Acta mater. VoL 46, No 9, pp. 3237-3245. 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PI!:Sl359-6454(98)0008-1 1359-6454/98s19.00+0.00 INTERFACIAL SHEAR DEBONDING PROBLEMS IN FIBER-REINFORCED CERAMIC COMPOSITES CHUN-HWAY HSUEH and P. F BECHeR Metals and Ceramics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831,US.A. (Received 3 September 1997; accepted in revised form 5 December 1997) Abstract--Toughening of fiber-reinforced ceramic composites by fiber pullout relies on mode II debonding at the fiber/matrix interface. This mode Il debonding has been analyzed using the strength-based and the energy-based criteria, in which the interfacial shear strength and the interface debond energy are respect- ely adopted to characterize the debonding behavior. Using the concept of griffith theory, an effective cumferential defect at the interface is defined to account for the stress intensity due to the presence of the fiber in the matrix and the fiber-pullout geometry. This effective circumferential defect is then used to de- of interfacial debonding vs fiber fracture for a bridging fiber behind the crack tip is established using the energy-based criterion. C 1998 Acta Metallurgica Inc. 1 INTRODUCTION hence raised as to whether the initial debond stress rtant toughening mechanism in fiber for fiber pullout in a fiber-reinforced ceramic ce inforced ceramic composites is pullout of fibers posite can be related to any defect at the interface from the matrix during matrix cracking [1, 2. This Considering two semi-infinite elastic materials relies on mode Il (i.e. shear) debonding at the fiber/ bonded at the interface, the crack propagation pro- matrix interface which can be analyzed using either blem has been analyzed by He and Hutchinson [18] the strength-based or the energy-based criterion. In When a crack reaches the interface. the crack he strength-based approach (3-61. interfacial either deflects into the interface or penetrates into debonding occurs when the maximum interfacial the next layer depending upon the ratio of the shear stress induced by loading reaches the inter- energy release rate due to debonding to that due to facial shear strength, ts. In the energy-based crack penetration. This criterion [18] has been used approach [7-12], a sharp mode Il crack propagating extensively to predict interfacial debonding vs fiber along the interface is considered and interfacial fracture for a crack propagating in a fiber-re- debonding occurs when the energy release rate due inforced ceramic composite. However, the crack o crack propagation, Gi, reaches the interface propagation problem in fiber-reinforced composites debond energy, I Based on the above two debond- is three-dimensional. For an embedded fiber of a ing criteria, the required loading stress on the fiber finite radius, there are three options when a matrix during fiber pullout to initiate interfacial debonding crack reaches the interface: the interface can (i.e. the initial debond stress ), ad, has been derived. debond, the fiber can fracture, or the crack can Assuming that t, and Ti are intrinsic material cumvent the fiber. The analysis by He and properties, the predicted dependence of the initial Hutchinson focuses on the case that the matrix debond stress on the other material properties (i.e. crack does not circumvent the fiber. However, when the embedded fiber length. elastic constants and the matrix crack circumvents the fiber, the matrix dimensions of the fiber and the matrix) has been crack is bridged by intact fibers, and the fiber-pull- compared for the above two debonding criteria [12 ]. out geometry can be used as a representative For a given composite system, interfacial debonding volume element for this case. Hence, the second can be strength-governed or energy-governed [13] issue is how the condition of interfacial debonding and both criteria have been supported by different vs fiber fracture is modified for the case of a brid xperimental results [14-16. The first issue con fib elationship The purpose of the present study address between ts and Ti. Also, for a monolithic ceramic, the above two issues. First, some of the existing sol- the tensile strength can be related to its defect size utions of the initial debond stresses based on the Griffith theory [17]. a question is strength and the energy criteria are summarized
INTERFACIAL SHEAR DEBONDING PROBLEMS IN FIBER-REINFORCED CERAMIC COMPOSITES CHUN-HWAY HSUEH and P. F. BECHER Metals and Ceramics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, U.S.A. (Received 3 September 1997; accepted in revised form 5 December 1997) AbstractÐToughening of ®ber-reinforced ceramic composites by ®ber pullout relies on mode II debonding at the ®ber/matrix interface. This mode II debonding has been analyzed using the strength-based and the energy-based criteria, in which the interfacial shear strength and the interface debond energy are respectively adopted to characterize the debonding behavior. Using the concept of Grith theory, an eective circumferential defect at the interface is de®ned to account for the stress intensity due to the presence of the ®ber in the matrix and the ®ber-pullout geometry. This eective circumferential defect is then used to derive the relation between the interfacial shear strength and the interface debond energy. Also, the condition of interfacial debonding vs ®ber fracture for a bridging ®ber behind the crack tip is established using the energy-based criterion. # 1998 Acta Metallurgica Inc. 1. INTRODUCTION An important toughening mechanism in ®ber-reinforced ceramic composites is pullout of ®bers from the matrix during matrix cracking [1, 2]. This relies on mode II (i.e. shear) debonding at the ®ber/ matrix interface which can be analyzed using either the strength-based or the energy-based criterion. In the strength-based approach [3±6], interfacial debonding occurs when the maximum interfacial shear stress induced by loading reaches the interfacial shear strength, ts. In the energy-based approach [7±12], a sharp mode II crack propagating along the interface is considered and interfacial debonding occurs when the energy release rate due to crack propagation, Gi, reaches the interface debond energy, Gi. Based on the above two debonding criteria, the required loading stress on the ®ber during ®ber pullout to initiate interfacial debonding (i.e. the initial debond stress), sd, has been derived. Assuming that ts and Gi are intrinsic material properties, the predicted dependence of the initial debond stress on the other material properties (i.e. the embedded ®ber length, elastic constants and dimensions of the ®ber and the matrix) has been compared for the above two debonding criteria [12]. For a given composite system, interfacial debonding can be strength-governed or energy-governed [13] and both criteria have been supported by dierent experimental results [14±16]. The ®rst issue considered in the present study is the relationship between ts and Gi. Also, for a monolithic ceramic, the tensile strength can be related to its defect size based on the Grith theory [17]. A question is hence raised as to whether the initial debond stress for ®ber pullout in a ®ber-reinforced ceramic composite can be related to any defect at the interface. Considering two semi-in®nite elastic materials bonded at the interface, the crack propagation problem has been analyzed by He and Hutchinson [18]. When a crack reaches the interface, the crack either de¯ects into the interface or penetrates into the next layer depending upon the ratio of the energy release rate due to debonding to that due to crack penetration. This criterion [18] has been used extensively to predict interfacial debonding vs ®ber fracture for a crack propagating in a ®ber-reinforced ceramic composite. However, the crack propagation problem in ®ber-reinforced composites is three-dimensional. For an embedded ®ber of a ®nite radius, there are three options when a matrix crack reaches the interface: the interface can debond, the ®ber can fracture, or the crack can circumvent the ®ber. The analysis by He and Hutchinson focuses on the case that the matrix crack does not circumvent the ®ber. However, when the matrix crack circumvents the ®ber, the matrix crack is bridged by intact ®bers, and the ®ber-pullout geometry can be used as a representative volume element for this case. Hence, the second issue is how the condition of interfacial debonding vs ®ber fracture is modi®ed for the case of a bridging ®ber. The purpose of the present study is to address the above two issues. First, some of the existing solutions of the initial debond stresses based on the strength and the energy criteria are summarized. Acta mater. Vol. 46, No. 9, pp. 3237±3245, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S1359-6454(98)00008-1 1359-6454/98 $19.00 + 0.00 3237
HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING Fig. 1. Schematic drawings showing (a) the fiber-pullout geometry and(b) the effective circumferential defect introduced at the interface to account for the presence of the fiber in the matrix and the fiber pullout geometry steno the relationship between the interfacial shear b2(+l)/4b2,/b ngth and the interface debond energy is derived This is achieved by using the concept of griffith theory to define an effective circumferential defect (b2-a2)(3b2-a2) at the interface to account for the stress intensity e matrix a the fiber-pullout geometry. Finally, using the b2-a2((1+)P1Em energy-based criterion, the condition of interfacial debonding vs fiber fracture for a bridging fiber is established. In fact, the bridging-fiber geometry considered here complements the geometry con +x(=)]) sidered in He and Hutchinsons model [18 C-Et where E and v are Youngs modulus and Poisson's ratio and the subscripts f and m denote the fiber and 2. SUMMARY OF EXISTING SOLUTIONS OF the matrix, respectively. The parameters, PI, P2 and Some of the existing solutions of the initial (D-vrQ1) debond stress based on the strength and the energy (D-VmQ1)Q2 (2a) criteria are summarized as follows 6(1+Vm)In(b/a) 2. 1. The strength-based debonding criterion P2=l(vmD-QI For the strength-based criterion, the shear lag model [19] has been used extensively to analyze the b2+a2+(3b2-a2)m initial debond stress. a fiber with a radius. a. is 4(b2-a2) embedded in a coaxial cylindrical shell of matrix with a radius, b, and is subjected to a tensile stress -(urD-Q (1++2)Em Go. at one end in its axial direction [Fig. I(a). The maximum interfacial shear stress induced by the loading stress can be derived. When this maximum (D-wQ1)Q2) nterfacial shear stress reaches ts, interfacial debond- ing initiates and the corresponding loading stress is P3ma?(Vm D-2D) defined as the initial debond stress, od. Depending b2-a2)(D-Vm Q1)Q2 upon the simplifications adopted in the shear lag model, the complexities in analyzing od vary. Whenwhere the fiber is infinitely long, the most rigorous solution (1-)Em +1m for the initial debond stress, ad, is given by [20] Er
Then the relationship between the interfacial shear strength and the interface debond energy is derived. This is achieved by using the concept of Grith theory to de®ne an eective circumferential defect at the interface to account for the stress intensity due to the presence of the ®ber in the matrix and the ®ber-pullout geometry. Finally, using the energy-based criterion, the condition of interfacial debonding vs ®ber fracture for a bridging ®ber is established. In fact, the bridging-®ber geometry considered here complements the geometry considered in He and Hutchinson's model [18]. 2. SUMMARY OF EXISTING SOLUTIONS OF INITIAL DEBOND STRESS Some of the existing solutions of the initial debond stress based on the strength and the energy criteria are summarized as follows. 2.1. The strength-based debonding criterion For the strength-based criterion, the shear lag model [19] has been used extensively to analyze the initial debond stress. A ®ber with a radius, a, is embedded in a coaxial cylindrical shell of matrix with a radius, b, and is subjected to a tensile stress, s0, at one end in its axial direction [Fig. 1(a)]. The maximum interfacial shear stress induced by the loading stress can be derived. When this maximum interfacial shear stress reaches ts, interfacial debonding initiates and the corresponding loading stress is de®ned as the initial debond stress, sd. Depending upon the simpli®cations adopted in the shear lag model, the complexities in analyzing sd vary. When the ®ber is in®nitely long, the most rigorous solution for the initial debond stress, sd, is given by [20] sd ts b2 1 m b2 ÿ a2 4b2 a2 ln b a ÿ b2 ÿ a2 3b2 ÿ a2 a2b2 b2 ÿ a2 a2 1 fP1Em Ef 2P2 1 b2 ÿ a2 a2 P1Em Ef ÿ P3 1=2 b2 ÿ a2P1Em a2Ef ÿ1 1 where E and n are Young's modulus and Poisson's ratio and the subscripts f and m denote the ®ber and the matrix, respectively. The parameters, P1, P2 and P3, are de®ned by P1 D ÿ fQ1 D ÿ mQ1Q2 , 2a P2 mD ÿ Q1 b4 1 mln b=a b2 ÿ a2 2 ÿ b2 a2 3b2 ÿ a2m 4 b2 ÿ a2 ÿ fD ÿ Q1 1 f 22 f Em 4Ef D ÿ mQ1Q2 ÿ1 , 2b P3 ÿa2m mD ÿ Q1 b2 ÿ a2 D ÿ mQ1Q2 , 2c where D b2 a2 b2 ÿ a2 m 1 ÿ fEm Ef , 3a Fig. 1. Schematic drawings showing (a) the ®ber-pullout geometry and (b) the eective circumferential defect introduced at the interface to account for the presence of the ®ber in the matrix and the ®berpullout geometry. 3238 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING
HSUEH and BECHER INTERFACIAL SHEAR DEBONDING 3239 the matrix and where can be defect-free. the stress (3b) intensity at this circumference is due to the fiber ullout geometry. To account for this stress inter Q2=1-=(hmD-Q1) sity, it is assumed in the present study that the pre- D-VmOl sence of the fiber in the pullout case is equivalent to the introduction of an effective circumferential defect at the interface which extends from the sur- face to a depth h [Fig. I(b)]. Intuitively, this depth 2. 2. The energy-based debonding criterion h. should be a function of the dimensions and elas Depending upon the simplifications adopted in tic constants of the fiber and the matrix which will the energy-based criterion, different solutions for od be derived in Section 5. Also, similar to the Griffith have also been derived. The classical solution [7, 8 theory, it is assumed that as obtained by (1)assuming an infinitely long jected to a uniform shear stress, t [Fig. 1(b)].When fiber embedded in a semi-infinite matrix, (2 )ignor- t reaches the interfacial shear strength, ts, crack cement of the fiber portion remaining bonde la- propagation occurs. Hence, ts can be related to ri ing the strain energy in the matrix 8], or the di and the length of the effective defect. h. the matrix [7(i.e assuming Em>>Er) and(3)ignor- The composite cylinder depicted in Fig. 1(b)is ing Poisson's effect (i.e. assuming vr=vm=0). semi-infinite long. The end of the composite, the Whereas the application of the classical solution is (effective)defect front and the surface are located limited by its oversimplification, the analysis by at ==0,2=I and z=[+h, respectively, with I Charalambides and Evans [10] is the sensible and approaching infinity. The condition of minimization nplest one, in which a composite cylinder model of the total free energy of the system is used to de- llr.Sed and Poisson's effect is ignored and the sol- rive the critical condition for propagation of the effective circumferential defect at the interface. To nalyze the energy, both stresses and displacements Erriaer+ the system are required and are derived as fol- The same result has been obtained elsewhere [21]. 3.1. Stresses and displacements in the defect region in which interfacial debonding at the crack fr t≤z≤I+h) with a constant frictional stress along the debonded interface is considered and equation(4) is a special With a constant interfacial shear stress. t. the case by setting the debonded length equal zero axial stresses in the fiber and the matrix at the sy using a classical shear lag model and consid. defect front, o rd and amd, can be obtained from the ering Poisson's effect, a more rigorous solution for stress transfer equation, such that ad has been derived by Gao et al. [9. Furthermore by modifying the shear lag model, the dependen (5a f aa on the embedded fiber length has also been obtained [12]. However, for a long embedded fiber length, regardless of the different formulations for omd E (b) gd in Refs [9, 10, 12, their numerical results are similar [22]. Hence, comparison of ad between the The axial stress distributions in the fiber and the strength-based and the energy-based criteria is matrix, ar and m, along the axial direction are made based on equations(1)and(4) od(t≤2≤t+h)(6a) 3. THE RELATIONSHIP BETWEEN Ts ANDTI o =1 Gmd(t≤z≤+h)(b) The concept of the Griffith theory [17] is adopted n the present study, to derive the relationship Using the defect front as the reference point, the between the interfacial shear strength, ts, and the axial displacement in the fiber and the matrix,w interface debond energy, Ti, for the fiber-pullout and wm, resulting from the axial stresses described geometry. In the Griffith theory, a monolithic cer- amic subjected to a uniform tension is considered. by equations(6a-b)are Crack propagation occurs at the existing crack tip, 2n」(s:51+) lated to the fracture energy and the crack size [171 In the fiber-pullout case [Fig. I(a). the fiber has (z-p)2 different material properties from the matrix and (≤z≤1+h)(7b) subjected to a tensile load. Interfacial debonding in- itiates at the circumference where the fiber enters The relative displacement between the fiber and the
Q1 2 fEm Ef a2m b2 ÿ a2 , 3b Q2 1 ÿ m mD ÿ Q1 D ÿ mQ1 : 3c 2.2. The energy-based debonding criterion Depending upon the simpli®cations adopted in the energy-based criterion, dierent solutions for sd have also been derived. The classical solution [7, 8] was obtained by (1) assuming an in®nitely long ®ber embedded in a semi-in®nite matrix, (2) ignoring the strain energy in the matrix [8], or the displacement of the ®ber portion remaining bonded to the matrix [7] (i.e. assuming Em>>Ef) and (3) ignoring Poisson's eect (i.e. assuming nf=nm=0). Whereas the application of the classical solution is limited by its oversimpli®cation, the analysis by Charalambides and Evans [10] is the sensible and simplest one, in which a composite cylinder model is used and Poisson's eect is ignored and the solution is sd 2 EfGi a a2Ef b2 ÿ a2Em b2 ÿ a2Em 1=2 : 4 The same result has been obtained elsewhere [21], in which interfacial debonding at the crack front with a constant frictional stress along the debonded interface is considered and equation (4) is a special case by setting the debonded length equal zero. By using a classical shear lag model and considering Poisson's eect, a more rigorous solution for sd has been derived by Gao et al. [9]. Furthermore, by modifying the shear lag model, the dependence of sd on the embedded ®ber length has also been obtained [12]. However, for a long embedded ®ber length, regardless of the dierent formulations for sd in Refs [9, 10, 12], their numerical results are similar [22]. Hence, comparison of sd between the strength-based and the energy-based criteria is made based on equations (1) and (4). 3. THE RELATIONSHIP BETWEEN tS AND GI The concept of the Grith theory [17] is adopted in the present study to derive the relationship between the interfacial shear strength, ts, and the interface debond energy, Gi, for the ®ber-pullout geometry. In the Grith theory, a monolithic ceramic subjected to a uniform tension is considered. Crack propagation occurs at the existing crack tip, and the tensile strength of the material can be related to the fracture energy and the crack size [17]. In the ®ber-pullout case [Fig. 1(a)], the ®ber has dierent material properties from the matrix and is subjected to a tensile load. Interfacial debonding initiates at the circumference where the ®ber enters the matrix and where can be defect-free. The stress intensity at this circumference is due to the ®berpullout geometry. To account for this stress intensity, it is assumed in the present study that the presence of the ®ber in the pullout case is equivalent to the introduction of an eective circumferential defect at the interface which extends from the surface to a depth h [Fig. 1(b)]. Intuitively, this depth, h, should be a function of the dimensions and elastic constants of the ®ber and the matrix which will be derived in Section 5. Also, similar to the Grith theory, it is assumed that the eective defect is subjected to a uniform shear stress, t [Fig. 1(b)]. When t reaches the interfacial shear strength, ts, crack propagation occurs. Hence, ts can be related to Gi and the length of the eective defect, h. The composite cylinder depicted in Fig. 1(b) is semi-in®nite long. The end of the composite, the (eective) defect front and the surface are located at z = 0, z = t and z = t + h, respectively, with t approaching in®nity. The condition of minimization of the total free energy of the system is used to derive the critical condition for propagation of the eective circumferential defect at the interface. To analyze the energy, both stresses and displacements in the system are required and are derived as follows. 3.1. Stresses and displacements in the defect region (tRzRt + h) With a constant interfacial shear stress, t, the axial stresses in the ®ber and the matrix at the defect front, sfd and smd, can be obtained from the stress transfer equation, such that sfd 2ht a 5a smd ÿ 2aht b2 ÿ a2 : 5b The axial stress distributions in the ®ber and the matrix, sf and sm, along the axial direction are sf 1 ÿ z ÿ t h sfd tRzRt h 6a sm 1 ÿ z ÿ t h smd tRzRt h: 6b Using the defect front as the reference point, the axial displacement in the ®ber and the matrix, wf and wm, resulting from the axial stresses described by equations (6a±b) are wf z ÿ t ÿ z ÿ t 2 2h sfd Ef tRzRt h 7a wm z ÿ t ÿ z ÿ t 2 2h smd Em tRzRt h: 7b The relative displacement between the ®ber and the HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING 3239
3240 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING matrix at the surface, th(=wrWm at z =I+ h), is free region, ut(=wrWm at 2=1), is h-t[a Er +(b--aEml 2ht[a-Er +(b--cEm] exp(Bi/a)+expAt/a) exp(Bt/a)-exp(-Bt/a (15) 3. 2. Stresses and displacements in the defect-free region(0≤x≤1) When I approaches infinity, u, becomes In this region, the interface remains bonded For a bonded interface, the stress transfer problem has 2ht[a Er +(b--aEml been analyzed and the differential equation govern- B(62-a )Er Em (16) ing the stress distribution in the fiber is [20] d-c (b-=a)Em 3.3. The energy balance condition a er Based on the energy criterion, energy terms are involved: (D)w, the work done by (62(1+n the load,(2) the elastic strain energy. The crack propagation criterion can be when the interface in the effective defect The solution of dr from equation(9)is subjected to subjected to an applied shear stress, t, and the the following two boundary conditions ective defect length h. advances a distance dh. 3.3.1. The work. In the defect region, the work af= Ofd (at2=D) (10a) done, h, due to the applied shear stress, t, at the 10b) interface is Whereas equation (10a) dictates the fiber stress at Wn=2a t(wr -Wm)dz (17) the defect front position, equation (10b)states that the axial stress is zero remote from shear loading. where wr and wm are given by equations(7a-b), re- Using the above boundary conditions, the solution spectively. Substitution of equations (5a-b)and of ar is (7a-b) into equation(17) yields ≤2≤1) 4h32a2+(b2-a2) where B is a dimensionless parameter given bp (11) The change in the work, dWi, in the defect region is hence a-Er+(b--aEr b2(1+m)E[(b2/(b2-a2)ln(b/a)-(3b2-a2)/4b2) The corresponding axial displacement of the fiber relative to the point at z =0 dWh=4nh Er +(b--a2)Em a )Er Em wr=BEr exp(Ba)/a(0s:s). For a constant t, equation(19) is equivalent to the (13) result using dWh=2rahtdun In the defect-free region, the change in the work Similarly, the average axial displacement dn in dw 2zahtdu aomd exp(Bz/a)+exp(-Bz/a (20) BEm exp(Bt/a)-exp(Br/a) ≤z≤D) Substitution of equation(16) into equation(20) Due to the shear load elative displac 4raht-a-Er +(b--a)em between the fiber and the matrix within the defect- dwI=b(b2-aErEm (21)
matrix at the surface, uh (=wfÿwm at z = t + h), is uh h2ta2Ef b2 ÿ a2Em a b2 ÿ a2EfEm : 8 3.2. Stresses and displacements in the defect-free region (0RzRt) In this region, the interface remains bonded. For a bonded interface, the stress transfer problem has been analyzed and the dierential equation governing the stress distribution in the ®ber is [20] d2 sf dz2 1 b2 ÿ a2Em a2Ef sf b2 1 m b2 b2 ÿ a2 ln b a ÿ 3b2 ÿ a2 4b2 : 9 The solution of sf from equation (9) is subjected to the following two boundary conditions: sf sfd at z t 10a sf 0 at z 0: 10b Whereas equation (10a) dictates the ®ber stress at the defect front position, equation (10b) states that the axial stress is zero remote from shear loading. Using the above boundary conditions, the solution of sf is sf exp bz=a ÿ exp ÿbz=a exp bt=a ÿ exp ÿbt=a sfd 0RzRt 11 where b is a dimensionless parameter given by b a2Ef b2 ÿ a2Em b2 1 mEf b2= b2 ÿ a2ln b=a ÿ 3b2 ÿ a2=4b2 1=2 : 12 The corresponding axial displacement of the ®ber relative to the point at z = 0 is wf asfd bEf exp bz=a exp ÿbz=a ÿ 2 exp bt=a ÿ exp ÿbt=a 0RzRt: 13 Similarly, the average axial displacement of the matrix can be derived, such that wm asmd bEm exp bz=a exp ÿbz=a ÿ 2 exp bt=a ÿ exp ÿbt=a 0RzRt: 14 Due to the shear load, the relative displacement between the ®ber and the matrix within the defectfree region, ut (=wfÿwm at z = t), is ut 2hta2Ef b2 ÿ a2Em b b2 ÿ a2EfEm exp bt=a exp ÿbt=a ÿ 2 exp bt=a ÿ exp ÿbt=a : 15 When t approaches in®nity, ut becomes ut 2hta2Ef b2 ÿ a2Em b b2 ÿ a2EfEm : 16 3.3. The energy balance condition Based on the energy criterion, the following energy terms are involved: (1) W, the work done by the load, (2) Ue, the elastic strain energy in the ®ber and the matrix, (3) Gi, the interface debond energy. The crack propagation criterion can be established by using the energy balance condition when the interface in the eective defect region is subjected to an applied shear stress, t, and the eective defect length, h, advances a distance dh. 3.3.1. The work. In the defect region, the work done, Wh, due to the applied shear stress, t, at the interface is Wh 2pa th t t wf ÿ wm dz 17 where wf and wm are given by equations (7a±b), respectively. Substitution of equations (5a±b) and (7a±b) into equation (17) yields Wh 4ph3t2 3 a2Ef b2 ÿ a2Em b2 ÿ a2EfEm : 18 The change in the work, dWh, in the defect region is hence dWh 4ph2 t2 a2Ef b2 ÿ a2Em b2 ÿ a2EfEm dh: 19 For a constant t, equation (19) is equivalent to the result using dWh=2pahtduh. In the defect-free region, the change in the work, dWt, is dWt 2pahtdut: 20 Substitution of equation (16) into equation (20) yields dWt 4paht2 b a2Ef b2 ÿ a2Em b2 ÿ a2EfEm dh: 21 3240 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING
HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING 3241 The total change in the work, dw(=dWh+dwu, equation(29) defines the relationship between the in the system is hence interfacial shear strength, ts. the interface debond dW=4n(,2, ah)Er+(b2-c)Em energy, Ti, and the effective defect length. h. which B)(b2-a2)E (22) will be examined further in Section 5 3.3.2. The elastic strain energy. In the defect 4. INTERFACIAL DEBONDING VS FIBER region, the elastic strain energy in the fiber and the matrIx The relationship between the debond stress, aa, and the interface debond d:(23) been defined by equation(4). When Gd is greater E than the fiber strength, o. fiber fracture occurs before interfacial debonding. However, construction where or and om are given by equations(6a-b), re- of the diagram of interfacial debonding vs fiber spectively. Substituting equations(5a-b)and(6a-b) fracture needs not only the interface debond into equation(23), Un becomes ergy, Ti, but also the fiber fracture energy, Tr[18] N 2/t2aEr+(b--4m.(24)o achieve this, the relationship between os and Tr is required which is derived as foll The relationship between as and Tr is a function When the defect front advances a distance dh, the of the shape and the size of the defect in the fiber change in the elastic strain energy, dUm in the When the fiber has a small(compared to the fiber defect region can be obtained by taking the deriva- radius) defect of size c and is subjected to a tive of equation (24)with respect to h, and the stress, o, the stress intensity factor at the crac result is Kl, can be expressed by a general equation a Er+(b--a)Em that (b2-a2)BEdh.(25) (31) It is noted that dUn-dw n/2, which is valid for an where i is a defect-geometry factor (=1.122 for a elastic system [9, 21, 23]. Similarly, it can be derived circumferential crack where c is the crack hat the change in the elastic strain energy, dUn in depth [24]=0.637 for an internal penny-shaped the defect-free region equals dw/2. Hence the total crack where c is the crack radius [24 and =0.34 change in the elastic strain energy in the syste or a thumb-nail faw extending from the surface to du(=dU,+dUn), is the interior where c is the crack radius [25). The d=2(+)+(62-a2)En corresponding strain energy release rate, Gr is (2-a)EEm0.②20 3.3.3. The interface debond energy. When the fiber fracture occurs when g reaches o. and the debond length advances a distance dh, the surface corresponding Gr reaches Ir. Combination of area of the debonded interface is increased by equations(31)and(32)gives 2radh. The change in the interfacial energy, dGi, is Erle dGi= 2ralid/ 27) 2Va(1-) Substitution of equation (33)into equation (4 3.3.4. The energy balance condition and solutions. yields a critical ratio for Ti/, such that The interfacial shear strength, ts, can be obtained from the energy balance condition, such that =c24x(1-a2Er+(b2-a2)Em] dw=dUe +dGi (at t=ts Substitution of equations(22),(26)and (27) into Interfacial debonding and fiber fracture occur when equation(28) yields Ti/Tr is smaller and greater than the critical ratio where 5. RESULTS (b--aErEm a-Er +(b-a)Em (30) using vr-=m=0.25,da→∞andb/a=10 to eluci- date the essential trends. Dimensionless parameters
The total change in the work, dW (=dWh+dWt), in the system is hence dW 4pt2 h2 ah b a2Ef b2 ÿ a2Em b2 ÿ a2EfEm dh: 22 3.3.2. The elastic strain energy. In the defect region, the elastic strain energy in the ®ber and the matrix, Uh, is Uh p 2 th t a2s2 f Ef b2 ÿ a2s2 m Em dz 23 where sf and sm are given by equations (6a±b), respectively. Substituting equations (5a±b) and (6a±b) into equation (23), Uh becomes Uh 2ph3t2 3 a2Ef b2 ÿ a2Em b2 ÿ a2EfEm : 24 When the defect front advances a distance dh, the change in the elastic strain energy, dUh, in the defect region can be obtained by taking the derivative of equation (24) with respect to h, and the result is dUh 2ph2 t2 a2Ef b2 ÿ a2Em b2 ÿ a2EfEm dh: 25 It is noted that dUh=dWh/2, which is valid for an elastic system [9, 21, 23]. Similarly, it can be derived that the change in the elastic strain energy, dUt, in the defect-free region equals dWt/2. Hence the total change in the elastic strain energy in the system, dU (=dUt+dUh), is dU 2pt2 h2 ah b a2Ef b2 ÿ a2Em b2 ÿ a2EfEm dh: 26 3.3.3. The interface debond energy. When the debond length advances a distance dh, the surface area of the debonded interface is increased by 2padh. The change in the interfacial energy, dGi, is hence dGi 2paGidh: 27 3.3.4. The energy balance condition and solutions. The interfacial shear strength, ts, can be obtained from the energy balance condition, such that dW dUe dGi at t ts: 28 Substitution of equations (22), (26) and (27) into equation (28) yields ts E*Gi a 1=2 h2 a2 h ab ÿ1=2 29 where E* b2 ÿ a2EfEm a2Ef b2 ÿ a2Em : 30 equation (29) de®nes the relationship between the interfacial shear strength, ts, the interface debond energy, Gi, and the eective defect length, h, which will be examined further in Section 5. 4. INTERFACIAL DEBONDING VS FIBER FRACTURE The relationship between the initial debond stress, sd, and the interface debond energy, Gi, has been de®ned by equation (4). When sd is greater than the ®ber strength, ss, ®ber fracture occurs before interfacial debonding. However, construction of the diagram of interfacial debonding vs ®ber fracture needs not only the interface debond energy, Gi, but also the ®ber fracture energy, Gf [18]. To achieve this, the relationship between ss and Gf is required which is derived as follows. The relationship between ss and Gf is a function of the shape and the size of the defect in the ®ber. When the ®ber has a small (compared to the ®ber radius) defect of size c and is subjected to a tensile stress, s, the stress intensity factor at the crack tip, KI, can be expressed by a general equation, such that KI ls pc p 31 where l is a defect-geometry factor (=1.122 for a circumferential crack where c is the crack depth [24], =0.637 for an internal penny-shaped crack where c is the crack radius [24] and =0.34 for a thumb-nail ¯aw extending from the surface to the interior where c is the crack radius [25]). The corresponding strain energy release rate, Gf, is Gf 1 ÿ 2 f K 2 I Ef : 32 Fiber fracture occurs when s reaches ss and the corresponding Gf reaches Gf. Combination of equations (31) and (32) gives ss 1 l EfGf pc 1 ÿ 2 f s : 33 Substitution of equation (33) into equation (4) yields a critical ratio for Gi/Gf, such that Gi Gf crit a cl2 b2 ÿ a2Em 4p 1 ÿ 2 f a2Ef b2 ÿ a2Em : 34 Interfacial debonding and ®ber fracture occur when Gi/Gf is smaller and greater than the critical ratio, respectively. 5. RESULTS Unless noted otherwise, the results are computed using nf=nm=0.25, t/a 4 1 and b/a = 10 to elucidate the essential trends. Dimensionless parameters HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING 3241