Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
The relationship of stress to strain in the damage regime for a brittle solid under compression
(USC Thesis Other)
The relationship of stress to strain in the damage regime for a brittle solid under compression
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
THE RELATIONSHIP OF STRESS TO STRAIN IN THL DAMAGE REGIME FOR A BRITTLE SOLID UNDER COMPRESSION by In n a A tschul A T h esis P re se n te d to th e FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial F ulfillm ent o f th e R e q u ire m e n ts for th e D egree MASTER OF SCIENCE (G eological Sciences) M ay 1995 U N IV ER SITY O F S O U T H E R N C A L IF O R N IA T H E G R A D U A T E S C H O O L U N IV E R S IT Y PA R K L O S A N O E L E S . C A L IF O R N IA S 0 0 0 7 This thesis, written by Inna Altachul under the direction of h**, Thesis Committee, and approved by all its members, has been pre sented to and accepted by the Dean of The Graduate School, in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE D tm * Date April __19l J995 TRESIS COMMITTEE ChairmmM, K u * x ......... ..... Acknowledgments This work was carried o u t un d er the guidance of Charles G. Sammis. His scientific, creative, and em otional support, as well as, sense of hum or m ake him an exceptional adviser and teacher. In the two years th at 1 have studied with Charlie, 1 have learned everything from how to predict earthquakes, to how to write a good thesis sentence, along with a dose of rock mechanics. I wish to extend my most earnest gratitude to Charlie lor his unvarying support and encouragem ent throughout the writing of this thesis. Greg N. Boitnott has also been instrum ental in the com pletion of this work. I wish to thank him for his expertise in the field of high pressure rock testing and for his help in the collection of laboratory d ata at New England Research. Finally, I wish to thank m em bers of my thesis comm ittee, Keiti Aki and Steven P. Lund, for their helpful com m ents on the text. Table of Contents A ck n o w led g em en t......................................................................................................... ii List of Figures and T a b l e s .............................................................................................iv A b s tr a c t.................................................................................................................................v C hapter One -Introduction ...........................................................................................1 C hapter Two— A Review of Damage Mechanics .......................... 6 C hpter Three— Development of the Stress/Strain Algorithm ........................IS The Stress/Strain C u r v e ...................................................................................IS Pure Elastic R e g im e ............................................................................................ 15 Non-Linear Elastic Regime .............................................................................. 17 Damage Regime .......................................................... 19 Brittle and Ductile Failure ..............................................................................20 Hysteresis ............................................................................................................. 21 Sub-critical Crack Growth . . 25 C hapter Four-Im plem entation of the Stress/Strain Algorithm .................. 27 Com ponents of the Stress/Strain Modeling Program .......................... 27 Stress/Strain Modeling Program ................................................................. 28 Stress/Strain Algorithm and Seismic W aveform M o d e lin g , 82 Chapter Five-Experim ental Stress/Strain D a t a .......................... 54 Experimental A p p a ra tu s .................................................................................. 54 Experimantal P ro c e d u re ........................................... 54 Experimental Results ...................................................................... 56 Modeling of Experimental R e s u lts ............................................................... 45 C hapter S ix -C o n c lu sio n ...............................................................................................57 References............................................................................................................................60 A ppendix.............................................................................................................................. 62 List of Figures and Tables Figure 1-1-Idealized Stress/Strain Curve.................................................................. 2 Figure 1-2— Stress Zones Around an U nderground Nuclear Explosion .... 4 Figure 2-1— Idealized Crack (Cross-Sectional and Plan V ie w s)..........................7 Figure 2-2-R elationship of Axial Stress to D a m a g e ........................................ 11 Figure 2-3-Failure Surfaces for Three Values of Initial D a m a g e ............... 12 Figure 3-1— Stress Region D ia g ra m ............................................................................14 Figure 3-2— Stress/Strain Curves at Two Different Confining Stresses . . . lb Figure 3-3— Young's Modulus of Cracked Solid ....................................................18 Figure 3-4-ldealized Hysteresis Loops from Figure 1-1 .................................. 22 Figure 3-5 -Subcritical Crack Growth Velocity vs. K[/K|C ................................ 25 Figure 4-1— Outline of Stress/Strain Modeling P r o g r a m ..................................30 Figure 4-2— Sample Param eter F i l e ............................................................................31 Figure 5-1-Schem atic Diagram of Experimental Set U p ..................................35 Figure 5-2-Stress/Strain Data (Experiment 1) ................................................... 37 Figure 5-3-Stress/Strain Data (Experiment 2) ................................................... 38 Figure 5-4— Stress/Strain Data (Experiment 3) ................................................... 39 Figure 5-5— Stress/Strain Data (Experiments 1 & 3 ) ..........................................40 Figure 5-6— Hysteresis Loops (Experiment 3) ...................................................... 42 Figure 5-7— Damage Regime (Experiment 3) ........................................................44 Figure 5-8— Modeling of Stress/Strain Curve for Experiment 1 ....................49 Figure 5-9— Modeling of Stress/Strain Curve for Experiment 2 ....................50 Figure 5-10— Modeling of Stress/Strain Curve for Experiment 3 ............... 51 Figure 5-11-M odeling of Hysteresis Loops for Experiment 1 .......................55 Table 5-1— Material Properties .................................................................................. 46 Table 5-2-Param eters Used to Model Data ......................................................... 46 iv Abstract The change in elastic properties of a solid due to active fracturing a ro u n d an underground nuclear explosion has a significant effect on the waveform of resultant seismic waves. In this work, an algorithm is developed, based on the dam age mechanics model of A sh b y a n d Sam m is [1990], that defines the stress/strain relationship for a brittle solid under com pression, lilastic behavior of the solid upon reversal of stress, such as during hysteresis, is also considered. Kxperimental stress/strain data were collected to verify the stress/strain algorithm and determ ine empirical constants. This data show the stress/strain algorithm to be generally effective in determ ining stress given a particular strain and confining stress. However, elastic behavior during hysteresis was not well modeled. More experim ental work is needed to determ ine the m echanical processes at work during hysteresis. v 1. Introduction The relationship of stress to strain for a brittle solid reflects a variety of processes that can be divided into five regimes: pure elastic, non-linear elastic, dam age, post-failure/brittle, and post-failure/ductile. These regimes can be seen in the idealized stress/strain curve in figure 1-1. The pure elastic regime is present at very low stresses, where the stress/strain curve is fully reversible, due to lack of any deform ation of the solid. As axial stress increases, shear stresses along cracks within the solid overcom e frictional forces, resulting in motion along the cracks. This regime is the non-linear elastic regime, since there is attenuation due to crack sliding, while there is no irreversible deform ation of the brittle material. The dam age regime, which follows the non-linear elastic regime, occurs at stresses high enough to nucleate and propagate new cracks. Within this regime the brittle solid is undergoing active fracturing. Following failure, the solid enters one of two regimes, post-failure/brittle or post failure/ductile. At low confining pressures the solid fails britally, entering the post-failure/brittle regime, where frictional processes dom inate. At high confining pressures, ductile failure occurs and the solid enters the post-failure/ductile regime. The stress/strain relationship is well defined within four of the five regimes: the pure elastic, non-linear elastic, and two post-failure regimes. A volum e of work exists on the pure elastic properties of brittle solids. Likewise, frictional processes dom inating the non-linear elastic regime are well described in Walsh [1965] and W alsh [1966]. The effects of cracks on the elastic properties of a solid have been studied by B udiansky a n d 1 Stress (MPa) Figure 1-1. Idealized Stress/Strain Curve 250 POST-FAILURE REGIMES Failure Stress 200 DAMAGE REGIME 150 Nucleation Stress 1 0 0 NON-LINEAR ELASTIC REGIME 50 PURE ELASTIC REGIME 0.005 0.006 0.001 0.002 0.003 Strain 0.004 2 O 'C onnell [1976]. Finally, Post-failure processes, while not necessarily well understood, seem to be adequately described by either frictional laws, in the case of brittle failure, or equations of state and flow laws, for ductile failure. However, little work has been done to understand the non- linearity within the dam age regime. The purpose of this work is to develop the relationship between stress and strain within the dam age regim e for a brittle solid. Work done most recently in this area by A sh b y a n d Sum m is [ 1990), describes a model that provides the m echanism for dam age accum ulation, an d shows how the model can be used to calculate failure surfaces in stress space. But, this work stops short of developing the stress/strain relationship within the dam age regime. In the following text this relationship will be developed based on the dam age m echanics model of A sh b y a n d Sam m is [1990]. The m otivation behind this aim is the ongoing work of seismologists to model seismic waveforms resulting from underground nuclear explosions an d the observation that non-linearity associated with the active fracture of rock affects the seismic waveform [for example: M inster a n d Day, 1986]. Models currently used to simulate seismic waveforms do not adequately describe the non-linear behavior of a brittle solid in the dam age regime. This is prim arily because theoretical work necessary to extrapolate laboratory d ata to larger scales did not, until recently, exist. Thus, there is still a need for an algorithm that will, independently of scale, describe the change in the elastic properties of a solid as that solid is being fractured. The five stress regimes described above correlate well with the zones arou nd an undergro und nuclear explosion (figure 1-2). The region 3 Figure 1-2. Stress Zones Around an Underground Nuclear Explosion PURE ELASTIC REGIME NON-LINEAR ELASTIC REGIME DAMAGE REGIME POST-FAILURE DUCTILE REGIME Rock flows or disintegrates Rock undergoes active brittle failure Motion occurs along pre-existing cracks Rock behaves elastically 4 im m ediately around the source undergoes high pressures and tem peratures, where rock is displaced to create a cavity and is locally vaporized or melted. This is the post-failure/ductile regime. Beyond this zone, rock undergoes active brittle failure, as small cracks propagate and interact to produce m ajor fractures, as in the dam age regime. Also within this zone, localized areas, where stresses have exceeded failure levels, fall in the post-failure/brittle regime. Further away from the source, where stresses are no longer high enough to cause fractures to propagate, attenuation occurs due to motion along pre-existing cracks; this region is in the non-linear elastic regime. Finally, at great distance from the source, rock behaves elastically and can be described by the pure elastic regime. Thus, it is possible to use the stress/strain relationship based on the dam age m echanics model [A shby a n d Sam m is, 1990] to develop an algorithm , which will determ ine the elastic properties of a solid as it is being fractured, that current waveform models require. The algorithm m ust be com putationally simple so as to be easily incorporated into the existing waveform modeling programs. This thesis attem pts to address the need for such an algorithm. The text is arranged as follows: Chapter Two contains a sum m ary of the dam age mechanics model of A sh b y a n d Sam m is [1990], on which this work is based. Chapter Three describes the developm ent of the stress/strain relationship from dam age m echanics principles. While, C hapter Four focuses on the im plem entation of this theory. C hapter Five describes collection of laboratory data and its implications for the stress/strain model. Lastly, Chapter Six contains a sum m ary of progress accom plished and recom m endations for further research. 5 2. A Review of Damage Mechanics The dam age m echanics model of A sh b y a n d Sam m is [1990] attem pts to characterize the evolution of microcrack damage, from nucleation through failure, in brittle solids under compression. The model was developed from analysis of nucleation, propagation, and interaction of cracks within solids in compression. The model considers these processes in relation to an idealized crack that is representative of all such inhomogeneities within the material. Under compression, a wide range of flaws within the solid m atrix can act as nuclei for new cracks. For example, small holes, poorly bonded particles, phases with differing strength properties can all facilitate the initiation of a new crack. Geometrically, these nuclei are boun ded by the spherical hole at one extreme, and the flat inclined crack at the other. The model developed by A sh b y a n d Sam m is [1990] deals with the flat inclined crack end-m em ber. However, the theoretical developm ent for a spherical hole would not be significantly different. Consider the crack in Figure 2-1. The cross-sectional view shows: the inclined crack, which is the original flaw, and the tensile extensions, or wing cracks. The length of the nucleating crack is taken to be 2a. The crack is inclined at an angle of y from the direction of m axim um stress. The wing cracks, each of length t , lie parallel to oj. W ithin this scope, damage is defined generally as the average concentration of cracks contained by the solid. Initial dam age is the concentration of original flaws within the material, and is defined by D = ^;t(aa)3 Nv 3 ( 2- 1 ) 6 Figure 2 -1 . Idealized Crack (C roas-Sectional and Plan Views) i r i i i n i w here Nv is the num ber of cracks per unit volume and a is a geometric constant related to v |/ (for cracks at 45° to ctj, a= 1/2). As the wing cracks extend an d new dam age is done, instantaneous dam age is defined by D= ^n (f +aa)3Nv 3 ( 2 - 2 ) Several forces contribute to the extension of the wing cracks: wedging, closing, an d crack interaction. Sliding occurring on the initial crack due to the rem ote stress field (oj, 0 3 ), results in a wedging effect at the crack tip causing it to open (as long as oi >> 0 3 ). Opposing this force is the confining stress, 0 3 , which acts to close the crack. In addition, due to crack interaction, the wedging force at the crack tip m ust be balanced w ithin the matrix by a m ean internal force, o '3 , which acts to open the crack. These forces com bine to define the stress intensity factor (Kj) at the crack tip K, = -AjO] yfna 2 3/2 n a ' f D 0" A / , 3/.2 -1 + P a . A 1 1+2 A / \2 -1 2k 2 2 ——a n Ai (2-4) The first term in the curly brackets describes the effects of wedging and crack-interaction, while the second term describes crack closure due to confining pressure, p is a constant set to ensure that K| will equal that for the initial crack when ( is zero, k is 0 3 / 0 1 , and the constants A iand A3 are defined as follows. A j = tiVP V3 / 7 \ X/1 (1 + P2) +P (2-4) 8 In tension, once nucleated, cracks propagate unstably until failure is reached, while in compression crack propagation is stable. For the crack to propagate in compression, the stress intensity at the crack tip (K[) m ust be greater than, or equal to, the critical stress intensity, or the fracture toughness of the material, (K[C ), which can be independently determ ined [for example: M eredith a n d A tkinson, 1985]. Thus, w hen a solid is subjected to a particular stress state, cracks will propagate only until K| has d ro p p ed to K|C . This implies that for any given set of stresses (o |, a d there is a specific am ount of damage that can be expected to result from them. This is the equilibrium damage. When Ki is set equal to K]C , the following expression for Sj in terms of equilibrium dam age can be derived. ( ■ - 1/3 x C> s ,= '_ D ' -l + fi a 3/2 ( 1/3 + S3C4 1+ / 7/J \/ C D 1- D 2/3 , 1 3 ( D V D0 / +S V J 3Cl (2-5) Here constants Ci through C4 are Si an d S3 are norm alized forms of a \ and ct3 defined as ~ „ _ a i sfna ^ K,c ( 2 -7 ) Figure 2-2 illustrates the relationship between Si and D. As dam age increases, axial stress first grows then decreases after reaching a peak stress. This peak is the failure stress. It is clear th at the physical explanation for this is the steady extension of microcracks within the solid, driven by an increasing axial stress. This process continues until there are enough microcracks to form a m ajor fracture and bring about failure, at which time the dam age continues to increase, b u t stress cannot be m aintained, and decreases. It is also expected that the peak, or failure, stress would increase with confining pressure, since it would take a larger wedging force a t the crack tip to counteract the greater closing force. Figures 2-2 and 2-3 illustrates this. In figure 2-2 for each increase in confining stress, 0 3 , the peak is found at a higher stress and damage. In figure 2-3, failure stress is shown to be directly proportional to the confining stress for a given initial dam age. Figure 2-3 also shows that m aterials with higher initial dam age fail at lower stresses. An im portant aspect of this dam age mechanics m odel is that it is scale independent. Because all calculations are based on a, the half-size of a characteristic flaw, and e, the length of a growing crack, it is possible to use this theory to take data collected in the laboratory and extrapolate it for use in the field. 10 Axial Stress (MPa) Figure 2-2. Relationship of Axial Stress to Damage 400 Sig3=20 MPa 350 300 250 Sig3=0 MPa 200 150 100 50 o = Failure Stress 0.6 0.2 0.4 0.8 Damage 11 Axial Stress (MPa) Figure 2 -3 . Failure Surfaces for Three Values of Initial Dam age 200 180 160 140 Do=0.01 1 2 0 00 80 6 0 - Do=0.03 40 Do=0.10 20 80 60 Confining Stress (MPa) 1 0 0 20 40 Confining 12 3. Development of the Stress/Strain Algorithm The dam age mechanics model, described in the previous section, provides some com ponents necessary' to define the non-linear stress/strain relationship in the dam age regime for a brittle solid under com pression. Most im portantly, it provides a constitutive relationship between axial stress and dam age (equation 2-5) that is scale independent. However, in ord er to draw a stress/strain curve using the following relationship °i = E(f ) u - n change in Young’s m odulus (E) with dam age m ust also be known. The following section considers prim ary factors that cause variations in the elastic m odulus throughout the stress/strain curve; am ong these are frictional im pedance of crack sliding, crack closure, dam age present in the solid, energy loss due to nucleation and propagation of cracks, hysteresis, an d sub-critical crack growth. The Stress/Strain Curve. Figure 1-1 shows a generalized stress/strain curve with two hysteresis loops. As a solid becomes dam aged under stress, it undergoes an increasing am ount of brittle deform ation an d exhibits a greater am ount of strain with each stress increm ent. Hence, as dam age accum ulates within a solid, the elastic m odulus decreases. This is evident from the progressive decrease in the slope of the stress/strain curve in figure 1-1, as Young's m odulus is simply the ratio of stress to strain. Analogous to the diagram of zones surrounding an undergroun d explosion (figure 1-2), figure 3-1 shows the different regimes in stress space; the elastic regime, which includes the pure elastic an d non-linear elastic regimes, the dam age regime, the post-failure/brittle and post- 13 Axial Stress (MPa) Figure 3 -1 . S tress Region Diagram 3500 POST-FAILURE POST-FAILURE BRITTLE REGIME DUCTILE REGIME 3000 2500 DAMAGE REGIME 2000 NON-LINEAR ELASTIC & 1500 PURE ELASTIC REGIMES 1000 Do = 0.095 a = 0.0005 Sigy = 2200 500 200 400 Confining Stress (MPa) 800 600 1000 1200 14 failure/ductile regimes. A stress/strain curve, w here the radial stress, or confining pressure, is kept constant, can be thought of as a slice through the stress-zone diagram (figure 3-1). Thus, in figure 3-2a the lower portion of the stress/strain curve falls in the elastic zone, which is b o u n d ed at higher stresses by nucleation, while the upper portion of the curve is in the dam age regime, which term inates in ductile failure. A stress/strain curve draw n for a lower confining stress (figure 3-2b), 100 MPa instead of 400 MPa, shows the dam age regime culm inating in brittle failure. W ithin all four zones of the stress-state diagram there are distinct processes that affect the elastic properties of the solid. This discussion will prim arily concern those processes operating in the dam age regime, with some m ention given to the processes responsible for features of the elastic portion of the stress/strain curve. Pure Elastic Regime. Essentially, the stress/strain curve prior to nucleation is linear and Young's m odulus is constant. However, there are several effects that modify this linear relationship somewhat. It has already been established that dam age within a solid causes its effective elastic m odulus to decrease, and that all solids contain an initial am ount of dam age. However, at the onset of loading, the solid responds with the elasticity of a pure solid, as if it contains no initial flaws. This is due to the presence of friction along crack faces, preventing them from sliding. Once axial stress is raised to a level (Aoi) at which these frictional forces can be overcom e, cracks an d o th er im perfections begin to affect the elasticity of the solid a n d the effective Young’s m odulus decreases. The feature on the idealized stress/strain curve (figure 1- 1 ) attributed to this process is the 15 Figure 3 -2 . Stress/Strain Curves at Two Different Confining S tr esse s a. Strata/Strain Curve at a Confining Straaa of 400 MPa I 2500 r Yield Stress 2000 ! - m a 2 e 1500 ■ ' Nucleation Stress 1000 DUCTILE FAILURE 500 0 02 0.01 0 0 3 0.05 0.04 0.06 Axial Strain b. Streea/Stratn Curve at a Confining Straaa of 100 MPa 1000 Failure Stress 900 800 700 600 f - £ c /> 500 s < 400 ' Nucleation Stress 300 BRITTLE FAILURE 200 100 0.005 001 Axial Strain 0.015 002 change in slope that occurs soon after the onset of loading. In actuality, this change in Young's m odulus is on the order of 10 to 15%. However, it is usually m asked by an increase in elasticity associated with crack closure. The effects of crack closure on elasticity are docum ented in Walsh 11965] and are not considered in this discussion. Non-Linear Elastic Regime. In the non-linear elastic regime, where attenuation of seismic waves occurs due to motion along cracks, the stress/strain curve is approxim ately linear. In this regime Young's m odulus is influenced by the presence of dam age within the solid. The effects of cracks on the elastic properties of solids have been developed by B udiansky a n d O 'C onnell [1976]. B udiansky a n d O 'C onnell [1976] take a self-consistent approach to deriving the effective moduli of cracked solids, including Young's m odulus. This self-consistent m ethod involves consideration of the potential energy of a cracked solid as being that of an uncracked solid m inus energy lost due to cracks. Crack energy loss is determ ined for a representative crack in an infinite solid and multiplied by a crack density factor. This analysis yields a som ewhat complex relationship for effective Young's m odulus and crack density involving an effective Poison's ratio. However, it is a linear relationship, and one that can be converted and used with the A sh b y a n d Sam m is [1990] definition of damage. Figure 3-5a shows the norm alized effective Poison's ratio, bulk modulus, shear m odulus, and Young's modulus, each plotted against crack density [from B u d ia n sky a n d O 'C onnell, 1976]. The norm alized effective elastic m odulus is re-plotted against dam age in Figure 3-3b. The latter graph yields the sim ple linear relationship 17 Figure 3 -3. Young’s Modulus of Cracked Solid a. Moduli ot Cracked Solid (from O'Connall and Budlanaky. 1974] 1 1 T ' T 1 --- * i/3 V ■ i/3 1/4 \ \ \ i " 0 n N sJ / 4 \ l / j \ X N . ± ---------L _ 1 I 1 \ E 1/3 X 1 .X 1 L l \ i- . € • N < o ') b. Chang* In Effective Younga Modulus with Oamago o.a 0.7 0,6 0.3 0 2 0.8 0.2 0 4 0.6 Damage E = Eo(1-1.25D) betw een effective Young's m odulus (E), the "undam aged" Young's m odulus (E0), and dam age (D). This relationship defines Young's m odulus within the non-linear elastic regime. However, once new cracks begin to nucleate from pre-existing flaws and the solid enters the dam age regime, this relationship is no longer adequate. The stress criteria for nucleation is defined by w here p is the coefficient of friction, 2 a is the size of the average flaw, and Kic is the fracture toughness of the solid. Because initial flaws in nature have at least some range of sizes, they nucleate new cracks at different stresses. Hence, the nucleation surface in nature is a range of nucleating stresses. The width of this range depends on the distribution of flaw sizes found in the solid. Damage Regime. W ithin the dam age regim e-the truly non-linear portion of the stress/strain curve-Y oung's m odulus steadily decreases as newly nucleated cracks extend throughout the solid. As was discussed earlier, the connection between increasing dam age and decreasing elasticity is clear. However, a satisfactory m ethod of determ ining the change in the m odulus is not as obvious. While empirical m ethods are theoretically simple, they require the collection of a great deal of data before they can be widely applied. Conversely, the derivation of Young’s m odulus from first principles-such as that of Sam m is a n d A sh b y [1986], w here Young's m odulus was calculated from the energy release rate of 12 19 crack form ation and propagation-results in a relationship that is too cum bersom e to be easily applied. A m edian approach involves use of the simple relationship derived by B udiansky a n d O 'C onnell [1976], which accounts for the decrease in Young's m odulus due to the presence of cracks, and an empirical factor that accounts for the additional decrease due to energy expended on the nucleation and propagation of cracks. While the relationship resulting from the self-consistent approach of B udiansky a n d O 'C onnell [1976| is quite useful for defining Young's m odulus within the non-linear elastic regime, it only accounts for the loss of potential energy due to the presence of cracks; it does not account for the loss of energy due to the active propagation of cracks. It is this portion of the calculation of Young's m odulus that Sam m is a n d A sh b y [1986] found too cum bersom e to be applicable. For this reason, the additional decrease in Young's m odulus due to the propagation of cracks will be determ ined empirically here. The effective Young's m odulus in the dam age regime will be com puted using the following equation E - E 0[ i - ( L 2 s + o q ( V 4 ) The em pirical constant £ will be determ ined from laboratory data. In the form ulation above, it is used in conjunction with cu rren t dam age so as to reflect energy loss due to the creation of that damage. Brittle an d Ductile Failure. The dam age regime is bounded at higher axial stresses by brittle failure, when radial stress is low, or ductile failure, w hen radial stress is high. The boundary condition for brittle failure is determ ined by the m axim um axial stress that can be com puted using equation 2-5 for a given radial stress. Figure 2-2 shows failure stress for three different confining stresses. The criteria for ductile failure is 20 defined by (0 l— a 2)2+(o2-o,) +(o,-o,) which reduces to when 0 2 =0 3 . oy is the yielding stress derived from hardness data <oy=H /3). Hysteresis. It is possible, while in the dam age regime, lor a solid to elastically behave as if it is in the non-linear elastic or pure elastic regimes. This occurs when loading is interrupted and axial stress is kept constant, or is decreased. When stress is not increasing, the solid, though containing damage, is not experiencing active fracturing (with the exception of a small am ount of sub-critical crack growth, which is described later). In these instances, Young's m odulus is defined by the undam aged m odulus or the Budiansky-O'Connell relationship. W hen axial stress is kept constant, Young's m odulus is that found using the Budiansky-O'Connell relationship and current damage. W hen stress begins to increase again, friction prevents the cracks from sliding, as in the pure elastic regime. Here, the undam aged Young's m odulus is invoked, as at the beginning of the stress/strain curve, for a span covering the small stress increm ent Act]. Once the change in stress is sufficient to overcom e friction, Young's m odulus is again determ ined by equation 3-4. W hen loading is interrupted and axial stress is decreased, then increased back to its previous level, energy is lost due to the imperfect elasticity of the solid; this is called hysteresis. Figure 1-1 shows two hysteresis loops along the stress/strain curve, one in the non-linear elastic 21 2 1 °v =2 Figure 3-4. Idealized H ysteresis Loops from Figure 1-1 a. Hyatareaia Loop 1 110 ■ ■ -■ — - -------------------- t r * 1 0 0 - NON-UNEAR , ELASTIC REGIME * * S i 7 0 1 - 60 50 0.001 0.0012 0.0014 0.0016 0.0016 0.0022 Strain b. Hyataraaia Loop 2 200 DAMAGE REGIME 190 180 160 150 140 0.0035 0.0037 0.0039 0.0041 0.0043 0.0045 0 0047 Strain regime a n d one in the dam age regime. These loops are magnified in figure 3-4. The initial portion of the loops, where stress has just begun to decrease, shows the effects of friction; the undam aged Young's m odulus is used. Once friction is overcome and cracks begin to affect elasticity, Young's m odulus is determ ined by the Budiansky-O'Conell relationship. Even though, stress is changing, cracks do not propagate and no energy is lost to this process, hence the solid behaves as in the non-linear elastic regime. The dam age used to determ ine Young’s m odulus is the equilibrium dam age corresponding to the highest axial stress experienced by the solid. W hen stress begins to increase, friction m ust be surm ounted once more. After the change in stress exceeds Acti, further increases in stress are accom m odated by sliding along pre-existing cracks in the solid. If the hysteresis loop is within the non-linear elastic regime, then this p art of the loop, where stress is increasing, simply overlaps the stress/strain curve W hen hysteresis is occurring within the dam age regime, this phase can be considered another occurrence of the non-linear elastic regime, to be followed by nucleation and entrance into the dam age regime. The change in nucleating stress is due to the change in dam age present within the solid. Furtherm ore, the new nucleating stress corresponds to the highest axial stress previously experienced by the solid; it is at this stress th at curren t dam age was incurred. Once axial stress has exceeded the new nucleating stress, or the highest stress previously reached, equation 3-4 is once m ore used to define Young's m odulus and the elastic behavior of the solid is that of the dam age regime. Sub-critical Crack Growth. Thus far, the effects of friction, dam age, propagation of cracks, and hysteresis on the stress/strain relationship 23 have been described. In conjunction with these processes, another operates throughout the stress/strain curve to alter Young's m odulus. This process is called sub-critical crack growth. Recalling that cracks propagate when the stress intensity at the crack tip (K[) either m atches or exceeds the critical stress intensity, or fracture toughness, (Kjc) of the solid, the nam e of this process suggests that cracks also propagate when stress intensity is less than K]C , or is sub-critical. Sub-critical crack growth is defined in term s of velocity of crack growth, which depends on stress intensity. As stress intensity at the crack tip approaches critical stress intensity for the solid, the velocity of sub- critical crack growth increases. This m eans that the am ount of additional damage, or total length of new cracks, resulting from this process depends on the ratio of Kj to K[C and the loading rate. When Ki/K[C approaches 1, sub-critical crack growth velocity for Westerly granite at room tem perature is on the order of 10 4-10 ^ m /s [ A tk in so n, 1984; A tkin so n a n d M ered ith, 19871 . However, these velocities are only reached when nucleation is imminent; throughout the pure elastic and most of the no n linear elastic regimes, where K]/K[C is less than 0.8, crack growth velocity is less than 10"? m /s for Westerly granite. Figure 3-6 shows crack velocity plotted against Kj/Kic for Westerly granite- The overall effect of sub-critical crack growth is to decrease the failure strength of the solid. The am ount of decrease depends on loading rate; for every order-of-m agnitude decrease in loading rate, the failure strength of a solid in com pression decreases about 3% [Paterson, 1978]. For rates of loading found in laboratory experim ents this is not a significant decrease. However, at geologic rates this effect becomes substantial. Sub- 24 lo g Crack Velocity (m/sec) Figure 3 -5 . Subcritical Crack Growth Velocity vs. KI/KIc -4 Westerly granite Tem perature = 20 deg. C Relative Humidity = 30% Crack Growth Index (m) = 39 -5 -6 -7 -9 -1 0 # Atkinson [1984] 0.65 0.6 0.7 0.75 0.8 KI/KIc 0.85 0.95 0.9 25 critical crack growth also acts to sm ooth the transition from the non-linear elastic to the dam age regimes by providing a m echanism that allows for the gradual increase in accum ulation of damage. A com plete review of work on sub-critical crack growth can be found in A tkin so n an M eredith [1987] and A tkin so n [1984], While, the relevance of sub-critical crack growth to tim e-dependent deform ation and failure is discussed in Costin [1987]. Within the context of this stress/strain algorithm, sub-critical crack growth is included for the purpose of generality; it allows the model to incorporate the effects of loading rate. And it is anticipated that its inclusion will provide for better matches of experim entally generated stress/strain curves because of its smoothing properties. For a laboratory stress/strain experim ent, the contribution of sub-critical crack growth to the total extension of the wing cracks is determ ined according to the following equation 8 ^ = 5o ' 1 \ ,fK, f V. (3-7) w here & £ is the change in wing crack length, 8c i is the increm ent of stress over which this change occurs, R/ is the rate of loading, Vc is crack growth velocity, and m is the crack velocity index, or simply the slope of a log/log plot of crack velocity vs. stress intensity. 26 4. Implementation of the Stress/Strain Algorithm Based on the stress/strain relationship previously described several com puter routines have been com posed to im plem ent various portions of the theory; one finds failure stress given confining stress and initial dam age, a second determ ines dam age given axial and radial stress, while an other routine determ ines the regime into which that stress state falls. These routines were com bined to form a program that models stress/strain curves from onset of loading through failure. Com parison of m odeled curves with laboratory generated ones allows for an experim ental check of the developed theory. Ultimately, the program that models stress/strain curves can be adjusted for use in seismic waveform modeling systems. Com ponents of the Stress/Strain Modeling Program . The initial program m ing aim was to im plem ent the dam age m echanics m odel and reproduce its results shown in A sh b y a n d Sam m is [ 1990]. Using the relationship of Si to D (equation 2-5) failure stresses were determ ined for different com binations of confining stress and initial damage. This was done by finding m axim um axial stress while increm enting dam age and calculating the corresponding stress. A recursive searching technique was used. Knowing that Dma\ (damage value corresponding to maximum stress) m ust lie between initial dam age and one (when dam age equals one, the volum e is completely filled with cracks), this span is traversed at dam age increm ents of 1CH to find m axim um stress. Then values of dam age between Dmax - 1 0 1 and Dmax + 1 0 * 1 are traversed at dam age increm ents of 10 2, and so on. This spanning of ever-narrow ing value 27 segm ents can be carried out an infinite num ber of times to arrive at a precise value for m axim um stress, and its corresponding dam age. However, in the interest of alacrity, this process is carried out only as m any times as necessary to determ ine the values at a desired precision level; num ber of repetitions corresponds to num ber of significant digits for found values. The resulting curves can be seen in figure 2-2. Despite the fact that an explicit relationship between axial stress and dam age exists, equation 2 - 5 cannot be solved for I>, hence a curve fitting technique is necessary. The m ethod described above lead naturally to a procedure for finding dam age given axial and radial stress. It was altered to search for the dam age value corresponding to the first axial stress value exceeding that which is given. Again, num ber of repetitions corresponds to the precision of resulting dam age value. Given an axial and confining stress, the regime into which they fall can be determ ined using nucleation, brittle failure, and ductile failure surfaces explicitly defined in A sh b y a n d Sam m is [1990). Nucleation criteria are easily checked by com paring current stress with that determ ined by equation 3-3. While, ductile failure is established by the use of equation 3-6. Satisfaction of criteria for brittle failure is determ ined by finding failure stress and corresponding damage, using the recursive m ethod described above, and com paring current dam age to Dmax. Brittle failure is determ ined using dam age instead of axial stress, which would be m ore direct, because as failure approaches, dam age changes m ore rapidly than stress. The rem aining aspects of the theory required to construct a stress/strain modeling program are calculation of stress intensity an d sub- 28 critical crack growth. Stress intensity, K], is specified by equation 2-3, which shows it to depend on oi, 0 3 , and current damage. Sub-critical crack growth, in turn, depends on K[. The change in wing crack length, bf, as a result of sub-critical crack growth is determ ined by equation 3-7. Before bt can be incorporated into damage, the length of the wing cracks prior to sub-critical extension m ust be determ ined (using equation 2 -2 ). Then new dam age is com puted by substituting (f+fiP) for (() in equation 2 - 2 . Stress/Strain Modeling Program . All com ponents necessary for construction of the stress/strain modeling program are now prepared. The FORTRAN code for this program appears in the Appendix. Figure 4-1 shows a schematic diagram outlining the sequence of processes used to generate a stress/strain curve based on the theory presented earlier. In the forem ost portion of the program all necessary param eters, variables, and constants are established. The first task executed is reading input from a param eter file. An example of this file is shown in figure 4-2. Param eters listed under the heading "Constants" are not expected to change dram atically, if at all, for different rock types and can be incorporated directly into the code. The next seven param eters, listed u n d er "Material Properties", can usually be found in existing literature and m ust be provided by the investigator. The two values under "Hmpirically Determ ined Factors", £ and Acri, are to be adjusted based on experim ental results. The rem ainder of the file provides experim ental param eters; these include: the desired precision of results, loading rate, strain increm ent, total strain accumulation, confining stress, and hysteresis loop specifications. Once the param eters are read in, variables are initialized, 29 Figure 4-1. Outline of Stress/Strain Modeling Algorithm Stress is calculated using undam aged Young's modulus. SETUP: Param eters read in, variables initialized, and constants computed BEGIN MAIN LOOP: Strain is incremented trom zero to total amount of strain J Stress is calculated using Budiansky-O'Connell Young's modulus Current stress greater than Ds1 S tress intensity and sub- critical crack growth are determined. Stress is calculated using Young's modulus adjusted with C (eq. 3-4) Has nucleation occurred ? Stress intensity and sub- critical crack growth are determined. Has ductile failure occurred? Has brittle failure occurred? Check for hysteresis. END OF LOOP CYCLE Yes Yes " . . . END OF ALGORITHM 30 Figure 4-2. Sample Parameter File KEY PHRASE VALUE (UNITS) TYPE OF ROCK granite * C onstants ALPHA BETA COEFFICIENT OF FRIC PI CONSTANT 3 0.7 0. 45 0. 6 3.14159265 2 * Material Properties INITIAL DAMAGE FRACTURE TOUGHNESS 1/2 LENGTH OF CRACK YIELD STRESS YOUNGS MODULUS {Eo) CRACK GRO. VELOCITY Vc/KI SLOPE (m) 0.093 1.0 0.0006 2200 69000 0.000001 39.0 (MPa m 1^ 2) (meters) (MPa) (MPa) (m/sec) * Empirically Determined Factors z DELTA SIGMA (dsig) 0.55 40 * Experimental Param eters DESIRED PRECISION LOADING RATE STRAIN INCREMENT TOTAL STRAIN ACCUM. CONFINING STRESS 6 0.62 0.00001 0.006 10 (MPa/sec) (MPa) NUMBER OF H LOOPS LOOP 1: BEGIN LOOP 1: END LOOP 2: BEGIN LOOP 2: END 2 0.00108 0.00078 0.00024 0.00013 com m only used constants com puted, and failure stress and dam age for the given confining stress are determ ined for later use. At this point, the program enters a loop, within which strain is increm ented from zero to the total am ount of accum ulated strain. For each strain, stress is initially calculated using either the undam aged Young’s m odulus, as in the pure elastic regime, or the Budiansky-O'Connell Young's modulus, as in the non-linear elastic regime, provided accum ulated stress has exceeded Aoi. Then, the condition for nucleation is tested. If nucleation has not occurred, stress intensity and sub-critical crack growth are determ ined and the rest of this loop cycle is skipped. If nucleation has occurred, the program moves on to evaluate damage. Damage is first determ ined based on current stress. This new dam age is used to adjust stress downward with empirical factor £. Then, dam age is again determ ined based on the adjusted axial stress. Stress intensity and sub-critical crack growth are com puted. Finally, the conditions for brittle an d ductile failure are tested. If either has occurred, the rest of the loop is not executed. W hen a hysteresis loop is encountered, execution of the main loop is paused while strain is decreased then increased within two auxiliary loops. On both sides of the loop, stress is calculated first with the undam aged Young’s m odulus, as crack sliding is im peded by friction, then with the Budiansky-O'Connell modulus. Upon com pletion of a hysteresis loop, the m ain loop is resum ed from the stress/strain position at which it was paused. Stress/Strain Algorithm and Seismic Waveform Modeling. This stress/strain modeling algorithm can provide the adjustm ents necessary 32 for m ore effective seismic waveform modeling program s. These algorithm s com m only utilize finite elem ent techniques that require the calculation of strains and stresses at each element. Given a state of stress, equations of m otion are used to calculate displacem ents, which in turn yield strain. A new stress state is determ ined from strain, using undam aged elastic constants, and the cycle begins again. During this last step, stress can be calculated from strain using the m ethod described above instead. This would allow the seismic waveform modeling algorithm s to account for the changing elasticity of a solid within the dam age regime. S. Experimental Stress/Strain Data Laboratory stress/strain data were collected in December of 1994 using facilities at New England Research (a private rock m echanics laboratory in White River Junction, Vermont). The main purpose of these experim ents was to test the stress/strain algorithm by com paring m odeled stress/strain curves to experim entally generated ones, and com paring calculated failure stresses to those found in the lab. The secondary aim was to determ ine em pirical constants which accounts for the decrease in Young's m odulus due to nucleation and propagation of cracks, and Aoi, which is the change in stress necessary to overcome friction along cracks. £ can be determ ined from change in slope of stress/strain curves in the dam age regime, while Ac t i can be determ ined from the shape of hysteresis loops. Experimental A pparatus. Figure 5-1 shows a schematic diagram of the experim ental set up. The experim ents were conducted using a hydraulically driven triaxial compression m achine with program m able servo-control. Samples used were right circular cylinders, 5.5 cm in diam eter and 10.5 cm in length. Argon gas was the compressive medium . In o rd er to keep the gas out, samples placed un d er com pression were jacketed with copper foil. Two load cells, one m ounted outside of the pressure vessel and one imm ediately above the sample, were used to m easure axial stress. Axial and radial strain gages were m ounted directly on the sam ple ,or its foil jacket, in the middle third of the sample. Two strain gages, placed on opposite sides of the sample, were used to m easure axial strain. A third strain gage was used to m easure radial strain. 34 Figure 5-1. Schem atic Diagram of Experimental Set Up Gi I HYDRAULIC RAM INTERNAL LOAD CELL ARGON GAS ARGON GAS HYDRAULIC RAM Gi 35 Experimental Procedure. The experim ents were conducted with three sam ples of Barre granite that were slowly loaded to failure in compression. The first and third samples were under a confining stress of 10 MPa, while the second sam ple was unconfined. A loading rate of 0.6 MPa/sec was used exclusively in experim ents two and three, and predom inantly in experim ent one. A loading rate of 0.3 MPa/sec was also used in experim ent one during several hysteresis loops. The reversal of axial stress during hysteresis was perform ed m anually in experim ent one, and autom atically in the other two experiments. The reason for this was that experim ent one contained single hysteresis loops of various sizes (15-60 MPa), while experim ents two and three contained sets of six loops all of one size (20 MPa); the precision needed to reverse stress six times at the sam e stress position could best be achieved by autom ating the process. Experimental Results. Data from the three experim ents is plotted in figures 5-2 through 5-4; figure 5-5 shows data from experim ents one and three together. The average of the two axial gages is taken as a m easure of axial strain. While only data from the internal load cell, which is considered to be less contam inated, is used as a m easure of axial stress. The first experim ent served as a test run to identify possible factors of interest, an d was used to design the rem aining two experiments. Based on this first experim ent three key observations were made. Loops of different sizes did not appear to have any difference in shape. Hence, smaller loops were used in the rem aining experim ents in the interest of speed. Likewise, hysteresis loops conducted at the slower loading rate did not ap p ear to differ from those conducted at the faster rate. Therefore, the faster rate was used in the rem aining experiments, again because of time 36 Axial Stress (MPa) Figure 5 -2 . Stress/Strain Data (Experiment 1) 300 Peak Stress = 280 MPa 250 Confining Stress = 10 MPa 200 150 100 50 0 0.001 0.002 0.004 0.005 0.006 0.003 Average Axial Strain Axial Stress (MPa) Figure 5-3. Stress/Strain Data (Experiment 2) 250 Peak Stress = 206 MPa Confining Stress = 0 MPa 200 150 100 50 0.004 0.005 0.001 0.002 Average Axial Strain 0.003 Strain Axial Stress (MPa) Figure 5-4. Stress/Strain Data (Experiment 3) 300 Peak Stress = 209 MPa 250 Confining Stress = 1 0 MPa 200 150 100 0.001 0.002 0.003 0.004 Average Axial Strain 0.005 0.006 Axial Stress (MPa) Figure 5 -5 . Stress/Strain Data (Experiments 1 & 3) 300 Peak Stress = 280 & 289 MPa 250 Confining Stress = 10 MPa 200 150 100 50 0.004 0.003 0.005 0.001 0.002 0.006 0 Average Axial Strain considerations. The third observation m ade was of the unexpected increase in strain, or decrease in Young's m odulus, appearing at the end of hysteresis loops. This ratcheting effect was observed to become m ore p ronounced at higher stresses. It was also observed that w hen several loops began at the same stress, each successive loop had less ratcheting. Based on these last observations, the next two experim ents were designed to focus on the ratcheting effect. In experim ents two and three, stress was reversed often to provide m any exam ples of hysteresis at various stresses. Sets of six hysteresis loops were executed each time to highlight the change in am o u n t of ratcheting with each successive loop, f igure 5-6 shows four sets of loops from experim ent three. The first set of loops, occurring well below nucleation, shows six loops superim posed directly one on top of another. This indicates that no additional strain occurred as a result of successive stress increases and decreases. In the fourth set of hysteresis loops, which occurs right at nucleation, the loops are not as well superim posed. In the ninth set, each successive loop has clearly shifted to the right, with the first few loops shifting m ore than the later ones. Finally, the twelfth set of loops, occurring only 10-20 MPa below failure, exhibits very distinct shifts to higher strains, as well as, a decrease in this shift with each successive loop. Figure 5-7 shows a close up of the upper third of the stress/strain curve for sam ple three. Here, the last five loop sets, and several single hysteresis loops, can be seen. The change in the am ount of ratcheting from set 9 to set 13 is dram atic. The last two single hysteresis loops are also particularly rem arkable in the great am ount of ratcheting they exhibit. 41 Figure 5-6. H ysteresis Loops (Experiment 3) a. 1 st Set of Hysteresis Loops 50-------------------------------------—■-------- ---------- ----- 40- 30 25 £ $ 0 0 6 0.0007 0.0006 Average AxiaJ Strain 000 0 9 0 001 b. 4tti Set of Hysteresis Loops 1701 - — -------------------------------- -------------------------------- 165 160 ■ « a. 2 | 155 - a s } 150- 145 0.003 0,0027 0.0028 Average Axial Strain 0.0029 Axial Stress (MPa) Axial Stress (MPa) Figure 5 -6 (continued). H ysteresis Loops (Experiment 3) c. 9th Set ot Hysteresis Loops 240- 235 ' > ■ 220 215 &03S 0.0042 0.0043 0.0041 Average Axial Strain 0.004 270 265 r 260 h 255- 250 h 245 f d. 12th Set of Hysteresis Loops 0.00475 0 00485 0.00495 Average Axial Strain 0.00505 Axial Stress (MPa) Figure 5 -7 . Damage Regime (Experiment 3) 290 280 270 260 set 13 250 set 12 240 set 11 230 set 10 220 210 set 9 200 0.006 0.0055 0.005 Average Axial Strain 0.004 0.0045 Modeling of Experimental Results. After completion of the experim ents an d processing of acquired data, an attem pt was m ade to m atch resulting stress/strain curves with those produced by the stress/strain model. In o rd er to generate model curves, appropriate values for m aterial constants had to be chosen first. Since laboratory data on Barre granite is not as a b u n d an t as it is for m ore com m only used materials, such as Westerly granite, some of the material constants were found from the data curves, while others were only approxim ately determ ined. Table 5-1 contains a listing of published material properties. Where available, data for Barre granite are shown. Otherwise, values for Westerly granite are listed. Table 5-2 shows the constants used to m odel the experim ental data. Initial dam age and half-length of pre-existing cracks were first taken to be that of Westerly granite, 0.08 and 0.0005 meters respectively. Later these values were adjusted to 0.093 (experiments one and two) and 0.080 (experim ent three) for initial damage and 0.0006 m eters for half-length of initial flaws. Increase of these material properties is supported by the fact that Barre granite is m ore coarse grained than Westerly granite [Boitnott, 1995], The new values were found by m atching shape of the overall curve an d failure stress for each experiment. It should be noted that for experim ent one a better fit of the curve shape was achieved with a value of 0.1 for initial damage. But, failure stress for this model was 10 MPa lower than in the experiment. It is possible that 0.1 initial dam age was, in fact, a m ore accurate value for sam ple one. In that case, it's higher failure stress would be attributed to structural inhomogeneities. However, the stress/strain algorithm does not take inhom ogeneities into account. It uses average m aterial properties to 45 Table 5-1. Material Properties M a te r ia l Property S o u r c e Initial Damage * 0.08 Ashby and Santmis [I 990] Fracture Toughness I MPa m l^ -) Ashby and Santmis [I 990] Half Length of Crack * 0.0005 meters Ashby and Santmis [I 990] Yield Stress 2200 MPa Ashby and Santmis [I960] Undamaged Young's Modulus *# 64 GPa A STM [1992; 1994] Crack Growth Velocity lOe-6 m/sec A tkinson [1984] Crack Growth Velocity Index 3 9 Atkinson [1984] * values adjusted later to fit data # Young's Modulus data is for Barre granite, all other values are for Westerly gran ite Table 5-2. Parameters Used to Model Data Model Parameter Models 1 & 4 Model 2 Model 3 Initial Damage * 0.093 0.093 0.086 Fracture Toughness (MPa m 1/2) 1 1 1 Half Length of Crack (meters) * 0.0006 0.0006 0.0006 Yield Stress (MPa) 2200 2200 2200 Undamaged Young's Modulus (GPa) * 69 65 69 Crack Growth Velocity (m/sec) IOe-6 IOe-6 IOe-6 Crack Growth Velocity Index 39 39 39 Empirical Constant C , * 0.55 0.50 0.30 Stress Increment A at (MPa) 40 40 40 * values adjusted to fit data 46 determ ine the general behavior of a solid body. This approach may som etim es be deficient in modeling the behavior of small rock samples. Nevertheless, it should be adequate when used to m odel the behavior of large rock volumes, such as those considered by seismic waveform m odeling algorithm s, since local inhomogeneities will be canceled out. A nother m aterial property that could affect failure stress is sub-critical crack growth velocity. However, because no changes in stress/strain behavior were observed when loading rate was decreased, it was assum ed that loading rate was sufficiently high to prevent sub-critical crack growth from significantly altering failure stress. As well, experim ental data is best fit by models in which failure is reached critically. Thus, a value of 10 6 m eters/sec. was chosen for the sub-critical crack growth velocity. This value allows for a 2 to 4% decrease in failure stress for every o rd er of m agnitude increase in loading rate as required by Paterson [1978]. It falls within the range of velocities, 10 ^ - 1 0 8 m eters/sec [ A tkin so n , 1984; A tk in so n a n d M eredith, 1987], shown in figure 8-5. And, it results in critical failure when a loading rale of 0.0 MPa/sec is used, as it was in the laboratory experim ents conducted for this study. Of the rem aining m aterial properties, the undam aged Young's m odulus was calculated from experim entally determ ined elastic moduli for Barre granite [ASTM, 1992; 1994] using the Budiansky-O'Connell relationship. The sub-critical crack growth velocity index, or AVc/AKi, was also taken from experim ental data [A tkinson , 1984; A tkin so n a n d M eredith, 1987], Yield stress was taken to be that of Westerly granite, 2200 MPa [A sh b y a n d Sam m is, 1990]. However, a m axim um confining stress of 10 MPa , that used in the laboratory, is not high enough for ductile failure to occur in 47 granite, hence finding a precise value for yield stress was not crucial. Likewise, a specific value for critical stress intensity is not necessary since cu rren t stress intensity is always calculated relative to this num ber. For, this reason a value of 1 MPa m<1/2> [A shby a n d Sam m is, 1990] was used. The resulting m odel curves are shown alone and com pared with laboratory data in figures 5-8 through 5-10. The overall fit of all three data sets is quite good. W hen plotted with data, model curves were offset to the right so that portions of the curves below nucleation that are linear were aligned. Model curves were not expected to m atch experim ental data at low stresses since the effects of crack closure were not included in the stress/strain model. The effects of crack closure are evident in all three data sets, especially in the unconfined experim ent (sample 2), w here a significant am ount of ’'strengthening" due to crack closure occurs below 10 MPa. In the other two experiments, this increase in Young’s m odulus occurred, prior to onset of axial loading, while the samples were allowed to reach equilibrium un der a confining stress of 10 MPa. Final adjustm ents to the shape of the model curve within the dam age regime were done with the use of empirical constant Values of 0.55, 0.5, and 0.3 were used to m atch data from experim ents one, two, and three respectively. This am ounts to additional decreases in Young's m odulus due to the nucleation and propagation of cracks of 24 and 40% of the decrease in Young's m odulus due to the presence of cracks. Recalling that the 1.25 Budiansky-O'Connell constant and the constants found here are always used in conjunction with dam age (see equation 3-4), these decreases gain significance with proximity to failure. Also, it should be noted that with adjustm ents to m aterial properties, such as the 48 Figure 5 -8 . Modeling of Stress/Strain Curve for Experiment 1 a. Streea/Strain Modal 1 3 0 0 ------------------------------ ■ ---------------------------------------------------- Failure S tress 250 h 200 h Q . 2 N ucleation S tress 100 0.001 0.002 0.003 Average Axial Strain 0.004 0.006 0 006 b. Model 1 With Data From Expert merit 1 300;--------------- . --------------- ■ -----—---------------------- Failure S tress 250 200 < s 0. 2 1 150 < 7 5 9 5 N ucleation S tress 100 50 0.002 0.003 Average Axial Strain 0.005 0 006 0.001 0.004 A xial Stress (M Pa) Axial Stress (MPa) Figure 5-9. Modeling of Stress/Strain Curve for Experiment 2 a. Stress/Strain Model 2 200 f - Failure S tress I80r- 160 i - i 140 [ 1 2 0 !- 100 N ucleation Sires: 80 60 40 2 0 - 0.001 0.002 Average Axial Strain 0.003 0.004 0.005 b. M odel 2 W H h Data From Experiment 2 200 Failure Stress 180 160 140 120 100 Nucleation S tress 80 60 40 0.001 0.003 ' Strain 0.004 0.002 Average Axial Strain 0.005 Figure 5-10. Modeling of Streas/Strain Curve for Experiment 3 a. Streaa/Straln Modal 3 3 0 0 ---------------------------- -------- ------------------- -------------------- — ------- Failure S tress 2 S 0 ■ 200 150 Nucleation Strei 100 50- 0.001 0.008 0.003 0.004 0.006 0.006 Average Ariel Strain b. Medal 3 K W a i Data From Experiment 3 300 Failure Stress 200 e Q . 3 150 5 5 I Nucleation Stress 100 50 0.002 0.003 0.004 0.005 0.006 Average Axial Strain 0.001 undam aged Young's m odulus and sub-critical crack growth velocity, failure stress can be m atched without the use of empirical constant However, these curves yield poorer fits to the data. Once it has been ascertained that the stress/strain algorithm provides a good fit to the overall shape of experimental stress/strain curves, attention is directed to analysis of hysteresis. The data presented here shows hysteresis loops to possess two distinct characteristics, aside from ratcheting. These are their extremely linear shape, indicating lack of attenuation, and their consistency of slope, indicating lack of dependence on damage. According to the stress/strain algorithm developed earlier, at the onset of a stress reversal, frictional forces act to im pede crack sliding an d the solid behaves elastically. During this phase, the stress/strain slope is defined by the undam aged Young's modulus. Once frictional forces have been overcome, crack sliding ensues and the Budiansky- O'Connell m odulus defines slope. Toward the end of the dam age regime, the change in slope of the stress/strain curve between these two regimes is quite dram atic, with Young's m odulus decreasing by nearly 50%. If both regimes were present within hysteresis loops, those loops near failure would be rhom bic in shape and have significant areas. Clearly this is not the case for the hysteresis loops in this data set. It appears that only one regime is present, and since loops throughout the stress/strain curve have the same slope, that regime m ust be the pure elastic. This would seem to indicate that Aoi, the stress range over which this regime dom inates, exceeds 60 MPa, the size of the largest hysteresis loop in the experim ents. At first glance, this appears to be a satisfactory conclusion. It allows for com pletely fiat hysteresis loops, spanning up to 60 MPa. As well, it 52 provides an explanation for the difference in peak stress between experim ents one and three, and the unusual shape of data curves near failure for experim ents two and three. The last two experiments, which contain a great num ber of hysteresis loops, particularly near failure, have stress/strain curves that are rounded near failure and appear to exhibit greater strength. If the stress range, over which crack sliding is impeded, exceeds the range of a hysteresis loop, when that loop has been carried o u t and stress increase along the main curve is resumed, frictional im pedance would still be in effect. This would result in an ap p aren t strengthening, allowing the solid to experience higher stresses before it fails, and would cause the stress/strain curve to appear m ore rounded near failure. However, if a value of 60 MPa is used for Ac t i to m odel a stress/strain curve with hysteresis, the results are very poor. The am ount of frictional im pedance that spills over onto the main curve far exceeds that which m ight explain the observations of strengthening and curvature of the curve. When the stress necessary to overcome friction is calculated independently with Mohr diagram techniques, it is determ ined to be 40 MPa at 10 MPa confining pressure, and O MPa when the solid is unconfined. These results are also not satisfactory. With Aai of 40 MPa, m odeling results similar to those described above for 60 MPa are found. Also, the hysteresis loop spanning 60 MPa, which occurs well within the dam age regime, would be expected to have substantial area. Yet, it appears as linear as the smaller loops. Furtherm ore, if Aoi were 0 MPa for the unconfined experiment, hysteresis loops in that experim ent would reside solely within the non-linear elastic regime and have slopes equal to 53 that of the stress/strain curve where they occur. Clearly, som ething other than a simple frictional effect is at work here. More experim ents are needed to define the behavior of brittle solids during hysteresis. In the m ean time, a model fit to the data from experim ent one was attained by using only the undam aged Young's m odulus during hysteresis. Aoi was set equal to 40 MPa, b u t only a fraction, based on damage, was allowed to affect the main curve following hysteresis. The result appears in figure 5-11. It was also found that a m ore precise value for undam aged Young's m odulus could be found from the slope of hysteresis loops. For experim ent one and three this value was 69 GPa, while for experim ent two it was 65 GPa. The third characteristic of experim ental hysteresis loops, ratcheting, can not be explained by the stress/strain theory developed earlier. That algorithm specifies that all hysteresis loops begin and end at the same point in stress/strain space. It assum es that no dam age is done within these loops, except th at resulting from sub-critical crack growth. Besides the already adopted assum ption that sub-critical crack growth did not play a significant role in these experiments, there are two other reasons why it can not account for the observed increase in damage. The first being that this extra dam age occurs only on the upw ard leg of the loop. If sub-critical crack growth was responsible for the ratcheting effect, it would be expected that the downward leg be equally affected, since the ratio Ki/K ic, on which sub-critical crack growth velocity depends, is the same over the course of the two legs. The other reason th at sub-critical crack growth is not a satisfactory explanation for the increase in dam age during hysteresis is that the am ount of increase varies throughout the curve, 54 Axial Stress (MPa) Axial Stress (MPa) Figure 5-11. Modeling of H ysteresis Loops for Experiment 1 a. Stress/Strain Model 4 300--------------- ■ --------------■ -------------- — Failure S tress / 150 Nucleation Stress 100 50 0.001 0.002 0 005 0.003 Average Axiel Strain 0.004 0.006 b. Modal 4 W W t Data Prom Experiment 1 3001 -------------- ■ --------------■ --------------■ --------------■ Failure Stress 250 200 150 Nucleation Stress 100 50 ■ 0.001 0.002 0.003 0.004 Averags Axial Strain 0.005 0.006 particularly within the dam age regime, and that it also varies with successive loops within a set. The ratio K[/K[C is always equal to one along the stress/strain curve within the damage regime, and it does not change betw een two hysteresis loops spanning the same stress range. It appears th at the observed ratcheting is the result of some process that affects only the upw ard portion of a hysteresis loop, and which is influenced by proximity to failure. As with the problem atic interpretation of o ther hysteresis mechanism s described above, further laboratory experim ents are needed to understand the nature of this ratcheting effect. 56 6. Conclusions and Recommendations for Further Research The goal of this work was to develop and im plem ent a stress/strain algorithm th at could be used in seismic waveform modeling program s. C hapters Three and Four describe this algorithm and its im plem entation. Laboratory data was collected to verify the stress/strain algorithm and supply the two empirical factors it utilizes. Chapter Five describes the experim ents and shows their results. Model fits of experim ental data dem onstrate the overall effectiveness of the stress/strain algorithm. Stress/strain behavior, outside of hysteresis, is shown to be well m atched by model results throughout the dam age regime. For all three experiments, both failure stress and change in Young's m odulus (or equivalently, curve shape) are reproduced by the models. Considering that material properties for Barre granite were not well defined, this result is very promising. It indicates that the algorithm can be used successfully even if material properties are not well known. Determ ination of empirical constants was less successful. A range of values, 0.30-0.55, were determ ined for empirical constant which was used to further reduce Young's m odulus within the dam age regime. One reason this constant could not be defined m ore precisely is the lack of available data on the material properties of Barre granite, and the fact that d ata that are available are poorly constrained. Another reason that a m ore precise value for £ could not be determ ined is the inherent assum ption in the stress/strain model th at m aterials are highly hom ogeneous. W hen inhomogeneities in a sample affect its behavior, these deviations wind up being attributed to the empirical constant. Thus, 57 the precision with which < ; can be determ ined depends on the uniform ity of the m aterial used The second empirical constant, Actj, used to represent the change in stress necessary to overcome frictional forces, could not be determ ined from experim ental results at all. Partially, this was the result of hysteresis loops that spanned changes in stress too small to overcome the effects of friction. For the confined experiments, in all cases but one, hysteresis loops did not exceed the 40 MPa stress range predicted by frictional theory as the m inim um stress change necessary for crack sliding to occur. However, it is not at all clear that if larger loops were used, the non-linear elastic regime, represented by crack sliding, could be observed during hysteresis along with the pure elastic regime. The non-linear elastic regime is not observed in the unconfined experiment, for which Aoi is calculated to be 0 MPa. Nor is it evident in the 60 MPa loop in experim ent one. This indicates that the role of frictional im pedance of crack sliding during hysteresis may be other than that specified by the stress/strain algorithm . Further experim ental work is required to determ ine the role, if any, of crack sliding during hysteresis. In the next round of experim ents, stress loops of different sizes, particularly large ones near failure, should be used. As well, experim ents should be conducted at several different confining stresses, in o rd er to determ ine the relationship of Aoq to < 7 3. Experimental results show another assum ption within the stress/strain algorithm regarding hysteresis to be unsubstantiated. This assum ption is th at no additional strain, or damage, occurs during hysteresis. Clearly, this can not be the case, as the ratcheting seen at the end of hysteresis loops in the dam age regime shows. However, a m echanical explanation for 58 this effect is not readily evident. As discussed earlier, sub-critical crack grow th can not satisfactorily explain all observations. Again, m ore laboratory data is needed to understand this phenom enon. In order to isolate any time d ep en d en t causes of ratcheting, experim ents with different loading rates are needed. Also, quantifying the relationships of ratcheting to dam age, and of ratcheting to repetition of hysteresis at the same stress may prove useful. In conclusion, the aim of this th esis-lo develop and im plem ent an effective stress/strain algorithm that could be used by seismic waveform m odeling p ro g ram s-h as been partially satisfied. The algorithm is effective in all aspects b u t that of hysteresis. Further experim ental work is necessary to clarify the m echanical processes at work during hysteresis. 59 References Ashby, M. F., and C. G. Sammis, The dam age mechanics of brittle solids in compression, PAGEOPH, 133, 489-521, 1990. ASTM Institute for Standards Research, Interlaboratory testing program for rock p ro p erties (ITP/RP): R ou n d o n e— Longitudinal a n d transverse p u lse velocities, u n co n fin ed com pressive strength, uniaxial elastic m od u lu s, a n d sp littin g tensile strength, Septem ber 30, 1992. ASTM Institute for Standards Research, Interlaboratory testing program for rock p ro p erties (ITP/RP): R ound tw o— C onfined com pression: elastic m o d u lu s a n d ultim ate strength, October, 1994. Atkinson, B. K., Subcritical crack growth in geological materials, /. G eophys. Res., 89, 1077-4114, 1984. Atkinson, B. K., and P. G. Meredith, The theory of subcritical crack growth with applications to minerals and rocks, in fra c tu re M echanics o f Rock, edited by B. K. Atkinson, pp. 111-167, Academic Press, London, 1987. Boitnott, G. N., Nonlinear rheology of rock at m oderate strains: Fundam ental observations of hysteresis in the deform ation of rocks, in P roceedings o f the 15th A n n u a l Seism ic Research S ym posium , edited by J. F. Lewkowicz and J. M. McPhetres, pp. 36-42, PL-TR-93-2 160, 1993. Boitnott, G. N., Fundam ental observations concerning hysteresis in the deform ation of intact and jointed rock with applications to nonlinear attenuation in the near source region, in Proceedings o f th e N um erical M odeling o f U nderground N uclear Test M onitoring S ym posium , edited by S. R. Taylor and J. R. Kamm, pp. 121-133, LA-UR-93-3839, 1993. Boitnott, G. N., Personal comm unication, 1995. Budiansky, B., and R. J. O’Connell, Elastic moduli of a cracked solid, int. J. Solids Structures, 12, 81-97, 1976. Costin, L. S., Tim e-dependent deform ation an d failure, in Fracture M echanics o f Rock, edited by B. K. Atkinson, pp. 167-215, Academic Press, London, 1987. Day, S. M., J. B. Minster, M. Tyron, and L. Yu, Nonlinear hysteresis in an endochronic solid, in Proceedings o f the N um erical M odeling o f U nderground N uclear Test M onitoring S ym p o siu m , edited by S. R. Taylor and J. R. Kamm, pp. 135-147, LA-UR-93-3839, 1993. 60 Meredith, P. G., and B. K. Atkinson, Fracture toughness and subcritical crack growth during high-tem perature tensile deform ation of Westerly granite and Black gabbro, Phys. Earth Planet. Int., 39, 33-51, 1985. Minster, J. B., and S. M. Day, Decay of wave fields near an explosive source due to high-strain, nonlinear attenuation, J. G eophys. Res., 91, 2113- 2122, 1986. O'Connell, R. J., and B. Budiansky, Seismic velocities in dry and saturated cracked solids, J. G eophys. Res., 79, 5412-5426, 1974. Paterson, M. S., E xperim ental Rock D eform ation—The Brittle fie ld , pp. 32, Springer-Verlag, New York, 1978. Sammis, C. G., and M. F. Ashby, The failure of brittle porous solids under compressive stress states, Acta. Metall., 34, 511-526, 1986. Sammis, C. G., Incorporating dam age mechanics into explosion source models, in Proceedings o f the 15th A n n u a l Seism ic Research S ym p o siu m , edited by J. F. Lewkowicz and J. M. McPhetres, pp. 349- 355, PL-TR-93-2160, 1993. Sammis, C. G., Incorporating dam age mechanics into explosion sim ulation models, in Proceedings o f the N um erical M odeling o f U nderground N uclear Test M onitoring Sym posium , edited by S. R. Taylor and J. R. Kamm, pp. 65-92, LA-UR-93-3839, 1993. Sammis, C. G., Modeling the dam age regime in nonlinear explosion source simulations, in Proceedings o f th e 16th A n n u a l Seism ic Research S ym p o siu m , edited by J. J. Cipar, J. F. Lewkowicz, and J. M. McPhetres, pp. 317-323, PL-TR-93-2160, 1994. Walsh, J. B., The effect of cracks on the uniaxial elastic com pression of rocks, J. G eophys. Res., 70, 399-411, 1965. Walsh, J. B., Seismic wave attenuation in rock due to friction, J. G eophys. Res., 71, 2591-2599, 1966. 61 Appendix I**#****#:!!:!!!***!!!***********************************************’ !'********** * This program computes axial stress for a given strain and confining stress. * Different stress/strain relationships are used for the three stress regimes prior to failure. * Equations 3-1, 3-2, and 3-4 are used in the pure elastic, non-linear elastic and damage * regimes respectively to determine stress. * During hysteresis only relationships for the elastic regimes are used. Program STRESS_STRAIN_HYST_T1ME_DAMAGE character*40 outfnam, parmfnam character*80 stringSO character*!3 string!3 integer yield, fail, nucl integer nn, pwr, nips, lp_ct integer tend, tbeg, iend(lOO), ibeg(lOO) real sigl, sig3, si, s3, sigl_base, dsig real al, be, mu, pi, c3 real di, Klc, a, sigy, Eo real z, fsig real cl, c2, c4, ba, al, a3 real ep, tot_strain, beg(lOO), end(lOO) real E, Eb, da, K 1, df, sf, Nv, Vc, Rl, m real daint real signu, sigyi. sigfa epn, epy, epf *** READING INPUT FROM PARAMETER FILE DAM.PARM' *** *** Note: Parameters are read in a particular order *** parmfnam=’ dam.parm’ open (file=parmfnam,unit=l 1) read (11 ,\a80)') string80 do while (string80( 1:5).ne.'ALPHA') read (1 L’(a80)') string80 enddo read (string80,'(20x,fl0.0)') al read (1 l,'(a80)') string80 do while (string80( 1:4).ne.'BETA') read (1 l,’(a80)') string80 enddo read (string80,'(20x,fl0.0)') be read ( I l,'(a80)') string80 do while (string80( 1:19).ne. COEFFICIENT OF FRIC) read {1 l,'(a80)') string80 enddo 62 read (string80,’(20x,fl0.0)') mu read (11 ,'(a80)') string80 do while (string80( 1:2).ne,'Pr) read (11 ,'(a80)’) stringSO enddo read (string80,'(20x,fl0.0)') pi read (I l,’(a80)') string80 do while (string80( 1 : lO).ne.CONSTANT 3') read (11 ,'(a80)’) string80 enddo read (string80,'(20x,fl0.0)') c3 read (11 ,’ (a80)') string80 do while (string80( 1:14).ne.’INITIAL DAMAGE’) read (11 ,'(a80)’) stringSO enddo read (stringSO,’(20x,f 10.0)') di read ( I L'(a80)’) string80 do while (string80( 1:18).ne. FRACTURE TOUGHNESS’) read (1 l,’(a80)') string80 enddo read (string80,'(20x,fl0.0)') Klc read (11 ,'(a80)’) string80 do while (string80( l:19).ne.1/2 LENGTH OF CRACK ) read (11 ,'(a80)') string80 enddo read (string80,’(20x,fl0.0)') a read (11 ,'(a80)') string80 do while (string80{ 1:12).ne.'YIELD STRESS') read (11 ,’(a80)’) stringSO enddo read (string80,’ (20x,fl0.0V) sigy read (11 ,'(a80)’) string80 do while (string80( 1:19).ne.'YOUNGS MODULUS (Eo)1 ) read (1 L'(a80)’) string80 enddo read (string80,'(20x.fl0.0)’) Eo read (11 ,'(a80)’) string80 do while (string80{ 1:19).ne.’ CRACK GRO. VELOCITY ) read (11 ,'(a80)') stringSO enddo read (string80,'(20x,fl0.0)’) Vc read (11 ,'(a80)') string80 do while (string80( 1:15).ne.'Vc/Kl SLOPE (m)') read (1 L'taSO)’) string80 enddo read (string80,’(20x,fl0.0)') m read (11 ,'(a80)’) string80 do while (string80( 1:2).ne.'z’) read (11 ,'(a80)’) string80 enddo read (string80,’ (20x,fI0.0)') 63 read ( I 1 ,'(a80)’) stringSO do while (string80( 1:18).ne. DELTA SIGMA (dsig)’) read (11 ,’ (a80)') string80 enddo read (string80,'(20x,fl0.0)’)fsig read (1 l,'(a80)') string80 do while (string80( 1:17).ne.'DESIRED PRECISION ) read( 11,’{a80)') string80 enddo read (string80,'(20x,ilO)') pwr read (1 l,'(a80)') string80 do while (string80(l:12).ne.'LOADIND RATE') read ( 1 l,'(a80)’) string80 enddo read (string80,'(20x,i IO)’) R 1 read (11 ,'(a80)’) stringSO do while (string80( 1: !6).ne.'STRAIN INCREMENT') read {1 l,'(a80)') stringSO enddo read (string80,'(20x,ilO)’) ep read (11 ,’(a80)’) string80 do while (string80(l:19).ne.'TOTAL STRAIN ACCUM. ) read (1 l,'(a80)’) string80 enddo read (string80,'(20x,ilO)') tot_strain read (1 l,'(a80)’) string80 do while (string80( 1:16).ne.'CONFINING STRESS') read( 11 ,'(a80)') stringSO enddo read (string80,'(20x,i 10 )') sig3 read (1 l,'(a80)') stringSO do while (string80( 1:17).ne.'NUMBER OF H LOOPS') read (11 ,’(a80)') string80 enddo read (string80,'(20x,ilO)') nips 5 format (a4,i2,a7) do i=l, nips write (string 13,5) LOOP', i, :BEGIN' read (1 l,'(a80)‘) string80 do while (string80(l:13).ne.stringl3) read (11,'(a80)') string80 enddo read (string80,'(20x,fl0.0)') beg(i) write (string 13,5) 'LOOP', i,': END ' read (1 l,'(a80)') stringSO do while (string80( 1:13).ne.string 13) read ( 11 ,'(a80)’) string80 enddo read (stringSO,’ (20x,f 10.0)’) end(i) enddo close (unit=l 1) 64 *** INITIALIZING VARIABLES *** yield=0 fai I = 0 nucl=0 nn=0 signu=0 sigyi=0 sigfa=0 epn=0 epy=0 epf=0 sig 1=0 sigl_base=0 da=di K 1 = 0 s3=sig3*( ((pi*a)**(0.5))/Klc) tend=anint{tot_strain/ep) tbeg=l do i=l,nlps iend(i)=anint(end(i)/ep) ibeg(i)=anint(beg(i)/ep) enddo lp_ct= I Call CONST (al,be,mu,pi,cl,c2,c4,ba,al,a3,a,di,Nv) Call FDAM (s3,pwr,ba,di,c 1 ,c2,c3,c4,df,sf) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *** WRITING HEADER TO OUTPUT FILE *** outfnam='damsht2.out' open(unit= 12,file=outfnam) write(12,'(4x,a6,f8.3)’) 's ig 3 -s ig 3 write(12,'(4x,a4,fl0.7)') 'S3= s3 write(12,'(4x,a4,fl0.7)') Do= di write(12,'(4x,a5,f9.7)’) 'Klc= Klc write( 12,'(4x,a3,fl 1.7)') a= a write{ 12,'(4x,a6,f8.3)') 'sigy= sigy write(l2,'(4x,a4,f 10.3)') ’Eo= Eo write(I2,'(4x,a3,fl 1.7)') ’ z= z write(12,'(4x,a4,fl0.7)') 'Df= df write(12,'(4x,a4,fl0.5)') ’Sf= \sf*(Klc/((pi*a)**(0.5)) write(12,*)'' write( 12,*)' ep Sigmal Damage K 1 ’ write(l2,*)'____________________________________________ ' format(lx,f7.5,4x,f 10.4,4x,fl0.6,4x,fl0.6) write( 12,25) write( 12,25) O, sig 1, da, K 1 65 *** MAIN LOOP*** 30 do i=tbeg,tend *** Change in Sig 1 is first determined in one of 2 ways: with Eo or Eh. *** K1 is calculated based on Sig I and damage. *** Damage due to sub-critical crack growth is calculated and added to total damage. if ((sigl-sigl_base).lt.fsig) then dsig=Eo*ep sigl=sigl+dsig Call STR„INT (al,ba,pi,di,a,cl,al ,sig 1 .sig3,da,K 1) Call DA_SCG (al.pi,a.K lc.Rl.Vcjn.KI .Nv.dsig.da) goto 40 else Eb=Eo*(l-(l.25*da)) dsig=Eb*ep sigl=sigl+dsig Call STR_INT (al,ba,pi,di,a,cl,al,sigl,sig3,da,Kl) Call DA_SCG {al,pi,a,Klc,Rl,Vc,m,Kl,Nv,dsig,da) endif *** The condition for nucleation is tested. if (nucl.eq.O) then s I =sig 1 *((pi*a)**(0.5))/Klc y=c2*(ba**( 1.5))+(s3*cl) if (sl.le.y) then goto 40 else nucl= 1 signu=sigl epn=ep*i endif endif *** At this point, nucleation HAS occurred, hence the current stress state is either in the *** damage or post-failure regimes. *** Therefore, current damage is determined. Sigl is adjusted accordingly. *** Then Da, Kl, and subcritical crack growth are determined again. Call FFAILN (sigl,s3,pwr,ba,di,pi,a,Klc,cl,c2,c3,c4,daint) *** It is asserted that additional damage has occurred. if (daint.gt.da) then da=daint sigl=sigl-dsig 66 E=Eo*(l-((l.25+z)*da)) dsig=E*ep sigl=sigl+dsig Call FFAILN (sigl,s3,pwr,ba,di,pi,a,Klc,cl,c2,c3,c4,daint) STR_INT (al,ba,pi,di,a,c 1 ,al ,sig t ,sig3,da,K 1) DA_SCG (al,pi,a,K lc,Rl,Vc,m.K 1,Nv.dsig.da) endif *** The condition for brittle failure is tested *** (comparison is made to 4 significant figures) if (int(da* 10000)/l.ge.int(df* 10000)/!) then print*, 'brittle failure noted' sigfa=sig 1 epf=ep*i write( 12.25) ep*i, sig 1. da, K 1 sigl=sig3*(( l+mu*mu)**(0.5)+mu)/((]+mu*mu)**{0.5)-nni) do j=i+l.tend write(12,25) ep*j, sigl, da. K1 enddo goto 100 endif *** The condition for ducti le fai lure is tested. if ((sigl-sig3).ge.sigy) then sigyi=sigl epy=ep*i do j=i,tend write(l2.25) ep*j, sigl, da, K1 enddo goto 100 endif *** END OF LOOP 40 write(12,25) ep*i, sigl, da, K1 *** Checking for the beginning of a hysteresis loop if (lp_ct.le.nips.and.i.eq.ibeg(lp_ct)) goto 50 enddo *** Asserting that the end of the desired stress/strain curve is reached if (i.eq.tend+l) goto 100 67 *** * * * 50 * * * *** 100 105 Hysteresis Loops: Decreasing Stress sigl_base=sigl do i=ibeg(lp_ct)-l,iend(lp_ct),-l if ((sigl_base-sigl ).lt.(fsig) then dsig=-Eo*ep sigl=sigl+dsig else Eb=Eo*(l-( 1.25*da)) dsig=-Eb*ep sig l=sig 1+dsig endif Call STR J N T (al,ba,pi,di,a.c 1 .a 1 .sig 1 ,sig3.da,K 1) Call DA_SCG (al.pi.a.Klc,Rl,Vc,ni,K 1 .Nv.dsig.da) write( 12,25) ep*i, sig 1, da, K I enddo Increasing Stress sigl_base=sigl do i=iend(lp_ct)+l,ibeg(lp_ct) if ((sig 1 -sig l_base).lt.(fsig) then dsig=Eo*ep sigl=sigl+dsig else Eb=Eo*(l-( 1.25*da)) dsig=Eb*ep sigl=sigl+dsig endif Call STR„INT (al,ba,pi,di,a,c 1 .a I ,sig 1 ,sig3,da,K 1) Call DA_SCG (al,pi,a,Klc,RI,Vc,m,Kl,Nv,dsig,da) write( 12,25) ep*i, sigl, da, K1 enddo tbeg=ibeg(lp_ct)+1 lp_ct=lp_ct+l sigl_base=0 goto 30 Writing end of output fi I e continue format(lx,a7,fl0.4,2x,a5,fl0.6) write(12,*)'_______________________________ 1 write( 12,*)'1 write( 12,105) 'signu= ’.signu, 'epn= \epn write(12,105) 'sigyi= '.sigyi, 'epy= r,epy write( 12,105) 'sigfa= \sigfa, 'epf= ',epf close(unit=12) end ************************************************************************ * This subroutine calculates frequently used constants. ************************************************************** * = ► * = * = * = * := ♦ ! Subroutine CONST (al,be,mu,pi,cl,c2,c4,ba,ai,a3,a,di,Nv) real al, be, mu, pi real c 1, c2, c4. ba real a l, a3, a, di, Nv al=(pi/((3.0/be)**(0.5)))*(( 1.0+mu**2.0)**(0.5) - mu) a3=(pi/((3.0/be)**<0.5)))*((1.0+mu**2.0)**(0.5) + mu) c 1 =a3/al c2=((pi**2)*(al**( 1.5)))/al c4=(2.0*(al**2.0)*(pi**2.0))/u 1 ba=be/al Nv=(3.0*di)/(4.0*pi*(al*a)**3) return end if***!!:!)!*!):**#******************:******************************************* *This subroutine calculates failure stress and damage for the given confining stress. $:|c:|tJtc)|f)|ta|c2 4f ) | c ? f e * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Subroutine FDAM (s3,pwr,ba,di,cl,c2,c3,c4,dmax,smax) integer pwr, i,j real s3, ba, di, c 1, c2, c3, c4 real smax, dmax, dfac, ri dfac=0 do i=l,pwr ri=real(i) do j= 1,20 da=( 10.0**(-ri))*j + dfac if (da.ge.l) goto 10 *** damage must be less than 1 if (da.lt.di) goto 9 *** damage must be greater than Do dd=(da/di)**(l. 0/3.0) c5= l+(c3*(di**(2.0/3.0)))/( 1,0-(da**(2.0/3.0)))*(dd-1 )**2.0 y=(c2*(dd-1.0+ba)**( 1,5)+s3*(cl*c5+c4*(dd-1.0)**2.0))/c5 *** record y as smax if y>smax or this is the first iteration if (y.gt.smax.or.i.eq. 1 .and.j.eq. 1) then smax=y dmax=da endif 9 enddo 10 dfac=dmax-( 10.0**(-ri)) 69 enddo return end * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * This subroutine calculates stress intensity. Subroutine STR_INT (al,ba.pi,di,a,cl,al,sigl,sig3,da,KI) real al, ba, pi, di, a real sig 1, sig3, da, K 1 real cl, a 1 real p i , p2, p3, p4, p3, dd dd = ((da/di)**( 1.0/3.0) - 1.0) pi = (al*sigl*((pi*a)**(0.5))) p2 = (pi*pi*(al 1 **( 1.5)))*(dd+ba)**( 1.5) p3 = (1 -0-c 1 *(sig3/sig 1)) p4 = 1.0+(2.0*dd**2.0)*( (di**(2.0/3.0)) /(1,0-(da**(2.0/3.0)))) p5 = 2.0*(sig3/sigl)*(al**2.0)*(pi**2.0)*(dd**2.0)/al K1 = (pl/p2) * ((p3*p4) - p5) return end * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * This subroutine calculates sub-critical crack growth. Subroutine DA_SCG (al,pi,a,Klc,Rl,Vc,m,Kl,Nv,dsig,da) real al, pi, a, Klc, Rl, Vc, m real Kl, Nv, dsig, da real 1 , dl 1=( ((3.0*da)/(4.0*Nv*pi))**( 1.0/3.0) - (al*a) ) if (l.lt.0) 1=0.0 dl=dsig*(1.0/Rl)*Vc*((Kl/K.lc)**(m)) da=(4.0/3.0)*pi*Nv * ((1+dl) + (al*a))**(3.0) return end 70 * This subroutine calculates damage for the given axial and confining stresses. Subroutine FFAlLN(sig 1 ,s3,pwr,ba,di,pi.a,K 1 c.c 1 ,c2,c3,c4,dmax) integer pwr, i,j real s I, s3, sig 1, pi, a, Klc real ba, di, cl, c2, c3, c4 real smax, dmax, dfac, ri, da, dd, y sl=sigl*((pi*a)**(0.5))/Klc *** Calculating current damage dfac=real(int(dmux* 10)/1 J/10.0 - 0.1 do i=l ,pwr ri=real(i) do j= 1,20 da=( 10.0**(-ri))*j + dfac *** damage must be less than 1 if (da.ge. 1) goto 10 *** damage must be greater than Do if (da.lt.di) goto 9 dd=(da/di)**(l.0/3.0) c5=l .0+(c3*(di**(2.0/3.0)))/( 1,0-(da**(2.0/3.0)))*(dd-1 )**2.0 y=(c2*(dd-1.0+ba)**( 1.5)+s3*(c 1 *c5+c4*(dd-1.0)**2.0))/c5 *** record highest damage encountered, just exceeding si if (y.ge.sl) then dmax=da goto 10 endif *** record y as smax if y>smax or this is the first iteration if (y.gt.smax.or.i.eq. 1 .and.j.eq. 1) then smax=y dmax=da endif 9 enddo 10 dfac=dmax-{ 10.0**(-ri)) enddo 100 return end 71 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (eg., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Aibor Ml 48106-1346 USA 313/761-4700 800/521-0600 UMI Number: 1378397 UMI Microform 1378397 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
The study of temporal variation of coda Q⁻¹ and scaling law of seismic spectrum associated with the 1992 Landers Earthquake sequence
PDF
Source Parameters Of The Joshua Tree Aftershock Sequence
PDF
Lateral variability in predation and taphonomic characteristics of turritelline gastropod assemblages from Middle Eocene - Lower Oligocene strata of the Gulf Coastal Plain, United States
PDF
A tectonic model for the formation of the gridded plains on Guinevere Planitia, Venus: Implications for the thickness of the elastic lithosphere
PDF
The Hall Canyon pluton: implications for pluton emplacement and for the Mesozoic history of the west-central Panamint Mountains
PDF
The characterization of Huntington Beach and Newport Beach through Fourier grain-shape, grain-size, and longshore current analyses
PDF
The effects of dependence among sites in phylogeny reconstruction
PDF
Quartz Grain-Shape Variation Within An Individual Pluton: Granite Mountain, San Diego County, California
PDF
Statistical analysis of the damage to residential buildings in the Northridge earthquake
PDF
A qualitative study on the relationship of future orientation and daily occupations of adolescents in a psychiatric setting
PDF
Occupational exposure to extremely low frequency electromagnetic fields as a potential risk factor for Alzheimer's disease
PDF
Complementarity problems over matrix cones in systems and control theory
PDF
Meta-analysis on the misattribution of arousal
PDF
The use of occupational therapists or interdisciplinary teams in the evaluation of assistive technology needs of children with severe physical disabilities in Orange County schools
PDF
Fault Development Under Simple Shear: Experimental Studies
PDF
Work for the masters degree
PDF
Trifluoromethanesulfonates (triflates) for organic syntheses
PDF
Fine motor skills of two- to three-year-old drug exposed children
PDF
The dream becomes a reality (?): nation building and the continued struggle of the women of the Eritrian People's Liberation Front
PDF
Comparison of evacuation and compression for cough assist
Asset Metadata
Creator
Altschul, Inna
(author)
Core Title
The relationship of stress to strain in the damage regime for a brittle solid under compression
School
Graduate School
Degree
Master of Science
Degree Program
Geological Sciences
Degree Conferral Date
1995-05
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
geology,geophysics,OAI-PMH Harvest
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
Sammis, Charles G. (
committee chair
), Aki, Keiiti (
committee member
), Lund, Steve P. (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c18-8763
Unique identifier
UC11356905
Identifier
1378397.pdf (filename),usctheses-c18-8763 (legacy record id)
Legacy Identifier
1378397-0.pdf
Dmrecord
8763
Document Type
Thesis
Rights
Altschul, Inna
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 au...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA
Tags
geology
geophysics