Y.-C. Chiang/ Engineering Fracture Mechanics 65(2000)15-28 indicate, respectively, the fiber and the matrix. The total axial stresses in Region I sats. crits f and m where E and V denote Youngs modulus and volume fraction, respectively. The subs Vor(=)+Mom(2)=0 here a(=)and om(z) denote the fiber and matrix tensile stresses. It is noted that this relationship is not readily satisfied in Region II(see Fig. 1). Therefore, a more rigorous analysis is needed to evaluate the stress/strain fields in Region II if the stress/strain field in this region needs to be considered in the modeling formulation. The boundary between Region I and Region Il has been given by Chiang et al For the bonded region(d<z) in the downstream Region I, the fiber, matrix and composite have the same displacements Ef=Em =fc=r Thus, the fiber and matrix stresses in the bonded region (d<z) become E E E For the debonded region(0<z la) in Region I, the fiber/matrix interface is resisted by a constant frictional stress, ts. The force equilibrium equation of the fiber in this region is given by where a is the fiber radius. The boundary condition at the crack plane z=0 is given by r(O)=2 m(0)=0 Solving Eqs. (2)and (6) with the boundary conditions given by Eqs. (7)and(8), the fiber and matrix stresses in the debonded zone, 0<z< ld. become Let w(a)and wm(z) denote the fiber and matrix displacements measured from the boundary z= oo and set w(oo)=wm(oo)=0. The stress-strain relationships of the fiber and the matrix are given by Ewhere E and V denote Young's modulus and volume fraction, respectively. The subscripts f and m indicate, respectively, the ®ber and the matrix. The total axial stresses in Region I satisfy Vfsf…z† ‡ Vmsm…z† ˆ s …2† where sf…z† and sm…z† denote the ®ber and matrix tensile stresses. It is noted that this relationship is not readily satis®ed in Region II (see Fig. 1). Therefore, a more rigorous analysis is needed to evaluate the stress/strain ®elds in Region II if the stress/strain ®eld in this region needs to be considered in the modeling formulation. The boundary between Region I and Region II has been given by Chiang et al. [7]. For the bonded region …ldRz† in the downstream Region I, the ®ber, matrix and composite have the same displacements ef ˆ em ˆ ec ˆ s Ec …3† Thus, the ®ber and matrix stresses in the bonded region …ldRz† become sD f ˆ Ef Ec s …4† sD m ˆ Em Ec s …5† For the debonded region …0Rz < ld† in Region I, the ®ber/matrix interface is resisted by a constant frictional stress, ts: The force equilibrium equation of the ®ber in this region is given by dsf dz ˆ ÿ…2=a†ts …6† where a is the ®ber radius. The boundary condition at the crack plane z ˆ 0 is given by sf…0† ˆ s Vf …7† sm…0† ˆ 0 …8† Solving Eqs. (2) and (6) with the boundary conditions given by Eqs. (7) and (8), the ®ber and matrix stresses in the debonded zone, 0Rz < ld, become sD f ˆ s Vf ÿ 2tsz a …9† sD m ˆ  Vf Vm 2tsz a …10† Let wf…z† and wm…z† denote the ®ber and matrix displacements measured from the boundary z ˆ 1 and set wf…1† ˆ wm…1† ˆ 0: The stress±strain relationships of the ®ber and the matrix are given by dwf dz ˆ sf Ef …11a† 18 Y.-C. Chiang / Engineering Fracture Mechanics 65 (2000) 15±28