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 parametersThe 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