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 theQ1 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