Engineering Fracture Mechanics PERGAMON Engineering Fracture Mechanics 65(2000)15-28 On crack-wake debonding in fiber reinforced ceramics Yih-Cherng Chiang ring, Chinese Culture University, No 55. Hua-Kang Road, Taipei, Taiwan Received 6 March 1999: received in revised form 4 September 1999: accepted 8 November 1999 Abstract In this paper, the effect of interfacial debonding in the wake of a crack on the stress for matrix cracking is studied for unidirectional fiber reinforced ceramics. A simple shear-lag model is adopted to calculate the stress and strain fields in the fiber and the matrix. a criterion for crack-wake debonding is proposed by treating the debonding process as a particular crack propagation problem. Then, by using an energy balance approach the formulation of the matrix cracking stress of an infinite fiber-bridged crack is derived. The theoretical results are compared with experimental data of Sic/borosilicate, SiC/LAS and C/borosilicate ceramic composites. C 2000 Elsevier Science Ltd. All rights reserved Keywords: Interfacial debonding: Matrix cracking: Ceramics; Composites 1. Introduction Studies, both from theoretical analyses and experimental observations, have indicated that interfacial properties are the key factors to developing a successful fiber reinforced brittle matrix composite. For analytical modeling, both the energy balance approach suggested by Aveston, Cooper and Kelly(ACK) [1, 2] and the fracture mechanics approach proposed by Marshall et al. [3] and McCartney [4] have been used to investigate the influence of interfacial properties on the matrix cracking stress. In these analyses, the fiber/matrix interface was assumed to be either perfectly bonded or unbonded but susceptible to weak frictional resistance. For the perfectly bonded composite, the fiber and matrix deform elastically above the crack plane and no relative fiber/matrix slippage occurs as the matrix crack propagates. The analytical expression of the matrix cracking stress by Aveston and Kelly [2] relates to the elastic properties and the geometrical constants of the fiber and the matrix; no specific interfacial property appears in the analytical expression of the matrix cracking stress. As the interface is unbonded and Fax:+886-2-2861-5241 E-mailaddress.johnycc(@gmail.gcn.net.tw(.-C.Chiang) 0013-794400/S- see front matter 2000 Elsevier Science Ltd. All rights reserved PI:S0013-7944(99)00130-7
On crack-wake debonding in ®ber reinforced ceramics Yih-Cherng Chiang* Department of Mechanical Engineering, Chinese Culture University, No. 55, Hua-Kang Road, Taipei, Taiwan Received 6 March 1999; received in revised form 4 September 1999; accepted 8 November 1999 Abstract In this paper, the eect of interfacial debonding in the wake of a crack on the stress for matrix cracking is studied for unidirectional ®ber reinforced ceramics. A simple shear-lag model is adopted to calculate the stress and strain ®elds in the ®ber and the matrix. A criterion for crack-wake debonding is proposed by treating the debonding process as a particular crack propagation problem. Then, by using an energy balance approach the formulation of the matrix cracking stress of an in®nite ®ber-bridged crack is derived. The theoretical results are compared with experimental data of SiC/borosilicate, SiC/LAS and C/borosilicate ceramic composites. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Interfacial debonding; Matrix cracking; Ceramics; Composites 1. Introduction Studies, both from theoretical analyses and experimental observations, have indicated that interfacial properties are the key factors to developing a successful ®ber reinforced brittle matrix composite. For analytical modeling, both the energy balance approach suggested by Aveston, Cooper and Kelly (ACK) [1,2] and the fracture mechanics approach proposed by Marshall et al. [3] and McCartney [4] have been used to investigate the in¯uence of interfacial properties on the matrix cracking stress. In these analyses, the ®ber/matrix interface was assumed to be either perfectly bonded or unbonded but susceptible to weak frictional resistance. For the perfectly bonded composite, the ®ber and matrix deform elastically above the crack plane and no relative ®ber/matrix slippage occurs as the matrix crack propagates. The analytical expression of the matrix cracking stress by Aveston and Kelly [2] relates to the elastic properties and the geometrical constants of the ®ber and the matrix; no speci®c interfacial property appears in the analytical expression of the matrix cracking stress. As the interface is unbonded and Engineering Fracture Mechanics 65 (2000) 15±28 0013-7944/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 1 3 0 - 7 www.elsevier.com/locate/engfracmech * Fax: +886-2-2861-5241. E-mail address: johnycc@gmail.gcn.net.tw (Y.-C. Chiang)
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 resisted by weak friction, extensive frictional slippage between the fiber and the matrix occurs as the matrix crack propagates. The analytical results of ACK [1], Marshall et al. [] and McCartney [4] have showed that the matrix cracking stress was closely related to the interfacial friction stress Budiansky, Hutchinson and Evans(BHE) [5] have re-examined the matrix cracking problem in a more rigorous manner. Two distinct situations of fiber-matrix interface considered in their analysis are as follows: (1) unbonded fibers but susceptible to frictional resistance which can be either high or low, and (2)weakly bonded fibers which may be debonded by the high transverse stress in front of the matrix crack-tip. In the case of large frictional resistance, no slippage occurs at the interface and BHE predicted the same result as Aveston and Kelly [2] for the perfectly bonded interface. On the other hand, if the interface is resisted by weak friction, the BhE model reduces to the ACK result. One feature of the BHE model that goes beyond the ACK models is that the bhe model provides the result to bridge between the no-slippage result of Aveston and Kelly [2] and the large-slippage result of ACK. If the fiber is weakly bonded to the matrix, the propagation of the matrix crack may cause interfacial debonding in front of the crack tip and, the debonding process may continue in the crack-wake due to the relative fiber /matrix displacement in the debonded region. Subsequently, the debonded interface may be either separated or resisted by the frictional force depending on transverse stress on the nterface and the characteristics of interface(e.g. the interfacial roughness). The mechanics of the crack tip interfacial debonding and its influence on the matrix cracking stress have been investigated by BHE, which indicated that a fairly small interfacial debonding toughness (about 1/5 of matrix fracture toughness) could inhibit the debonding process by the matrix crack-tip transverse stress. If the weakly bonded interface could be debonded by the transverse tensile stress in the matrix crack tip, the debonded length was assumed unchanged in the BHE model as the crack continued propagating. The possible debonding process in the crack-wake was not considered in their modeling Many researches [6] have indicated that the interfacial debonding in the crack-wake plays an important role in the fracture behavior of fiber reinforced brittle matrix composites. However, the heoretical analysis of the effects of crack-wake interfacial debonding on matrix cracking is still lacking The intent of this paper is to investigate the influences of crack-wake debonding on stress for matrix cracking. In this paper, the debonding of the fiber/matrix interface in the crack-wake is treated particular crack propagation problem by which the interfacial debonding criterion can be derived and subsequently, the debonding length can be determined. Then, an energy balance approach is adopted to calculate the steady-state matrix cracking stress of an infinite fiber-bridged crack. For purely frictional interface, the theoretical results of the present analysis reduce to the ACK results. 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.. Downstream stresses c The composite with fiber volume fraction Ve is loaded by a remote uniform stress a normal to a long rack plane, as shown in Fig. 1. The free body diagram of the unit cell in the downstream Region I is llustrated in Fig. 2, where the fiber closure traction a/Vt that causes debonding between the fiber and the matrix over a distance ld and the crack opening displacement u(O). In the debonded region, the fiber/ matrix interface is resisted by a constant frictional shear stress, Ts. The Youngs modulus of composite Ec can be approximated by the rule of mixtures Ec= vrEf + vmem
resisted by weak friction, extensive frictional slippage between the ®ber and the matrix occurs as the matrix crack propagates. The analytical results of ACK [1], Marshall et al. [3] and McCartney [4] have showed that the matrix cracking stress was closely related to the interfacial friction stress. Budiansky, Hutchinson and Evans (BHE) [5] have re-examined the matrix cracking problem in a more rigorous manner. Two distinct situations of ®ber±matrix interface considered in their analysis are as follows: (1) unbonded ®bers but susceptible to frictional resistance which can be either high or low, and (2) weakly bonded ®bers which may be debonded by the high transverse stress in front of the matrix crack-tip. In the case of large frictional resistance, no slippage occurs at the interface and BHE predicted the same result as Aveston and Kelly [2] for the perfectly bonded interface. On the other hand, if the interface is resisted by weak friction, the BHE model reduces to the ACK result. One feature of the BHE model that goes beyond the ACK models is that the BHE model provides the result to bridge between the no-slippage result of Aveston and Kelly [2] and the large-slippage result of ACK. If the ®ber is weakly bonded to the matrix, the propagation of the matrix crack may cause interfacial debonding in front of the crack tip and, the debonding process may continue in the crack-wake region due to the relative ®ber/matrix displacement in the debonded region. Subsequently, the debonded interface may be either separated or resisted by the frictional force depending on transverse stress on the interface and the characteristics of interface (e.g. the interfacial roughness). The mechanics of the cracktip interfacial debonding and its in¯uence on the matrix cracking stress have been investigated by BHE, which indicated that a fairly small interfacial debonding toughness (about 1/5 of matrix fracture toughness) could inhibit the debonding process by the matrix crack-tip transverse stress. If the weakly bonded interface could be debonded by the transverse tensile stress in the matrix crack tip, the debonded length was assumed unchanged in the BHE model as the crack continued propagating. The possible debonding process in the crack-wake was not considered in their modeling. Many researches [6] have indicated that the interfacial debonding in the crack-wake plays an important role in the fracture behavior of ®ber reinforced brittle matrix composites. However, the theoretical analysis of the eects of crack-wake interfacial debonding on matrix cracking is still lacking. The intent of this paper is to investigate the in¯uences of crack-wake debonding on stress for matrix cracking. In this paper, the debonding of the ®ber/matrix interface in the crack-wake is treated as a particular crack propagation problem by which the interfacial debonding criterion can be derived and, subsequently, the debonding length can be determined. Then, an energy balance approach is adopted to calculate the steady-state matrix cracking stress of an in®nite ®ber-bridged crack. For purely frictional interface, the theoretical results of the present analysis reduce to the ACK results. Three dierent 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 ®ber volume fraction Vf is loaded by a remote uniform stress s normal to a long crack plane, as shown in Fig. 1. The free body diagram of the unit cell in the downstream Region I is illustrated in Fig. 2, where the ®ber closure traction s=Vf that causes debonding between the ®ber and the matrix over a distance ld and the crack opening displacement u 0: In the debonded region, the ®ber/ matrix interface is resisted by a constant frictional shear stress, ts: The Young's modulus of composite Ec can be approximated by the rule of mixtures Ec VfEf VmEm 1 16 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 I I Crack-til Debonded debonding interface Crack plane Crack-wake debonding Fig. 1. Schematic representation of crack-tip and crack-wake debonding. Matrix Fiber Matrix u(0) Crack Plane
Fig. 1. Schematic representation of crack-tip and crack-wake debonding. Fig. 2. A simple shear-lag model. Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28 17
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 indicate, respectively, the fiber and the matrix. The total axial stresses in Region I sats. crits f and m where E and V denote Youngs modulus and volume fraction, respectively. The subs Vor(=)+Mom(2)=0 here a(=)and om(z) denote the fiber and matrix tensile stresses. It is noted that this relationship is not readily satisfied in Region II(see Fig. 1). Therefore, a more rigorous analysis is needed to evaluate the stress/strain fields in Region II if the stress/strain field in this region needs to be considered in the modeling formulation. The boundary between Region I and Region Il has been given by Chiang et al For the bonded region(d<z) in the downstream Region I, the fiber, matrix and composite have the same displacements Ef=Em =fc=r Thus, the fiber and matrix stresses in the bonded region (d<z) become E E E For the debonded region(0<z la) in Region I, the fiber/matrix interface is resisted by a constant frictional stress, ts. The force equilibrium equation of the fiber in this region is given by where a is the fiber radius. The boundary condition at the crack plane z=0 is given by r(O)=2 m(0)=0 Solving Eqs. (2)and (6) with the boundary conditions given by Eqs. (7)and(8), the fiber and matrix stresses in the debonded zone, 0<z< ld. become Let w(a)and wm(z) denote the fiber and matrix displacements measured from the boundary z= oo and set w(oo)=wm(oo)=0. The stress-strain relationships of the fiber and the matrix are given by E
where E and V denote Young's modulus and volume fraction, respectively. The subscripts f and m indicate, respectively, the ®ber and the matrix. The total axial stresses in Region I satisfy Vfsf z Vmsm z s 2 where sf z and sm z denote the ®ber and matrix tensile stresses. It is noted that this relationship is not readily satis®ed in Region II (see Fig. 1). Therefore, a more rigorous analysis is needed to evaluate the stress/strain ®elds in Region II if the stress/strain ®eld in this region needs to be considered in the modeling formulation. The boundary between Region I and Region II has been given by Chiang et al. [7]. For the bonded region ldRz in the downstream Region I, the ®ber, matrix and composite have the same displacements ef em ec s Ec 3 Thus, the ®ber and matrix stresses in the bonded region ldRz become sD f Ef Ec s 4 sD m Em Ec s 5 For the debonded region 0Rz < ld in Region I, the ®ber/matrix interface is resisted by a constant frictional stress, ts: The force equilibrium equation of the ®ber in this region is given by dsf dz ÿ 2=ats 6 where a is the ®ber radius. The boundary condition at the crack plane z 0 is given by sf 0 s Vf 7 sm 0 0 8 Solving Eqs. (2) and (6) with the boundary conditions given by Eqs. (7) and (8), the ®ber and matrix stresses in the debonded zone, 0Rz < ld, become sD f s Vf ÿ 2tsz a 9 sD m Vf Vm 2tsz a 10 Let wf z and wm z denote the ®ber and matrix displacements measured from the boundary z 1 and set wf 1 wm 1 0: The stress±strain relationships of the ®ber and the matrix are given by dwf dz sf Ef 11a 18 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 Substituting Eqs. (4),(5)and(9),(10)into Eqs. (1la) and(Ilb), the fiber and the matrix displacements in the debonded zone, 0<z<ld, are obtained by integrating Eqs. (lla) and (1lb) (a Er Ec VREf (12) wm(2) 令、f(G-2) Then, the relative displacement u(=) between the fiber and the matrix in the debonded zone, 0<z<ld, is obtained by ()=)-m((4-27_E(a-2) (14) ameMe 2. 2. Upstream stresses The upstream region III(see Fig. 1)is so far away from the crack tip that the stress and strain fields re also uniform. Thus, fiber and matrix have the same displacements and the fiber and the matrix stresses are given by Ef These stresses are the same as those of the bonded region in the downstream region I, given by Eqs.(4 3. Interfacial debonding criterion There are two different approaches to the fiber-matrix interfacial debonding problem in the crack wake region, 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 [8, 9]. 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 interface attains the interfacial debonding toughness [6, 10-12]. Following the arguments of Refs. [6, 11, 12] 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 volume V is loaded with tractions T and Ts, on the surfaces Sr and Sp with corresponding displacements dw and du, respectively. As the crack grows by dA along the fractional surface Se, an energy balance relation can
dwm dz sm Em 11b Substituting Eqs. (4), (5) and (9), (10) into Eqs. (11a) and (11b), the ®ber and the matrix displacements in the debonded zone, 0Rz < ld, are obtained by integrating Eqs. (11a) and (11b) wf z z 1 sf Ef dz ld 1 s Ec dz ÿ ld ÿ zs VfEf ÿ l 2 d ÿ z2 ts aEf 12 wm z z 1 sm Em dz ld 1 s Ec dz ÿ Vf ÿ l 2 d ÿ z2 ts aVmEm 13 Then, the relative displacement u z between the ®ber and the matrix in the debonded zone, 0Rz < ld, is obtained by u z jwf z ÿ wm zj ld ÿ zs VfEf ÿ Ec ÿ l 2 d ÿ z2 ts aVmEmEf 14 2.2. Upstream stresses The upstream region III (see Fig. 1) is so far away from the crack tip that the stress and strain ®elds are also uniform. Thus, ®ber and matrix have the same displacements and the ®ber and the matrix stresses are given by sU f Ef Ec s 15 sU m Em Ec s 16 These stresses are the same as those of the bonded region in the downstream region I, given by Eqs. (4) and (5). 3. Interfacial debonding criterion There are two dierent approaches to the ®ber±matrix interfacial debonding problem in the crackwake region, 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 ®ber/matrix interface reaches the shear strength of interface [8,9]. 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 interface attains the interfacial debonding toughness [6,10±12]. Following the arguments of Refs. [6,11,12] 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 volume V is loaded with tractions T and ts, on the surfaces ST and SF with corresponding displacements dw and du, respectively. As the crack grows by dA along the fractional surface SF, an energy balance relation can be Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28 19
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 expressed as [6 Tdv ds=2ydA+ tsdu ds+di where y is the free surface energy, j tsdu ds represents the work of friction and U is the stored strain energy of the body. For an elastic system, U is equal to Td ds- Substituting Eq(18)into Eq(17), the fracture criterion is obtained as Td ds τduds 2dA 28A If the traction T consists of n concentrated forces Pl, .. Pn and the corresponding displacements △1,…,An,Eq1.(19) then becomes 2=2∑ t du ds (20) da 28A For the interfacial debonding problem(see Fig. 2), the debonding process can be regarded as one propagating along the fiber/matrix interface. Thus, we have 2y equal to the critical strain energy re rate of interface d, A=2rald, ds= 2radz and Pi= P=taa/vf, which is the fiber force at the plane. In Eq(20), u(=)is given by Eq (14)and Ai=-wf(0)is given by Eq. (12). Then, the debonding criterion of Eq(20)becomes P awo) 1 4 au(=) Taking the derivatives of wr(O)and u(=) with respect to ld, Eq (21)becomes →ow τ.δu Fig 3 Schematic representation of a general case of a crack body
expressed as [6] ST Tdw ds 2gdA Sf tsdu ds dU 17 where g is the free surface energy, tsdu ds represents the work of friction and U is the stored strain energy of the body. For an elastic system, U is equal to dU 1 2 ST Tdw ds ÿ 1 2 SF tsdu ds 18 Substituting Eq. (18) into Eq. (17), the fracture criterion is obtained as 2g @ 2@A ST Tdw ds ÿ @ 2@A SF tsdu ds 19 If the traction T consists of n concentrated forces P1, ... ,Pn and the corresponding displacements D1, ... ,Dn, Eq. (19) then becomes 2g 1 2 X ST Pi @Di @A ÿ @ 2@A SF tsdu ds 20 For the interfacial debonding problem (see Fig. 2), the debonding process can be regarded as one crack propagating along the ®ber/matrix interface. Thus, we have 2g equal to the critical strain energy release rate of interface zd, A 2pald, ds 2padz and Pi P pa2s=Vf, which is the ®ber force at the crack plane. In Eq. (20), u z is given by Eq. (14) and Di ÿwf 0 is given by Eq. (12). Then, the debonding criterion of Eq. (20) becomes zd ÿ P 4pa @wf 0 @ld ÿ 1 2 ld 0 ts @u z @ld dz 21 Taking the derivatives of wf 0 and u z with respect to ld, Eq. (21) becomes Fig. 3. Schematic representation of a general case of a crack body. 20 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28
Y.-C. Chiang/ Engineering Fracture Mechanics 6.5(2000)15-28 ,、+一E aVm EmEr/avmt VEcts 4VFETEc The debonding length ld is then given by aVmEmo avmEm Ersd Eτ2 For the case of a purely frictional interface (i.e, Sd=0), the debond g length la equals the frictional slipping length and it is expressed as lasavmEmo The expression of Eq. (24)is consistent with the length at which the fiber stress at the crack plane transfers back to the matrix in the ACK model. It also can be seen from Eq .(23) that the inclusion of nterface bonding will decreases the debonding length To initiate the crack-wake debonding process (i.e, la>0), the interface debonding energy Sd should satisfy aLmemar VERE 4. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [5] [=-9(+xR后广()2m veLd where Sm is the critical strain energy release rate of the matrix, Gm is the matrix shear modulus and the radius of the matrix R can be expressed as a/vf. The contribution of the shear energy term in Eq (26)was neglected in the ACK model. It was verified that this negligence is well accepted for the slipping length larger than a few fiber radii [5]. Following the ACK model, the contribution of shear energy is neglected in the present analysis. Substituting the fiber and matrix stresses of Eqs. (9),(10)and (4),(5)and the debonding length of Eq (23)into Eq(26), the energy balance equation leads to the form of A1()2+A2(0)a+A3(0)=0 (28a)
l 2 d ÿ aVmEms VfEcts ld aVmEmEf Ect2 s " aVmEms2 4V 2 f EfEc ! ÿ zd # 0 22 The debonding length ld is then given by ld aVmEms 2VfEcts ÿ aVmEmEfzd Ect2 s s 23 For the case of a purely frictional interface (i.e., zd 0), the debonding length ld equals the frictional slipping length and it is expressed as ld aVmEms 2VfEcts 24 The expression of Eq. (24) is consistent with the length at which the ®ber stress at the crack plane transfers back to the matrix in the ACK model. It also can be seen from Eq. (23) that the inclusion of interface bonding will decreases the debonding length. To initiate the crack-wake debonding process (i.e., ld > 0), the interface debonding energy zd should satisfy zd < aVmEms2 4V 2 f EfEc 25 4. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [5] 1 2 1 ÿ1 Vf Ef ÿ sU f ÿ sD f 2 Vm Em ÿ sU m ÿ sD m 2 dz 1 2pR2Gm ld ÿld R a ats r 2 2pr dr dz Vmzm 4Vfld a zd 26 where zm is the critical strain energy release rate of the matrix, Gm is the matrix shear modulus and the radius of the matrix R can be expressed as a=V 1=2 f : The contribution of the shear energy term in Eq. (26) was neglected in the ACK model. It was veri®ed that this negligence is well accepted for the slipping length larger than a few ®ber radii [5]. Following the ACK model, the contribution of shear energy is neglected in the present analysis. Substituting the ®ber and matrix stresses of Eqs. (9), (10) and (4), (5) and the debonding length of Eq. (23) into Eq. (26), the energy balance equation leads to the form of A1 ss2 A2 ss A3 s 0 27 where A1 s VmEm VfEfEc ld 28a Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28 21
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 A3(a) t()(1E) eve For the purely fractional interface (i.e, Sd=0), Eq (27)is reduced to the ACK result and expressed as 6VftsEYEcsm When interfacial bonding exists between the fiber and the matrix interface (i.e., Sa >0). the matrix cracking stress, cr, can be obtained by using the root finding technique to solve Eq. (27)with the equirement of Eq (25) 5. Results and discussion e The ceramic composite systems of SiC(SCS-6)/borosilicate. SiC(Nicalon )/LAS and C/borosilicate are ed for the theoretical study and their material properties are listed in Table I The influences of interfacial bonding toughness and frictional shear stress on the matrix cracking stresses are illustrated in Fig. 4, in which the matrix cracking stresses of SiC/borosilicate composite are plotted as a function of frictional shear stress, ts, for different relative critical strain energy release rate, Sa/sm. The prediction for a perfectly bonded interface by the bhe model is also plotted in Fig. 4 for the purpose of comparison. He and Hutchinson [13] indicated that the critical strain energy release rate of the interface Sa should be less than one-fourth of the critical strain energy release rate of the matrix 4 otherwise the matrix crack propagates into the fiber rather than deflecting along the fiber/matrix interface. Therefore, the maximum relative critical strain energy release rate, Sa/5m, is chosen as 0.25 in the present analysis If the interface is resisted by frictional shear stress only (i. e, Sa/sm=0), the present analysis is consistent with the ACK result. For the case of maximum facial bonding toughness (i.e Table I Properties of composite systems SiC/borosilicate C/ borosilicate SiC/LAS Er 400 Gpa 380 Gpa 200 Gpa 63 Gpa 8.92J 892Jm2 47J 6-8 MPa 10-25 MPa 1-2 MPa 35-40×10-6°C-(ong)2.6×10-6°-( radia) 0.1×10-6°C-1 3.2×10-6°C-1 3.2×10-6C-1 500° 500°C -675°Cto-1200°C SCS-6AVcO Data from Ref [15] Nicalon Data from Refs. [15 and 18
A2 s ÿ 2ts aEf l 2 d 28b A3 s 4 3 ts a 2 VfEc VmEmEf l 3 d ÿ 4Vfzd a ld ÿ Vmzm 28c For the purely fractional interface (i.e., zd 0), Eq. (27) is reduced to the ACK result and expressed as scr 6V 2 f tsEfEczm aVmE 2 m !1=3 29 When interfacial bonding exists between the ®ber and the matrix interface (i.e., zd > 0), the matrix cracking stress, scr, can be obtained by using the root ®nding technique to solve Eq. (27) with the requirement of Eq. (25). 5. Results and discussion The ceramic composite systems of SiC(SCS-6)/borosilicate, SiC(Nicalon)/LAS and C/borosilicate are used for the theoretical study and their material properties are listed in Table 1. The in¯uences of interfacial bonding toughness and frictional shear stress on the matrix cracking stresses are illustrated in Fig. 4, in which the matrix cracking stresses of SiC/borosilicate composite are plotted as a function of frictional shear stress, ts, for dierent relative critical strain energy release rate, zd=zm: The prediction for a perfectly bonded interface by the BHE model is also plotted in Fig. 4 for the purpose of comparison. He and Hutchinson [13] indicated that the critical strain energy release rate of the interface zd should be less than one-fourth of the critical strain energy release rate of the matrix zm, otherwise the matrix crack propagates into the ®ber rather than de¯ecting along the ®ber/matrix interface. Therefore, the maximum relative critical strain energy release rate, zd=zm, is chosen as 0.25 in the present analysis. If the interface is resisted by frictional shear stress only (i.e., zd=zm 0), the present analysis is consistent with the ACK result. For the case of maximum interfacial bonding toughness (i.e., Table 1 Properties of composite systems SiCa /borosilicateb C/ borosilicateb SiCc /LASd Ef 400 Gpa 380 Gpa 200 Gpa Em 63 Gpa 63 Gpa 85 Gpa a 70 mm 4 mm 8 mm zm 8.92 J/m2 8.92 J/m2 47 J/m2 ts 6±8 MPa 10±25 MPa 1±2 MPa af 3:5 ÿ 4:0 10ÿ6 8Cÿ1 (long.) 2:6 10ÿ6 8Cÿ1 (radial) 0:1 10ÿ6 8Cÿ1 3:1 10ÿ6 8Cÿ1 am 3:2 10ÿ6 8Cÿ1 3:2 10ÿ6 8Cÿ1 1:5 10ÿ6 8Cÿ1 DT ÿ5008C ÿ5008C ÿ6758C to ÿ12008C a SCS-6 AVCO. b Data from Ref. [15]. c Nicalon. d Data from Refs. [15 and 18]. 22 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28
Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 Sa/sm=0. 25), the crack-wake debonding can occur from Eq.(25) and the predicted matrix cracking stress is smaller than that for a perfectly bonded interface given by the BhE model, as shown in Fig 4 The BHE result for a perfectly bonded interface has been commonly cited to be the upper-bound estimation of matrix cracking stress. However, from the present analysis, it appears that this upper bound prediction may overestimate the stress for matrix cracking ig. 4 also indicates that the higher frictional shear stress has less influence on the matrix cracking stress. Singh [14 has studied the influence of frictional shear stress on the stress for matrix cracking by using fiber coating to obtain different frictional shear stress. Singh reported that the matrix crack stresses of SiC(SCS-6)/zircon composites are independent of the different frictional shear stresses and he concluded that the ACK formulation cannot be applied to this composite system. However, if there some degree of interfacial bonding between the fiber and the matrix in this composite system, the present result shows that the matrix cracking stresses will become insensitive to the frictional shear stress, as shown in Fig. 4. The SiC/zircon composite system, which has the same fiber and matrix thermal expansion coefficients (i.e, no interfacial compressive stress), may have a bonded interface which renders the composite having high frictional shear stress(15-39 MPa). The experimental results of Singh can be well interpreted by the present model if interface bonding exists in the Sic/zircon composite The debonding lengths of SiC/borosilicate composite are plotted in Fig. 5 as a function of frictional hear stress, ts, for different relative critical strain energy release rate, sa/sm. the debonding length is shown to decrease as Sd/5m and ts increase. Similar to the matrix cracking stress predictions in Fig. 4, the influence of ts on the debonding length decreases as 5a/sm increases The matrix cracking stresses and the debonding lengths of SiC/borosilicate composite are respectively, plotted in Figs. 6 and 7 as a function of relative critical strain energy release rate, Sa/5m, for lifferent frictional shear stress, ts. The composite with higher Sa/5m and ts has higher matrix cracking 600 500 BHE 0.25 0.2 上400 0.15 0.05 200 ACK t MPa) Fig. 4. Matrix cracking stress vs. frictional shear stress at different sa/5m for SiC/ borosilicate composite
zd=zm 0:25), the crack-wake debonding can occur from Eq. (25) and the predicted matrix cracking stress is smaller than that for a perfectly bonded interface given by the BHE model, as shown in Fig. 4. The BHE result for a perfectly bonded interface has been commonly cited to be the upper-bound estimation of matrix cracking stress. However, from the present analysis, it appears that this upper bound prediction may overestimate the stress for matrix cracking. Fig. 4 also indicates that the higher frictional shear stress has less in¯uence on the matrix cracking stress. Singh [14] has studied the in¯uence of frictional shear stress on the stress for matrix cracking by using ®ber coating to obtain dierent frictional shear stress. Singh reported that the matrix crack stresses of SiC(SCS-6)/zircon composites are independent of the dierent frictional shear stresses and he concluded that the ACK formulation cannot be applied to this composite system. However, if there is some degree of interfacial bonding between the ®ber and the matrix in this composite system, the present result shows that the matrix cracking stresses will become insensitive to the frictional shear stress, as shown in Fig. 4. The SiC/zircon composite system, which has the same ®ber and matrix thermal expansion coecients (i.e., no interfacial compressive stress), may have a bonded interface which renders the composite having high frictional shear stress (15±39 MPa). The experimental results of Singh can be well interpreted by the present model if interface bonding exists in the SiC/zircon composite. The debonding lengths of SiC/borosilicate composite are plotted in Fig. 5 as a function of frictional shear stress, ts, for dierent relative critical strain energy release rate, zd=zm: The debonding length is shown to decrease as zd=zm and ts increase. Similar to the matrix cracking stress predictions in Fig. 4, the in¯uence of ts on the debonding length decreases as zd=zm increases. The matrix cracking stresses and the debonding lengths of SiC/borosilicate composite are, respectively, plotted in Figs. 6 and 7 as a function of relative critical strain energy release rate, zd=zm, for dierent frictional shear stress, ts: The composite with higher zd=zm and ts has higher matrix cracking Fig. 4. Matrix cracking stress vs. frictional shear stress at dierent zd=zm for SiC/borosilicate composite. Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28 23
Y.-C. Chiang/ Engineering Fracture Mechanics 6.5(2000)15-28 10 5/=m=0 10 t MPa) Fig. 5. Debonding length vs frictional shear stress at different sa/5m for SiC/borosilicate composite, the fiber diameter d= 2a. stress, as shown in Fig. 6. However, the higher Sd/sm and ts also result in shorter debonding length. The composite with shorter debonding length will have smaller fiber pull-out length which results ie interfacial debonding process is the prerequisite for the process of fiber pull-out. It implies that the smaller work of fracture. This inverse relation between matrix cracking stress and work of fracture 500 IOMPa 0.0 0.3 5/m Fig. 6. Matrix cracking stress vs. a/5m at different frictional shear stress for SiC/borosilicate composite
stress, as shown in Fig. 6. However, the higher zd=zm and ts also result in shorter debonding length. The interfacial debonding process is the prerequisite for the process of ®ber pull-out. It implies that the composite with shorter debonding length will have smaller ®ber pull-out length which results in a smaller work of fracture. This inverse relation between matrix cracking stress and work of fracture is Fig. 5. Debonding length vs. frictional shear stress at dierent zd=zm for SiC/borosilicate composite, the ®ber diameter d 2a: Fig. 6. Matrix cracking stress vs. zd=zm at dierent frictional shear stress for SiC/borosilicate composite. 24 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28