J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 residual radial compression due to thermal ex- This type of measurement also has its limitations pansion mismatch between the fibers and matrix is The measurement of slip length is difficult to make assumed. Using SLA [6,7], analytical models are and in many cases the fiber slips at both ends, developed for the pushout test. Three expressions hence making it difficult to relate the slip length (depending on type of experimental data collected) one measurement. In addition, accounting for wo mecheressing experimental data to find the more than one slip zone in a theoretical model is anical properties of the interface are Imost intractable due to the nonlinear nature of given below the frictional interface 2. 1. Fiber displacement 2.3. Maximum pi In the first case, the pushout force, P on the fiber is measured as a function of the displacement The third type of experimental here the maximum pushout force, Pmax, is mea Au, on the surface of the fiber below the matrix sured as a function of specimen thickness, L. The surface due to interfacial slip. The SLA equation Is SLA model equation [5] is same as Eq (3), except given by [5] L=L, and hence is given by Ag=(1-2vk)o∫P kEf 1,(00+k(P/xr2) (1) This type of measurement does require several samples of different thickness as opposed to only one in the previous two cases. But, the slip length is predetermined; that is, it is equal to the length of Er(1+Vm)+Em(1-ve) (2) the specimen. Also, as will be shown later that this type of measurement is not influenced by indentor Youngs modulus of matrix are Er and Em, re- radius and type. Since this model seems to have the pectively while L is the specimen length. The least limitations of the three sets of possible ex Poisson's ratio of fiber and matrix are v, Vm, re- perimental data, this SLa model will be the used in spectively with u being the coefficient of friction. this study for comparison with the BEM model Residual radial stress at the interface is o and re stands for the fiber radius However, in an analytical study of [8]. the 3. Boundary element method problem statement above pushout force vs. displacement curve(eq (1)) was found to be highly influenced by many An axisymmetric model is composed of a solid extrinsic factors of the test such as type and radius cylindrical fiber of radius, rr, and length L, and a of indentor This observation cannot be ignored in hollow cylinder of matrix of internal radius, rf, analyzing experimental data and outer radius, Im, and length, L. The load due to a hard flat indentor is simulated by a uniform 2.2. Pushout force displacement of the fiber over 60-90% of the fiber radius, and the corresponding force on the fiber is The second type of experimental measurements is where the pushout force, P, on the fiber is calculated by integrating the axial stresses on the measured as a function of the length of the inter fiber over the loading area The boundary conditions for the BEM model facial slip zone, Ls. The SLA equation [5] is given (Fig. 3)are as follows. At the top surface(==0)of the composite, the fiber is subjected to a uniform vertical displacement, la, over the indentor radius therwise it is tractresidual radial compression due to thermal ex￾pansion mismatch between the ®bers and matrix is assumed. Using SLA [6,7], analytical models are developed for the pushout test. Three expressions (depending on type of experimental data collected) used in regressing experimental data to ®nd the two mechanical properties of the interface are given below. 2.1. Fiber displacement In the ®rst case, the pushout force, P on the ®ber is measured as a function of the displacement, Duz on the surface of the ®ber below the matrix surface due to interfacial slip. The SLA equation is given by [5] Duz ˆ rf…1 ÿ 2mfk†r0 kEf P 2pr2 f lr0  ÿ 1 2lk ln r0 ‡ k …P=pr2 f† r0  ; …1† where k ˆ Emmf Ef…1 ‡ mm† ‡ Em…1 ÿ mf† : …2† YoungÕs modulus of matrix are Ef and Em, re￾spectively while L is the specimen length. The PoissonÕs ratio of ®ber and matrix are mf; mm, re￾spectively with l being the coecient of friction. Residual radial stress at the interface is r0 and rf stands for the ®ber radius. However, in an analytical study of [8], the above pushout force vs. displacement curve (Eq. (1)) was found to be highly in¯uenced by many extrinsic factors of the test such as type and radius of indentor. This observation cannot be ignored in analyzing experimental data. 2.2. Pushout force The second type of experimental measurements is where the pushout force, P, on the ®ber is measured as a function of the length of the inter￾facial slip zone, Ls. The SLA equation [5] is given by P ˆ pr2 f r0 k e 2lkLs=rf … † ÿ 1 : …3† This type of measurement also has its limitations. The measurement of slip length is dicult to make and in many cases the ®ber slips at both ends, hence making it dicult to relate the slip length as one measurement. In addition, accounting for more than one slip zone in a theoretical model is almost intractable due to the nonlinear nature of the frictional interface. 2.3. Maximum pushout force The third type of experimental measurements is where the maximum pushout force, Pmax, is mea￾sured as a function of specimen thickness, L. The SLA model equation [5] is same as Eq. (3), except L ˆ Ls, and hence is given by Pmax ˆ pr2 f r0 k e 2lkL=rf … † ÿ 1 : …4† This type of measurement does require several samples of di€erent thickness as opposed to only one in the previous two cases. But, the slip length is predetermined; that is, it is equal to the length of the specimen. Also, as will be shown later that this type of measurement is not in¯uenced by indentor radius and type. Since this model seems to have the least limitations of the three sets of possible ex￾perimental data, this SLA model will be the used in this study for comparison with the BEM model. 3. Boundary element method problem statement An axisymmetric model is composed of a solid cylindrical ®ber of radius, rf, and length L, and a hollow cylinder of matrix of internal radius, rf, and outer radius, rm, and length, L. The load due to a hard ¯at indentor is simulated by a uniform displacement of the ®ber over 60±90% of the ®ber radius, and the corresponding force on the ®ber is calculated by integrating the axial stresses on the ®ber over the loading area. The boundary conditions for the BEM model (Fig. 3) are as follows. At the top surface (z ˆ 0) of the composite, the ®ber is subjected to a uniform vertical displacement, ua, over the indentor radius a; otherwise it is traction free, that is, J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 17