SUN and SINGH: MULTIPLE MATRIX CRACKING 1659 Frictional Sliding Debonding Tip ■ Fiber Matrix (b) (MPa) De bonded zone Bonded zone 800· 400 200 Distance From Crack Surface (z) Fig. 2.(a)Schematic of interfacial debonding and fiber pullout. (b) Stress distributions in the fiber, matrix and at the interface For a partially bonded and debonded interface, matrix far from the debonded region, respectively the stress distribution in the cracked area is depen- p is the shear-lag parameter determined by the dent on the debond length Ld, which is determined following formula by the interfacial shear strength ty. Figure 2(a) p2 4ECG shows a cylindrical unit of composite with a debonded interface of length Ld. The stresses in the debonded region(0 <2< La follow equations (4- where Gm is the shear modulus of matrix and (6). According to the shear-lag solution given by BHE [16, the stresses beyond the debonded region 2In Vr+Vm(3-vr (z>Ld) determined by the folle exp ons The above expressions are fundamentally identical (8) along the interface from the matrix crack surface to the far field beyond the debonded region is show in Fig. 2(b). The interfacial shear stress is shown to am()=0m-o have its maximum value at the debonded tip (=La if ta is larger than t (9) The mode II debond shear stress, ta is equal to he interfacial shear strength tu for a bonded inter- face with zero coefficient of friction as estimated by r(z)= BEH [12] (BHE[2(11) where of and am are the stresses in fiber andFor a partially bonded and debonded interface, the stress distribution in the cracked area is depen￾dent on the debond length Ld, which is determined by the interfacial shear strength tu. Figure 2(a) shows a cylindrical unit of composite with a debonded interface of length Ld. The stresses in the debonded region (0 < z < Ld) follow equations (4)± (6). According to the shear-lag solution given by BHE [16], the stresses beyond the debonded region (z>Ld) are determined by the following expressions [12]: sf…z†ˆs1 f ‡ " Vm Vf   s1 m ÿ2tf Ld r # exp ÿ r…z ÿ Ld† r  …8† sm…z†ˆs1 m ÿ " s1 m ÿ Vf Vm   2tf Ld r # exp ÿ r…z ÿ Ld† r  …9† ti…z†ˆ r 2   " Vm Vf   s1 m ÿ2tf Ld r # exp ÿ r…z ÿ Ld† r  …10† where s1 f and s1 m are the stresses in ®ber and matrix far from the debonded region, respectively. r is the shear-lag parameter determined by the following formula r2 ˆ 4EcGm VmEmEfj , where Gm is the shear modulus of matrix and j ˆ ÿ 2 ln Vf ‡ Vm…3 ÿ Vf† 2V2 m , The above expressions are fundamentally identical for AK [6], BEH [12], HJ [10], and Sutcu and Hillig [17] (SH) models without considering the re￾sidual stresses in composite. The stress distribution along the interface from the matrix crack surface to the far ®eld beyond the debonded region is shown in Fig. 2(b). The interfacial shear stress is shown to have its maximum value at the debonded tip (z = Ld) if td is larger than tf. The mode II debond shear stress, td is equal to the interfacial shear strength tu for a bonded inter￾face with zero coecient of friction as estimated by BEH [12], tBEH d ˆ  4GmGd rj s (BHE[12]) …11† Fig. 2. (a) Schematic of interfacial debonding and ®ber pullout. (b) Stress distributions in the ®ber, matrix and at the interface. SUN and SINGH: MULTIPLE MATRIX CRACKING 1659