Int J. Solids Structures Vol. 33, No. 8, pp, Printed in Grear Bri 020-7683(95)0067 MATRIX CRACKING AND DEBONDING OF CERAMIC MATRIX COMPOSITES FRANCOIS HILD and ALAIN BURR laboratoire de mecanique et Technologie, E. N.S. de Cachan/C N R 61 Avenue du president wilson, F-94235 Cachan Cedex x. Franicersite Paris 6 FREDERICK A. LECKIE epartment of Mechanical and Environmental Engineering, University of California Santa Barbara, CA 93106-5070 U.S.A Received 7 July 1994; in revised form 15 March 1995) chanical model. The cracking and debonding proces loops and crack closure. These feature num Mechanics by the introduction of internal variable icromechanical analysis. The evolution laws of the internal variables can be determined b lbining the experimental data with micromechanical modeling the infuences of residual stress fields due to processing are also included. Comparisons are made between theoretical predictions and the results of experiments performed on layered materials INTRODUCTION It is known that the formation of matrix cracks and the subsequent matrix-fiber interface sliding are the source of the nonlinear stress-strain curves observed when loading con tinuous fibers as well as ceramic-matrix composites. Matrix cracking and debonding reduce the secant Youngs modulus E, and induce inelastic strains upon complete unloading, a i and hysteresis loops, 8E(Beyerley et al., 1992; Pryce and Smith, 1992) In this paper, the effects of matrix cracking and interface sliding are studied within the framework of Continuum Damage Mechanics. Constitutive laws are derived by introducing internal variables which are identified in a micromechanical description of matrix cracking and debonding. The effects of residual stresses introduced during processing are also included. The growth laws for the internal state variables in terms of the associated forces are determined from experiments which involve unloading-reloading sequences. The procedures developed in the study are used to define a model which describes the tensile behavior of a layered laminate made of alternating layers of alumina and unidirectional carbon/ epoxy prepreg tapes 2. CONTINUUM MECHANICS FORMULATION In this section, a Continuum Mechanics formulation(Germain, 1973; Lemaitre and Chaboche, 1985)is attempted using the framework of the thermodynamics of irreversible processes(Bataille and Kestin, 1979; Germain et al., 1983). The first step in establishing such a model is to identify the internal state variables which define the condition of the material. The second is to determine the expression of the state potential in terms of the tate variables and the third to define the evolution laws of the internal variables 2. 1. Degradation mechanisms Upon loading, a composite or laminate which consists of a brittle matrix sup by a stronger material usually forms a pattern of multiple cracking. In fiber-reinforced systems and layered architectures, matrix cracking is usually accompanied by interfacial
In!. J. Solids Swucrures Vol. 33, No. 8, pp. 120%1220, 1996 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. AU rights rcscrvod 0020-7683(!J5)OtNl67-4 002%7683/96 $15.00 + .OO MATRIX CRACKING AND DEBONDING OF CERAMIC-MATRIX COMPOSITES FRANCOIS HILD and ALAIN BURR Laboratoire de Mecanique et Technologie, E.N.S. de Cachan/C.N.R.S./Universitt Paris 6, 61, Avenue du President Wilson, F-94235 Cachan Cedex, France and FREDERICK A. LECKIE Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106-5070, U.S.A. (Received 7 July 1994 ; in revisedform 15 March 1995) Abstract-The effects of matrix cracking and debonding which occur in ceramic-matrix composites are described by a micromechanical model. The cracking and debonding processes induce loss of stiffness, inelastic strains, hysteresis loops and crack closure. These features are analysed within the framework of Continuum Mechanics by the introduction of internal variables identified in the micromechanical analysis. The evolution laws of the internal variables can be determined by combining the experimental data with micromechanical modeling. The influences of residual stress fields due to processing are also included. Comparisons are made between theoretical predictions and the results of experiments performed on layered materials. 1. INTRODUCTION It is known that the formation of matrix cracks and the subsequent matrix-fiber interface sliding are the source of the nonlinear stress-strain curves observed when loading continuous fibers as well as ceramic-matrix composites. Matrix cracking and debonding reduce the secant Young’s modulus E, and induce inelastic strains upon complete unloading, Zi,, and hysteresis loops, 6~ (Beyerley et al., 1992; Pryce and Smith, 1992). In this paper, the effects of matrix cracking and interface sliding are studied within the framework of Continuum Damage Mechanics. Constitutive laws are derived by introducing internal variables which are identified in a micromechanical description of matrix cracking and debonding. The effects of residual stresses introduced during processing are also included. The growth laws for the internal state variables in terms of the associated forces are determined from experiments which involve unloading-reloading sequences. The procedures developed in the study are used to define a model which describes the tensile behavior of a layered laminate made of alternating layers of alumina and unidirectional carbon/epoxy prepreg tapes. 2. CONTINUUM MECHANICS FORMULATION In this section, a Continuum Mechanics formulation (Germain, 1973; Lemaitre and Chaboche, 1985) is attempted using the framework of the thermodynamics of irreversible processes (Bataille and Kestin, 1979; Get-main et al., 1983). The first step in establishing such a model is to identify the internal state variables which define the condition of the material. The second is to determine the expression of the state potential in terms of the state variables and the third to define the evolution laws of the internal variables. 2.1. Degradation mechanisms Upon loading, a composite or laminate which consists of a brittle matrix supported by a stronger material usually forms a pattern of multiple cracking. In fiber-reinforced systems and layered architectures, matrix cracking is usually accompanied by interfacial 1209
F. Hild et al debonding and sliding. At high values of stress, fiber breakage occurs which is accompanied by fiber pull-out and is mainly driven by the normal stresses or strains in the fiber directions (Hild et aL., 1994). These two mechanisms need different treatments Since the matrix cracking process occurs at load levels significantly lower than the fiber breakage mechanism. it is assumed that these mechanisms are uncoupled The matrix cracks which normally occur at low values of stress are usually aligned with the principa stress or strain directions. These cracks cause stiffness reduction when the stress is tensile In addition it is the closure of the cracks which indicates the onset of increased stiffness hen the specimen is subsequently loaded in compression to be able to predict crack closure, the most natural choice of variable is the crack opening displacement A Debonding followed by sliding gives rise to inelastic strains and hysteresis loops. To model these phenomena, different models have been proposed( Cox, 1952; Aveston et al 1971: Hutchinson and Jensen. 1990: Hsueh. 1993). They all use an elementary cell of length 2L, characterizing the average crack spacing, and consisting of two different materials(I and (2), as shown in Fig. I. There is a crack of size 2a at the center and a friction length 2lF. Thermal mismatch between parts(1)and (2)develops during processing. The self- balanced residual stress field is-P,Ey/E and -p,Ey/E in parts (1)and(2), respectively (Fig. 2), where E is Young,'s modulus of the unbroken composite, E, that of part(1)and E2 that of part(2). When a crack appears, part of these residual stresses are relieved within the slipping region 2. 2. State potential The model can describe the reduction in stiffness due to matrix cracking, the inelastic strains due to friction, the size of the hysteresis loops and the closure phenomenon when the loading is compressive. The details of the unloading and reloading process are fairly complex, but the introduction of the crack opening displacement A simplifies the calcu lations. To characterize the state of the composite, four different quantities are required These are the overall strain a, the friction length 2lF, the cell length 2L and the crack opening displacement A. The expression of the strain distribution in the friction zone in part(2)is defined by a function F which depends on the details of the interfacial behavior, and is given by Se2(z)=dE2F(z), the difference between the strain field when friction occurs and the strain field with no friction, where z is the direction normal to the crack(Fig. 1). The crack opening displacement due to slip As is then given by F (1) with X 2W Fig. 1. Elementary cell of size 2L x 2W containing a crack of size 2a. A friction zone is characterized
1210 F. Hild et al. debonding and sliding. At high values of stress, fiber breakage occurs which is accompanied by fiber pull-out and is mainly driven by the normal stresses or strains in the fiber directions (Hild et al., 1994). These two mechanisms need different treatments. Since the matrix cracking process occurs at load levels significantly lower than the fiber breakage mechanism, it is assumed that these mechanisms are uncoupled. The matrix cracks which normally occur at low values of stress are usually aligned with the principal stress or strain directions. These cracks cause stiffness reduction when the stress is tensile. In addition, it is the closure of the cracks which indicates the onset of increased stiffness when the specimen is subsequently loaded in compression. To be able to predict crack closure, the most natural choice of variable is the crack opening displacement A. Debonding followed by sliding gives rise to inelastic strains and hysteresis loops. To model these phenomena, different models have been proposed (Cox, 1952 ; Aveston et ul., 197 1 ; Hutchinson and Jensen, 1990 ; Hsueh, 1993). They all use an elementary cell of length 2L, characterizing the average crack spacing, and consisting of two different materials (1) and (2), as shown in Fig. 1. There is a crack of size 2a at the center and a friction length 21,. Thermal mismatch between parts (1) and (2) develops during processing. The selfbalanced residual stress field is -p,E,/E and -pzEz/E in parts (1) and (2), respectively (Fig. 2), where E is Young’s modulus of the unbroken composite, E, that of part (1) and E2 that of part (2). When a crack appears, part of these residual stresses are relieved within the slipping region. 2.2. State potential The model can describe the reduction in stiffness due to matrix cracking, the inelastic strains due to friction, the size of the hysteresis loops and the closure phenomenon when the loading is compressive. The details of the unloading and reloading process are fairly complex, but the introduction of the crack opening displacement A simplifies the calculations. To characterize the state of the composite, four different quantities are required. These are the overall strain E, the friction length 21,, the cell length 2L and the crack opening displacement A. The expression of the strain distribution in the friction zone in part (2) is defined by a function F which depends on the details of the interfacial behavior, and is given by &&) = &2F(~), the difference between the strain field when friction occurs and the strain field with no friction, where z is the direction normal to the crack (Fig. 1). The crack opening displacement due to slip As is then given by with Z 2w -1 ) Fig. 1. Elementary cell of size 2L x 2 W containing a crack of size 2~2. A friction zone is characterized by a length 21
Matrix cracking of ceramic-matrix com IF Matrix d/(1-f) 男|(-p2E2/E E/E Fig. 2. Motion of the unbroken part(2)with respect to the broken part (1)with no external loac by an amount a over a length lF. Axial stresses in layers(1)and(2)with a constant shear strength F(z)d where f is the volume fraction of part (1) The free energy density for a given state is calculated by performing two elastic calculations. Two"cut and paste"steps are used to evaluate the elastic energies following approaches introduced by Volterra(1907), and used to analyse the elastic behavior of homogeneous and isotropic media by considering the elastic properties of a cut cylinder (Volterra, 1907: Love, 1927), as well as inclusions in an infinite medium(Eshelby, 1957) or to study creeping materials( Cocks and Leckie, 1987). The first step consists of calculating the elastic energy when the unbroken part(2)is moved with respect to the broken part (1) by an amount As over a length lF(Fig. 2)with no external load. This displacement As give rise to a self-balanced stress field along a length lF in parts(I)and (2). By integration over Ia, and averaging over the total length L, the elastic energy density associated with this process is given by( see Appendix W =AS Ei(-NE2F F F(zd The crack opening displacement A, induces an overall inelastic strain a due to slip, expressed
Matrix cracking of ceramic-matrix composites I 1211 Matrix Crack - Slip I c No Slip (2) I -2 0 lF Fig. 2. Motion of the unbroken part (2) with respect to the broken part (1) with no external load by an amount A over a length IF. Axial stresses in layers (1) and (2) with a constant shear strength. F(z) dz, wheref is the volume fraction of part (1). The free energy density for a given state is calculated by performing two elastic calculations. Two “cut and paste” steps are used to evaluate the elastic energies following approaches introduced by Volterra (1907) and used to analyse the elastic behavior of homogeneous and isotropic media by considering the elastic properties of a cut cylinder (Volterra, 1907; Love, 1927), as well as inclusions in an infinite medium (Eshelby, 1957), or to study creeping materials (Cocks and Leckie, 1987). The first step consists of calculating the elastic energy when the unbroken part (2) is moved with respect to the broken part (1) by an amount A, over a length IF (Fig. 2) with no external load. This displacement A, gives rise to a self-balanced stress field along a length I, in parts (1) and (2). By integration over Z,, and averaging over the total length L, the elastic energy density associated with this process is given by (see Appendix) : G = A: fE, (1 -f)Ez F’ - s 2L1, E p2 ’ with F2 (z) dz. (2) The crack opening displacement A, induces an overall inelastic strain c1 due to slip, expressed as
1212 F. Hild et al fE1△ L The second step consists of an elastic loading of a cracked system with friction prevented The presence of a crack results in a stiffness reduction defined by an internal damage variable D(Budiansky and O'Connell, 1976; Chaboche, 1982), so that the elastic energy density is given by v=:E(-D)(E-)2 This damage variable D depends upon the crack density Ia/4LW, as well as the elastic properties of the two components. In the following treatment, the exact microscopic descrip- tion is not needed. The total free energy density is the sum of the two components of energy For convenience the free energy can be expressed in a more compact form by using four statc variables, which arc the total strain E and three intcrnal variables, namely the damage variable d modeling the loss of stiffness due to the cracking mechanism, the damage variable d=fE/(1-nE2lF/L which defines the size of the slip zone related to the crack spacing, and the inelastic strain a due to slip. The free energy density in terms of the new internal =E(1-D)(-2)2+E F The crack opening strain a is similar to a kinematic hardening variable, since eqn(3)shows that the opening displacement As can be related to the inelastic strain, and the associated force corresponds to the back- stress induced by the slipping mechanism the forces associ ated with the previous variables are respectively given by E(I-D)(E-a) 0D2 (E-x) (6b) 0+E quation(6a) can be rewritten in terms of a total crack opening which consists of the contribution of the elastic opening augmented by the opening due to slip so that the crack closure condition is given by
1212 F. Hild et al .fE, As “=EL. (3) The second step consists of an elastic loading of a cracked system with friction prevented. The presence of a crack results in a stiffness reduction defined by an internal damage variable D (Budiansky and O’Connell, 1976; Chaboche, 1982), so that the elastic energy density is given by I//~ = ;E(l -D)(G-cc)‘. (4) This damage variable D depends upon the crack density na2/4LW, as well as the elastic properties of the two components. In the following treatment, the exact microscopic description is not needed. The total free energy density is the sum of the two components of energy. For convenience the free energy can be expressed in a more compact form by using four state variables, which are the total strain E and three internal variables, namely the damage variable D modeling the loss of stiffness due to the cracking mechanism, the damage variable d =fE,/(l -.f)E&/L which defines the size of the slip zone related to the crack spacing, and the inelastic strain a due to slip. The free energy density in terms of the new internal variables is II/ = ;E(I -D)(~-cc)~+;E* $ 0 , with The crack opening strain CI is similar to a kinematic hardening variable, since eqn (3) shows that the opening displacement A, can be related to the inelastic strain, and the associated force corresponds to the back-stress induced by the slipping mechanism. The forces associated with the previous variables are respectively given by 6 = z = E(1 -D)(E-u) a+ E* U 2 ‘=-z=2 d 0 &fk -d+E*! au d’ (6~) Equation (6a) can be rewritten in terms of a total crack opening which consists of the contribution of the elastic opening augmented by the opening due to slip so that the crack closure condition is given by
Matrix cracking of ceramic-matrix composites l213 OD +a=0. Lastly, the growth laws of the internal variables can be established from the results of materials science, but these in themselves can be major undertakings, as is evidenced by the work of Curtin(1991)describing the fragmentation process in a model where constant shear stress along the interface is assumed. This analysis requires substantial information bout the statistical failure properties of part (1). Another method which is more practical and in keeping with the aim of this approach is to deduce the evolution of the state variables from experimental data 23. Evolution laws pro When deriving the expressions for the elastic energy it was not necessary to have se information about the conditions of slip and or debond properties of the fiber matrix interface. To proceed further it is necessary to have information about the interface d in the following it is assumed that the stress of the interface has a constant value t, rer the friction length IF. It can be shown that &e2=fi/(l-fE, F(z)=(F-z)/a, so that F=lo/2a, F=1/aand E*=4E/3 It will be demonstrated that the evolution laws can be deduced from the measurements luring a loading-unloading-reloading sequence. Reversed motion of fibers relative to the matrix has been studied by numerous authors(Marshall and Oliver, 1987; McMeeking and Evans, 1990; Pryce and Smith, 1992). The main results are now summarized. After reaching a maximum stress value oM, when the maximum friction length is /FM, the load is reversed(Fig. 3). Upon unloading by Bou, in the range 0<|zl <lFu, where IFu is the unloading friction length, both the relative sliding direction and the frictional shear stress reverse. Between lFu <Izl IFM, the shear stress remains unchanged from that prevailing during the loading process. After reaching a minimum value of load is reversed again( Fig 3), causing reloading. Upon reloading by dog, sliding is confined too<z lfR The expression for total strain is fE, lEM 2(M-P,)2-0m+SoR EE=(OM-0m+SaR)+ (8a) and the crack opening displacement due to slip during the sequence is given by No SIi Fig 3. Depiction of the friction length, IF upon loading, the unloading friction length, lFU, and the reloading friction length, p
Matrix cracking of ceramic-matrix composites 1213 E(p_DD) +a = 0. 0) Lastly, the growth laws of the internal variables can be established from the results of materials science, but these in themselves can be major undertakings, as is evidenced by the work of Curtin (1991) describing the fragmentation process in a model where constant shear stress along the interface is assumed. This analysis requires substantial information about the statistical failure properties of part (1). Another method which is more practical and in keeping with the aim of this approach is to deduce the evolution of the state variables from experimental data. 2.3. Evolution laws When deriving the expressions for the elastic energy it was not necessary to have precise information about the conditions of slip and/or debond properties of the fibermatrix interface. To proceed further it is necessary to have information about the interface and in the following it is assumed that the stress of the interface has a constant value z, over the friction length fF. It can be shown that Be2 =fi/(l -f)& E;(z) = (IF-z)/a, so that P = 1,/2a, F = lS/3a2 and E” = 4E/3. It will be demonstrated that the evolution laws can be deduced from the measurements during a loading-unloading-reloading sequence. Reversed motion of fibers relative to the matrix has been studied by numerous authors (Marshall and Oliver, 1987; McMeeking and Evans, 1990; Pryce and Smith, 1992). The main results are now summarized. After reaching a maximum stress value 0 M, when the maximum friction length is ZFM, the load is reversed (Fig. 3). Upon unloading by 6au, in the range 0 < IzI 6 IF”, where lFu is the unloading friction length, both the relative sliding direction and the frictional shear stress reverse. Between lFu ,< Izl < &,,, the shear stress remains unchanged from that prevailing during the loading process. After reaching a minimum value of stress when 60, = cr,, the load is reversed again (Fig. 3), causing reloading. Upon reloading by do,, sliding is confined to 0 < IzI < Z,,. The expression for total strain is fE, IFM 2(a, -p,)2 -0; +64 EC= (GM-cm+&)+ (1_f)E2 L 4(Gl-Pl) and the crack opening displacement due to slip during the sequence is given by ZFR 1 - (2) I 7 . Matrix Crack-- t 2a 1 (1) / -----------T T Slip No Slip I- (2) Fig. 3. Depiction of the friction length, IF upon loading, the unloading friction length, /,,, and the reloading friction length, IFR
1214 F hild et al △、(oM,Gmn,O0g) lFM2(oM-p1)2-Gm+0σ (1-E The crack opening displacement As is also related to the total strain by EL E For a cracked system, a loss of stifness is involved. The effect of this change of Youngs modulus can also be interpreted in terms of effective stress(rabotnov, 1963; Lemaitre and Chaboche, 1978). The microscopic or"effective"stress, o=o/(1-D), is higher than the macroscopic stress a because of the presence of a crack. Therefore, the stress-strain relation ship becomes (10) Equation(10)is consistent with eqn (6a). Upon loading, the evolution of the irreversible strain a is given by dM-p1(1-D) 2E(1-D) The variation of a, da=a-Co with respect to minimum or maximum value, %o(cor responding to a maximum loading or minimum unloading level characterized by do)is related to the stress variation, dd=ddp, by d °x-4E(1-DM-p1(1-D)5gn(60 (11b) Instead of tracking the complicated pattern of friction and reverse friction, it is found to be convenient to introduce the opening strain into the calculations The expression of the maximum hysteresis loop width, dE, is given by d 42E(1-D)M-p1(1-D) (12) Equation( 12)shows the role of residual stresses on the evolution of the maximum hysteresis loop width. When the residual stresses vanish, a simple relationship exists between the inelastic strain and the maximum hysteresis loop width(Fig. 4) E(1-D) Fig. 4. Stress, a, versus strain, a. during a loading-unloading-reloading sequence
1214 F. Hild rt ul. The crack opening displacement A, is also related to the total strain by For a cracked system, a loss of stiffness is involved. The effect of this change of Young’s modulus can also be interpreted in terms of effective stress (Rabotnov, 1963 ; Lemaitre and Chaboche, 1978). The microscopic or “effective” stress, 0 = a/( 1 -D), is higher than the macroscopic stress 0 because of the presence of a crack. Therefore, the stress-strain relationship becomes d e-cc = E(l _D). Equation (10) is consistent with eqn (6a). Upon loading, the evolution of the irreversible strain a is given by “Idl,--,(1-D) x=- 2 E(l-D) (lla) The variation of X, Sa = CI -a,, with respect to minimum or maximum value, CI” (corresponding to a maximum loading or minimum unloading level characterized by a,) is related to the stress variation, 66 = ?r - bO, by & = ” (d# 4 E(l-@[a,-p,(l-D)] Sign (66). (1 lb) Instead of tracking the complicated pattern of friction and reverse friction, it is found to be convenient to introduce the opening strain into the calculations. The expression of the maximum hysteresis loop width, &, is given by & = (” a& 42E(1-fI)[a,-p,(l-D)]’ (12) Equation (12) shows the role of residual stresses on the evolution of the maximum hysteresis loop width. When the residual stresses vanish, a simple relationship exists between the inelastic strain and the maximum hysteresis loop width (Fig. 4) : Fig. 4. Stress. ~7, versus strain, E, during a loading-unloading-reloading sequence
Matrix cracking of ceramic-matrix composites 1215 (13) where ein is equal to a when dm=0. when dm=0, the inelastic strains are given by n=4E(1-D-p1(-D (14) and the macroscopic closure stress, reached when a=dc, is given by solving eqn(7b) 2D [aM-p1(1-D)1+(1+ D OM-P(1-D)2D Equation(15)defines the role of the residual stress field but as the damage parameter D increases, the effect of the initial residual stress field becomes less important. Furthermore when the damage parameter d is very small (i.e. t is large), the closure condition is given by ac 0, and the elastic strains are small as well(therefore Ec a 0) Equations(11)-(15)are only valid when saturation is not reached. Beyond that point, hese evolution laws need to be altered in order to account for matrix cracking saturation (L=lF). Lastly, a means of determining the variable D is to measure the initial unloading modulus, which is equal to E=E(1-D). Similarly, the initial reloading modulus is equal to E= E(I-D), as shown in Fig 4 2. 4. Identification from macroscopic quantities By using eqn(6a)and the definitions of the internal variables, it is possible to measure some of the internal variables macroscopically. To get the necessary additional information, a complete loading-unloading-reloading sequence is needed. The useful information is the maximum applied strain eM, the inelastic strain corresponding to a complete unloading ein, the maximum hysteresis loop width SE and the macroscopic Young's modulus E=E(1-D shown in Fig. 4. Three state variables, EM, Ein and D, can be used in a phenomenological model where it is assumed that the unloading process is linear(Fig 4). The expression of he corresponding state potential y is written as(Lemaitre and Chaboche, 1985) =E(1-D-Em) (16) The stress is obtained by partial differentiation of t with respect to the total strain Z 0=T=E(1-D)(E-Ein) (17a) and the forces associated with the inelastic strain Ein =a a=0), and with the macroscopi damage D are respectively given by y=-aD=iE(6-Ein) (17c) Within the framework of Continuum Damage Mechanics, Y is the strain energy release rate density( Chaboche, 1978). In this study, the evolution of the two internal variables Ein and D is obtained from experiments by plotting their evolution against the respective associated forces, which are assumed to be the driving forces. The measurement of the inelastic strains is easy to carry out since it involves complete unloading. The measurement
Matrix cracking of ceramic-matrix composites 1215 26s = Ein, (13) where Ei, is equal to CL when &,, = 0. When B,,, = 0, the inelastic strains are given by d 2[&, --p, (1 -@I’ -a: sin =4E(l-D)[a,-p,(l-D)] (14) and the macroscopic closure stress, reached when g = a,, is given by solving eqn (7b) : DC -= - i 1-2$a,-p&D)] GM . (15) Equation (15) defines the role of the residual stress field but as the damage parameter D increases, the effect of the initial residual stress field becomes less important. Furthermore, when the damage parameter d is very small (i.e. z is large), the closure condition is given by ~‘c z 0, and the elastic strains are small as well (therefore 8, z 0). Equations (1 I)-(1 5) are only valid when saturation is not reached. Beyond that point, these evolution laws need to be altered in order to account for matrix cracking saturation (L = Z,). Lastly, a means of determining the variable D is to measure the initial unloading modulus, which is equal to E” = E( 1 -D). Similarly, the initial reloading modulus is equal to _?? = E(l -D), as shown in Fig. 4. 2.4. Identljication from macroscopic quantities By using eqn (6a) and the definitions of the internal variables, it is possible to measure some of the internal variables macroscopically. To get the necessary additional information, a complete loading-unloading-reloading sequence is needed. The useful information is the maximum applied strain &, the inelastic strain corresponding to a complete unloading Ei,, the maximum hysteresis loop width 6~ and the macroscopic Young’s modulus B = E( 1 -D) shown in Fig. 4. Three state variables, z M, ZZin and D, can be used in a phenomenological model where it is assumed that the unloading process is linear (Fig. 4). The expression of the corresponding state potential $ is written as (Lemaitre and Chaboche, 1985) I) = $(l -LS)(E--i”)2. (16) The stress is obtained by partial differentiation of $ with respect to the total strain B : a$ d = ag = E(1 -d)(E-Ei”), (174 and the forces associated with the inelastic strain Ei, = cc(a = 0), and with the macroscopic damage 6. are respectively given by aiJ b= -- dEi, (17b) (l7c) Within the framework of Continuum Damage Mechanics, P is the strain energy release rate density (Chaboche, 1978). In this study, the evolution of the two internal variables E,, and d is obtained from experiments by plotting their evolution against the respective associated forces, which are assumed to be the driving forces. The measurement of the inelastic strains is easy to carry out since it involves complete unloading. The measurement
1216 F. Hild et al. of the damage variable can be performed through the measurement of the unloading modulus E= E(1-D) In this approach the internal variables are obtained from macroscopic measurements Since the residual stresses are unknown but constant additional information is needed to evaluate them, i.e. the maximum hysteresis loop width 8t. From the knowledge of the four previous quantities, the expressions for the damage variables are given by D≈EMD-gD 26E Em-En-268 (18a) d√(n+268)b (18b) and the residual stress in part(1)is -1(5M-En-26 (18c) Equations (18)allow us to quantify the microscopic variables by using macroscopic quan- tities. To complete the evolution laws, the driving forces need to be known. For the damage variable D, the natural driving force is its associated force Y. The role played by y is simila to that played by the energy release rate g in Linear Elastic Fracture Mechanics. Similarly the force X associated with the inelastic strain can be its driving force. However, egns. the driving force of the damage variable d is assumed to be its associated force y. Lastly, show that the macroscopic stress can also be taken and therefore will be chosen for the sake of simplicity In the following the previous results are applied to a layered material. It is, however, worth noting that these results can also be applied to fiber-reinforced systems. the definition of the damage variables and the opening strain may change slightly, but the essential features obtained in this paper are identical. In particular, the method of deriving the free energy density through two elastic steps can be conserved 3. ANALYSIS OF EXPERIMENTS ON A LAYERED MATERIAL The previous model will be used to predict the behavior of a layered material subject to tension. The material is constructed by alternating three alumina plates and two unidi ectional carbon/ epoxy prepreg tapes. The laminate is put in a Kapton vacuum bag at room temperature, then hot pressed at a moderate pressure of 350 kPa and at a temperature of 135C for 90 min(Lange et al., 1992: Sherman, 1992). The tensile stress-strain response is shown in Fig. 5. Because of the discrete process of matrix cracking, there are stress drops at each new break T 0.00300060.0090.012 Fig. 5. Experimental stress-strain curve of a layered material subject to tension fafter Sherman(1992)
1216 F. Hild et ~1. of the damage variable can be performed through the measurement of the unloading modulus E = E( 1 - D). In this approach the internal variables are obtained from macroscopic measurements. Since the residual stresses are unknown but constant, additional information is needed to evaluate them, i.e. the maximum hysteresis loop width 6% From the knowledge of the four previous quantities, the expressions for the damage variables are given by and the residual stress in part (1) is -p’~(~~-l)(~~-4”-26~). E (184 Equations (18) allow us to quantify the microscopic variables by using macroscopic quantities. To complete the evolution laws, the driving forces need to be known. For the damage variable D, the natural driving force is its associated force Y. The role played by Y is similar to that played by the energy release rate Y in Linear Elastic Fracture Mechanics. Similarly, the driving force of the damage variable d is assumed to be its associated force y. Lastly, the force X associated with the inelastic strain can be its driving force. However, eqns (11) show that the macroscopic stress can also be taken and therefore will be chosen for the sake of simplicity. In the following the previous results are applied to a layered material. It is, however, worth noting that these results can also be applied to fiber-reinforced systems. The definition of the damage variables and the opening strain may change slightly, but the essential features obtained in this paper are identical. In particular, the method of deriving the free energy density through two elastic steps can be conserved. 3. ANALYSIS OF EXPERIMENTS ON A LAYERED MATERIAL The previous model will be used to predict the behavior of a layered material subject to tension. The material is constructed by alternating three alumina plates and two unidirectional carbon/epoxy prepreg tapes. The laminate is put in a Kapton vacuum bag at room temperature, then hot pressed at a moderate pressure of 350 kPa and at a temperature of 135°C for 90 min (Lange et al., 1992 ; Sherman, 1992). The tensile stress-strain response is shown in Fig. 5. Because of the discrete process of matrix cracking, there are stress drops at each new break. 400 g 300 V 3 200 2 100 0 0 0.003 0.006 0.009 0.012 Strain Fig. 5. Experimental stress-strain curve of a layered material subject to tension [after Sherman (1992)]
Matrix cracking of ceramic-matrix composites 0.9 0.7 0.5 0.3 0.1 024681012 Strain Energy Release Rate Density(MPa) Fig. 6. Experimental and fitted evolution of the macroscopic damage variable as a function of strain energy release rate density This first part of the identification used macroscopic data only. The evolution of the macroscopic damage D and the inelastic strains E are plotted against their respective associated forces, Y and 0. The evolution of the macroscopic damage d against the associated force Y is shown in Fig. 6. The damage variable saturates so that a convenient way of describing this growth law is the exponential form The parameter D corresponds to the saturation value of the macroscopic damage D and Y, is a normalizing strain energy release rate density. For this particular system the following D2=079±001 F1=0.21±0.02 third order polynomial describes well the experimental measurements 0.0016 0012 吕0.0008 0.0004 2 0 100150200250300350400450 Stress(MPa) Fig. 7. Experimental and fitted evolution of the inelastic strain as a function of stress
Matrix cracking of ceramic-matrix composites 1217 0.9 0.7 0.5 0.3 0.1 0 2 4 6 8 10 12 Strain Energy Release Rate Density (MPa) Fig. 6. Experimental and fitted evolution of the macroscopic damage variable as a function of strain energy release rate density. This first part of the identification used macroscopic data only. The evolution of the macroscopic damage d and the inelastic strains e, are plotted against their respective associated forces, P and 8. The evolution of the macroscopic damage d against the associated force P is shown in Fig. 6. The damage variable saturates so that a convenient way of describing this growth law is the expanential form B=b,{l-exp(-g)}. (19) The parameter 6, corresponds to the saturation value of the macroscopic damage d and Y, is a normalizing strain energy release rate density. For this particular system the following values are obtained : 6, = 0.79 f 0.01 9, = 0.21 kO.02 (20) Figure 7 shows the evolution of the inelastic strain as a function of the applied stress 8. A third order polynomial describes well the experimental measurements with Fig. I. 100 150 200 250 300 350 400 450 Stress (MPa) Experimental and fitted evolution of the inelastic strain as a function (21) of stress
F. Hild et al En=-0.003±0.001 S1=21±6GPa S,=2.4+0.4GPa S3=1.7±0.2GPa The agreement between experiments and identification is good in terms of stress-strain rcsponsc as well. The identifications performed in this subsection constitute the basis of the identification of the microscopic model. So far, the identification consists of choosing the type of function describing the experiments, and of fitting the parameters of each function In the following, the evolution laws of the microscopic quantities, namely the damage measures of d and D, the inelastic strain a and the residual stress -p, will be determined The residual stress is important since it defines the onset of the first matrix crack in composite system(Budiansky et al, 1986). The first step is to compute the residual stress Pr. By using eqn(18c), and the experimental data concerning EM, ein and the maximum hysteresis loop width, aE, the residual stress-Pr can be calculated at each experimental data point. It is found that the value of -, varies, and since the residual stress is constant. the average for -P, is identified to be a tensile stress of +18 MPa(i.e a residual stress of the order of 25 MPa). this value is consistent with independent observations by removing one alumina layer that found a tensile residual stress of the order of 20 MPa in the alumina layer (Sherman, 1992). It is worth remembering that the basis of identification of the residual stresses is such that saturation is not observed macroscopically (i.e. the damage variable still increases) Since d and Ein are easy to measure, these measurements will be used to establish their evolution laws as a fit of those experimental data. By using eqn(18c), the evolution law of the maximum hysteresis loop width, SE, is then completely known and given by Figure 8 shows the comparison between the experiments and the predictions: the agreement is good These results are compared with those obtained when residual stresses are assumed to be negligible and correspond to the dashed lines in Fig. 8. In that case, eqn(13)applies and leads to an overestimate of the maximum hysteresis loop width. The knowledge of the parameter -p, is therefore crucial to the evolution of the maximum hysteresis loop width and the inelastic strain The values of the damage variables can be determined by using eqns(18a, b). Mea suring the initial unloading Youngs modulus gives directly the value of D, which can be compared with the prediction. Figure 9 shows the comparison between the experiments 00004 Predictions(-p,≠0) Predictions (-p,=0) 0.0003 0.0002 0.000l 00.000200004000060.0008 Inelastic Strain Fig 8. Experimental and predicted evolution of the maximum hysteresis loop width as a function of inelastic strain, when-P, =0 MPa and-P,=18 MPa
1218 F. Hild et al. E,] = -0.003+0.001 S, = 21 f6GPa S7 = 2.4f0.4GPa S, = 1.7f0.2GPa. (22) The agreement between experiments and identification is good in terms of stress-strain response as well. The identifications performed in this subsection constitute the basis of the identification of the microscopic model. So far, the identification consists of choosing the type of function describing the experiments, and of fitting the parameters of each function. In the following, the evolution laws of the microscopic quantities, namely the damage measures of d and D, the inelastic strain a and the residual stress -p, will be determined. The residual stress is important since it defines the onset of the first matrix crack in a composite system (Budiansky et al., 1986). The first step is to compute the residual stress -p,. By using eqn (18c), and the experimental data concerning F~,, F,, and the maximum hysteresis loop width, AZ, the residual stress -p, can be calculated at each experimental data point. It is found that the value of -p, varies, and since the residual stress is constant, the average for -p, is identified to be a tensile stress of + 18 MPa (i.e. a residual stress of the order of 25 MPa). This value is consistent with independent observations by removing one alumina layer that found a tensile residual stress of the order of 20 MPa in the alumina layer (Sherman, 1992). It is worth remembering that the basis of identification of the residual stresses is such that saturation is not observed macroscopically (i.e. the damage variable still increases). Since D and 8,” are easy to measure, these measurements will be used to establish their evolution laws as a fit of those experimental data. By using eqn (18c), the evolution law of the maximum hysteresis loop width, CL?, is then completely known and given by 1 26C = C,” 2 l+_---PI [ E(& -C,,, - 2&) 1 2 (23) - l Figure 8 shows the comparison between the experiments and the predictions ; the agreement is good. These results are compared with those obtained when residual stresses are assumed to be negligible and correspond to the dashed lines in Fig. 8. In that case, eqn (13) applies and leads to an overestimate of the maximum hysteresis loop width. The knowledge of the parameter -p, is therefore crucial to the evolution of the maximum hysteresis loop width and the inelastic strain. The values of the damage variables can be determined by using eqns (18a, b). Measuring the initial unloading Young’s modulus gives directly the value of D, which can be compared with the prediction. Figure 9 shows the comparison between the experiments Fig. 8. ?? Experiments 0 0.0002 0.0004 0.0006 0.0008 Inelastic Strain Experimental and predicted evolution of the maximum hysteresis loop width of inelastic strain, when -p, = 0 MPa and --p, = 18 MPa. as a function