Journal of the European Ceramic Society 15(1995) Printed in Great Britain. All right 09552219(95)00095 Toughness of Damage Tolerant Continuous Fibre Reinforced Ceramic Matrix Composites Bent F. Sorensen@& Ramesh talreja Materials Departrnent, Rise National Laboratory, 4000 Roskilde, Denmark sChool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA (Received 30 March 1995; revised version received 24 May 1995; accepted 25 May 1995) abstract I potential energy g stress A simple shear-lag model is used to estimate the Ts interfacial sliding shear stress toughness of a unidirectional fibre reinfored ceramic strain energy density matrix composite. In the model, which includes the residual axial stresses, the stress transfer across Subscript fibre/matrix interface is by a constant shear stress. db debonding estimates from the model experimental measurements for four composites and m matrix good agreement is found. The effects uf distributed mux value at specimen separation and localized energy uptake on fracture stability are mc matrix cracking discussed, and it is concluded that for most appli- u value at ultimate tions the toughness, rather than the pull-out energy Fs full sliding along the interface absorption, should be maximized. Superscript ACK fully and multiply cracked matrix Nomenclature ini initiation of matrix cracks fibre radius value after specimen separation a cross-section area Initial state B specimen width I-IV stages of damage crack length e Young' s modulus f fibre volume fraction 1 Introduction G (critical) energy-release-rate L Monolithic ceramics are brittle, flaw-sensitive and of s sliding length low energy absorption capacity before and during P pull-out length fracture. However, a class of ceramic composites applied load with continuous reinforcement is emerging, which s spacing of m cracks has much more attractive damage and fracture toughness(ene r unit volume) of com- behaviour. Materials in this class are damage toler- ant,in the sense that the first mode of damage displacement (matrix cracking) does not lead to final fracture W energy dissipated(per unit volume) due to Instead, the strength of the materials is controlled frictional sliding by the strength of the reinforcement. This flaw energy dissipated (per unit area) due to fibre insensitive behaviour is a remarkable property of materials that are made of brittle constituents. As ywoF work of fracture a consequence, fracture toughness is not a proper 8 matrix crack opening parameter to characterize such materials. Rather, 4 specimen elongation the stress at the onset of multiple matrix cracking, axial strain the failure stress, and total energy ' uptake until eu ultimate tensile strain failure constitute a set of property parameters
Journal of the European Ceramic Sociery 15 (1995) 1047-1059 0 1995 Elsevier Science Limited 0955-2219(95)00095-X Printed in Great Britain. All rights reserved 0955-2219/95/$9.50 Toughness of Damage Tolerant Continuous Fibre Reinforced Ceramic Matrix Composites Bent F. SOrensena 1E Ramesh Talrejab “Materials Department, Riser National Laboratory, 4000 Roskilde, Denmark ‘School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA (Received 30 March 1995; revised version received 24 May 1995; accepted 25 May 1995) Abstract A simple shear-lag model is used to estimate the toughness of a unidirectionaljbre reinfored ceramic matrix composite. In the model, which includes the residual axial stresses, the stress transfer across the jibre/matrix interface is by a constant shear stress. The estimates from the model are compared to experimental measurements for four composites and good agreement is found. The efsects of distributed and localized energy uptake on fracture stability are discussed, and it is concluded that for most applications the toughness, rather than the pull-out energy absorption, should be ma.ximized. Nomenclature B ; f G L 1, 1P P S u V Ki w, %/OF s A E eu fibre radius cross-section area specimen width crack length Young’s modulus fibre volume fraction (critical) energy-re:lease-rate specimen length sliding length pull-out length applied load spacing of matrix cracks toughness (energy per unit volume) of composite displacement energy dissipated (per unit volume) due to frictional sliding energy dissipated ‘(per unit area) due to fibre pull out work of fracture matrix crack open.ing specimen elongation axial strain ultimate tensile strain IT potential energy (T stress 7s interfacial sliding shear stress 4 strain energy density Subscript db debonding f fibre m matrix max value at specimen separation mc matrix cracking U value at ultimate stress FS full sliding along the interface Superscript ACK ini res * 0 I-IV fully and multiply cracked matrix initiation of matrix cracks residual value after specimen separation initial state stages of damage 1 Introduction Monolithic ceramics are brittle, flaw-sensitive, and of low energy absorption capacity before and during fracture. However, a class of ceramic composites with continuous reinforcement is emerging, which has much more attractive damage and fracture behaviour. Materials in this class are damage tolerant, in the sense that the first mode of damage (matrix cracking) does not lead to final fracture. Instead, the strength of the materials is controlled by the strength of the reinforcement. This flawinsensitive behaviour is a remarkable property of materials that are made of brittle constituents. As a consequence, fracture toughness is not a proper parameter to characterize such materials. Rather, the stress at the onset of multiple matrix cracking, the failure stress, and total energy ‘uptake until failure constitute a set of property parameters. 1047
1048 B. F. Sorensen, R. talreja Most theoretical works on continuous fibre reinforced ceramics focus on two topics: progression P pation due to fibre pull out. 4/ Multiple matri cracking is of importance since it leads to non linear constitutive behaviour that must be taken into account when these materials are used. see e.g. ref. 8. It is equally important to point out that in general there is no correlation between the ER proportional limit, i.e. the stress level at which u=o de multiple matrix cracking causes non-linearity in the stress-strain curve and the fatigue limit. The energy dissipation due to pull out is important for assessment of the final fracture behaviour. A central part of all these models is the role of the interface(see ref. 10 for a recent review ) The energy uptake by distributed mechanisms (i.e. prior to localization) has not been addressed sufficiently in the previous works. experimentally a remarkably non-linear stress-strain response has been found for several continuous fibre-reinforced ceramics composites.-4 However, the details of -di =g,b dc the associated evolution of damage have been studied only recentl on the basis of ecent findings, this paper develops a model p Quasi defined here as toughness, i.e. the total energy uptake until failure in volume-distributed mechan isms. This energy comprises both the recoverable and non-recoverable parts. It is of major impor- tance, because this is the maximum energy that can be absorbed per unit volume without causing fracture of a component, i.e.(the component will be damaged but will retain its integrity such that it can still continue to carry loads 2 Measurements of energy for material Fig. 1. Energy measures for materials:(A)the toughness, U characterization B) the critical energy release rate, G, and (C)the work of fracture, wor 2.1 Energy quantities Firstly, for clarity, we start off reviewing some where o and e are stress and strain, respectively energy concepts and quantities that are often used and e, is the failure strain (onset of localization to characterize the fracture behaviour of materials. It is the area under the stress-strain curve, and For simplicity the considerations are limited to the has the sI unit J/m case where a specimen is loaded by a single tension The energy release rate is defined in fracture force.Figure I illustrates three commonly used mechanics by the decrease of the potential energy quantities: the toughness, the critical energy per new crack area during a quasi-static crack release rate(fracture toughness) and the work of Increment fracture 1 dl The toughness20 is the ability of a material to uptake energy (per unit volume) prior to failure, e. the strain energy density to failure 20 where I is the potential energy, and c and B are the length and the width of the crack, respectively The critical value of G is equivalent to the energy U=de)dE, absorbed per unit new crack area, during an incre- mental(stable) crack growth. The SI unit for G is
1048 B. F. Smensen, R. Talreja Most theoretical works on continuous fibre reinforced ceramics focus on two topics: progression of multiple matrix crackinglm3 and energy dissipation due to fibre pull out.47 Multiple matrix cracking is of importance since it leads to nonlinear constitutive behaviour that must be taken into account when these materials are used, see e.g. ref. 8. It is equally important to point out that in general there is no correlation between the ‘proportional limit’, i.e. the stress level at which multiple matrix cracking causes non-linearity in the stress-strain curve and the fatigue limit.’ The energy dissipation due to pull out is important for assessment of the final fracture behaviour. A central part of all these models is the role of the interface (see ref. 10 for a recent review). The energy uptake by distributed mechanisms (i.e. prior to localization) has not been addressed sufficiently in the previous works. Experimentally, a remarkably non-linear stress-strain response has been found for several continuous fibre-reinforced ceramics composites. 9-14 However, the details of the associated evolution of damage have been studied only recently. 15-19 On the basis of these recent findings, this paper develops a model for the energy uptake by distributed mechanisms, defined here as toughness, i.e. the total energy uptake until failure in volume-distributed mechanisms. This energy comprises both the recoverable and non-recoverable parts. It is of major importance, because this is the maximum energy that can be absorbed per unit volume without causing fracture of a component, i.e. (the component will be damaged but will retain its integrity such that it can still continue to carry loads). 2 Measurements of energy for material characterization 2.1 Energy quantities Firstly, for clarity, we start off reviewing some energy concepts and quantities that are often used to characterize the fracture behaviour of materials. For simplicity the considerations are limited to the case where a specimen is loaded by a single tension force. Figure 1 illustrates three commonly used quantities: the toughness, the critical energy release rate (fracture toughness) and the work of fracture. The toughness20 is the ability of a material to uptake energy (per unit volume) prior to failure, i.e. the strain energy density to failure2’; C” U = s O(E) de, (1) 0 4-I = GIG B de (4 (b) YWOF =s E dA OA P A tA f 1 1 Fig. 1. Energy measures for materials: (A) the toughness, U, (B) the critical energy release rate, G, and (C) the work of fracture, -ywOF. where (T and E are stress and strain, respectively, and l U is the failure strain (onset of localization). It is the area under the stress-strain curve, and has the SI unit J/m3. The energy release rate is defined in fracture mechanics by the decrease of the potential energy per new crack area during a quasi-static crack increment21 G=-L!!Z B dc ’ where 17 is the potential energy, and c and B are the length and the width of the crack, respectively. The critical value of G is equivalent to the energy absorbed per unit new crack area, during an incremental (stable) crack growth. The SI unit for G is
Damage tolerant ceramic matrix composites 1049 J/m. Alternatively, the critical value of G is called and strain states at any point in the material. For the fracture toughness. Sometimes in the literature continuum mechanics to be applicable for materials the term toughness is used loosely when fracture experiencing distributed damage, the test volume ghness is meant. This may cause confusion In (gage length of extensometer or strain gauge)must our paper toughness is defined by(1)and fracture be much larger than any microstructural dimen- sions(e.g. matrix crack spacing), such that the The work of fracture is defined by the total measured volume behaviour is scale-independent, energy absorped (during a quasi-static fracture and the definitions apply. Moreover, the test process) by the specimen per unit fracture area, volume must have uniform stress and strain fields i.e. the area under the load-displacement curve In bending tests the normal stress distribution is per cross-sectional area(see e.g. ref. 22), non-uniform (going from tension on one side to compression on the other ), and so is the damage P(△) sketched in Fig. 2. The true (3) A stress-strain curve cannot be determined since the stress and strain distributions across the specimen where A is the cross-sectional area of the specimen, depth are not known (when damage takes place, P is the load and 4 is the elongation. ywor has the the distribution of normal stress is not linear, but SI unit J/m2 depends on the type of damage mechanism and From the definitions it follows that the quanti- degree of damage ). Furthermore, at the peak load, ties mentioned are suited for characterizing differ localized fibre failure takes place at the tensile ent phenomena. U relates the energy uptake(prior surfacc, multiple matrix cracking takes place to localization) to the volume, and is the proper the central region of the specimen, and the com- measure of the fracture initiation energy. G is pression side may still be undamaged. From the a measure related to characterizing phenomena bending load-displacement curve it is not possible appearing with growth of a single crack. Work of to distinguish between the efects of distributed mech- fracture relates the total energy absorped(fracture anisms (matrix cracking, fibre/matrix debond energy) to the full crack area ng and interfacial sliding) and localized(fibre breakage and pull out ). Furthermore, a shear 2.2 Distributed versus localized energy absorption stress field is present in bending tests. Thus, the of uniaxial fibre composites damage evolution and failure may be due to shear When unidirectional fibre composites are loaded or compressive stress components, rather than in the fibre direction a range of distributed damage tension mechanisms occur before final failure. The frac It is unfortunate that bending tests have been so ture occurs as a localized phenomenon, since fibre widely used for characterizing ceramic composites fracture and pull out only take place at a single since this has led to incorrect conclusions concern matrix crack, namely where the specimen sepa- ing the matcrial bchaviour. For instance, it has rates into two pieces. Throughout this paper sometimes been suggested that the tail (the down- distinction will be made between distributed energy going part)of the load-displacement curve(mea- uptake and energy absorped during localization. surable in displacement controlled experiments Distributed energy uptake should be measured by only) represents the energy dissipation due to fibre a quantity that reflects that the energy uptake is by distributed phenomena, such as the area under the stress-strain curve, U. The energy dissipation during localization also involves distributed and localized mechanisms. It is of importance to mea- sure and calculate these contributions separately, at tension surface since they do not scale in the same manner with specimen geometry, as will be elaborated later 2.3 Choice of test method-bending or pure tension Having recognised the importance of distingu ⊥战A ing between distributed and localized mechanisms we now proceed with a discussion of test methods for measurements of the toughness. Materials having fig. 2. A schematic illustration of various damage mech distributed damage are characterized by non- nisms operating simultaneously during a bending experiment linear constitutive laws. Per definition a consti- The damage evolution is not uniform throughout the depth tutive law is the relationship between the stress of the specimen
Damage tolerant ceramic matrix composites 1049 J/m2. Alternatively, the critical value of G is called the fracture toughness. Sometimes in the literature the term toughness is used loosely when fracture toughness is meant. This may cause confusion. In our paper toughness is defined by (1) and fracture toughness by (2). The work of fracture is defined by the total energy absorped (during a quasi-static fracture process) by the specimen per unit fracture area, i.e. the area under the load-displacement curve, per cross-sectional area (see e.g. ref. 22), A s ““f’(A) dA YWOF = __ o-4 ’ (3) where A is the cross-sectional area of the specimen, P is the load and A is the elongation. ywor has the SI unit J/m2. From the definitions it follows that the quantities mentioned are suited for characterizing different phenomena. 6’ relates the energy uptake (prior to localization) to the volume, and is the proper measure of the fracture initiation energy. G is a measure related to characterizing phenomena appearing with growth of a single crack. Work of fracture relates the total energy absorped (fracture energy) to the full crack area. 2.2 Distributed versus 1oc:alized energy absorption of uniaxial fibre composites When unidirectional fibre composites are loaded in the fibre direction a range of distributed damage mechanisms occur before final failure. The fracture occurs as a localized phenomenon, since fibre fracture and pull out only take place at a single matrix crack, namely where the specimen separates into two pieces. Throughout this paper distinction will be made between distributed energy uptake and energy absorped during localization. Distributed energy uptake should be measured by a quantity that reflects .that the energy uptake is by distributed phenomena, such as the area under the stress-strain curve, i!J. The energy dissipation during localization also involves distributed and localized mechanisms. It is of importance to measure and calculate these contributions separately, since they do not scale in the same manner with specimen geometry, as will be elaborated later. 2.3 Choice of test method - bending or pure tension Having recognised the importance of distinguishing between distributed and localized mechanisms, we now proceed with a discussion of test methods for measurements of the toughness. Materials having distributed damage are characterized by nonlinear constitutive laws. Per definition a constitutive law is the relationship between the stress and strain states at any point in the material. For continuum mechanics to be applicable for materials experiencing distributed damage, the test volume (gage length of extensometer or strain gauge) must be much larger than any microstructural dimensions (e.g. matrix crack spacing), such that the measured volume behaviour is scale-independent, and the definitions apply. Moreover, the test volume must have uniform stress and strain fields. In bending tests the normal stress distribution is non-uniform (going from tension on one side to compression on the other), and so is the damage evolution, as sketched in Fig. 2. The true stress-strain curve cannot be determined, since the stress and strain distributions across the specimen depth are not known (when damage takes place, the distribution of normal stress is not linear, but depends on the type of damage mechanism and degree of damage). Furthermore, at the peak load, localized fibre failure takes place at the tensile surface, multiple matrix cracking takes place in the central region of the specimen, and the compression side may still be undamaged. From the bending load-displacement curve it is not possible to distinguish between the eflects of distributed mechanisms (matrix cracking, fibre/matrix debonding and interfacial sliding) and localized (fibre breakage and pull out). Furthermore, a shear stress field is present in bending tests. Thus, the damage evolution and failure may be due to shear or compressive stress components, rather than tension.23 It is unfortunate that bending tests have been so widely used for characterizing ceramic composites, since this has led to incorrect conclusions concerning the material behaviour. For instance, it has sometimes been suggested that the tail (the downgoing part) of the loadclisplacement curve (measurable in displacement controlled experiments only) represents the energy dissipation due to fibre Onset of localized fibre failure at tension surface Onset of 4--- matrix cracking \I at tension surface A Locus of fibre breakage ‘, and pull out moves Fig. 2. A schematic illustration of various damage mechanisms operating simultaneously during a bending experiment. The damage evolution is not uniform throughout the depth of tile specimen
1050 rensen r talre pull out. This is incorrect, since matrix cracking, the matrix before matrix cracking are proportional ibre/matrix debonding, fibre sliding and fibre fail- to the axial strain of the composite. This approach ure continue to take place during this part of the also makes it easier to compare behaviour of bending response, as the failure locus moves unidirectional and cross-ply composites. For the across the depth of the specimen towards the com- SiC/CAS II composite shown in Fig. 3, U was pression side measured to be 3. 1 MJ/m, which is almost 40 Uniaxial tension experiments, on the other times that of the pure matrix material. However, hand,allow an unambiguous interpretation, since due to loading rate cffcct h the damage mechanisms initiate and develop at stress-strain behaviour, the value for SIC/CAS II identifiable stress or strain levels, 5-19 and take can vary from 1.7 MJ/m(at 0.01 MPa/s)to 5 4 place uniformly over the entire volume of the gage MJ/m3(500 MPa/s) section of the specimen The non-linear stress-strain curve reflects the Unfortunately, tensile tests are difficult to various stages of damage( see ref. 19 for a detailed perform, since an accurate alignment is needed to discussion). At low strain the material response is prevent bending effects. 4 Also, although tensile linearly elastic and reversible( Stage I). At highe tests are performed with much care, it is not strain matrix cracks initiate and evolve into fairly unusual that failure takes place at tabs so the true regularly spaced multiple matrix cracks( Stage in) strength of the composite cannot always be mea- This takes place over a certain strain range, rather sured. However, if there is no bending in the test than at a specific strain value. Fibre/mat section, then the relationship between stress, strain debonding also takes place during Stage Il, and and damage mechanisms can readily be identified. facilitates frictional sliding along the fibre/matrix interface. These damage processes cause the non 2.4 Experimental characterization of damage in lincar part of the strcss-strain curve. The corre- uniaxial tension sponding changes in the overall unloading Fi ental stress-strain curve modul of a hot pressed unidirectional Sic-fibre-rein- be modelled by a continuum damage model forced calcium alumino silicate(denoted CAS II Increasing the applied stress further leads to a se from Corning Inc, NY, USA)glass-ceramic ond linear response( Stage III), where the stiffness matrix composite, tested in uniaxial tension along is mainly due to the fibres. However, significant fibres at a strain rate of 4 x 104 min hysteresis appears during unloading due to effects Throughout this paper we will refer the evolution from interfacial friction. For some specimens a of damage to the axial strain of the composite small amount of distributed fibre fracture( Stage rather than the composite stress, since the axial IV)may take place just prior to failure. Final fail stress component in the fibres resulting from the ure occurs by localized fibre fracture. Recalling external load and the axial stress component in the preceding discussion, the area under the stress-strain curve represents material has absorbed per unit volume of the composite up to failure. The energy quantity mea sured by U is the energy that has been absorbed Matrix by distributed mechanisms. Mechanisms operating (MPa) Initiation during the localized fracture, such as fibre fracture and pull out do not contribute to this quantity 3 Model of Energy Uptake 3.1 Basic consideratior 200 In the fol III IV nechanisms of a unidirectional ceramic fibre rein- 100 forced ceramic composite is estimated, based on the damage mechanisms and 000.250507510 above. The tensile stress-strain curve represents re quasI-static Fig. 3. The non-inear stress-strain curve of SiCCAS I mea- ing in equilibrium with the stresses in the material, ores (strain rate and therefore, according to the first law of thermo- 0/ min). applied stress as function of strain. The stages of dynamics, the work of the external force must be damage evolution are indicated(after ref. 19) equal to the change of the energy in the specimen
1050 B. F. Swensen, R. Talreja pull out. This is incorrect, since matrix cracking, fibre/matrix debonding, fibre sliding and fibre failure continue to take place during this part of the bending response, as the failure locus moves across the depth of the specimen towards the compression side. Uniaxial tension experiments, on the other hand, allow an unambiguous interpretation, since the damage mechanisms initiate and develop at identifiable stress or strain levels,‘5-‘9 and take place uniformly over the entire volume of the gage section of the specimen. Unfortunately, tensile tests are difficult to perform, since an accurate alignment is needed to prevent bending effects. 24 Also, although tensile tests are performed with much care, it is not unusual that failure takes place at tabs,19 so the true strength of the composite cannot always be measured. However, if there is no bending in the test section, then the relationship between stress, strain and damage mechanisms can readily be identified. 2.4 Experimental characterization of damage in uniaxial tension Figure 3 shows an experimental stress-strain curve of a hot pressed unidirectional SiC-fibre-reinforced calcium alumino silicate (denoted CAS II from Corning Inc., NY, USA) glass-ceramic matrix composite, tested in uniaxial tension along fibres19 at a strain rate of 4 X lOA min. Throughout this paper we will refer the evolution of damage to the axial strain of the composite rather than the composite stress, since the axial stress component in the fibres resulting from the external load and the axial stress component in Fibre Failure 0.0 0.25 0.5 0.75 1.0 (%I Fig. 3. The non-linear stress-strain curve of SiCKAS II measured in uniaxial tension along fibres (strain rate of 4 X lO?min). Applied stress as function of strain. The stages of damage evolution are indicated (after ref. 19). the matrix before matrix cracking are proportional to the axial strain of the composite. This approach also makes it easier to compare behaviour of unidirectional and cross-ply composites. For the SiCKAS II composite shown in Fig. 3, U was measured to be 3.1 MJ/m3, which is almost 40 times that of the pure matrix material. However, due to loading rate effects on the monotonic stress-strain behaviour, 25 the value for SiCKAS II can vary from 1.7 MJ/m3 (at 0.01 MPa/s) to 5.4 MJ/m3 (500 MPa/s). The non-linear stress-strain curve reflects the various stages of damage (see ref. 19 for a detailed discussion). At low strain the material response is linearly elastic and reversible (Stage I). At higher strain matrix cracks initiate and evolve into fairly regularly spaced multiple matrix cracks (Stage II). This takes place over a certain strain range, rather than at a specific strain value. Fibre/matrix debonding also takes place during Stage II, and facilitates frictional sliding along the fibre/matrix interface. These damage processes cause the nonlinear part of the stress-strain curve. The corresponding changes in the overall unloading modulus and Poisson’s ratio in Stage II can be modelled by a continuum damage model.* Increasing the applied stress further leads to a second linear response (Stage III), where the stiffness is mainly due to the fibres. However, significant hysteresis appears during unloading due to effects from interfacial friction. For some specimens a small amount of distributed fibre fracture (Stage IV) may take place just prior to failure. Final failure occurs by localized fibre fracture. Recalling the preceding discussion, the area under the stress-strain curve represents the energy that the material has absorbed per unit volume of the composite up to failure. The energy quantity measured by ti is the energy that has been absorbed by distributed mechanisms. Mechanisms operating during the localized fracture, such as fibre fracture and pull out do not contribute to this quantity. 3 Model of Energy Uptake 3.1 Basic considerations In the following the energy uptake by distributed mechanisms of a unidirectional ceramic fibre reinforced ceramic composite is estimated, based on the damage mechanisms and scenario identified above. The tensile stress-strain curve represents the response of the material to a quasi-static loading in equilibrium with the stresses in the material, and therefore, according to the first law of thermodynamics, the work of the external force must be equal to the change of the energy in the specimen
Damage tolerant ceramic matrix composites In many cases the stress-strain curve does not 3. 2 Basis of unit cell model show non-linearity immediately before failure, In this paper the strain densities and frictional slid indicating that the presence of Stage IV (distributed ing are calculated from a simple shear-lag model fibre failure)can be neglected. Thus, our analysis Similar concerns only energy uptake in Stages L, II and stresses are added( details are given in the appendix II, with Stage Ill extended until failure. In Stage I The model is a simple one-dimensional analysis, i.e the energy uptake per unit volume of the compos- only the changes in the axial stresses and strains ite, the area under the stress-strain curve up to E= are considered, and the interfacial friction shear Emen, is denoted U. According to the principle of stress Ts is assumed to be constant throughout the virtual work (or the first law of thermodynamics) experiment. This is a good assumption for the the energy uptake U is equal to the increase in accuracy of modelling presented here since it has strain energy density in the matrix and fibres, p been found that the effects of fibre poisson's con nd qr, respectively, where superscript 0 and i traction and roughness more or less cancel out. 26.27 denote the start and end of Stage I, respectively Initially, i.e. in the virgin state, the composite is free from external stresses, but has axial residual stresses, ores and o mes in the fibre and matrix, due Ul=ode=Im+i-oom-e (4) to a thermal expansion mismatch between fibre and matrix. Before fibre/matrix debonding has taken In Stage II the external work per unit volume is, place, there is no slip between fibre and matrix at again, according to the principle of virtual work, the interface. During multiple matrix cracking the equal to the sum of the changes in strain energy fibre and matrix debond during the loading, the density of fibre and matrix, the fracture energy of matrix retracts and, relatively, the fibres slide along multiple matrix cracks, fibre/matrix debonding the interface monotonically(Fig 4). It is assumed and the energy dissipation due to interfacial fric- that the fibres slide symmetrically inside the matrix tional sliding in Stage II. Denoting superscript II cylinder from both ends, such that there is sliding for the end of Stage II (e= fm ACk), the energy along the entire interface except at the middle of uptake in Stage II is the matrix cylinder(0<x<s/2, Fig. 4), where sticking friction is assumed due to symmetry(the condition TAI =o de=m +d- m-1+Um+UJb+wJl. (5) 3.3 Geometrical considerations where Wsstands for the frictional energy dissipa A unit cell model is depicted in Fig. 4. The cylin drical volume cell represents a single fibi tion per unit volume in Stage Il, Ume is the energy radius, a, surrounded by a matrix cylinder. The consumed due to the formation of matrix crack and Uab is the debond energy per unit volume. length of the cell is s, corresponding to the matrix Assuming that the deformation of fibre and matrix crack spacing, and the outer radius of the matrix cylinder is such that the fibre volume fraction is f continues in Stage Ill, the cncrgy uptake in Stage The cross-section area Ar of the fibre is d e=wu U=σd∈=φ+φ-φ-φ+W,(0 The interfacial surface area A is A where superscript Ill stands for the end stat when localization sets in(∈=∈u), and w is the frictional energy dissipation in Stage Ill. This gives the total energy uptake as U=U+Um+UⅢ=φⅢ+φ-Φ。-φ? Umc Uab+ Wsl where Ws is the total energy dissipation per unit Matrix volume due to frictional sliding in Stages II and t←a→0m III. Note that u can be calculated from the stress and strain states in the initial and end states. The frictional energy dissipation can also be calculated from the strain distributions in the end state assuming the fibre sliding to occur monotonically Fig. 4. The axisymmetric unit cell used in the analysis
Damage tolerant ceramic matrix composites 1051 In many cases the stress-strain curve does not show non-linearity immediately before failure, indicating that the presence of Stage IV (distributed fibre failure) can be neglected. Thus, our analysis concerns only energy u.ptake in Stages I, II and III, with Stage III extended until failure. In Stage I the energy uptake per unit volume of the composite, the area under the stress-strain curve up to E = E i”i, is denoted U’. According to the principle of vrtual work (or the first law of thermodynamics) the energy uptake U’ is equal to the increase in strain energy density in the matrix and fibres, @,,,I and @f’, respectively, where superscript 0 and I denote the start and end of Stage I, respectively E 1,“: U’= ade=@;+@;-@0,-@Of. s (4) 0 In Stage II the external work per unit volume is, again, according to the principle of virtual work, equal to the sum of the changes in strain energy density of fibre and matrix, the fracture energy of multiple matrix cracks, fibreimatrix debonding and the energy dissipat:ion due to interfacial frictional sliding in Stage II. Denoting superscript II for the end of Stage II (E = gmCACK), the energy uptake in Stage II is where II’,,” stands for the frictional energy dissipation per unit volume in Stage II, U,, is the energy consumed due to the formation of matrix cracks and U,, is the debond energy per unit volume. Assuming that the deformation of fibre and matrix continues in Stage III, the energy uptake in Stage III is where superscript III stands for the end state, when localization sets in (E = E,), and WsllI1 is the frictional energy dissipation in Stage III. This gives the total energy uptake as &I, + udb + Kl, (7) where I+‘,, is the total energy dissipation per unit volume due to frictional sliding in Stages II and III. Note that U can be calculated from the stress and strain states in the initial and end states. The frictional energy dissipation can also be calculated from the strain distributions in the end state, assuming the fibre sliding to occur monotonically. 3.2 Basis of unit cell model In this paper the strain densities and frictional sliding are calculated from a simple shear-lag model, similar to Aveston et al.;’ the effect of residual stresses are added (details are given in the Appendix). The model is a simple one-dimensional analysis, i.e. only the changes in the axial stresses and strains are considered, and the interfacial friction shear stress 7s is assumed to be constant throughout the experiment. This is a good assumption for the accuracy of modelling presented here, since it has been found that the effects of fibre Poisson’s contraction and roughness more or less cancel out.26,27 Initially, i.e. in the virgin state, the composite is free from external stresses, but has axial residual stresses, afres and (T c in the fibre and matrix, due to a thermal expansion mismatch between fibre and matrix. Before fibre/matrix debonding has taken place, there is no slip between fibre and matrix at the interface. During multiple matrix cracking the fibre and matrix debond during the loading, the matrix retracts and, relatively, the fibres slide along the interface monotonically (Fig. 4). It is assumed that the fibres slide symmetrically inside the matrix cylinder from both ends, such that there is sliding along the entire interface except at the middle of the matrix cylinder (O<x<s/2, Fig. 4), where sticking friction is assumed due to symmetry (the condition for full slip is given in the Appendix). 3.3 Geometrical considerations A unit cell model is depicted in Fig. 4. The cylindrical volume cell represents a single fibre of radius, a, surrounded by a matrix cylinder. The length of the cell is s, corresponding to the matrix crack spacing, and the outer radius of the matrix cylinder is such that the fibre volume fraction isf. The cross-section area A, of the fibre is Af =ra2. (8) The interfacial surface area A, is A, = 2z-as, (9) S I\ II \\ II - -..-.-Ju- II Matrix , \ r------- L&L ,_-sA 1 ,1 I I I TV_ 7-7 Fibre / p % s/2 Fig. 4. The axisymmetric unit cell used in the analysis