.5 Test for repeatability of experiment on different specimens at 19.35 M P a .................................................................................................................... 45 4.6 Creep curves for hand-mixed Z R C ........................................................ 46 4.7 Isochronous Stress-Strain curves for hand-mixed Z R C .................... 47 4.8 Comparison of creep of hand-mixed and vacuum-mixed ZRC at 7.53 M P a ...........................................................................................................48 4.9 Creep curves for vacuum-mixed L V C .................................................. 49 4.10 Isochronous Stress-Strain curves for vacuum-mixed LVC ............. 50 5.1 Five-parameter model fit for vacuum-mixed ZRC 58 j I 5.2 Powerlaw model fit for vacuum-mixed Z R C 59 j l i 5.3 Five-parameter model fit for hand-mixed Z R C 60 | I l 5.4 Powerlaw model fit for hand-mixed Z R C 61 j I 5.5 Five-parameter model fit for vacuum-mixed L V C 62 ! 5.6 Powerlaw model fit for vacuum-mixed L V C 63 ! I 5.7 Comparison of fits of five-parameter and Powerlaw models . . . . 64 ‘ 5.8 Comparison of long-term creep prediction of five-parameter and Powerlaw models fit on short-time creep d a t a ........................................ 65 5.9 Logarithmic projection of ZRC five-parameter and Powerlaw models 66 5.10 Step stress response prediction for ZRC from five-parameter model 67 5.11 Step stress response prediction for ZRC from powerlaw model . . 68 5.12 Ramp stress response for ZRC from five-parameter m o d el............ 69 5.13 Ramp stress response prediction for ZRC from powerlaw model . 70 5.14 Stress relaxation prediction from five-parameter model by numeri cal inversion....................................................................................................... 71 A.l Stainless Steel M aster.............................................................................. 75 B .l Load Cell C alibration.............................................................................. 77 C .l Axial Extensometer C a lib ra tio n ........................................................... 79 I List of Tables 5.1 5-Param eter Model Material Constants for Zimmer Cements 5.2 Powerlaw Model Material Constants for Zimmer Cements . . A b stra ct t _ A 5-parameter linear viscoelastic model of the form e = — e T 1) I j t +t<1 - e" r 2 ) and a nonlinear powerlaw model of the form e — + Acrmtn were used to model the creep response exhibited by Zimmer regular cement and | Zimmer low viscosity cement. The models followed the experimentally observed creep behavior well. The creep models were then used to predict the material response to ramp loading and to step increases and decreases in stress. The I agreement between the predicted and experimentally observed responses was very ' ! good. Attem pts to predict stress relaxation from the 5-parameter creep model j l using linear viscoelastic theory were, however, unsuccessful. Stress relaxation of j Zimmer cements is a phenomenen distinct from its creep and has to be modeled accordingly. ix _____ i Chapter 1 Introduction 1.1 Historical Background Polymethylmethacrylate (PMMA) materials have enjoyed widespread use in the fabrication of dentures since the late 1930’s. As used then, the system consisted of two basic components: an acrylic powder and a liquid monomer. The addition j of the liquid to the power formed a dough-like mass which, when placed in a mold under heat and pressure, set into the desired shape. Incorporating a tertiary amine into the liquid monomer yielded a self-curing resin, wherein the amine chemically activated an initiator (usually benzoyl peroxide) in the powdered phase. The ^ development of technology relating to suitable polymerization systems and their manipulation made possible the utilization of these materials in many in-vivo and ex-vivo prosthetic devices such as denture liner, repair, restorative materials, in the investigation of aneurysms, and the formation of cranial plates [1,2]. Favorable experience with a number of these devices suggested the implementation of acrylics as a fixation medium for orthopaedic applications such as the repair of pathological 1 J fractures, obtaining immediate stabilization of spinal fixation, and, especially, J fixation of the femoral component of the total hip replacement [3,4]. The clinical efficacy of these procedures before the advent of acrylic cements was not encouraging. The basic problem was one of preventing slippage between the femoral stem of the prosthesis and the medullary canal. The cement has been instrum ental in alleviating this problem. It modifies the prosthesis-biologic interface; i.e. a continuous bridging is afforded which is im portant in that the distribution of stresses transm itted between the prosthesis and bone is made more uniform and continuous. Prior to this procedure, stress concentrations of the order j ! of 25 megapascals had.been estimated to exist when bone resorption had led to the loosening of the femoral component [5], Stresses of this magnitude usually led to the perm anent deformation of the bone which, in turn, caused the failure I of the prosthesis. i The metallic femoral component, acrylic cement, and bone have been modeled j as a composite structure by McNeice and Amstutz, as shown in figure 1.1 [5]. The stem of the prosthesis fits in the medullary canal inside the cancellous bone. The cement fills the region between the cement and the bone. When no slippage j occurs at the interfaces of the components of this structure, a uniform transferal j of stresses exists between the m etal prosthesis and bone. However, if loosening occurs at the stem-cement interface, the stress distribution changes radically, and ; i I the tensile stresses in the cement can increase from about 7 megapascals to as much j as 21 megapascals [5]. Like most other brittle materials, bone cement is weaker ! in tension than in compression [6] and tensile stresses of this order can initiate cracks in the acrylic component, leading to gross loosening of the metal prosthesis and, possibly, fatigue failure of the metallic stem. Deformation strain lines have 2 been observed, frequently associated with cracks in the cement, indicating that plastic deformation and cracking are intimately related [7]. New alloys with improved mechanical strength and fatigue properties have significantly reduced the incidence of stem failure. Subsequently, attention has been focused upon bone cement as the weakest link in total hip replacement procedures. Several methods have been suggested for the improvement of the mechanical properties of cement. Fiber reinforcement of cement has been pro posed to develop a material with mechanical properties matching that of bone. Unreinforced cement is significantly weaker than compact bone, with tensile and compressive strengths approximately 30 and 70 megapascals, respectively, while those for wet femoral cortical bone are 123 and 170 megapascals [9]. Reinforce ment with m etal wire [8], and graphite and aramid fibers [6] have been shown to increase the compressive strength, shear strength and fracture toughness of j bone cement. Reinforcing cement may reduce stress concentration effects where J j bone defects are replaced by a material with identical mechanical properties and j I strain-rate sensitivity [3]. However, fiber-reinforced cements have inferior intru sion characteristics, which prevents them from completely filling crevices in the I bone. This limits their potential for use in clinical situations [10]. Although the primary component of most bone cements is polymethylmetha crylate, data available in the literature [11] on the mechanical properties of the fully dense forms like lucite and plexiglas cannot be used for the porous, com- j j posite m aterial used as bone cement. Porosity, generated from the mixing of the powder and the liquid monomer, can be reduced by mixing the components un der a partial vacuum, centrifuging the mixed components and/or by curing the m aterial under pressure. Many researchers have studied the effect of porosity on 3 the mechanical properties of bone cement, methods of reducing the porosity, and the improvement in mechanical properties thus produced [12-27]. While most re searchers [12,16,17,19,20] have found dram atic increases in the tensile strength, fatigue strength, and fracture toughness of bone cement as a result of porosity reduction, others [22,23] have reported th at porosity reduction by centrifugation i does not alter the static or cyclic fracture properties of bone cement. An accurate mathem atical model relating stress to strain, i.e. a constitutive ; equation, is essential for calculating the stress levels that exist in the layer of bone i ! cement surrounding a prosthesis. W ith an appropriate viscoelastic constitutive ; model, the effects of modifications of the mechanical properties of cement or mod ifications of the geometry of the prosthesis on the stress distribution in the cement maybe calculated. This will help in avoiding areas of high stress concentrations which will reduce the incidence of loosening failure of the artificial joint. I ! The goal of the present investigation was to develop a model of the viscoelastic behavior of bone cement. In conjunction with existing finite element programs I such as AD IN A or MARC, which accept viscoelastic m aterial models, this consti tutive model could then be used to analyze the effects of variations in the prop- i erties of the acrylic cement or the prosthesis geometry on the stress distribution j in the implant structure. j i 1.2 Chemistry and Mechanical Properties of Acrylic Bone Cement j i Bone cement consists primarily of PMMA powder and monomer methacrylate liq- j I i uid which are usually mixed in the ratio 2 g of powder to 1 ml of monomer. There ; 4 I are a number of commercial bone cements available for clinical use. The varia tions in the formulations are usually in the incorporation of a PMMA-copolymer to increase creep resistance and in the choice of the radiopaque pigment. I Zimmer bone cement consists of two components.The first is packaged as an I J ampule containing 20 ml of liquid of the following composition [28]: Methyl methacrylate monomer 97.25% v/v N,N-dimethyl-p-toluidine 2.75% v /v ^ Hydroquinone 75±10 ppm - - . Formula: C H 2 = C — C O O C H j j i n , ! i M ethylm ethacrylate (m onom er) j Hydroquinone is an inhibitor added to prevent the premature polymerizationv which may occur under conditions such as heat, light, or chemical reagents. N,N- i I dimethyl-p-toluidine is a promoter added to accelerate cold-curing when the two I components are mixed together. The liquid component is sterilized by membrane j filtration [29]. ■ The second component is a polythene pouch containing 40 g of finely divided white powder having the following composition [28]: Poly (methyl methacrylate) 89.25% w/w Barium Sulfate (BaSCh), U.S.P 10.00% w/w Benzoyl Peroxide 0.75% w/w ! 5 Formula: CHj CH3 » I I - c h 2- c - c h 2- c - I ! COOCHj COOCHj PMMA When the two components are mixed together, the liquid MMA swells and starts to dissolve the PMMA powder. At the same time, polymerization of the monomer liquid is initiated by the activator, benzoyl peroxide, and the poly merization occurs by the free radical (addition) polymerization process. Benzoyl peroxide reacts with a monomer molecule to form a monomer radical which will then attack another monomer molecule to produce a dimer. The process contin ues until long-chain molecules are produced. As the polymerization proceeds, the m ixture of monomer and powder increases in viscosity and eventually becomes a brittle, hard solid. The final product is a two-phase material, consisting of the original spherical particles of PMMA embedded in a matrix of in situ polymerized MMA, as shown in figure 1.2 [29]. The following reaction scheme illustrates the free radical polymerization of PMMA. 6 Initiation: COOOOC—^ o ) COOCHi • + c h 2= c CH, + 2COz COOCHj I C H ,—C- + heat CH, (R-) (M) (RM •) Propagation: RM • + M RMM* + M RM„* + M RMM + heat RMMM + heat RM,l1. l +heat Termination: Combining two free radicals or addition of a terminator. Polymerization during curing increases the degree of polymerization, increas ing the molecular weight of the cured material. The number average molecular weight (Mn) increases from 44,000 to 51,000 and the weight average molecular weight increases from 198,000 to 242,000. However, the molecular weight distri bution does not change significantly after curing, as shown in figure 1.3 [14]. The polymerization reaction is exothermic, as shown in figure 1.4 [30]. After a tim e following the mixing of the powder and the monomer, the polymerizing m aterial is in its dough-like phase, during which it may be kneaded or worked by hand for 8-10 minutes after the initial mixing, at which point the material sets into a rigid, brittle mass. As shown in figure 1.4, the setting time is arbitrarily designated as the time corresponding to the point at which the tem perature of the dough is midway between the mixing (ambient) tem perature and the maximum tem perature achieved. The working time is the interval between the dough and the set time. Vacuum-mixing of the mixture, if desired, is begun before the advent of dough tim e and is carried on for up to 2 minutes. The vacuum-mixed mass is injected during the early stages of working time, while it is still fluid enough to completely fill the region between the stem and the bone. A number of intrinsic and extrinsic factors affect the properties of bone cement. The intrinsic factors are: 1. Composition of monomer and powder; 2. Powder particle size, shape and distribution: degree of polymerization; and : i \ 3. Liquid/powder ratio. The extrinsic factors are: I i i (a) Mixing environment, especially temperature; [ (b) Mixing technique; and { (c) Curing environment: tem perature, pressure, contacting surface (tissue, I i air, water etc.). In the absence of laminations, additive variations affect the properties of ce- \ i ment more than manipulation variables. Hence, the composition of monomer and j powder is im portant in determining the properties of the final cured product. J Residual organic and inorganic additives tend to adversely affect the mechani cal properties of cement. Haas et al. [14] have found that cements containing barium sulfate had 10% lower tensile strength, transverse strength, and modulus 8 of rupture compared to cements without barium sulfate. Presence of styrene as a copolymer in the powder improves mixing properties and increases creep re sistance [3]. A bimodal distribution of powder particles comprised of small and large spherical beads in the size range 5000 - 75000 nm produces a smoother pow der, such th at less monomer is required for wetting, packing, and doughing, thus reducing tissue toxicity, heat of polymerization, and dimensional changes. Haas and colleagues [14] also found th at varying the powder to liquid ratio from 2/1 to 3/1 reduced the dough time, setting time, and handling time, as well as the peak tem perature; increasing the tem perature of the mixing bowl from 24°C to 32°C shortened the dough time, setting time, and handling time with no significant effect on the peak tem perature. Lee et al.[32] have determined that the ultim ate compressive strength of test specimens made from cement beaten at 4.33 Hz was on average 10% less than that of test specimens made from cement beaten at 1 Hz. The difference probably arises due to the greater porosity introduced in the cement at higher beating speeds. Application of pressure during curing of cement in the clinical situation is desirable to reduce the size of the voids, in addition to the benefits of producing better flow of the cement into interstices for better seating of the prosthesis. Infrared studies [33] have shown th at there is no crosslinking present in the cured material. Residual monomer, however, is present. Bauer et al. [33] have determined th at the monomer content decreases from approximately 33% when mixing the dough to 3.3% after 1 hour and 2.7% after 20 hours on storage in air at room tem perature. The monomer is toxic to the tissue surrounding the implant and also causes lowering of the blood pressure. High concentrations of monomer m ust, therefore, be avoided within the body. This would dictate that the surgeon 9 delay the insertion of cement as much as possible during the working time. Lee et al. [32] have found that cement specimens prepared from dough placed in specimen molds after 2.5 minutes showed an 11% reduction in ultim ate compressive strength compared to specimens placed in the molds after 1.5 minutes. Hence, the surgeon ; has to optimize the insertion tim e so that there is no serious deterioration in the mechanical properties at an acceptable toxicity level. Residual monomer is also detrim ental to the mechanical properties as can be seen from figure 1.4 [33]. Bone cement is weaker than conventionally processed acrylic due to the intrin sic defects present in the m aterial. To improve the properties of the cement, the possible sources of these flaws must be considered. Six possible sources are: 1. Porosity which is intrinsic to the initial two-phase system and can also be acquired during mixing; 2. Particle-m atrix interfaces which can be a potential source of weakness due to poor adhesion between the dispersed phase and the m atrix phase. Poor adhesion may be caused by too high a polymerization rate relative to the ini- I tial particle molecular weight which prevents sufficient diffusion of monomer into the particles before curing. It may also be caused by the retention of a suspension agent from the original bead polymerization; i I 3. Residual stresses which arise from compositional and therm al gradients set up due to the difference in the rates of monomer absorption into the beads i and polymerization. These gradients, subsequent to curing and differential cooling, result in a complex stress state; I 4. Low molecular weight products like benzoyl peroxide have poor physical ! properties and liquid residues plasticize the cement; 10 5. Inorganic and organic additions like antibiotics, pigments, and radioopaque fillers act as defects in the composite structure and fracture may originate at their site; 6. At 37°C the water sorption of bone cements is 2-4%. In its actual use, bone cement is in contact with body fluids and these may be expected to have the same plasticizing effect that water has on the composite. Thus the structural integrity of acrylic cement is affected by these intrinsic defects. All these sources of defects, except water, represent a trade-off of prop erties to ensure satisfactory mixing and curing at the outset. These properties m ust now be recouped to ensure longer service lives of prosthetic devices. The following modifications can be useful in reducing the detrim ental effects of the intrinsic defects:[2] | 1 1. Increasing the powder/monomer ratio to an optim um level to reduce shr- j inkage, porosity, and tem perature while retaining sufficient working time; 2. Using appropriate coupling agents to improve adhesion of organic and inor ganic additions; 3. Adding porous semicrystalline powders to increase particle-matrix inter facial strength by encouraging imbibition of monomer and/or to reduce j residual stresses by decreasing tem perature via the latent heat of fusion. 1.3 Low Modulus Cement The cured m aterial is a brittle, relatively porous polymer composite which pro- ( duces stress shielding of the proximal femur [34]. This situation worsens with < i i ! time, such that the stresses in the distal cement become even greater, leading to greater incidence of fracture of the distal cement. W eightman et al. [34] suggested that the high modulus and lack of ductility of the PMMA cements might limit proximal load transfer while a more flexible cement should allow greater axial movement of the stem realtive to bone and thus increase proximal load transfer. They developed an alternate formulation using polyethylmethacrylate as the pow der phase and n-butyl methacrylate as the monomer, all other components being j the same as conventional surgical PMMA. This cement has a lower modulus and creep resistance. The intention was th at this would promote stress transfer to the proximal femur. 1.4 Modeling History Chwirut [35] has measured the long-term compressive creep deformation of several acrylic cements and concluded that bone cement behaves approximately as a linear viscoelastic material. He proposed a powerlaw model for the creep of the form e = K a mtn and an exponential model of the form e = A tneh < T to fit the experimental results. He suggested th at this kind of empirical model can be used to predict long-term creep behavior from short-term experiments. Pal et al. [36] measured stress relaxation and suggested that creep and stress relaxation might play an im portant role in the long-term stability of total hip replacements. However, no attem pt was made in either study to produce a single m athematical model which could accurately predict the response of the cement to different loading conditions. The development of such a constitutive model was the goal of the present study. i 12 METAL PROSTHESIS BONE Figure 1.1 Composite Structure of Total Hip Replacement (Reprinted from Amstutz and McNeice t53) Monomer liquid Polymer powder Opacifying agent (B aS04) Pore Figure 1.2 Two-dimensional representation of bone cement structure after curing. (Reprinted from reference 29) t 14 RELATIVE MASS O F MOLECULAR WEIGHT M MOLECULAR W EIG H T DISTRIBUTION 80 70 60 50 40 PO W D ER CURED MATERIAL 30 20 1 0 0 1 0 0 MOLECULAR W EIG HT M » 1Q1 1000 10000 Figure 1.3 Molecular Weight Distribution of commercial bone cement powder and cured product. (Reprinted from reference 14) 15 t CD m a x ( T m a x ~~ A m bient) o < u CL .£ < D I — Dough Ambient Begin mix time Figure 1.4 Schematic graph showing the exothermic temperature changes occurring in acrylic cements during the curing process. (JReprinted from reference 30) 16 0.5 140 130 120 i- t— 1 0 0 2.4 2.6 2.8 2.0 2.2 MONOMER CONTENT - PERCENT Figure 1.5 Dependence of indentation resistance on monomer concentration.of the cement. The numbers at the experimental points indicate the length of storage of the cured material (in hours) in air. (Reprinted from reference 33) 17 Chapter 2 Mathematical Models of Viscoelasticity 2.1 Introduction The fundamental characterization of the mechanical behavior of engineering m at erials is typically done through a stress-strain curve. For polymeric materials, factors such as the rate of loading, duration of loading, and frequency of loading in cyclic experiments strongly affect the mechanical behavior. Hence, a stress- strain curve at a single stress level or loading rate is not sufficient for the com plete characterization of the material. A preliminary step in the development of a viscoelastic constitutive model is to establish the nature of the relationship be tween stress and strain. T hat is, we must establish whether the m aterial is linear viscoelastic or nonlinear viscoelastic. 18 The Boltzman equations relate single components of stress and strain for linear viscoelastic materials as shown below e(t) = I D(t — t) < r (r)dr (2.1) J — OO cr{t) = f E(t — r ) e (r)dr (2.2) J — O O Isochronous stress-strain curves from constant stress creep experiments is the m ethod commonly used to evaluate the linearity of the viscoelastic response of a material. The stress history in a creep experiment can be expressed as cr (f) = 0, t < 0; a(t) = < r 0, t > 0 (2.3) and equation 2-1 gives e(t) = a0D(t ) (2.4) If isochronous data obtained at any given time t = t0 produces a linear stress- strain curve, the m aterial is said to be linearly viscoelastic [37]. M athematical modeling of the stress-strain behavior of linear polymers can be done by the two following methods: 1) Equation of state formulations and 2) Linear viscoelastic models. Strain is usually represented as a function of stress, time, and tem perature. This can be represented as e = F(cr,T,t) (2.5) Acrylic bone cement is used at the constant physiological tem perature and, hence, we need not be concerned about the effect of tem perature on strain. There fore, we can write £ = F(<T,i) (2.6) 19 Assuming the separability of the stress and time effects, we can represent the relationship between stress, strain, and tim e as £ = f 2 M < T )&(t) (2-7) i= 1 2.2 Equation of State Models The equation of state approach assumes th at the response of the material depends on the present value of certain state variables explicitly. For example, the strain may be assumed to be related to stress and time by an expression of the form e = - |- + Aamtn (2.8) E q where Eq, A, m, and n are m aterial parameters. The first term accounts for the initial instantaneous elastic deformation and the second term for the subsequent creep deformation. The value of m is greater than one and n is generally less than one. This form of expression is called the Bailey-Norton law and is intended for ; I the modeling of primary and secondary creep [38]. One m ajor drawback of the equation of approach is its inability to deal with variable stress problems. Considering the creep portion of equation 2-8 we can express the creep strain as ec = Aamtn (2.9) Differentiating equation 2-9 with respect to time gives us (e°)= — = nA amtn~x (2.10) Equation 2-10 is the time-hardening formulation. It states th at in a variable stress situation, the creep strain rate depends on stress and time. In deriving 2-10, 20 it should be noted th at the time derivative of the stress has been neglected even though we are looking for a variable stress formulation. This limits the equation of state approach to step changes in stress that are of long duration. The strain hardening formulation is obtained by eliminating time from equation 2-10. First, equation 2-9 is solved for time to give t = (2.11) K A<rmJ This is substituted into equation 2-10 to give (ec)= A~na~ (ec) < ' " (2.12) Given a family of creep curves at different stresses, we can predict the response of a m aterial to increases in stresses using the strain rate predicted by the time- hardening or the strain-hardening formulations. But this involves the additional experimental work of determining whether the m aterial is strain-hardening or time-hardening. In either case, the model does not deal well with decreases in stresses because the model predicts that, as long as a tensile stress is applied, the strain rate is always positive. This is contrary to the usually observed behavior of m aterials which show a negative strain rate for a period of time after a reduction in stress. This point will be discussed further in chapter 5. 2.3 Linear Viscoelastic Models The stress-strain relationship of linear viscoelastic materials can be represented in an integral form by using the Boltzman superposition principle (equations 2-1 and 2-2). We can also represent this by means of a linear differential equation 21 which in its most general form can be written as Pa = Qe (2.13) where P and Q are linear differential operators with respect to time. This is equivalent to describing the mechanical behavior of viscoelastic materials by models consisting of elastic springs which obey Hook’s law and viscous dashpots which obey Newton’s law of viscosity. A spring and a dashpot can be combined in two ways: 1) in series, which gives us the Maxwell model or, 2) in parallel which gives us the Kelvin model. 2 .3 .1 T h e M a x w e l l M o d e l The Maxwell model (Figure 2.1) consists of a spring and a dashpot in series [39]. The extension of the spring is given by a — Et\ (2-14) while the extension of the dashpot is given by » = (2.15) Since both elements are connected in series, the total elongation is given by e = £i + €2 (2.16) Combining equations 2-14, 2-15, and 2-16, we get * = + t (2.17) dt E dt r] In a stress relaxation experim ent,^ = 0 and equation 2-17 becomes 1 _ Therefore — = - - d t (2.19) a r] Integrating this expression recognizing the fact that at t = 0, a = < r0 , the initial stress, we get er = a 0e~r (2.20) where r = -^is called the relaxation time. Thus, equation 2-20 states that the stress decays exponentially, with a characteristic time constant r. For a creep experim ent,^ = 0 and equation 2-17 becomes % = - < 2-21> dt T ) i.e. the model predicts that newtonian flow would be observed which is not generally true for viscoelastic materials. This is one drawback of the model. Also, a single exponential decay term may not be adequate to represent stress relaxation behavior, nor does the stress necessarily decay to zero at infinite time. 2 .3 .2 T h e K e l v i n - V o i g t M o d e l The Kelvin or Voigt model (figure 2.2) consists of a spring and dashpot in par allel [39]. At all times, the elongation of the two elements is the same,with the total applied force split between the spring and the dashpot [40]. The applicable equations are < rx = Ee (2.22) and < 7 2 = n j t (2.23) 23 Since,<t = oq + cr2 , we have < j = E c + t (2-24) dt For stress relaxation,^ = 0 and the model predicts th at the m aterial would behave like an elastic solid which is not an adequate representation of general viscoelastic behavior. For creep,a = cr0 , a constant and solving equation 2-24, we get * = 9 < i - < r# ') < 2-25) i.e. the creep response is retarded elastic with a characteristic retardation time t' = 2 L E ' 2 .3 .3 T h r e e a n d F i v e P a r a m e t e r S o l i d s In order to adequately represent the mechanical behavior of linear viscoelastic polymers, a combination of a number of Maxwell elements in parallel or a number of Voigt elements in series may be used [41,42,43]. The simplest possible model which can adequately represent both creep and stress relaxation is the 3-parameter model (figure 2-3) [42] which has a spring in series with a Voigt element. The strain of the Voigt element is represented by equation 2-25 and the strain of the spring is represented by equation 2-14. Since both elements experience the same constant stress, , in creep, the total strain is the sum of the strains of the spring and the Voigt element, i.e. + <2 '2 6 ) Jh o £j\ where,rj = Jr- is the retardation time. Adding one more voigt element to the 3-parameter model, we get the 5-parameter model with the total strain,e , under 24 constant stress,cr0 , given by « = f r + | r ( l - e -n ) + g ( l - e“ * ) (2.27) where ri = ^ and r 2 — ^ are retardation times. 2 .3 .4 S t r e s s R e l a x a t i o n If < j> is a function of time, its Laplace transform is defined as J ro o f < f> e~ std t (2.28) o Borel’s theorem states that a convolution integral may be transformed as / f(T)g(t - r)dr = f g{r)f(t - r)dr = g(s)f{s) (2.29) Jo Jo Therefore, equations 2-1 and 2-2 may be transformed to give e = D(s)Ws (2.30) and a = E(s)es (2.31) respectively. Combining equations 2-30 and 2-31 gives the result D (s)E{s) = ^ (2.32) 5 which, when inverted back to the tim e domain, becomes f D(t — r)E(r)dT = t (2.33) Jo The derivative of an integral is defined as T t £ f m = I o % dt + m ( 2 - 3 4 ) 25 Taking the derivative of equation 2-31, we get [* D(t - r)E(T)dr = 1 (2.35) Jo d_ ft dt i.e. j f 9D{fdt T) E{T)ir + E ^t)D(s>) = 1 (2-36) This is a Volterra integral equation of the second kind which, in principle, can be used to determine E (t) when D(t) is known and hence the stress response at constant strain. 2 .3 .5 R a m p L o a d in g In ram p loading, the stress is given by cr =a t (2.37) From equations 2-1 and 2-27, we get e(t)= a t[ - L + j r + ± - £ - n - e- * ) - - p - n - e-* )] (2.38) J u o j l j \ H /2 JZ /\Z H /2 Z For very fast loading rates (instantaneous), there is no time for creep to occur and the stress-strain relationship is given by €C0 =* * [tH (2‘39) C jQ If the loading rate is very slow (quasistatic), such th at each dashpot relaxes out for each increment of stress, the strain response becomes e(t) =a t[— + -p- + -p-] (2.40) H/q Hi 2 For any stress ramp in finite time, the strain response will lie in between the instantaneous and quasistatic limits. 26 2 .3 .6 M u l t i p l e S t r e s s H i s t o r i e s For step increase and decrease in stress, = (2.41) 2=0 Knowing the form of D (t), we can predict the response to step changes in stress. However, this formulation is based on the assumption that linear superposition holds, both for increases and decreases in stress, which may not apply for a given material. 27 Figure 2.1 The Maxwell model (Reprinted from reference 39) cr Figure 2.2 The Kelvin model (Reprinted from reference 39) 29 —o — a / w — 1 f € ' €r Figure 2.3 Three-parameter model (Reprinted from reference 4 2) Chapter 3 Experimental Procedure Our experimental approach towards finding m athematical models to describe the mechanical behavior of surgical PMMA consisted of the following steps. First, creep experiments were run at five different constant stress levels in order to determine the m aterial constant values of the 5-parameter and powerlaw models which fit these results. The ability of the model to predict the m aterial response to ramp-loading, step-loading and constant strain (stress relaxation) was done by comparing the experimental results against the model prediction in each instance. Vacuum-mixed specimens of both Zimmer regular cement (ZRC) (Zimmer, Inc., Warsaw, Indiana) and Zimmer low-viscosity cement (ZLV) (Zimmer, Inc., Warsaw, Indiana) were made by emptying the contents of a 20 ml ampule contain ing the monomer into a cartridge sealed at one end. Forty grams of powder were then added to the monomer. The components were added in this sequence so as to prevent the boiling of the monomer when vacuum was applied. The cartridge was then placed in a Miller cement cannister (Zimmer, Inc., Warsaw, Indiana). Vacuum was applied to the cannister and the contents were mixed for 45 seconds 31 using the paddle in the cannister with steady strokes. The paddle was then raised above the fluid m ixture and the m ixture was allowed to stand undisturbed in vacuum for 30 seconds. The cartridge was then removed from the cannister after releasing the vacuum. A nozzle was attached to the cannister and the contents of the cartridge were injected into silicone rubber molds (Appendix A) using an injector. The cement was allowed to harden in the molds for 45 minutes. The specimens were then pushed put of the molds and stored dry at room tem pera ture for at least 8 weeks before being subjected to experimentation. A schematic diagram of the vacuum-mixing apparatus is given in figure 3-1. Creep tests were conducted in a servo-hydraulic MTS machine (Material Test ing Systems, Inc., Chicago, Illinois). The MTS machine can be operated under one of three control modes: 1) load, 2) strain, or 3) stroke. Creep, step-loading, and ramp-loading experiments were conducted under load control while stress relax ation experiments were conducted under strain control. Specimens were m ounted in self-aligning tensile grips to minimize bending loads on the specimen. The specimens were held at each end in curved brass plates with sandpaper glued to the inner surface to prevent slippage of the specimen. The experimental setup is shown in figure 3.2. In creep tests, the load was applied in 0.1 seconds and then held constant for 10,000 seconds. Creep was measured on the same specimen at 5 different stress levels with 21 hours of recovery between successive loadings. The specimen was always loaded up in stress in successive experiments as per Turner [44] who showed that the residual unrecoverable strain in the specimen did not affect the results of an experiment as long as the specimen was allowed to recover for a period of tim e four times longer than the time th at the specimen was subjected to loading 32 and as long as the specimen was subjected to successively higher stresses. Creep tests were also conducted on different specimens at the same load to evaluate the amount of variation between specimens. Ramp-loading tests on vacuum-mixed Zimmer regular cement were conducted by increasing the load from zero to maximum at a constant rate over 500 seconds and monitoring the strain response. Step-loading experiments were conducted by applying the initial load in 0.1 seconds and allowing the specimen to creep for 1250 seconds. The load was then increased to approximately three times the initial value and the specimen allowed to creep for a further 1250 seconds. The load was then reduced to a level interm ediate to the first two and the strain response was monitored for an additional 1250 seconds. Stress relaxation experiments were conducted under strain control. Ideally, stress relaxation is monitored after an instantaneously applied strain value that is held constant during relaxation. However, with the brittle PMMA specimens, attem pts to apply such very rapid initial strains typically resulted in specimen fracture. The control program was, therefore, modified to apply the strain at a constant rate over a 25 second interval. The stress relaxation response for a ramp strain compared to an instantaneous strain is nearly identical after a period about ten times the duration of application of the strain [45]. Therefore, the relaxation response in these specimens was compared to the theoretical prediction based on instantaneous loading only after 500 seconds of relaxation. In all experiments, the load on the specimen was recorded on the MTS chart I recorder using the output from a 500 Kg load cell. Details of the load cell cali bration are given in appendix B. 33 Strain was recorded on a Gould X-Y chart recorder as the output from a semiconductor strain transducer attached to the specimen by rubber bands. Be fore the start of each experiment, the width of the jaws of the strain transducer was electronically set to 1 cm. The output voltage was calibrated to read strain of the specimen directly , defined as change in gage length divided by original length. Details of the strain transducer calibration are shown in appendix C. The schematic of the measurement of experimental outputs is shown in figure 3.3. 34 VACUUM TU8ING STERILE FILTER CHARCOAL FILTER CARTRIDGE PRESS TO OPERATE PUMP T Z T BOWL fcr PRESSURIZED AIR OUTLET AIR INLET Figure 3*1 Zimmer vacuum-mixing system. 35 36 CH ART RECO RDER LO N G ITU D IN A L T R A N S D U C E R LO N G ITU D IN A L T R A N SD U C E R CO N D ITIO N ER LOAD CONTROLLER SP E C IM E N Figure 3.3 Schematic of the measurement of experimental outputs 37 Chapter 4 Results Figure 4.1 shows the creep behavior of vacuum-mixed zimmer regular cement at 5.59, 10.75, 15.48, 19.35, and 21.93 M Pa over 10,000 seconds. All curves showed I an initial elastic response and a subsequent retarded elastic response with the creep gradually decreasing with time at the lower 4 stress levels. At 21.93 MPa, however, a tendency towards a constant creep rate is seen. The assessment of linearity of this m aterial is shown in figure 4.2 where isochronous stress-strain data points have been plotted at 10, 500, and 10000 seconds along with the linear regression lines for the 5 data points at a given value of time. It is seen th at vacuum-mixed ZRC behaves approximately as a linear viscoelastic m aterial in creep thus making it amenable to modeling by linear viscoelastic formulations. To test the repeability of the creep experiment at a given stress level, the experiments at 5.59 M Pa and 21.93 M Pa were repeated on the same specimen, 72 hours apart. Figures 4.3 and 4.4 show the results from the experiments at 5.59 M Pa and 21.93 M Pa respectively. At the lower stress level, there was a 5% difference in strain between the two runs at 10 seconds but this difference 38 decreased with increasing time, going to zero in 10,000 seconds. At 21.93 M Pa, the maximum difference between the two strain readings occurred at 10,000 seconds and was 2.3%. To check for variations in creep behavior between two different specimens made from the same batch of cement, a creep test was run on a new specimen at 19.35 MPa. As shown in figure 4.5, the two specimens showed almost identical creep behavior indicating that specimen-to-specimen variations within a given batch of cement were negligible. The creep behavior of hand-mixed ZRC at 5.81, 10.75, 14.84, 18.71, and 21.29 M Pa is shown in figure 4.6. The creep behavior was similar to th at observed for vacuum-mixed ZRC with the initial elastic jum p in strain and the subsequent retarded elastic portion of decreasing strain rate. There was no apparent tendency towards a steady creep rate at any stress level. Figure 4.7 shows the isochronous stress-strain data points for hand-mixed ZRC at 10, 500, and 10,000 seconds along with the linear regression lines for these data points. It was seen that hand-mixed ZRC also behaved approximately as a linear viscoelastic material in creep. The creep behavior of hand-mixed ZRC at 7.53 M Pa is compared to that of vacuum-mixed ZRC at the same stress level in figure 4.8. Almost identical creep responses are observed indicating that the reduction in porosity obtained by vacuum-mixing was not significant enough to affect the creep response of the material. Figure 4.9 shows the creep behavior of vacuum-mixed LVC at 5.59, 10.75, 15.91, 18.92, and 21.50 MPa. The behavior was similar to that observed in vacuum-mixed ZRC. A tendency towards attaining a steady creep rate at the highest stress level employed, 21.50 MPa, is again observed. The similarity of 39 creep behavior between ZRC and LVC indicated th at the propietary additives in LVC which serve to increase the low-viscosity working tim e stage of the cement did not affect the creep response of the cement in any way. Figure 4.10 shows the isochronous stress-strain data points for vacuum-mixed LVC at 10, 500, and 10,00 seconds along with the corresponding linear regression lines. We observed that vacuum-mixed LVC was also approximately linear viscoelastic in creep and could be analyzed by the appropriate m athem atical formulations. 40 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 Time (Thousand Seconds) Figure 4.1 Creep curves for vacuum-mixed ZFC 21.93 MPa 19.35 MPa 15.48 MPa 10.75 MPa 5.59 MPa H 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 4 8 12 16 20 STRESS MPa Figure 4.2 Isochronous Stress-Strain curves for vacuum-mixed ZRC. 10000 Sec 500 10 Sec t 1 - - - - - - - - - - 1 - - - - - - - - - - 1 - - - - - - - - - - 1 - - - - - - - - - - 1 - - - - - - - - - - r - - - - - - - - 1 - - - - - - - - - - 1 --------- 1 - - - - - - - - - - r Strain Figure 4.3 0.005 0.004 0.003 Run 2 0.002 Run 1 0.001 10 8 6 4 2 0 Time (Thousand Seconds) Test for repeatability of experiment on same specimen at 5.59 MPa (vacuum-mixed ZRC) ► f t . w 0.02 0.019 0.018 0.017 0.018 0.015 0.014 0.013 0.012 Strain 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 2 4 6 8 10 Time (Thousand Seconds) Figure 4.4 Test for repeatability of experiment on same specimen at 21.93 MPa (vacuum-mixed ZRC) Run 2 Run 1 4 5 . 0 .0 2 -I......................................... : --------------------------------------- 0.019 - 0.018 - 0.017 - 0.016 - 0.015 - 0.014 - 0 013 - Specimen 2 z 0012 — * | o.OII - Specimen'1 I o.oi - 9 ; 0.009 - 9 0.008 - 0.007 - 0.008 - 0.005 - 0.004 - 0.003 - 0.002 - 0.001 - o 4----------1 ---------- 1 ---------- 1 ---------- 1 ---------- 1 ---------- 1 r ~ 1 ---------- 1 ---------- 0 2 4 6 8 10 Time (TTTousarid Seconds) Figure 4.5 Test for repeatability of experiment on different specimens at 19.35 MPA (vacuum-mixed ZRC) Specimen 2 Specimen' 1 z tt I n 21.29 MPa, 18.71 MPa 14.84 MPa 10.75 MPa 5.81 MPa Time (Thousand Seconds) Figure 4.6 Creep curves for hand-mixed ZRC J S k C T \ STRAIN 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 4 8 12 16 20 24 Stress MPa Figure 4.7 Isochronous Stress-Strain curves for hand-mixed ZRC i t * . 10000 Se r 1 ------- 1 ------- 1 ------- 1 -------1 -------1 ------- 1 --------r~— i------ r 0.01 0.009 0.008 0.007 0.006 Strain 0.005 0.004 0.003 0.002 0.001 Time (Thousand Seconds) Figure 4.8 Comparison of creep of hand-mixed and vacuum-mixed ZRC at 7.53 MPa .t* 00 STRAIN 0.018 0.017 21.50 MPa 0.016 0.015 0.014 18.92 MPa 0.013 0.012 0.011 15.91 MPa 0.009 o.ooa 10.75 MPa 0.007 0.006 0.005 0.004 5.59 MPa 0.003 0.002 0.001 Time (Thousand Seconds) a* Figure 4.9 Creep curves for vacuum-mixed LVC 0.02 0.019 0.018 0.017 0.01* 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 10000 S^c I 4 T 8 T I 1 12 Stress MPa _T _ 16 _ r _ 20 24 Figure 4.10 Isochronous Stress—Strain curves for vacuum-mixed LVC U 1 o Chapter 5 Discussion The creep behavior of PMMA was not substantially affected by the reduction in porosity brought about by vacuum-mixing or by the prolonging of the low- viscosity working tim e by incorporating propietary additives. A comparison of the values of the m aterial constants obtained from a nonlinear, least-squares 5- param eter model (equation 2-27) and powerlaw (equation 2-8) model fits of the creep behavior of vacuum-mixed and hand-mixed ZRC and vacuum-mixed LVC [ further reinforced this assertion. Figures 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6 show the 5- param eter model fit for vacuum-mixed ZRC, powerlaw model fit for vacuum-mixed ZRC, 5-param eter model fit for hand-mixed ZRC, powerlaw model fit for hand- mixed ZRC, 5-param eter model fit for vacuum-mixed ZLV, and powerlaw model fit for vacuum-mixed ZLV respectively. Tables 5-1 and 5-2 show the values of the various m aterial constants obtained from the model fits for the three cements. There was no significant difference in the values of the param eters obtained from a 5-param eter model fit for the various cements. The value of the maximum I 51 | ____I Cement Eo Ei n e 2 t2 Vacuum-Mixed ZRC Hand-Mixed ZRC Vacuum-Mixed LVC 2220 2134 2354 13408 11307 12092 115 109 122 4140 3919 3434 5737 5682 7752 Table 5.1: 5-Parameter Model M aterial Constants for Zimmer Cements Cement E 0 ....... T....... T O n Vacuum-Mixed ZRC 2366 157260 1.33 0.31 Hand-Mixed ZRC 2245 66264 1.06 0.32 Vacuum-Mixed LVC 2477 470582 1.66 0.33 Table 5.2: Powerlaw Model M aterial Constants for Zimmer Cements residual indicated the maximum deviation away from ideal linear viscoelastic be havior over all 5 stress levels. This value was the lowest for hand-mixed ZRC, indicating th at, of the three cements tested, hand-mixed ZRC was the most linear viscoelastic. We did see a variation in the value of the param eters obtained from a powerlaw model fit of creep behavior of the three cements as shown in Table 5-2. This variation arose from the attem pt by the model to fit the creep behavior at all stress levels with the lowest possible value of the maximum residual. The factor A c t™ accounts for the spacing between the curves. To follow variations in this spacing between different stress levels, the model tries to optimize the value of A and m. But, small increases in the value of m lead to large decreases in the value of A. Since, the variations in the creep behavior of the three cements occurred in the spacing between the creep curves, the values of the param eters obtained from the powerlaw fit for the three cements were also different. A comparison of the value of the maximum residual between tables 5-1 and 5-2 showed that the maximum residual increased for hand-mixed ZRC in going from the 5-param eter model fit to the powerlaw model fit while there was a 35% 52 reduction in this value for the other two cements due to this change in the model employed. This fact reinforced our earlier statem ent th at hand-mixed ZRC most closely conformed to ideal linear viscoelastic behavior. In Figures 5.1 and 5.5, we saw that the 5-parameter model overpredicted the creep at the lowest level and underpredicted the creep at the highest level. This “fanning out” of the creep behavior of the vacuum-mixed cements was a result of two factors: 1) error in following the initial elastic deformation consistently across the different stress levels due to relative motion between the surface of the specimen and the jaws of the transducer, and 2) the tendency of the vacuum-mixed cements to attain a steady creep rate at the highest stress level. The powerlaw model could follow these variations more closely than the 5-parameter model could and hence the value of the maximum residual was reduced by using the powerlaw model for the vacuum-mixed cements. Hand-mixed ZRC was highly linear viscoelastic and did not attain a constant creep rate at the highest stress level. The 5-param eter model followed the gradually decreasing strain rates better and hence we saw the slight i increase in the value of the maximum residual when a powerlaw fit was used for this cement. The way the two models attem pt to follow the creep behavior at a given stress level is illustrated in Figure 5.7. It was seen th at the powerlaw model most closely conformed to the actual creep behavior. The prim ary difference between the two models was that at large values of time, the powerlaw model predicts a constant creep rate and the 5-parameter model predicts constant strain as shown in Figure 5.8. Acrylic bone cement did not attain a constant strain value when subjected to loading for this time period. Figure 5.9 shows the result of trying to I predict the strain of vacuum-mixed ZRC at 20,000 seconds from a 5-param eter ! L. 53 and a powerlaw model fit done over the first 2000 seconds. The powerlaw model predicted the strain at 20,000 seconds much better than the 5-param eter model due to the abovementioned difference in the nature of the two models. Thus, the 5-param eter model could be used to make predictions of the creep within the timeframe over which the model was fit. Extrapolating the fit to longer times could be done only with the powerlaw model. Since the creep behavior of the three cements was substantially the same, all further analyses were done with vacuum-mixed ZRC specimens since vacuum- mixing enabled us to produce consistently uniform specimens with a low incidence of m ajor defects. A creep model was an idealized approximation of actual m aterial behavior. The validity of this type of modeling could be established by determining how well the models predicted the material response to different loading histories. Figure 5- 10 shows the actual response of the material to multiple stress histories along with the 5-parameter model prediction. We observed the tendency to overpredict the strain response at the low stress level and underpredict the strain response at the high stress level, the maximum difference being about 10%. At the intermediate stress level, after the step decrease in stress, the prediction agreed well with the observed behavior. The form of the experimental and prediction curves at the high and low stress levels was the same, with nearly identical creep rates at any instant of time. The error at the low stress level may be attributed to the error in the measurement of the initial elastic jum p in strain. The deviation at the high stress level was consistent with the findings of Turner [44] who observed a similar deviation between the values of the observed and predicted strains after an increment in load, the deviation being larger, larger the increment in stress. 54 The corresponding prediction using the powerlaw model is shown in Figure 5.11. As mentioned in chapter 2, as long as a m aterial is under tensile stress, an equation of state model will predict a positive strain rate after a step decrease in stress, which is contrary to the behavior exhibited by vacuum-mixed ZRC. To predict the response of the m aterial to the reduction in stress, the response to the m agnitude of the stress reduction was subtracted from the maximum creep at the end of the second interval, i.e. simple superposition. This m ethod of analysis probably contributed to the discrepancy between the experimentally observed and predicted values of strain after the drop in load. At the high and low stress levels, however, the nonlinear model predicted the creep response much better than the 5-parameter model. Figure 5-12 shows the response of the material to a stress ram p of 18.7 M Pa in 485 seconds with the corresponding prediction from the 5-parameter model. The lower solid line indicates the instantaneous solution with the strain response equal to . The upper solid line is the quasi-static solution with the strain response equal to J r -f Jj- -f J r . At short times, i.e. low stresses, we again observed the overprediction of strain response by the model, the data points falling below the instantaneous solution limit. W ith increasing time (higher stress), however, the model predicted the m aterial response better and the predicted value was identical to the experimental value at 485 seconds. The corresponding plot with the powerlaw prediction is shown in Figure 5.13 with the quasi-static solution line the same as th at in Figure 5-9. Two predictions were obtained from the powerlaw model by following slightly different methods of 55 analysis. Recalling equation 2-11 e = ~ + A a mtn (5.1) Eo For a stress ramp,cr =cr t . Therefore, a .m , , e = — + A a tm+n (5.2) E q This was designated as the powerlaw equation of state solution. The Leaderman form of the superposition integral which is not based on the assumption of a linear model is given by e(t) = e0 + / D(t — T ) ? 4 - ~ d T (5.3) Jo or Considering only the creep portion of the strain response and comparing equa tions 5-1 and 5-3, we obtain D(t - t ) = A(t - r ) n (5.4) and f ( a ) = (a r)m (5.5) with = (5.6) O T Hence, equation 5-3 becomes /T f m ft e(t) = — + Amo I ( t - T ) n T m - x d r (5.7) E q J o This was designated as the powerlaw integral solution. The response curve was determined from equation 5-7 by numerical integration. It was seen from Figure 5-13 th at the powerlaw model predicted the strain response to the stress ram p better than the 5-parameter model with the powerlaw 56 integral solution following the experimental results more closely than the powerlaw equation of state. This once again indicated that vacuum-mixed ZRC tended towards nonlinear behavior which was predicted better by the powerlaw model. The powerlaw integral solution was a more rigorous analysis than the powerlaw equation of state and hence it followed the strain response better. As shown in chapter 2, the stress relaxation of a material can be predicted from the creep model by a numerical inversion procedure (equation 2-33). Stress j i relaxation tests were performed at four different strain levels 0.00315, 0.00347, 0.00588, and 0.00718, to check how well the prediction agreed with the experi m ental results. As seen from Figure 5-14, the model predicted th at the rate of stress relaxation would be higher at higher strain levels while the experimental results showed th at the material relaxed at a constant rate from 500 seconds after the application of the strain, i.e. the m aterial did not relax as a linear spring and dashpot model. These results demonstrated that the relaxation characteristics of Zimmer ce ment were an independent material property that cannot be predicted from the creep properties using simple linear viscoelastic analysis. This phenomenen should be analyzed separately and modeled accordingly. 57 Strain 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 Tit) 0.001 o □ n □ a t - i -------- 1 -------- r -------- 1 ------- r 6 8 Time (Thousand Seconds) 21.93 MPa 19.35 MPa 15.4 8 MPa 10.75 MPa 5.59 MPa 10 Figure 5.1 Five-parameter model fit for vacuum-mixed ZRC L n o o Strain 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.009 0.008 - 6 0.007 0.006 0.005 0.004 0.003 0.002 0.001 v 21.93 MPa 19.35 MPa 15.4 8 MPa 10.75 MPa 5.59 MPa Time (Thousand Seconds) F i g u r e 5 .2 P ow erlaw m odel f i t f o r vacuum-mixe|hi>ZtRC U i 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 S t r a i n 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 2 4 6 6 10 Time (Thousand Seconds) F i g u r e 5*3 F i v e - p a r a m e t e r m odel f i t f o r h a n d -m ix e d ZRC o <21.29 MPa 18.71 MPa 14.84 MPa lQ.-35'lSEa n----n— — ti 5. 81 MPa ~T--------------r-------------r— --------------------- 1 — . , • ---------------------- - .. Strain 0.019 0.018 21.29 0.01 / 0.016 18.71 0.015 0.014 0.013 14.84 0.012 0.011 0.009 10.75 7 0.008 0.007 0.006 0.005 n - f a 5 . 81 0.004 0.003 0.002 0.001 _ 1 - 2 4 6 Time (Thousand Seconds) F i g u r e 5 .4 P ow erlaw m odel f i t f o r h a n d -m ix e d ZRC a\ MPa MPa MPa MPa MPa Strain 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 2 4 6 8 10 Time (Thousand Seconds) F i g u r e 5 .5 F i v e - p a r a m e t e r m odel f i t f o r vacuum -m ixed LYCt .& -• ' 2 1 .5 0 MPa 4 1 8 .9 2 MPa 1 5 .9 1 MPa 1 0 .7 5 MPa 5 .5 9 MPa < 7 > to Strain 0.019 - 0.018 - 0.017 - 0.016 0.015 0.014 0.013 0.012 0.011 0.009 0.008 - 0.006 - 0.005 0.004 - 0.003 - 0.002 -tF ’ 21.50 MPa 18.92 MPa 15.91 MPa 10.75 MPa a n 5.59 MPa Time (Thousand Seconds) F i g u r e 5 .6 P ow erlaw m odel f i t f o r vacuum -m ixed LVC c r > u> t o ( - t r < m z 0 .0 0 9 5-parameter model O.OOB Powerlaw model 0 .0 0 7 0 .0 0 6 0 .0 0 5 0 .0 0 4 0 .0 0 3 0.002 0.001 0 0 1000 2000 3000 4000 5000 TIME (SECONDS) F i g u r e 5 .7 C o m p ariso n o f f i t s o f 5 - p a r a m e te r and P ow erlaw m o d e ls Z H > 2 H (J 0.009 Powerlaw model 0 . 0 0 B 5-parameter model 0.0 0 7 0 .0 0 6 0 .005 0 .0 0 3 0.001 0 0 5000 10000 15000 20000 TIME (SECONDS) F ig u r e 5 .8 C o m p ariso n o f l o n g - t e r m c r e e p p r e d i c t i o n o f 5 - p a r a m e te r and P ow erlaw m o d els f i t on s h o r t - t i m e c r e e p d a t a JOG E H K h Z Powerlaw model _ - 2 5-parameter model -3 1 2 3 4 5 LOG TIME (SECONDS) F i g u r e 5 .9 L o g a r ith m ic p r o j e c t i o n o f ZRC 5 - p a r a m e te r an d P ow erlaw m o d els i n ( - c < h z 0.02 0.0 1 5 22.37 MP^ o.oi 13.12 MPa 0 .0 0 5 7.31 MPa 1000 3000 2000 TIME (SECONOS) Figure 5.10 Step stress response prediction for ZRC from 5-parameter model 4000 0 ) H H < H z 0.02 0 .015 22.37 MPa o.oi 13.12 MPa 0.0 0 5 7.31 MPa 4000 2000 3000 0 10 0 0 TIME (SECONDS) Figure 5.11 Step stress response prediction for ZRC from powerlaw model Strain 0 1 Quasistatic solutig 0 .0075 Instantaneous solution 0 .0 0 2 5 0 0 100 aoo 500 Time (Seconds) Figure 5.12 Ramp stress response for ZRC from 5-parameter model ci 0.01 Strain 0.0 0 7 5 - 0 .0 0 5 0.0025 100 Quasistatic soluti. Instantaneous solution 200 300 Time (Seconds) r 400 500 F i g u r e 5 .1 3 Ramp s t r e s s r e s p o n s e p r e d i c t i o n f o r ZRC fro m p o w erlaw m o d el Stress M Pa Figure 5 20 15 10 5 0 0 1000 2000 3000 4000 fiOOO Time (Seconds) .14 Stress relaxation prediction from five-parameter model by numerical inversion -j i — * Chapter 6 I ' Conclusions The creep response exhibited by Zimmer regular cement does not depend on the i m ethod of preparation of the specimens, i.e. the reduction in porosity effected by vacuum-mixing the cement does not make the m aterial more creep resistant. i The incorporation of propietary additives to prolong the working time, as in ZLV, j t i also does not affect the creep behavior of the cement after polymerization. In | all instances, the m atrix and the embedded m aterial are primarily PMMA with ' fillers and other additives randomly dispersed throughout the volume. While . these inclusions may act as sources of failure within the m aterial, they do not j influence the creep behavior of the m aterial and the creep response exhibited by j I bone cement is th at of PMMA. j Both the 5-parameter and powerlaw models predict the creep behavior well, | with the powerlaw model being better able to handle the inherent nonlinearity of the creep response of vacuum-mixed Zimmer cements. The justification for this kind of modeling is evidenced from the ability of the models to predict material response to step increases and decreases in stress and to stress ramps. However, beyond the tim e range for which the parameters were derived, the 5-parameter model predicts constant strain. In contrast, the powerlaw model predicts finite strain rate, which conforms better with actual m aterial behavior. Hence, to pre dict long-term creep from short-term experiments, we should use the powerlaw model. Stress relaxation of surgical PMMA does not proceed as predicted from the creep properties using linear viscoelastic theory. Stress relaxation should be con sidered as a distinct m aterial property and modeled accordingly. 73 Appendix A Fabrication of Silicone Molds Silicone rubber molds for the injection molding of the cement specimens were formed from stainless steel master, with the dimensions shown in figure A -l. The molds were m ade by mixing 10 parts by weight of silicone rubber (Silastic E RTV rubber, Dow Corning) with 1 part by weight of the curing agent (Silastic E curing agent, Dow corning) thoroughly. The mixture was then deaerated in a vacuum chamber until the m ixture stopped bubbling. The m ixture was then poured into ! a cylindrical plastic container 10 cm long, coated with mold release, in which the stainless steel m aster was held at the center by guide wires. The container was then sealed and allowed to stand for 48 hours. The plastic container was then split open and the flexible, solid silicone rubber mold was removed. The ends of the mold were cut to expose the stainless steel m aster and the m aster was then pushed out of the mold. The mold was subsequently used in the fabrication of cement specimens. UNIT: INCH (mm) 1.250 (32) 0 .200 1.001 (4.68 i.0 3 ) 0 .3 7 3 ±.005 (9.5 i.1 ) > O o o 1.000 1.005 (25.4 i.1) 4 .500 (114) Figure A-l Stainless Steel Master 1 Appendix B ! t j Load Cell Calibration The load cell (MTS, Model $ 661.13A-03, range 0-5000 N) was calibrated by stacking calibrated weights on the load cell and recording the output voltage. Figure B-l shows the calibration curve for the load cell. The output voltage was linear, with a standard error of 0.04%, over the calibration range of zero to 583 N. (fl-H I " 0 < H C T M C 0 1.5 0.5 300 400 W eight (Newtons) Figure B-l Load Cell Calibration 600 'j -j Appendix C Construction and Calibration of the Strain Transducer The active element of the strain transducer consisted of a fiat section of spring steel with four identical semiconductor strain gages attached in the form of a I W heatstone bridge. The strain transducer was calibrated by driving a 90° wedge, attached to a , micrometer, between the jaws of the transducer. The resultant output voltage j 1 was linear, with a standard error of 0.7%, over the calibration range zero to 0.2 | i mm. The calibration curve is shown in figure C -l. I I I W ith the bridge balanced and no strain on the specimen, the voltage output was zero. An output directly proportional to the separation of the transducer jaws was produced when the specimen underwent creep. 1 78 U ) h r 0 < H C U C O 2.5 2 5 1 5 0 0.2 0 . 15 0.1 Displacement (mm) 0 . 05 0 Figure C-l Axial Extensometer Calibration References 1. R. W hiting and P.H. Jacobsen, J. Mat. Sci., 20, 929, 1985. 2. R.P. Kusy, J. Biomed. Mat. Res., 12, 271, 1978. 3. S. Saha and S. Pal, J. Biomed. Mat. Res., 17, 1041, 1983. 4. T.A. Freitag and S.L. Cannon, J. Biomed. Mat. Res., 10, 805, 1976. 5. G.M. McNeice and H.C. Amstutz, Proc. of the 5th International Congress of Biomechanics, U. of Jyvaskyla, 1975. 6. S. Saha and S. Pal, J. of Biomechanics, 17, 467, 1984. 7. D. F. Gibbons and K.A. Buran, Implant Retrieval: M aterial and Biological Analysis, Proceedings of a conference held at the National Bureau of Stan dards, Gaithersburg, Maryland, A. Weinstein, D. F. Gibbons, S. Brown, and W. Ruff, Eds. 8. J.P. Taitsm an and S. Saha, J. Bone and Jt. Surg., 59A, 419, 1977. 9. S. Saha and S. Pal, J. Biomed. Mat. Res., 18, 435, 1984. 10. R.P. Robinson, T.M. Wright, and A.H. Burstein, J. Biomed. Mat. Res., 15, 203, 1981. 11. J. Marin, Y. Pao, and G. Cuff, Trans, of the ASME, 12, 705, 1951. 12. R.L. Wixson, E.P. 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Creator Narayan, Sridhar Shankar (author)

Core Title Mechanical characterization of acrylic bone cement

Contributor Digitized by ProQuest (provenance)

Degree Master of Science

Degree Program Chemical Engineering

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

Tag engineering, biomedical,OAI-PMH Harvest

Language English

Advisor Salovey, Ronald (committee chair), McKellop, Harry A. (committee member), Shing, Katherine S. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-315416

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