《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_INTERFACIAL SHEAR DEBONDING PROBLEMS IN

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 respect￾ively adopted to characterize the debonding behavior. Using the concept of Grith theory, an e€ective cir￾cumferential 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 e€ective circumferential defect is then used to de￾rive 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-re￾inforced 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 inter￾facial 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 debond￾ing 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 di€erent experimental results [14±16]. The ®rst issue con￾sidered 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 Grith theory [17]. A question is hence raised as to whether the initial debond stress for ®ber pullout in a ®ber-reinforced ceramic com￾posite can be related to any defect at the interface. Considering two semi-in®nite elastic materials bonded at the interface, the crack propagation pro￾blem 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-re￾inforced 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 cir￾cumvent 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-pull￾out 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 brid￾ging ®ber. The purpose of the present study is to address the above two issues. First, some of the existing sol￾utions 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 Grith theory to de®ne an e€ective 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 con￾sidered 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 debond￾ing 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 ˆ  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 ‡ f†P1Em Ef ‡ 2P2   1 ‡ b2 ÿ a2 a2  P1Em Ef ÿ P3 1=2 …b2 ÿ a2†P1Em 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 ˆ …D ÿ fQ1† …D ÿ mQ1†Q2 , …2a† P2 ˆ  …mD ÿ Q1†  b4…1 ‡ m†ln…b=a† …b2 ÿ a2† 2 ÿ b2 ‡ a2 ‡ …3b2 ÿ a2†m 4…b2 ÿ a2†  ÿ …fD ÿ Q1† …1 ‡ f ‡ 22 f †Em 4Ef  ……D ÿ mQ1†Q2† ÿ1 , …2b† P3 ˆ ÿa2m…mD ÿ Q1† …b2 ÿ a2†…D ÿ mQ1†Q2 , …2c† where D ˆ b2 ‡ a2 b2 ÿ a2 ‡ m ‡ …1 ÿ f†Em Ef , …3a† Fig. 1. Schematic drawings showing (a) the ®ber-pullout geometry and (b) the e€ective circumferential defect introduced at the interface to account for the presence of the ®ber in the matrix and the ®ber￾pullout 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 ˆ 2 fEm Ef ‡ a2m b2 ÿ a2  , …3b† Q2 ˆ 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, di€erent 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) ignor￾ing the strain energy in the matrix [8], or the displa￾cement of the ®ber portion remaining bonded to the matrix [7] (i.e. assuming Em>>Ef) and (3) ignor￾ing Poisson's e€ect (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 e€ect is ignored and the sol￾ution is sd ˆ 2 EfGi a a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†Em     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 consid￾ering Poisson's e€ect, 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 di€erent 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 Grith 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 Grith theory, a monolithic cer￾amic subjected to a uniform tension is considered. Crack propagation occurs at the existing crack tip, and the tensile strength of the material can be re￾lated to the fracture energy and the crack size [17]. In the ®ber-pullout case [Fig. 1(a)], the ®ber has di€erent material properties from the matrix and is subjected to a tensile load. Interfacial debonding in￾itiates 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 ®ber￾pullout geometry. To account for this stress inten￾sity, it is assumed in the present study that the pre￾sence of the ®ber in the pullout case is equivalent to the introduction of an e€ective circumferential defect at the interface which extends from the sur￾face to a depth h [Fig. 1(b)]. Intuitively, this depth, h, should be a function of the dimensions and elas￾tic constants of the ®ber and the matrix which will be derived in Section 5. Also, similar to the Grith theory, it is assumed that the e€ective defect is sub￾jected 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 e€ective defect, h. The composite cylinder depicted in Fig. 1(b) is semi-in®nite long. The end of the composite, the (e€ective) 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 de￾rive the critical condition for propagation of the e€ective circumferential defect at the interface. To analyze the energy, both stresses and displacements in the system are required and are derived as fol￾lows. 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 ˆ 2ht a …5a† smd ˆ 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 ˆ  1 ÿ z ÿ t h  sfd …tRzRt ‡ h† …6a† sm ˆ  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 ˆ  z ÿ t ÿ …z ÿ t† 2 2h  sfd Ef …tRzRt ‡ h† …7a† wm ˆ  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 ˆ h2t‰a2Ef ‡ …b2 ÿ a2†EmŠ a…b2 ÿ a2†EfEm : …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 di€erential equation govern￾ing the stress distribution in the ®ber is [20] d2 sf dz2 ˆ 1 ‡ …b2 ÿ a2†Em 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 ˆ sfd …at z ˆ t† …10a† sf ˆ 0 …at z ˆ 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 ˆ  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 ˆ a2Ef ‡ …b2 ÿ a2†Em b2…1 ‡ m†Ef‰…b2=…b2 ÿ a2††ln…b=a† ÿ ……3b2 ÿ a2†=4b2†Š  1=2 : …12† The corresponding axial displacement of the ®ber relative to the point at z = 0 is wf ˆ 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 ˆ 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 defect￾free region, ut (=wfÿwm at z = t), is ut ˆ 2ht‰a2Ef ‡ …b2 ÿ a2†EmŠ b…b2 ÿ a2†EfEm exp…bt=a† ‡ exp…ÿbt=a† ÿ 2 exp…bt=a† ÿ exp…ÿbt=a† : …15† When t approaches in®nity, ut becomes ut ˆ 2ht‰a2Ef ‡ …b2 ÿ a2†EmŠ b…b2 ÿ a2†EfEm : …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 e€ective defect region is subjected to an applied shear stress, t, and the e€ective 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 ˆ 2pa …t‡h t t…wf ÿ wm† dz …17† where wf and wm are given by equations (7a±b), re￾spectively. Substitution of equations (5a±b) and (7a±b) into equation (17) yields Wh ˆ 4ph3t2 3 a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm : …18† The change in the work, dWh, in the defect region is hence dWh ˆ 4ph2 t2 a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm 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 ˆ 2pahtdut: …20† Substitution of equation (16) into equation (20) yields dWt ˆ 4paht2 b a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm 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 ˆ 4pt2  h2 ‡ ah b  a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm 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 ˆ p 2 …t‡h t a2s2 f Ef ‡ …b2 ÿ a2†s2 m Em   dz …23† where sf and sm are given by equations (6a±b), re￾spectively. Substituting equations (5a±b) and (6a±b) into equation (23), Uh becomes Uh ˆ 2ph3t2 3 a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm : …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 deriva￾tive of equation (24) with respect to h, and the result is dUh ˆ 2ph2 t2 a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm 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 ˆ 2pt2  h2 ‡ ah b a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm 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 ˆ 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 ˆ dUe ‡ dGi …at t ˆ ts†: …28† Substitution of equations (22), (26) and (27) into equation (28) yields ts ˆ  E*Gi a 1=2 h2 a2 ‡ h ab ÿ1=2 …29† where E* ˆ …b2 ÿ a2†EfEm a2Ef ‡ …b2 ÿ a2†Em : …30† equation (29) de®nes the relationship between the interfacial shear strength, ts, the interface debond energy, Gi, and the e€ective 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 ˆ 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 ˆ …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 ˆ 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 ˆ a cl2 …b2 ÿ a2†Em 4p…1 ÿ 2 f †‰a2Ef ‡ …b2 ÿ a2†EmŠ : …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 eluci￾date the essential trends. Dimensionless parameters HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING 3241

3242 HSUEH and BECHER INTERFACIAL SHEAR DEBONDING Ef/em=5 3目oz 0.4 =0.5 5 Fig. 2. The normalized axial stress, ar/ard, along the fiber in the effective defect-free region for are used in the following figures. To normalize the quickly along the fiber length(within a distance of debond stress, ts is sufficient in the strength cri- about four times the fiber diameter when the fiber terion; however, Ti Youngs modulus, and a spatial and the matrix have similar Youngs modulus)to dimension are required in the energy criterion zero. Lower ratios of E Em require a shorter length When a uniform shear stress is applied at the to reach zero stress. This indicates that low ratios nterface in the effective defect region, the calcu- of ErEm are more effective in load transfer from lated normalized axial stress, o ard, along the fiber the fiber to the matrix n the defect-free region is shown in Fig. 2 for The calculated normalized initial debond stresses rla=100 at different ratios of Er/Em. The results od/ts and da/(ETio\y normalized initial 62. as functions of the are not sensitive to i/a when t/a>30. The axial Y oung's modulus ratio stress in the fiber, ar. is ord at z=t and decreases both criteria show that th 6 30 10 10 Youngs modulus ratio, Ef/ em Fig. 3. The normalized initial debond stresses, Oats and a/(ET/a) as functions of the Young's modulus ratio, Er/Em, for vr=Vm=0.25. b/a=10 and t/a

are used in the following ®gures. To normalize the debond stress, ts is sucient in the strength cri￾terion; however, Gi, Young's modulus, and a spatial dimension are required in the energy criterion. When a uniform shear stress is applied at the interface in the e€ective defect region, the calcu￾lated normalized axial stress, sf/sfd, along the ®ber in the defect-free region is shown in Fig. 2 for t/a = 100 at di€erent ratios of Ef/Em. The results are not sensitive to t/a when t/a>30. The axial stress in the ®ber, sf, is sfd at z = t and decreases quickly along the ®ber length (within a distance of about four times the ®ber diameter when the ®ber and the matrix have similar Young's modulus) to zero. Lower ratios of Ef/Em require a shorter length to reach zero stress. This indicates that low ratios of Ef/Em are more e€ective in load transfer from the ®ber to the matrix. The calculated normalized initial debond stresses, sd/ts and sd/(E*Gi/a) 1/2, as functions of the Young's modulus ratio, Ef/Em, are shown in Fig. 3. Both criteria show that the normalized initial Fig. 2. The normalized axial stress, sf/sfd, along the ®ber in the e€ective defect-free region for nf=nm=0.25, b/a = 10 and t/a = 100 when the interface in the e€ective defect region is subjected to a uniform shear stress. Fig. 3. The normalized initial debond stresses, sd/ts and sd/(E*Gi/a) 1/2, as functions of the Young's modulus ratio, Ef/Em, for nf=nm=0.25, b/a = 10 and t/a 4 1. 3242 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING

HSUEH and BECHER INTERFACIAL SHEAR DEBONDING 5 b/a=100 o>o=9N目oz 10 Youngs modulus ratio, E/E Fig. 4. The normalized effective defect length, h/a, as a function of EEm for vr=Vm=0.25 and debond stress increases with the increase in Ed Em. Also, the curve in Fig. 5 becomes flatter as b/a For convenience, the ratio of aa/ts to oa/(eTi/a) Increases is defined as z. Assuming that the strength-based and the energy-based criteria predict the same in- the effective defect length 6. CONCLUDING REMARKS can be related to z, such that Both the strength-based and the energy-based cri- teria have been adopted to analyze the required (35) loading stress on the fiber, ad, to initiate mode II nterfacial debonding during fiber pullout. The The solution of equation(35) yields interfacial shear strength, ts, and the interface debond energy, Ti. are used in the strength-based + (36) and the energy-based criteria, respectively,to characterize interfacial debonding. The relationship etween ts and Ti is derived in the present study The normalized effective defect length, h/a, as a (equation (29)] by adopting the approach of function of Er/Em is shown in Fig. 4 at different Griffith theory. However, instead of using an exist- ing defect in Griffith theory for a monolithic cer- increase in either Er/ Em or b/a. When the fiber and amic. an effective circumferential defect at the the matrix have similar Youngs modulus, the effec- nterface which extends from the loading surface to tive defect length is in the order of the fiber radius a depth, h [Fig. 1(b)], is assumed in the present ha≈ I when ErEm≈1) study. It is noted that the effective defect is not a Using equation (34), the diagram of interfacial real defect. Instead, it is introduced to account for debonding vs fiber fracture is constructed in Fig. 5, the stress intensity due to the presence of the fiber n which the Dundurs'parameter, z, is defined in the matrix and the fiber-pullout geometry. The as[26] effective defect length. h. can be determined Er(1-=)-Em( equation(36) by equating the initial debond stres- Er(1-=)+Em(1 (37) ses derived from the two debonding criteria. When the fiber and the matrix have the similar young It is noted that both the relative defect size cla, in modulus (e.g. for ceramic composites), h is in the the fiber and the defect-geometry factor, i, are order of the fiber radius. The normalized effective olved in defining the diagram. Interfacial defect length, h/a, increases with the increase in onding and fiber fracture occur when c T /alr either the Young's modulus ratio of fiber to matrix below and above the curve, respectively, in Fig. 5. Er Em, or the radius ratio of matrix to fiber b/a This critical ratio decreases with the increase in a.(Fig. 4). For the material design, the usefulness of

debond stress increases with the increase in Ef/Em. For convenience, the ratio of sd/ts to sd/(E*Gi/a) 1/2 is de®ned as w. Assuming that the strength-based and the energy-based criteria predict the same in￾itial debond stress, sd, the e€ective defect length can be related to w, such that h2 a2 ‡ h ab 1=2 ˆ w: …35† The solution of equation (35) yields h a ˆ ÿ 1 2b ‡  1 4b2 ‡ w2 s : …36† The normalized e€ective defect length, h/a, as a function of Ef/Em is shown in Fig. 4 at di€erent ratios of b/a. It is shown that h/a increases with the increase in either Ef/Em or b/a. When the ®ber and the matrix have similar Young's modulus, the e€ec￾tive defect length is in the order of the ®ber radius (i.e. h/a11 when Ef/Em11). Using equation (34), the diagram of interfacial debonding vs ®ber fracture is constructed in Fig. 5, in which the Dundurs' parameter, a, is de®ned as [26] a ˆ Ef…1 ÿ 2 m† ÿ Em…1 ÿ 2 f † Ef…1 ÿ 2 m† ‡ Em…1 ÿ 2 f † : …37† It is noted that both the relative defect size, c/a, in the ®ber and the defect-geometry factor, l, are involved in de®ning the diagram. Interfacial debonding and ®ber fracture occur when cl2 Gi/aGf is below and above the curve, respectively, in Fig. 5. This critical ratio decreases with the increase in a. Also, the curve in Fig. 5 becomes ¯atter as b/a increases. 6. CONCLUDING REMARKS Both the strength-based and the energy-based cri￾teria have been adopted to analyze the required loading stress on the ®ber, sd, to initiate mode II interfacial debonding during ®ber pullout. The interfacial shear strength, ts, and the interface debond energy, Gi, are used in the strength-based and the energy-based criteria, respectively, to characterize interfacial debonding. The relationship between ts and Gi is derived in the present study [equation (29)] by adopting the approach of Grith theory. However, instead of using an exist￾ing defect in Grith theory for a monolithic cer￾amic, an e€ective circumferential defect at the interface which extends from the loading surface to a depth, h [Fig. 1(b)], is assumed in the present study. It is noted that the e€ective defect is not a real defect. Instead, it is introduced to account for the stress intensity due to the presence of the ®ber in the matrix and the ®ber-pullout geometry. The e€ective defect length, h, can be determined [equation (36)] by equating the initial debond stres￾ses derived from the two debonding criteria. When the ®ber and the matrix have the similar Young's modulus (e.g. for ceramic composites), h is in the order of the ®ber radius. The normalized e€ective defect length, h/a, increases with the increase in either the Young's modulus ratio of ®ber to matrix, Ef/Em, or the radius ratio of matrix to ®ber, b/a (Fig. 4). For the material design, the usefulness of Fig. 4. The normalized e€ective defect length, h/a, as a function of Ef/Em for nf=nm=0.25 and t/a 4 1. HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING 3243

HSUEH and BECHER INTERFACIAL SHEAR DEBONDING 0.10 Fiberfracture 0.08 Interfacial debond 0.04 20 10 0.02 0.00 0.5 Dundurs' parameter, a Fig. 5. The diagram of interfacial debonding vs fiber fracture for the bridging-fiber geometry knowing the effective defect length in a fiber-re- Hutchinsons model requires the initial defect sizes inforced ceramic composite is comparable to know- in both the fiber and the matrix and the present ing the defect size in a monolithic ceramic model requires the initial defect size and shape in of ina ng a two-dimensional geometry, a dia- the fiber to define the critical ratio of T /r gram of interfacial debonding vs fiber fracture has The residual stress is not included in the present been constructed previously based on the ratio of analysis. Including the residual stress in the analy the interface debond energy to the fiber fracture sis, the difficulties in analyzing the present problem energy, Ti/Tr[18]. The fiber and the matrix are rep- would be significantly compounded and effects of resented by two semi-infinite planes. Hence, when the residual stress are hoped to be addressed in a the matrix crack reaches the interface. it will either future work. However. using the strength-based deflect into the interface or penetrate the fiber. One approach, effects of the residual stress on the initial modification of this two-dimensional approach is to debond stress in a fiber-pullout geometry have been consider a finite width for the fiber, which is sand- analyzed in a previous study [28]. Also, effects of wiched between two semi-infinite planes of matrix. the residual stress on the competition between It is found that the critical ratio of Ti / Tr in defining interface debonding and crack penetration have the condition of interfacial debonding vs fiber frac- been obtained in He and Hutchinsons modified ture is a function of not only the Dundurs model [29]. Whereas interface debonding is affected parameter [18 but also the fiber width [27]. by the residual stress normal to the interface, crack Considering a bridging fiber behind the crack tip, penetration is affected by the residual stress parallel the critical ratio of T/ Tr is derived in the present to the interface. It is concluded that the residual study. It is found that this critical ratio decreases stress has substantial importance in layered ma with the increase in the Dundurs'parameter, a. terials because the residual stress normal to the Also, the curve in the diagram of interfacial interface is usually zero and the incidence of crack debonding vs fiber fracture becomes flatter when penetration is dominated by the residual the radius ratio of matrix to fiber b/a, increases allel to the interface. However, in fiber-reinforced ( Fig. 5). It is noted that He and Hutchinsons composites, effects of residual stress on interface model considers the case where the crack reaches debonding vs fiber fracture will generally be mini- the interface and the crack either deflects along the mum because of the coupling between residual interface or penetrates the fiber. The present model radial and axial stresses [29] considers the case where the crack reaches the inter- face, circumvents the fiber leaving an intac Icknowledgements-The authors thank E. Y. Sun and A ging fiber and is then followed by either A. Wereszczak for reviewing the manuscript. Researc debonding or fiber fracture. Thus. the interfa onsored by the U.S. Department of Energy, Division of model is an attempt to supplement He and contract DE-AC05-960R22464 with Lockheed Martin utchinson's model. It is also noted that He and Energy Research Corp

knowing the e€ective defect length in a ®ber-re￾inforced ceramic composite is comparable to know￾ing the defect size in a monolithic ceramic. Considering a two-dimensional geometry, a dia￾gram of interfacial debonding vs ®ber fracture has been constructed previously based on the ratio of the interface debond energy to the ®ber fracture energy, Gi/Gf [18]. The ®ber and the matrix are rep￾resented by two semi-in®nite planes. Hence, when the matrix crack reaches the interface, it will either de¯ect into the interface or penetrate the ®ber. One modi®cation of this two-dimensional approach is to consider a ®nite width for the ®ber, which is sand￾wiched between two semi-in®nite planes of matrix. It is found that the critical ratio of Gi/Gf in de®ning the condition of interfacial debonding vs ®ber frac￾ture is a function of not only the Dundurs' parameter [18] but also the ®ber width [27]. Considering a bridging ®ber behind the crack tip, the critical ratio of Gi/Gf is derived in the present study. It is found that this critical ratio decreases with the increase in the Dundurs' parameter, a. Also, the curve in the diagram of interfacial debonding vs ®ber fracture becomes ¯atter when the radius ratio of matrix to ®ber, b/a, increases (Fig. 5). It is noted that He and Hutchinson's model considers the case where the crack reaches the interface and the crack either de¯ects along the interface or penetrates the ®ber. The present model considers the case where the crack reaches the inter￾face, circumvents the ®ber leaving an intact brid￾ging ®ber and is then followed by either interface debonding or ®ber fracture. Thus, the present model is an attempt to supplement He and Hutchinson's model. It is also noted that He and Hutchinson's model requires the initial defect sizes in both the ®ber and the matrix and the present model requires the initial defect size and shape in the ®ber to de®ne the critical ratio of Gi/Gf. The residual stress is not included in the present analysis. Including the residual stress in the analy￾sis, the diculties in analyzing the present problem would be signi®cantly compounded and e€ects of the residual stress are hoped to be addressed in a future work. However, using the strength-based approach, e€ects of the residual stress on the initial debond stress in a ®ber-pullout geometry have been analyzed in a previous study [28]. Also, e€ects of the residual stress on the competition between interface debonding and crack penetration have been obtained in He and Hutchinson's modi®ed model [29]. Whereas interface debonding is a€ected by the residual stress normal to the interface, crack penetration is a€ected by the residual stress parallel to the interface. It is concluded that the residual stress has substantial importance in layered ma￾terials because the residual stress normal to the interface is usually zero and the incidence of crack penetration is dominated by the residual stress par￾allel to the interface. However, in ®ber-reinforced composites, e€ects of residual stress on interface debonding vs ®ber fracture will generally be mini￾mum because of the coupling between residual radial and axial stresses [29]. AcknowledgementsÐThe authors thank E. Y. Sun and A. A. Wereszczak for reviewing the manuscript. Research sponsored by the U.S. Department of Energy, Division of Materials Sciences, Oce of Basic Energy Sciences, under contract DE-AC05-960R22464 with Lockheed Martin Energy Research Corp. Fig. 5. The diagram of interfacial debonding vs ®ber fracture for the bridging-®ber geometry. 3244 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING

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