J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 where [x] and ly consist of unknown and known Since the results are limited to cases where the quantities, respectively, matrices [A] and [B] con- whole fiber has slipped, only the slip interface tain rearranged corresponding coefficients of [H conditions are discussed. The simulation of slip and [G], and [el consists of known coefficients interface is performed by iteration method. It ssumed that the two bo 4. 1. Temperature effects before any pushout force is being applied. So the first iteration is for perfect bond(continuity of the temperature effects. the radial and axial stresses and displacements at the composite consists of two sub-regions-fiber and interface)case. After the first iteration, the ratio of matrix For an elastic body with sub-regions un- local tangential stress to local normal stress is dergone uniform temperature change, if the coef- calculated to see if there is any violation of stress ficient of thermal expansion a is different for the state. If the ratio is greater than the coefficient of two sub-regions, then there would be strain and friction, A, then the originally assumed sticking stress at the interface of the sub-regions caused by condition is violated. and Coulomb's friction this temperature mismatch between two different condition o,=+uo, is applied instead. Since the materials. This mismatch results in additional unknowns in the matrix equations are in the global tractions at the surface. system, we need to represent the local tractions in terms of their corresponding global values using (q)r=(q)le+1=2△n(q (26) the following transformation equations where the subscript T and E stand for thermo- ∫4 cosO sin 8 elastic and elastic, respectively, E and v are the 0 e -coset Young's modulus and Poisson's ratio, respectively, where 0 is the contact angle between the unit AT is the temperature change, and n are th components of the unit outward normal at g. Then tangential vector and the positive radial axis, tt and tn are local tractions in tangential and normal directions, respectively. Because 0c is perpendicu- integral identity with uniform temperature change lar to the radial axis in the pushout model,the can be written as: absolute value of local tractions ft and In equal to Cui(p)=-/T, (p,q)u(q)ds(q) absolute value of global tractions ty and tr If 2/t>u, the slip occurs between the node Uu(p, q)a)ds(q) pair. So in the next iteration, the Coulomb slip condition is applied instead of stick condition aE△T That is for global coordinates (P,q)ny(q)dS(q).(27) In Eq.(26), I: should be against the relative axial 4.2. Interface conditions movement of node pairs. Then for matrix, if 0. the minus sign should be used in the The interface is modeled using the Coulomb relation of t, and t, that is, t'=-urh, and vice w. The element mesh along the interface is the versa. same for the two bodies. that is the two bodies or displacement between node pair a and b, have the same element length. Each node on fiber have following relationships in the global coordi and its corresponding node on matrix form a node nates pair a and b corresponding to interface node on For u-u>o, u=u, u=u+Aus,(30) fiber and matrix, respectively. Both nodes in a node pair have the same unit outward normal but where Aus is the amount of slip between node a with opposite sign and bwhere [x] and [y] consist of unknown and known quantities, respectively, matrices [A] and [B] con￾tain rearranged corresponding coecients of [H] and [G], and [E] consists of known coecients. 4.1. Temperature e€ects For incorporating the temperature e€ects, the composite consists of two sub-regions ± ®ber and matrix. For an elastic body with sub-regions un￾dergone uniform temperature change, if the coef- ®cient of thermal expansion a is di€erent for the two sub-regions, then there would be strain and stress at the interface of the sub-regions caused by this temperature mismatch between two di€erent materials. This mismatch results in additional tractions at the surface: ‰tj…q†ŠT ˆ ‰tj…q†ŠE ‡ aE 1 ÿ 2m DT nj…q†; …26† where the subscript T and E stand for thermo￾elastic and elastic, respectively, E and m are the YoungÕs modulus and PoissonÕs ratio, respectively, DT is the temperature change, and nj are the components of the unit outward normal at q. Then from Eq. (13), the three-dimensional boundary integral identity with uniform temperature change can be written as: Cijui…p† ˆ ÿ Z S Tij…p; q†uj…q†dS…q† ‡ Z S Uij…p; q†tj…q†dS…q† ‡ aEDT 1 ÿ 2m Z S Uij…p; q†nj…q†dS…q†: …27† 4.2. Interface conditions The interface is modeled using the Coulomb law. The element mesh along the interface is the same for the two bodies, that is, the two bodies have the same element length. Each node on ®ber and its corresponding node on matrix form a node pair a and b corresponding to interface node on ®ber and matrix, respectively. Both nodes in a node pair have the same unit outward normal but with opposite sign. Since, the results are limited to cases where the whole ®ber has slipped, only the slip interface conditions are discussed. The simulation of slip interface is performed by iteration method. It is assumed that the two bodies are sticking together before any pushout force is being applied. So the ®rst iteration is for perfect bond (continuity of radial and axial stresses and displacements at the interface) case. After the ®rst iteration, the ratio of local tangential stress to local normal stress is calculated to see if there is any violation of stress state. If the ratio is greater than the coecient of friction, l, then the originally assumed sticking condition is violated, and CoulombÕs friction condition rt ˆ lrr is applied instead. Since the unknowns in the matrix equations are in the global system, we need to represent the local tractions in terms of their corresponding global values using the following transformation equations: tt tn   ˆ coshc sinhc sinhc ÿ coshc   tr tz   ; …28† where hc is the contact angle between the unit tangential vector and the positive radial axis, tt and tn are local tractions in tangential and normal directions, respectively. Because hc is perpendicu￾lar to the radial axis in the pushout model, the absolute value of local tractions tt and tn equal to absolute value of global tractions tz and tr; If jtz=trj P l, the slip occurs between the node pair. So in the next iteration, the Coulomb slip condition is applied instead of stick condition. That is for global coordinates: tz ˆ ltr: …29† In Eq. (26), tz should be against the relative axial movement of node pairs. Then for matrix, if ua z ÿ ub z > 0, the minus sign should be used in the relation of t b r and t b z , that is, t b z ˆ ÿlt b r , and vice versa. For displacement between node pair a and b, we have following relationships in the global coordi￾nates: For ua z ÿ ub z > 0; ua r ˆ ub r ; ua z ˆ ub z ‡ Dus; …30† where Dus is the amount of slip between node a and b. 20 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25