COMPOSITES SCIENCE AND TECHNOLOGY ELSEⅤIER Composites Science and Technology 61(2001)1743-1756 www.elsevier.com/locate/compscitech On fiber debonding and matrix cracking in fiber-reinforced ceramics Yih-Cherng Chiang Department of Mechanical Engineering, Chinese Culture University, No. 55. Hud-Kang Road, Taipei, Taiwan Received 30 May 2000; received in revised form 12 April 2001; accepted 7 June 2001 Abstract The relationships between debonding in the wake of a crack and the critical stresses for propagating a fiber-bridged matrix crack in fiber-reinforced ceramics have been studied. By adopting a shear-lag model which includes the matrix shear deformation in the bonded region and friction in the debonded region, the relationship between the fiber-closure traction and the debonded length is obtained by treating the interfacial debonding as a particular crack propagation problem along the interface By using an energy balance approach, the formulation of the critical stress for propagating a fiber-bridged matrix crack can then be derived. The conditions for attaining no-debonding and debonding during matrix cracking are discussed in terms of the two interfacial properties of debonding toughness and interfacial shear stress. The theoretical results are compared with experimental data of Sic/bor- licate, SIC/LAS and C/borosilicate ceramic composites. 2001 Elsevier Science Ltd. All rights reserved Keywords: A. Ceramic-matrix composites: B Debonding: B Matrix cracking 1. Introduction the distributed spring model [3-5] and the continuous distributions of dislocation loops model [6]. An assumed The properties of the fiber/matrix interface have been constant interfacial shear stress was adopted to perform identified as a key factor to develop a successful fiber- he fiber and matrix stress calculations in the analyses of reinforced brittle-matrix composite from both theore- ACK [1], BHE [2], Marshall et al. [3]. McCartney [4] tical analyses and experimental studies. For example, as and Chiang et al. [5], in which BhE [2] and Chiang et al the fibers are weakly constrained with ceramics, the [5] further considered the matrix shear deformation in composite starts to form a matrix crack at a relatively the no-slipping region. The results by the constant low stress but it exhibits toughness on account of fiber interfacial shear stress model show that the composite bridging as matrix cracking occurs On the other hand, with the higher interfacial shear stress results in the if the fibers are strongly coupled with the matrix, the higher matrix cracking stress. The inclusion of the composite initiates a matrix crack at higher stress but it matrix shear deformation in the analyses of BHE [21 may fail catastrophically because of fiber fracture as the and Chiang et al. [5] solves the discontinuity problem of matrix cracks. The constraint between the fiber and the the interfacial shear stress at the slipping crack tip that matrix is also related to the ease of slipping or debond- occurs in the models without consideration of the ing and fiber pull-out, which is associated with the work matrix shear deformation [1, 3, 4. The BHE [2] results of fracture for composite failur indicate that the effect of the matrix shear deformation Regarding the coupling between the fiber and the on the matrix cracking stress becomes more profound as matrix, the interface can be categorized as a frictional the interfacial shear stress is increasing. For weakly and bonded interface. For the frictionally constrained frictional interface the BHE model will reduce to the interface in a brittle- matrix composite, several modeling ACK model. On the other hand, if the interfacial shear approaches have been adopted for predicting the critical stress is high enough to prevent relative slippage stress to propagate a fiber-bridged matrix crack. These between the fiber and the matrix, the bhe model pre- approaches include the energy-balance approach [1, 2], dicts the same result of Aveston and Kelly [7] for the perfectly bonded interface. Hence, the bhe model Tel+886-2-2861-0511x458;fax:+886-2-2861-5241 could provide the results to bridge the Aveston and mail address: ycchiang @staff. pccu. edu. tw (Y-C. Chiang) Kelly result for the perfectly bonded interface and the 0266-3538/01/ S.see front matter C 2001 Elsevier Science Ltd. All rights reserved. PII:S0266-3538(01)00078-1
On fiber debonding and matrix cracking in fiber-reinforced ceramics Yih-Cherng Chiang* Department of Mechanical Engineering, Chinese Culture University, No. 55, Hua-Kang Road, Taipei, Taiwan Received 30 May 2000; received in revised form 12 April 2001; accepted 7 June 2001 Abstract The relationships between debonding in the wake of a crack and the critical stresses for propagating a fiber-bridged matrix crack in fiber-reinforced ceramics have been studied. By adopting a shear-lag model which includes the matrix shear deformation in the bonded region and friction in the debonded region, the relationship between the fiber-closure traction and the debonded length is obtained by treating the interfacial debonding as a particular crack propagation problem along the interface. By using an energybalance approach, the formulation of the critical stress for propagating a fiber-bridged matrix crack can then be derived. The conditions for attaining no-debonding and debonding during matrix cracking are discussed in terms of the two interfacial properties of debonding toughness and interfacial shear stress. The theoretical results are compared with experimental data of SiC/borosilicate, SiC/LAS and C/borosilicate ceramic composites. # 2001Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Debonding; B. Matrix cracking 1. Introduction The properties of the fiber/matrix interface have been identified as a key factor to develop a successful fiberreinforced brittle-matrix composite from both theoretical analyses and experimental studies. For example, as the fibers are weakly constrained with ceramics, the composite starts to form a matrix crack at a relatively low stress but it exhibits toughness on account of fiber bridging as matrix cracking occurs. On the other hand, if the fibers are strongly coupled with the matrix, the composite initiates a matrix crack at higher stress but it may fail catastrophically because of fiber fracture as the matrix cracks. The constraint between the fiber and the matrix is also related to the ease of slipping or debonding and fiber pull-out, which is associated with the work of fracture for composite failure. Regarding the coupling between the fiber and the matrix, the interface can be categorized as a frictional and bonded interface. For the frictionally constrained interface in a brittle-matrix composite, several modeling approaches have been adopted for predicting the critical stress to propagate a fiber-bridged matrix crack. These approaches include the energy-balance approach [1,2], the distributed spring model [3–5] and the continuous distributions of dislocation loops model [6]. An assumed constant interfacial shear stress was adopted to perform the fiber and matrix stress calculations in the analyses of ACK [1], BHE [2], Marshall et al. [3], McCartney [4] and Chiang et al. [5], in which BHE [2] and Chiang et al. [5] further considered the matrix shear deformation in the no-slipping region. The results by the constant interfacial shear stress model show that the composite with the higher interfacial shear stress results in the higher matrix cracking stress. The inclusion of the matrix shear deformation in the analyses of BHE [2] and Chiang et al. [5] solves the discontinuity problem of the interfacial shear stress at the slipping crack tip that occurs in the models without consideration of the matrix shear deformation [1,3,4]. The BHE [2] results indicate that the effect of the matrix shear deformation on the matrix cracking stress becomes more profound as the interfacial shear stress is increasing. For weakly frictional interface the BHE model will reduce to the ACK model. On the other hand, if the interfacial shear stress is high enough to prevent relative slippage between the fiber and the matrix, the BHE model predicts the same result of Aveston and Kelly [7] for the perfectly bonded interface. Hence, the BHE model could provide the results to bridge the Aveston and Kelly result for the perfectly bonded interface and the 0266-3538/01/$ - see front matter # 2001Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00078-1 Composites Science and Technology 61 (2001) 1743–1756 www.elsevier.com/locate/compscitech * Tel.: +886-2-2861-0511x458; fax: +886-2-2861-5241. E-mail address: ycchiang@staff.pccu.edu.tw (Y.-C. Chiang)
1744 Y -C. Chiang/ Composites Science and Technology 61(2001)1743-1756 ACK result for extensive slipping interface. As for the debonding process in the crack-wake region was not distributed spring approach, the including of the matrix considered in their modeling. The effects of crack-wake hear deformation by Chiang et al. [5] could solve the debonding on the matrix cracking stresses have been problem occurred in the analyses of Marshall et al. [] investigated by Chiang[ 8] for bonded composite in and McCartney [4 that the fiber axial stress is vanishing which the debonded region is resisted by a constant as the fiber is approaching to the matrix crack tip frictional stress. The Chiang result shows that the com- As the fiber is bonded to the matrix, the interfacial posite with the higher debonding toughness results in debonding may be initialized by the high transverse the higher matrix cracking stress. Geo et al. [9] and ensile stress in front of the matrix crack tip. And, the Hutchinson and Jensen [10] have the adopted Lame debonding process may continue in the crack-wake approach and Coulomb frictional law, by which the region due to the relative fiber-matrix displacement Poisson contraction effects can be included in the mod- bove the crack plane (see Fig. 1). Accordingly, the eling, to analyze the debonding process in the bonded debonded interface may be either separated or resisted composites with the friction in the debonded region by frictional stress depending on the transverse stress on The result by the Lame approach shows that the inter the interface and the characteristics of interface(e.g. the facial shear stress varies along the debonded length interface roughness). The matrix cracking problem of rather than being constant value. Recently, Chiang [11] perfectly bonded interface has been analyzed by Aves- has also adopted the Lame approach and Coulomb ton and Kelly [7]. The analytical expression of the frictional law to evaluate the Poisson contraction effects matrix cracking stress for perfectly bonded composite on the matrix cracking stresses for bonded composites by Aveston and Kelly [7]relates to the elastic properties, with friction matrix fracture toughness and the geometrical constants The advantages of the Lame approach to the constant of the fiber and the matrix; no specific interfacial prop- interfacial shear stress model are that the tractions and erty appears in the formulation of the matrix cracking displacements of the fiber and the matrix are continuous stress. The mechanics of the crack-tip debonding and its at the interface and the compressive stress on the inter- on the matrix cracking stress have been inve face induced from the thermal residual stress and pois gated by the BHE [2]. The result indicated that a fairly son contraction can be assessed. However, the Lame small interfacial debonding toughness(about 1 /5 of approach possesses the same problem, that the com- matrix fracture toughness) could prohibit debonding puted interfacial shear stress is discontinuous at the process from the crack-tip transverse tensile stress. bhe debonding crack tip, as the constant interfacial shear model assumed that the debonded length caused by the stress model that does not consider the matrix shear crack-tip transverse tensile stress was unchanged as the deformation. This is because the interfacial shear stress crack continually propagating. The possible interfacial caused by the matrix shear deformation in the bonded Downstream Transient-++-Upstream Crack-tip debonding Crack-wake Fig. I. Schematic representation of crack-tip and crack-wake debonding
ACK result for extensive slipping interface. As for the distributed spring approach, the including of the matrix shear deformation by Chiang et al. [5] could solve the problem occurred in the analyses of Marshall et al. [3] and McCartney [4] that the fiber axial stress is vanishing as the fiber is approaching to the matrix crack tip. As the fiber is bonded to the matrix, the interfacial debonding may be initialized by the high transverse tensile stress in front of the matrix crack tip. And, the debonding process may continue in the crack-wake region due to the relative fiber-matrix displacement above the crack plane (see Fig. 1). Accordingly, the debonded interface may be either separated or resisted by frictional stress depending on the transverse stress on the interface and the characteristics of interface (e.g. the interface roughness). The matrix cracking problem of perfectly bonded interface has been analyzed by Aveston and Kelly [7]. The analytical expression of the matrix cracking stress for perfectly bonded composite by Aveston and Kelly [7] relates to the elastic properties, matrix fracture toughness and the geometrical constants of the fiber and the matrix; no specific interfacial property appears in the formulation of the matrix cracking stress. The mechanics of the crack-tip debonding and its influence on the matrix cracking stress have been investigated by the BHE [2]. The result indicated that a fairly small interfacial debonding toughness (about 1/5 of matrix fracture toughness) could prohibit debonding process from the crack-tip transverse tensile stress. BHE model assumed that the debonded length caused by the crack-tip transverse tensile stress was unchanged as the crack continually propagating. The possible interfacial debonding process in the crack-wake region was not considered in their modeling. The effects of crack-wake debonding on the matrix cracking stresses have been investigated by Chiang[8] for bonded composite in which the debonded region is resisted by a constant frictional stress. The Chiang result shows that the composite with the higher debonding toughness results in the higher matrix cracking stress. Geo et al. [9] and Hutchinson and Jensen [10] have the adopted Lame´ approach and Coulomb frictional law, by which the Poisson contraction effects can be included in the modeling, to analyze the debonding process in the bonded composites with the friction in the debonded region. The result by the Lame´ approach shows that the interfacial shear stress varies along the debonded length rather than being constant value. Recently, Chiang [11] has also adopted the Lame´ approach and Coulomb frictional law to evaluate the Poisson contraction effects on the matrix cracking stresses for bonded composites with friction. The advantages of the Lame´ approach to the constant interfacial shear stress model are that the tractions and displacements of the fiber and the matrix are continuous at the interface and the compressive stress on the interface induced from the thermal residual stress and Poisson contraction can be assessed. However, the Lame´ approach possesses the same problem, that the computed interfacial shear stress is discontinuous at the debonding crack tip, as the constant interfacial shear stress model that does not consider the matrix shear deformation. This is because the interfacial shear stress caused by the matrix shear deformation in the bonded Fig. 1. Schematic representation of crack-tip and crack–wake debonding. 1744 Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756
Y -C. Chiang/ Composites Science and Technology 61(2001)1743-1756 745 region near the debonding crack tip cannot be evaluated This approximation is consistent with the rule-of- by the Lame formula. Since the matrix shear deformation mixtures of Eq (1). Then, the fiber and matrix stresses was identified as an important factor on the matrix at the far-field end become(for L->oo) cracking problem, the shear lag model adopted by bhe E [2]is applied in the present paper to perform the stress Or(L)=Ea and strain calculations in the fiber and the matrix. In this paper, the crack-wake interlace debonding process om(L)=Ema which the criterion of interfacial debonding in the crack-wake can be derived and. thereafter the debon- The fiber and matrix axial stresses at the crack plane ded length can be determined. Then, an energy balance (i.e, ==0) are given by approach is adopted to evaluate the critical stress for propagating a fiber-bridged matrix crack. The condi- or(0) (6) tions to achieve no-debonding and debonding as matrix cracking are discussed in terms of the interfacial prop- erties of debonding toughness and the interfacial shear m(0)=0 stress. Three different composite systems, of which experimental data are already available in the literature, are used for case studies 2. Fiber/matrix stress analysis Matrix Fiber Matrix 2. Downstream stresses The composite with fiber volume fraction Vr loaded IA L by a remote uniform stress o normal to a semi-infinite crack plane is shown in Fig. 1. The effective axial Youngs modulus of composite E is approximated by the rule-of-mixture dQ小 E= VrEr +ver where E and v denote Youngs modulus and volume v(0 fraction, and the subscripts f and m indicate the fiber nd the matrix, respectively. The downstream region (see Fig. 1)is sufficiently behind the crack-tip so that the T=O/Ve stress and strain fields are uniform with respect to the crack plane. Thus, the total axial stresses satisfy Vror(z)+mom()=o where od=)and mz)denote the fiber and matrix axial stresses at the z location. as shown in Fig. 2. It is noted that this relationship is not readily satisfied in the tran ient region (see Fig. 1). Therefore, a more rigorous analysis is needed to evaluate the stress-strain field in transient region if the stress-strain field in this region needs to be considered in the modeling formulation(e. g Chiang et al. 5D) Neglecting the initial residual stresses and the poisson ffects the fiber and matrix strains at the far -field end (i.e.z→∞) al to composite strain R Er Em =E
region near the debonding crack tip cannot be evaluated by the Lame´ formula. Since the matrix shear deformation was identified as an important factor on the matrix cracking problem, the shear lag model adopted by BHE [2] is applied in the present paper to perform the stress and strain calculations in the fiber and the matrix. In this paper, the crack–wake interface debonding process is treated as a particular crack propagation problem by which the criterion of interfacial debonding in the crack–wake can be derived and, thereafter, the debonded length can be determined. Then, an energy balance approach is adopted to evaluate the critical stress for propagating a fiber-bridged matrix crack. The conditions to achieve no-debonding and debonding as matrix cracking are discussed in terms of the interfacial properties of debonding toughness and the interfacial shear stress. Three different composite systems, of which experimental data are already available in the literature, are used for case studies. 2. Fiber/matrix stress analysis 2.1. Downstream stresses The composite with fiber volume fraction Vf loaded by a remote uniform stress normal to a semi-infinite crack plane is shown in Fig. 1. The effective axial Young’s modulus of composite E is approximated by the rule-of-mixtures E ¼ VfEf þ VmEm ð1Þ where E and V denote Young’s modulus and volume fraction, and the subscripts f and m indicate the fiber and the matrix, respectively. The downstream region (see Fig. 1) is sufficiently behind the crack-tip so that the stress and strain fields are uniform with respect to the crack plane. Thus, the total axial stresses satisfy VffðzÞ þ VmmðzÞ ¼ ð2Þ where f(z) and m(z) denote the fiber and matrix axial stresses at the z location, as shown in Fig. 2. It is noted that this relationship is not readily satisfied in the transient region (see Fig. 1). Therefore, a more rigorous analysis is needed to evaluate the stress–strain field in transient region if the stress–strain field in this region needs to be considered in the modeling formulation (e.g. Chiang et al. [5]). Neglecting the initial residual stresses and the Poisson effects, the fiber and matrix strains at the far-field end (i.e. z ! 1) is equal to composite strain "f ¼ "m ¼ " ¼ E ð3Þ This approximation is consistent with the rule-ofmixtures of Eq. (1). Then, the fiber and matrix stresses at the far-field end become (for L ! 1) fðLÞ ¼ Ef E ð4Þ mðLÞ ¼ Em E ð5Þ The fiber and matrix axial stresses at the crack plane (i.e., z ¼ 0) are given by fð0Þ ¼ Vf ð6Þ mð0Þ ¼ 0 ð7Þ Fig. 2. A composite-cylinder model. Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756 1745
Y -C. Chiang/Composites Science and Technolog y 61(2001)1743-1750 y means of the composite-cylinder model adopted where Gm is the matrix shear modulus and w is the axial by BhE [2 ], the fiber and matrix axial stresses and the displacement measured from the far-field end (i interfacial shear stress in the downstream region can z= L). From Eq (13), the shear stress tr: is given by then be determined. The free body diagram of the com posite-cylinder model is illustrated in Fig. 2, where the fiber closure traction o/Vf that causes interfacial Tr(r, z) aTiz) (15) debonding between the fiber and matrix over a distance ld and the crack opening displacement v(O). In the debonded length, the fiber/matrix interface is resisted by Substituting eq .(15) into Eq.(14), the interfacial a constant frictional shear stress Ts. The radius of the shear stress, T (), in the bonded length can be expressed matrix cylinder is given by in terms of the relative displacement between the fiber and the matrix. where a is fiber radius. Following BHE [2], the model aln(r/a) can be further simplified by defining an effective radius R(a<r< R) such that the matrix axial load to be where wr=w(a, z)and wm =w(R, 2)are, respectively, concentrated at R and the region between a and r car- fiber and matrix axial displacemen es only the shear stress. The expression of R is given by be expressed in terms of the fiber and matrix axial BHE [2]as stresses: 2InVr+Vm(3-vr) (9) dwr or Consider the equilibrium of the axial force ac ne element of length dz in the debonded fiber leads to dw,m the following differential equation (18) -(2/a)Ts Substituting Eqs. (16-18)into Eq (10), and applying the boundary condition of Eq. (4), and requiring the Solving Eqs.(2)and (10) with the boundary condi- fiber axial stress continuity at z=ld, leads to the fiber tions given by Eqs.(6) and (7), the fiber and matrix and matrix stresses in the bonded length stresses in the debonded length (i.e. 0<z<la) become aP()= oa()= (12)c m(=vRt._E Em +=σ Consider the equilibrium of the radial force acting on rential element dz(dr)(rde)in出4ma=(-2)1==m the following differential equation where aE+=0 (13) Recall that the matrix in the domain asr<r carries p (22) V Vm EmErIn(R)a only the shear stress, the stress-strain relation is, then The fiber and matrix displacements in the debonded (14) region are obtained by integrating Eqs. (17)-(18)from L to z. where the fiber and matrix axial stresses from L to
By means of the composite-cylinder model adopted by BHE [2], the fiber and matrix axial stresses and the interfacial shear stress in the downstream region can then be determined. The free body diagram of the composite-cylinder model is illustrated in Fig. 2, where the fiber closure traction /Vf that causes interfacial debonding between the fiber and matrix over a distance ld and the crack opening displacement v(0). In the debonded length, the fiber/matrix interface is resisted by a constant frictional shear stress s. The radius of the matrix cylinder is given by R ¼ a ffiffiffiffiffi Vf p ð8Þ where a is fiber radius. Following BHE [2], the model can be further simplified by defining an effective radius R (a < R < R) such that the matrix axial load to be concentrated at R and the region between a and R carries only the shear stress. The expression of R is given by BHE [2] as ln R a ¼ 2lnVf þ Vmð3 VfÞ 4V2 m ð9Þ Consider the equilibrium of the axial force acting on the element of length dz in the debonded fiber, leads to the following differential equation df dz ¼ ð2=aÞs ð10Þ Solving Eqs. (2) and (10) with the boundary conditions given by Eqs. (6) and (7), the fiber and matrix stresses in the debonded length (i.e. 04z < ld) become D f ðzÞ ¼ Vf 2s a z ð11Þ D mðzÞ ¼ Vf Vm 2s a z ð12Þ Consider the equilibrium of the radial force acting on the differential element dz(dr)(rd) in the domain a < r < R of the bonded matrix region (i.e. z5ld), leads to the following differential equation @rz @r þ rz r ¼ 0 ð13Þ Recall that the matrix in the domain a4r < R carries only the shear stress, the stress-strain relation is, then, given by rz ¼ Gm @w @r ð14Þ where Gm is the matrix shear modulus and w is the axial displacement measured from the far-field end (i.e. z ¼ L). From Eq. (13), the shear stress rz is given by rzðr; zÞ ¼ aiðzÞ r ð15Þ Substituting Eq. (15) into Eq. (14), the interfacial shear stress, i(z), in the bonded length can be expressed in terms of the relative displacement between the fiber and the matrix: iðzÞ ¼ Gmðwm wf Þ alnðR=aÞ ð16Þ where wf=w(a,z) and wm ¼ w R; z are, respectively, the fiber and matrix axial displacements, which can be expressed in terms of the fiber and matrix axial stresses: dwf dz ¼ f Ef ð17Þ dwm dz ¼ m Em ð18Þ Substituting Eqs. (16–18) into Eq. (10), and applying the boundary condition of Eq. (4), and requiring the fiber axial stress continuity at z=ld, leads to the fiber and matrix stresses in the bonded length D f ðzÞ ¼ VmEm VfE 2sld a e ðzldÞ=a þ Ef E ð19Þ D mðzÞ ¼ 2Vf sld Vma Em E e ðzldÞ=a þ Em E ð20Þ D i ðzÞ ¼ 2 VmEm VfE 2sld a e ðzldÞ=a ð21Þ where ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2GmE VmEmEf ln R=a s ð22Þ The fiber and matrix displacements in the debonded region are obtained by integrating Eqs. (17)–(18) from L to z, where the fiber and matrix axial stresses from L to 1746 Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756
Y -C. Chiang/Composites Science and Technology 61(2001)1743-1756 747 Id is given by Eqs. (19)and(20)and the fiber and matrix These stresses are the same as those of the bonded axial stresses from Id to z is given by Eqs. (11)and (12) at the far-field end in the downstream region by Eqs. (4)and (5) (z) 2Is 3. Interfacial debonding criterion Er There are two different approaches to the problem of Ve 2r、l E fiber/matrix interface debonding, namely, the shear Je-pe-la)/a stress approach and the fracture mechanics approach The shear stress approach is based upon a maximum (G-2) (ld-2)+=l shear stress criterion in which interfacial debonding occurs as the shear stress in the fiber /matrix interface reaches the shear strength of interface [12, 13]. On the PVrErE E other hand, the fracture mechanics approach treats (23) interfacial debonding as a particular crack propagation problem in which interfacial debonding occurs as the strain energy release rate of the fiber/matrix interface achieves the debonding toughness [9, 14. Following the arguments of Gao et al. [9] and Stang and Shah[ 14] that 2V the fracture mechanics approach is preferred to the Em shear stress approach for the interfacial debonding pro blem, the fracture mechanics approach is also adopted 2VrTsla_Em ale-pe-lal/a + Em in the present analy A general case of a cracked body is schematically shown in Fig. 3, in which a crack body is loaded by mh(6--a1+ tractions T and Ts, on the surfaces Sr and Sp with cor- responding displacements dw and dv, respectively. A 7(L-l) the crack grows dA along the fractional surface SF,an energy balance relation can be expressed as [9] (24) Tads=2ydA+T,dvds +dU The relative displacement v(z)between the fiber and matrix is, then, given by where y is the free surface energy, JTsdvds represents the work of friction and U is the stored strain energy of ()=|wr(2)-wmc the body. For an elastic body, U is equal to .EEa-2)+ dU=( 1/2) Tduds-(1/2)Dvds (29) 2Eτs pVmemEr PUre 2. 2. Upstream stresses The upstream region(see Fig. 1)is so far away from the matrix crack tip such that the stress and strain field are also uniform. Thus. the fiber and matrix stresses are ts d Ts dy dA o()=E (26) om()=E (27) Fig 3. A general case of a crack body
ld is given by Eqs. (19) and (20) and the fiber and matrix axial stresses from ld to z is given by Eqs. (11) and (12): wf ðzÞ ¼ ðz L f Ef dz ¼ 1 Ef (ðld z Vf 2sz a dz þ ðL ld " VmEm VfE 2sld a e ðzldÞ=a þ Ef E # dz ) ¼ s aEf l 2 d z2 VfEf ð Þ ld z þ 2s Ef ld aVmEm VfEfE E ðL ldÞ ð23Þ wmðzÞ ¼ ðz L m Em dz ¼ 1 Em (ðld z 2Vf s Vma zdz þ ðL ld 2Vf sld Vma Em E e ðzldÞ=a þ Em E dz ) ¼ Vf s aVmEm l 2 d z2 2Vf s VmEm ld þ a E EðL ldÞ ð24Þ The relative displacement v(z) between the fiber and matrix is, then, given by vðzÞ ¼ wf ðzÞ wmðzÞ ¼ Es VmEfEma l 2 d z2 þ VfEf ð Þ ld z 2Es VmEmEf ld þ a VfEf ð25Þ 2.2. Upstream stresses The upstream region (see Fig. 1) is so far away from the matrix crack tip such that the stress and strain fields are also uniform. Thus, the fiber and matrix stresses are given by U f ð Þ¼ z Ef E ð26Þ U mðzÞ ¼ Em E ð27Þ These stresses are the same as those of the bonded region at the far-field end in the downstream region, given by Eqs. (4) and (5). 3. Interfacial debonding criterion There are two different approaches to the problem of fiber/matrix interface debonding, namely, the shear stress approach and the fracture mechanics approach. The shear stress approach is based upon a maximum shear stress criterion in which interfacial debonding occurs as the shear stress in the fiber/matrix interface reaches the shear strength of interface [12,13]. On the other hand, the fracture mechanics approach treats interfacial debonding as a particular crack propagation problem in which interfacial debonding occurs as the strain energy release rate of the fiber/matrix interface achieves the debonding toughness [9,14]. Following the arguments of Gao et al. [9] and Stang and Shah [14] that the fracture mechanics approach is preferred to the shear stress approach for the interfacial debonding problem, the fracture mechanics approach is also adopted in the present analysis. A general case of a cracked body is schematically shown in Fig. 3, in which a crack body is loaded by tractions T and s, on the surfaces ST and SF with corresponding displacements dw and dv, respectively. As the crack grows dA along the fractional surface SF, an energy balance relation can be expressed as [9] ð ST Tdwds ¼ 2 dA þ ð SF sdvds þ dU ð28Þ where g is the free surface energy, Ð sdvds represents the work of friction and U is the stored strain energy of the body. For an elastic body, U is equal to dU ¼ ð1=2Þ ð ST Tdwds ð1=2Þ ð SF sdvds ð29Þ Fig. 3. A general case of a crack body. Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756 1747