J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 The conditions. or(rm, z)=om(rr, z), Lo <z<Ls d (12) Im (r1,z)=i(r,2) (rr, z)l=uor (rr, z)l, Lo are for the slip zone with (rr, 3)<0, L0 <z<L (13) Fig. 3. Schematic of the pushout test for boundary element To be satisfied in the stick zone are dr(rr, z)=om(e, z), Ls <z<L, dr(rm, z)=om(r, z), Ls<z<L l(r,0)=ln,0≤r≤a (rr, z)=ur(r, 2), Ls <z<L, The traction free conditions for the fibers are u!(r1,z)=2(r1,z),L、<z<L (0)=0,a<r≤r such tha o(,0)=0,0≤r≤r (6)an(,2)<0,L<z<L (15) and for the matrix are lor(r, z)l< uor(r, z)l, Ls <z<L, d2(r,0)=0,r≤r≤rm, (7) where Lo is the length of the open zone and ls-l dm(,0)=0,r≤r≤rm is the length of the slip zone. In the slip zone, the condition of positive dis- At the bottom(z=L)of the composite, only the sipation has also to be met, that is, the slippage is matrix is constrained in axial direction beyond the in the same direction as the shear stress as the pushout hole of radius ro, that is pushout force is increased [91 2(r,L)=0,0≤F≤rm dm(r,D)=0,r≤r≤rm 4. Analysis by boundary element method (r,L)=0,r≤r≤10<r Based on linear elasticity, BEM formulation [10-12] is applied to the fiber and matrix region eparately. The governing three-dimensional (,L)=0,0≤r≤r, (9) boundary, integral equation for linearly elastic o(r,L)=0,0≤r≤r single body for a source point p is The fiber-matrix interface at r=r is modeled ac Ty (p, q)u, a)ds(q) cording to the Coulomb friction law as: (,2)=m(r,z)=0.0≤z<L + Uy(p, q)t (a)ds(a), Ty and Uy are three-dimensional fundam for the open zone where the crack is open, solutions for traction and displacement, respec (r,z)-u(r,z)>0.0≤z<L tively, u and t, are boundary displacement and (11) traction, respectively,uf z…r; 0† ˆ ua; 0 6 r 6 a < rf: …5† The traction free conditions for the ®bers are: rf z…r; 0† ˆ 0; a < r 6 rf; rf rz…r; 0† ˆ 0; 0 6 r 6 rf; …6† and for the matrix are: rm z …r; 0† ˆ 0; rf 6 r 6 rm; rm rz…r; 0† ˆ 0; rf 6 r 6 rm: …7† At the bottom (z ˆ L) of the composite, only the matrix is constrained in axial direction beyond the pushout hole of radius r0, that is, um z …r; L† ˆ 0; r0 6 r 6 rm; rm rz…r; L† ˆ 0; rf 6 r 6 rm; rm z …r; L† ˆ 0; rf 6 r 6 r0 < rm: …8† and rf z…r; L† ˆ 0; 0 6 r 6 rf; rf rz…r; L† ˆ 0; 0 6 r 6 rf: …9† The ®ber±matrix interface at r ˆ rf is modeled ac￾cording to the Coulomb friction law as: rf rr…rf;z† ˆ rm rr…rf;z† ˆ 0; 0 6 z < L0; rf rz…rf;z† ˆ rm rz…rf;z† ˆ 0; 0 6 z < L0; …10† for the open zone where the crack is open, um r …rf;z† ÿ uf r…rf;z† > 0; 0 6 z < L0: …11† The conditions: rf rr…rf;z† ˆ rm rr…rf;z†; L0 < z < Ls; rf rz…rf;z† ˆ rm rz…rf;z†; L0 < z < Ls; uf r…rf;z† ˆ um r …rf;z†; L0 < z < Ls; jrf rz…rf;z†j ˆ ljrf rr…rf;z†j; L0 < z < Ls; …12† are for the slip zone with rm rr…rf;z† < 0; L0 < z < Ls: …13† To be satis®ed in the stick zone are: rf rr…rf;z† ˆ rm rr…rf;z†; Ls < z < L; rf rz…rf;z† ˆ rm rz…rf;z†; Ls < z < L; uf r…rf;z† ˆ um r …rf;z†; Ls < z < L; uf z…rf;z† ˆ um z …rf;z†; Ls < z < L; …14† such that: rf rr…rf;z† < 0; Ls < z < L; jrf rz…rf;z†j < ljrf rr…rf;z†j; Ls < z < L; …15† where L0 is the length of the open zone and Ls ÿ L0 is the length of the slip zone. In the slip zone, the condition of positive dis￾sipation has also to be met, that is, the slippage is in the same direction as the shear stress as the pushout force is increased [9]. 4. Analysis by boundary element method Based on linear elasticity, BEM formulation [10±12] is applied to the ®ber and matrix region separately. The governing three-dimensional boundary integral equation for linearly elastic single body for a source point p is Cijui…p† ˆ ÿ Z S Tij…p; q†uj…q†dS…q† ‡ Z S Uij…p; q†tj…q†dS…q†; …16† Tij and Uij are three-dimensional fundamental solutions for traction and displacement, respec￾tively; uj and tj are boundary displacement and traction, respectively, Fig. 3. Schematic of the pushout test for boundary element method. 18 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25