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
Y -C. Chiang/Composites Science and Technolog y 61(2001)1743-1750 Substituting Eq(29)into Eq(28), the fracture cri- whe terion is obtained Al 2 Did 2aA/.T,dvds Vrer 2ts/d If the traction T consists of n concentrated forces P, Er (37) Pn and the corresponding displacements△1,…,△n, Eq (30)becomes aVeR 4VrEte 3a2VmEmEr d pavmEmEr dA 20 I, dvds (31) 4VrEt2 4rs P-vmEm Er (38) For the interfacial debonding problem(see Fig. 2), the debonding process can be regarded as one crack propa- Note that the debonded length ld is the function of t gating along the fiber-matrix interface. Thus, we have applied stress o so that Eq (35)is a nonlinear equation 2y=the debonding toughness 5d, A=2ald, ds= 2adz that can be solved by using the numerical root-finding and Pi=P=rao/vr, which is the fiber load at the method crack plane. In Eq (31), A; equals -wr(0) given by eq (23)and v(z)is given by Eq. (25), leading to the follow ing deboning criterion 5. Discussions P awr(o) I[ dv(z) Two classes of matrix cracking problem correspond ing to the"frictionless"and""frictional"'interface in the debonded length are discussed in the following Taking the derivatives of wd()and v(z) with respect to Id, the debonding criterion of Eq(32)can be expressed 5.1. Frictionless interface(i.e ts=0) If the net pressure on the fiber/matrix interface Vm Emer+ negative and the interface roughness is negligible, the *AVErE prier o-su=0 (33) debonded length will not be resisted by friction and memEr vr. avmEm ats consequently, the interfacial shear stress Ts becomes zero. Depending on the debonding toughness of the interface, two distinguish interfacial debonding modes will occur as matrix cracking. If the interfacial debond- 4. Matrix cracking stress ing toughness is sufficiently high, the first matrix crack may propagate with no relative displacement between The energy relationship to evaluate the steady-state the fiber and the matrix and we classify the case as the matrix cracking stress is expressed as[2] perfectly bonded interface. Conversely, if the interfacial debonding toughness is not sufficiently high, the fiber/ (q2-q)+(-a) matrix interface may be extensively debonded as matrix cracking and we classify the case as the extensively debonded interface 2trdrdz 2TR-G 5.1.I. Perfectly bonded interface (i.e. ld=0) Vm Sm+(vrla The critical debonding stress od is obtained by sub g Is q.(33) where Sm is the matrix fracture toughness. Substituting 4VEE 2 the downstream stresses given by eqs. (I1)and (12)and (39) Eqs. (19)and(20)and the upstream stresses of Eqs. (26) and(27)into Eq. (34), leads to the form of By substituting ts =0 and ld=0 into Eq.(35) the critical matrix cracking stress ome is given by A1a2+A2a+A3=0 (35) pUrEE
Substituting Eq. (29) into Eq. (28), the fracture criterion is obtained 2 ¼ @ 2@A ð ST Tdwds @ 2@A ð SF sdvds ð30Þ If the traction T consists of n concentrated forces P1, ..., Pn and the corresponding displacements 1, ..., n, Eq. (30) becomes 2 ¼ 1 2 X ST Pi @i @A @ 2@A ð SF sdvds ð31Þ For the interfacial debonding problem (see Fig. 2), the debonding process can be regarded as one crack propagating along the fiber-matrix interface. Thus, we have 2 =the debonding toughness d, A ¼ 2 ald, ds ¼ 2 adz and Pi ¼ P ¼ a2 =Vf , which is the fiber load at the crack plane. In Eq. (31), i equals wf ð0Þ given by Eq. (23) and v(z) is given by Eq. (25), leading to the following debonging criterion d ¼ P 4 a @wf ð0Þ @ld 1 2 ðld 0 s @vðzÞ @ld dz ð32Þ Taking the derivatives of wf(0) and v(z) with respect to ld, the debonding criterion of Eq. (32) can be expressed as E2 s aVmEmEf l 2 d þ E2 s VmEmEf s VfEf ld þ aVmEm 4V2 fEfE2 as 2 VfEf d ¼ 0 ð33Þ 4. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [2] 1 2 ð1 1 Vf Ef U f D f 2 þ Vm Em U m D m 2 dz þ 1 2 R2Gm ð1 1 ðR a aD i r 2 2 rdrdz ¼ Vm m þ 4Vf ld a d ð34Þ where m is the matrix fracture toughness. Substituting the downstream stresses given by Eqs. (11) and (12) and Eqs. (19) and (20) and the upstream stresses of Eqs. (26) and (27) into Eq. (34), leads to the form of A12 þ A2 þ A3 ¼ 0 ð35Þ where A1 ¼ VmEm VfEfE ld þ a ð36Þ A2 ¼ 2s Ef ld a þ 2 ld ð37Þ A3 ¼ 4VfE2 s 3a2VmEmEf l 3 d þ 4VfE2 s aVmEmEf l 2 d þ 4VfE2 s 2VmEmEf 4Vf d a ld Vm m ð38Þ Note that the debonded length ld is the function of the applied stress so that Eq. (35) is a nonlinear equation that can be solved by using the numerical root-finding method. 5. Discussions Two classes of matrix cracking problem corresponding to the ‘‘frictionless’’ and ‘‘frictional’’ interface in the debonded length are discussed in the following. 5.1. Frictionless interface (i.e. ts=0) If the net pressure on the fiber/matrix interface is negative and the interface roughness is negligible, the debonded length will not be resisted by friction and, consequently, the interfacial shear stress s becomes zero. Depending on the debonding toughness of the interface, two distinguish interfacial debonding modes will occur as matrix cracking. If the interfacial debonding toughness is sufficiently high, the first matrix crack may propagate with no relative displacement between the fiber and the matrix and we classify the case as the perfectly bonded interface. Conversely, if the interfacial debonding toughness is not sufficiently high, the fiber/ matrix interface may be extensively debonded as matrix cracking and we classify the case as the extensively debonded interface. 5.1.1. Perfectly bonded interface (i.e. ld=0) The critical debonding stress sd is obtained by substituting s=0 and ld=0 into Eq. (33): d ¼ 4V2 f EfE aVmEm d 1=2 ð39Þ By substituting s ¼ 0 and ld ¼ 0 into Eq. (35) the critical matrix cracking stress mc is given by mc ¼ VfEfE aEm m 1=2 ð40Þ 1748 Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756
Y-C. Chiang/ Composites Science and Technology 61(2001)1743-1756 1749 The expression of Eq.(40)is the same as the results of the frictionless interface of which the critical debor Aveston and Kelly [7] for a perfectly bonded interface. stress and the critical matrix cracking stress are unre For ensuring perfect bonding as matrix cracking, the lated to the interfacial shear stress, both the interfacial critical matrix cracking stress me should be less than debonding toughness and the interfacial shear stress the critical debonding stress ad, leading to the following affect the crack-wake debonding as matrix cracking. If equirement he combination of the interfacial debonding toughne and the interfacial shear stress is sufficiently high, the m (41) first matrix crack will propagate in a manner indis tinguishable with the perfectly bonded interface descri bed in Section 5.L.1. on the other hand. as the 5.1.2. Extensively debonded interface(i.e Id ->oo) combination of the interfacial debonding toughness and By substituting ts =0 into Eq.(33)the critical the interfacial shear stress is not sufficiently high, the debonding ad for extensive debonding is the same debonded length will be resisted by frictional stress and as Eq (39), which is independent of debonded length ld. we categorize the case as the restrictively debonded Accordi the critical matrix cracking stress ome for interface extensive debonding is given by substituting ts=0 into Eq.(35) 5.2.1. Perfectly bonded interface (i.e. ld=0) The critical debonding stress od of a perfectly bonded (4Vrsa/ald + VmSm (mEm/VrErEd+(avmEm/pVrErE (42) Interface can be obtained by substituting la=0 into Eq (33) For raising debonding as matrix cracking, the critical debonding stress od given by Eq.(39)should be less vE(1+、 4p-VmEm Er(5d than the critical matrix cracking stress omc given (42), leads to pvm (43) By substituting ld=0 into Eq (35), the critical matrix cracking stress ome of perfectly bonded interface is the same as Eq. (40) for frictionless interface, which is Note that Eq.(43)is independent of the debonded independent of interfacial shear stress ts length ld and consistent with Eq. (41). As the debonded For ensuring perfect bonding as the matrix cracking, length ld ->0, the critical matrix cracking stress omc the critical debonding stress od should be less than the given by Eq.(42)approaches to that of a perfectly critical matrix cracking stress omc, leading to the rela bonded interface given by Eq (40). On the other hand, tionship of as the debonded length ld ->oo, the critical matrix cracking stress ome approaches to the critical debonding stress od given by Eq(39). Under the requirement of sd >/5m Is aE MEnE/幼 (45) Eq(43), the interface tends to be extensively debonded because the critical matrix cracking stress omc can reach the lowest value of Eq(42)as ld ->oo. In the analysis For the case of purely frictional interface (i.e. sa=0), of BHE [2] for weakly bonded fiber with frictionless Eq. (45)reduces to the expression interface (i.e. ts =0). the debonded length caused by the crack-tip transverse tensile stress was assumed to remain ne same as the fiber-bridged crack continuously pro- Is> pEmEr pagating. From the present analysis, the frictionless interface will be extensively debonded in the crack-wake region right after the crack-tip debonding which is equivalent to the Bhe [2] result 5.2. Frictional interface (i.e. ts >0) 5.2.2. Restrictively debonded interface (i.e. ld>0) For the case of the restrictively debonded interface. The interfacial shear stress Ts in the debonded length the expression of the debonded length ld can b may be caused by initially compressive pressure that obtained by rearranging Eq (33) occurs during fabrication of the composite(e.g. different thermal expansion of fiber and matrix). Besides, the a(mEmo I interfacial shear stress ts may be induced by interface la=vEts P) a aVmEm Er (47) roughness during relative fiber/matrix slippage. Unlike
The expression of Eq. (40) is the same as the results of Aveston and Kelly [7] for a perfectly bonded interface. For ensuring perfect bonding as matrix cracking, the critical matrix cracking stress mc should be less than the critical debonding stress d, leading to the following requirement d > Vm 4Vf m ð41Þ 5.1.2. Extensively debonded interface (i.e. ld ! 1) By substituting s ¼ 0 into Eq. (33) the critical debonding stress d for extensive debonding is the same as Eq. (39), which is independent of debonded length ld. Accordingly, the critical matrix cracking stress mc for extensive debonding is given by substituting s ¼ 0 into Eq. (35): mc ¼ ð Þ 4Vf d=a ld þ Vm m ð Þ VmEm=VfEfE ld þ ð Þ aVmEm= VfEfE 1=2 ð42Þ For raising debonding as matrix cracking, the critical debonding stress d given by Eq. (39) should be less than the critical matrix cracking stress mc given by Eq. (42), leads to d 0) The interfacial shear stress s in the debonded length may be caused by initially compressive pressure that occurs during fabrication of the composite (e.g. different thermal expansion of fiber and matrix). Besides, the interfacial shear stress s may be induced by interface roughness during relative fiber/matrix slippage. Unlike the frictionless interface of which the critical debonding stress and the critical matrix cracking stress are unrelated to the interfacial shear stress, both the interfacial debonding toughness and the interfacial shear stress affect the crack–wake debonding as matrix cracking. If the combination of the interfacial debonding toughness and the interfacial shear stress is sufficiently high, the first matrix crack will propagate in a manner indistinguishable with the perfectly bonded interface described in Section 5.1.1. On the other hand, as the combination of the interfacial debonding toughness and the interfacial shear stress is not sufficiently high, the debonded length will be resisted by frictional stress and we categorize the case as the restrictively debonded interface. 5.2.1. Perfectly bonded interface (i.e. ld=0) The critical debonding stress d of a perfectly bonded interface can be obtained by substituting ld ¼ 0 into Eq. (33) d ¼ VfEs VmEm 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4 2VmEmEf aE d 2 s ! s ð44Þ By substituting ld ¼ 0 into Eq. (35), the critical matrix cracking stress mc of perfectly bonded interface is the same as Eq. (40) for frictionless interface, which is independent of interfacial shear stress s. For ensuring perfect bonding as the matrix cracking, the critical debonding stress d should be less than the critical matrix cracking stress mc, leading to the relationship of d > Vm 4Vf m s 2 aE VfEmEf 1=2 1=2 m ð45Þ For the case of purely frictional interface (i.e. d ¼ 0), Eq. (45) reduces to the expression s5 Vm 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi EmEf aVfE r 1=2 m ð46Þ which is equivalent to the BHE [2] result. 5.2.2. Restrictively debonded interface (i.e. ld>0) For the case of the restrictively debonded interface, the expression of the debonded length ld can be obtained by rearranging Eq. (33) ld ¼ a 2 VmEm VfEs 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 2 þ aVmEmEf E2 s d s ð47Þ Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756 1749
Y -C. Chiang/ Composites Science and Technology 61(2001)1743-1756 Table I for the case studies and their material properties are lis- operties of composite systems ted in Table 1 SiCa/borosilicate SiCb/LAS C borosilicate From Eqs.(41)and(43), the threshold crack-wake debonding toughness between perfectly bonded and 400 GPa 200 GPa 380 GPa extensively debonded for frictionless interface is given 63 GPa 63 GPa 47 1-2 MPa (49) △T 1.5×10-°/C -675°C-1200°C-500°C From Eq. (49), the threshold debonding toughness, Sa/sm, is related to the volume fraction and the elastic a SCs-6 Avco moduli of materials. Because the SiC/borosilicate and b Nicalon Data from Barsoum et al. [16]- C/borosilicate composites have similar elastic moduli Data from Barsoum et al. [16] and Phillips [19] (see Table 1), the Sic/borosilicate and SiC/LAS com posites are chosen for showing the values of the thresh old debonding toughness given by Eq. (49). The The requirement of the debonded length ld >0 for the threshold debonding toughness, sa/sm, as a function of restrictively debonded interface yields the following the fiber volume fraction is plotted in Fig 4, where lationship threshold debonding toughness decreases as the fiber ts aE volume fraction increases. As the fiber volume fraction 2 eveNer (48) attains 0.6, the threshold debonding toughness reduces to a value between 0.6 and 0.7. This value is quite large which is consistent with Eq(45) compared to threshold crack-tip debonding toughness. Substituting Eq (47)into Eq (35), the critical matrix He and Hutchinson [15] indicated that the relative ng stress amc can n be solved numerically with the debonding toughness should be less tha requirement of Eq (48) fourth, otherwise the tip of matrix crack would propa gate into the fiber rather than deflect along the fiber/ matrix interface. BHE [2] also reported that a fairly 6. Theoretical results and experimental comparisons small relative debonding toughness(Sd/Sm 1/5)could reduce the crack-tip debonding as matrix cracking. The ceramic composite systems of SiC(SCS-6)/bor- Hence, if the weakly bonded Sic/borosilicate and Sic silicate, SiC(Nicalon)/LAS and C/borosilicate are used LAS composites are frictionless and can be debonded in No debonding debonding SIC/LAS 0.5 iC/borosilicate 0 Fiber volume fraction Fig. 4. Threshold debonding toughness for frictionless interface
The requirement of the debonded length ld>0 for the restrictively debonded interface yields the following relationship: d < Vm 4Vf m s 2 aE VfEmEf 1=2 1=2 m ð48Þ which is consistent with Eq. (45). Substituting Eq. (47) into Eq. (35), the critical matrix cracking stress mc can be solved numerically with the requirement of Eq. (48). 6. Theoretical results and experimental comparisons The ceramic composite systems of SiC(SCS-6)/borosilicate, SiC(Nicalon)/LAS and C/borosilicate are used for the case studies and their material properties are listed in Table 1. From Eqs. (41) and (43), the threshold crack–wake debonding toughness between perfectly bonded and extensively debonded for frictionless interface is given by d m ¼ Vm 4Vf ð49Þ From Eq. (49), the threshold debonding toughness, d/ m, is related to the volume fraction and the elastic moduli of materials. Because the SiC/borosilicate and C/borosilicate composites have similar elastic moduli (see Table 1), the SiC/borosilicate and SiC/LAS composites are chosen for showing the values of the threshold debonding toughness given by Eq. (49). The threshold debonding toughness, d/ m, as a function of the fiber volume fraction is plotted in Fig. 4, where the threshold debonding toughness decreases as the fiber volume fraction increases. As the fiber volume fraction attains 0.6, the threshold debonding toughness reduces to a value between 0.6 and 0.7. This value is quite large compared to threshold crack-tip debonding toughness. He and Hutchinson [15] indicated that the relative debonding toughness should be less than about onefourth, otherwise the tip of matrix crack would propagate into the fiber rather than deflect along the fiber/ matrix interface. BHE [2] also reported that a fairly small relative debonding toughness ( d= m 1=5) could reduce the crack-tip debonding as matrix cracking. Hence, if the weakly bonded SiC/borosilicate and SiC/ LAS composites are frictionless and can be debonded in Table 1 Properties of composite systems SiCa /borosilicate SiCb/LASd C/ borosilicatec Ef 400 GPa 200 GPa 380 GPa Em 63 GPa 85 GPa 63 GPa m 0.25 0.25 0.25 a 70 mm 8 mm 4 mm m 8.92 J/m2 47 J/m2 8.92 J/m2 s 6–8 MPa 1–2 MPa 10–25 MPa f 3.6 106 / C 3.1 106 / C 0.1 106 / C m 3.2 106 / C 1 .5 106 / C 3.2 106 / C T 500 C 675 C–1200 C 500 C a SCS-6 AVCO. b Nicalon. c Data from Barsoum et al. [16]. d Data from Barsoum et al. [16] and Phillips [19]. Fig. 4. Threshold debonding toughness for frictionless interface. 1750 Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756
Y -C. Chiang/ Composites Science and Technology 61(2001)1743-1756 front of the matrix crack-tip, it will be extensively By setting the debonding toughness sd =0, debonded in the crack-wake region continuously sent analysis reduces to the case of purely frictional From Eq.(45)and(48), the threshold condition for interface. The comparison between the ACK [1] and the crack-wake debonding for the frictional interface is present analyses is shown in Fig. 6, where the critical given by matrix cracking stresses are plotted as a function of the interfacial shear stress for the SiC/borosilicate compo- sas plm Vr 2 pVeM Er are almost identical as the interfacial shear stress ranges from 0 to around 50 MPa. As the interfacial shear stress Compared to Eq (49) for frictionless interface, the is getting higher, the difference between the two analyse threshold debonding condition for the frictional inter- becomes more obvious. The difference represents the face given by Eq.(50)is related to the interfacial shear matrix shear deformation effect on the matrix cracking stress Ts in addition to the interfacial debonding tough- stress. As the interfacial shear stress t, reaches 176 MPa, ness. The threshold debonding toughness, sa/Sm, vS the the critical matrix cracking stress by solving Eq(35) fiber volume fraction at different interfacial shear stres- numerically is 494 MPa, which is equivalent to the value ses for the SiC/borosilicate composite are illustrated in given by eq (40)for perfectly bonded interface. At the Fig 5, where the presence of the interfacial shear stress interfacial shear stress Ts beyond 176 MPa, the calcu will lower the threshold debonding toughness. For the lated debonded length becomes negative and the calcu small interfacial shear stress, the minimum threshold lated critical matrix cracking stress becomes invalid debonding toughness, Sa/Sm, occurs at the largest fiber The influences of interfacial debonding toughness and volume fraction On the other hand, the composite with interfacial shear stress on the critical matrix cracking the high interfacial shear stress(e.g. Is-100 MPa in stresses are illustrated in Fig. 7, where the critical matrix Fig. 5) has lowest threshold debonding toughness, Sa/ cracking stresses are plotted as a function of the inter Sm, at medium fiber volume fraction. With sufficiently facial shear stress for different relative debonding high interfacial shear stress, the crack-wake debonding toughness for the SiC/borosilicate composite. Following can be prohibited at the relatively small debonding the crack-tip debonding analysis of BHE [2], the max toughness(e.g. Sa/5m =0. 175 for ts 100 MPa at Vr= imum relative debonding toughness, sa/5m, is chosen as 0. 2 in Fig. 5), which is smaller than the crack-tip 0.2 in the present analysis. Fig. 7 indicates that the cri debonding toughness(Sa/Sm 1/5) tical matrix cracking stress increases quickly from the For purely frictional interface, a well-known for- frictionless interface (i.e. ts =0)to the slightly frictional mulation of critical matrix cracking stress for extensive interface (i.e. ts =0+) and this phenomenon is more slipping composite has been derived by ACK[1] profound for higher debonding toughness. As the inter oVer et facial shear stress ranges from 0 to about 140 Mpa, the E2 (51) higher interfacial debonding toughness results in the higher critical matrix cracking stress. When the inter SiC/borosilicate ts=OMPa 0 0 Fiber Volume fraction Fig. 5. Threshold debonding toughness for frictional interface
front of the matrix crack-tip, it will be extensively debonded in the crack–wake region continuously. From Eq. (45) and (48), the threshold condition for crack–wake debonding for the frictional interface is given by d ¼ Vm 4Vf m s 2 aE VfEmEf 1=2 1=2 m ð50Þ Compared to Eq. (49) for frictionless interface, the threshold debonding condition for the frictional interface given by Eq. (50) is related to the interfacial shear stress s in addition to the interfacial debonding toughness. The threshold debonding toughness, d/ m, vs. the fiber volume fraction at different interfacial shear stresses for the SiC/borosilicate composite are illustrated in Fig. 5, where the presence of the interfacial shear stress will lower the threshold debonding toughness. For the small interfacial shear stress, the minimum threshold debonding toughness, d/ m, occurs at the largest fiber volume fraction. On the other hand, the composite with the high interfacial shear stress (e.g. s=100 MPa in Fig. 5) has lowest threshold debonding toughness, d/ m, at medium fiber volume fraction. With sufficiently high interfacial shear stress, the crack–wake debonding can be prohibited at the relatively small debonding toughness (e.g. d= m ¼ 0:175 for s ¼ 100 MPa at Vf ¼ 0:2 in Fig. 5), which is smaller than the crack-tip debonding toughness ( d= m 1=5). For purely frictional interface, a well-known formulation of critical matrix cracking stress for extensive slipping composite has been derived by ACK [1]: mc ¼ 6VfEfEs aVmE2 m m 1=3 ð51Þ By setting the debonding toughness d ¼ 0, the present analysis reduces to the case of purely frictional interface. The comparison between the ACK [1] and the present analyses is shown in Fig. 6, where the critical matrix cracking stresses are plotted as a function of the interfacial shear stress for the SiC/borosilicate composite. The results by the ACK [1] and the present analyses are almost identical as the interfacial shear stress ranges from 0 to around 50 MPa. As the interfacial shear stress is getting higher, the difference between the two analyses becomes more obvious. The difference represents the matrix shear deformation effect on the matrix cracking stress. As the interfacial shear stress s reaches 176 MPa, the critical matrix cracking stress by solving Eq. (35) numerically is 494 MPa, which is equivalent to the value given by Eq. (40) for perfectly bonded interface. At the interfacial shear stress s beyond 176 MPa, the calculated debonded length becomes negative and the calculated critical matrix cracking stress becomes invalid. The influences of interfacial debonding toughness and interfacial shear stress on the critical matrix cracking stresses are illustrated in Fig. 7, where the critical matrix cracking stresses are plotted as a function of the interfacial shear stress for different relative debonding toughness for the SiC/borosilicate composite. Following the crack-tip debonding analysis of BHE [2], the maximum relative debonding toughness, d/ m, is chosen as 0.2 in the present analysis. Fig. 7 indicates that the critical matrix cracking stress increases quickly from the frictionless interface (i.e. ts =0) to the slightly frictional interface (i.e. s ¼ 0þ) and this phenomenon is more profound for higher debonding toughness. As the interfacial shear stress ranges from 0 to about 140 Mpa, the higher interfacial debonding toughness results in the higher critical matrix cracking stress. When the interFig. 5. Threshold debonding toughness for frictional interface. Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756 1751
1752 Y -C. Chiang/ Composites Science and Technology 61(2001)1743-1756 facial shear stress is larger than 140 Mpa, the critical These results are consistent with the discussion in matrix cracking stresses for different debonding tough- Section 5.2.1 ness approach the value of a perfectly bonded interface Fig. 9 plots the trend of critical matrix cracking stress iven by Eq (40) and the debonded length as a function of relative The distributions of the debonded lengths corre- debonding toughness. The composite with higher inter ponding to Fig. 7 are plotted in Fig. 8, where the facial debonding toughness has higher critical matrix debonded length decreases as the interfacial debondin cracking stress while the higher interfacial debonding toughness and the interfacial shear stress increase. A: toughness also results in a shorter debonded length. the he interfacial shear stress approaches 0, the debonded interfacial debonding process is the prerequisite for the length approaches infinity, and, consequently, the fiber pull-out, which is associated with the work of critical matrix cracking stress given by Eq(35)approa- fracture. It implies that the composite with higher ches the critical debonding stress od given by Eq. (39). interfacial debonding toughness will lead to a smaller SiC/borosilicate 800 700 ACK 苏 100 0 180 Interfacial shear stress(MPa) Fig. 6. Comparison of ACK [I] and present results for purely frictional interface for SiC/borosilicate at Vr=0.5 SiC/borosilicate 600 400 苏 e300 1001201 160180200 Interfacial Shear Stress(MPa) Fig. 7. Umc VS. T, at different sa/tm for SiC/borosilicate at Vr=0
facial shear stress is larger than 140 Mpa, the critical matrix cracking stresses for different debonding toughness approach the value of a perfectly bonded interface given by Eq. (40). The distributions of the debonded lengths corresponding to Fig. 7 are plotted in Fig. 8, where the debonded length decreases as the interfacial debonding toughness and the interfacial shear stress increase. As the interfacial shear stress approaches 0, the debonded length approaches infinity, and, consequently, the critical matrix cracking stress given by Eq. (35) approaches the critical debonding stress d given by Eq. (39). These results are consistent with the discussion in Section 5.2.1. Fig. 9 plots the trend of critical matrix cracking stress and the debonded length as a function of relative debonding toughness. The composite with higher interfacial debonding toughness has higher critical matrix cracking stress while the higher interfacial debonding toughness also results in a shorter debonded length. The interfacial debonding process is the prerequisite for the fiber pull-out, which is associated with the work of fracture. It implies that the composite with higher interfacial debonding toughness will lead to a smaller Fig. 7. mc vs. s at different d/ m for SiC/borosilicate at Vf ¼ 0:5. Fig. 6. Comparison of ACK [1] and present results for purely frictional interface for SiC/borosilicate at Vf ¼ 0:5. 1752 Y.-C. Chiang / Composites Science andTechnology 61 (2001) 1743–1756