Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 Substituting Eqs. (4),(5)and(9),(10)into Eqs. (1la) and(Ilb), the fiber and the matrix displacements in the debonded zone, 0<z<ld, are obtained by integrating Eqs. (lla) and (1lb) (a Er Ec VREf (12) wm(2) 令、f(G-2) Then, the relative displacement u(=) between the fiber and the matrix in the debonded zone, 0<z<ld, is obtained by ()=)-m((4-27_E(a-2) (14) ameMe 2. 2. Upstream stresses The upstream region III(see Fig. 1)is so far away from the crack tip that the stress and strain fields re also uniform. Thus, fiber and matrix have the same displacements and the fiber and the matrix stresses are given by Ef These stresses are the same as those of the bonded region in the downstream region I, given by Eqs.(4 3. Interfacial debonding criterion There are two different approaches to the fiber-matrix interfacial debonding problem in the crack wake region, namely, the shear stress approach and the fracture mechanics approach. The shear stress approach is based upon a maximum shear stress criterion in which interfacial debonding occurs as the shear stress in the fiber/matrix interface reaches the shear strength of interface [8, 9]. On the other hand the fracture mechanics approach treats interfacial debonding as a particular crack propagation problem in which interfacial debonding occurs as the strain energy release rate of interface attains the interfacial debonding toughness [6, 10-12]. Following the arguments of Refs. [6, 11, 12] that the fracture mechanics approach is preferred to the shear stress approach for the interfacial debonding problem, the fracture mechanics approach is also adopted in the present analysis a general case of a cracked body is schematically shown in Fig. 3, in which a volume V is loaded with tractions T and Ts, on the surfaces Sr and Sp with corresponding displacements dw and du, respectively. As the crack grows by dA along the fractional surface Se, an energy balance relation candwm dz ˆ sm Em …11b† Substituting Eqs. (4), (5) and (9), (10) into Eqs. (11a) and (11b), the ®ber and the matrix displacements in the debonded zone, 0Rz < ld, are obtained by integrating Eqs. (11a) and (11b) wf…z† ˆ …z 1 sf Ef dz ˆ …ld 1 s Ec dz ÿ …ld ÿ z†s VfEf ‡ ÿ l 2 d ÿ z2
ts aEf …12† wm…z† ˆ …z 1 sm Em dz ˆ …ld 1 s Ec dz ÿ Vf ÿ l 2 d ÿ z2
ts aVmEm …13† Then, the relative displacement u…z† between the ®ber and the matrix in the debonded zone, 0Rz < ld, is obtained by u…z† ˆ jwf…z† ÿ wm…z†j ˆ …ld ÿ z†s VfEf ÿ Ec ÿ l 2 d ÿ z2
ts aVmEmEf …14† 2.2. Upstream stresses The upstream region III (see Fig. 1) is so far away from the crack tip that the stress and strain ®elds are also uniform. Thus, ®ber and matrix have the same displacements and the ®ber and the matrix stresses are given by sU f ˆ Ef Ec s …15† sU m ˆ Em Ec s …16† These stresses are the same as those of the bonded region in the downstream region I, given by Eqs. (4) and (5). 3. Interfacial debonding criterion There are two di€erent approaches to the ®ber±matrix interfacial debonding problem in the crack￾wake region, namely, the shear stress approach and the fracture mechanics approach. The shear stress approach is based upon a maximum shear stress criterion in which interfacial debonding occurs as the shear stress in the ®ber/matrix interface reaches the shear strength of interface [8,9]. On the other hand, the fracture mechanics approach treats interfacial debonding as a particular crack propagation problem in which interfacial debonding occurs as the strain energy release rate of interface attains the interfacial debonding toughness [6,10±12]. Following the arguments of Refs. [6,11,12] that the fracture mechanics approach is preferred to the shear stress approach for the interfacial debonding problem, the fracture mechanics approach is also adopted in the present analysis. A general case of a cracked body is schematically shown in Fig. 3, in which a volume V is loaded with tractions T and ts, on the surfaces ST and SF with corresponding displacements dw and du, respectively. As the crack grows by dA along the fractional surface SF, an energy balance relation can be Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28 19