J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 In the matrix equation, uf, and u, are solved done because the computational time required in independently, and their solutions would yield the BEM is prohibitive not for individual runs but amount of slip because for the number of runs required to extract (31) the mechanical properties of the interface. To In summary, for Coulomb slip interface condition total of 30 data points provided in[14](used in[13] the following relationship apply in the range of specimen thickness of up to 2045 u=u t=-tt um is used (32) multiple regressio lysis [5] of the So u can be expressed in terms of u,, u, and u, can experimental data from Table 2 to Eq(4)gave the be solved independently and all the unknown residual radial stress oo=-75 MPa and coefficient tractions in the slip region can be expressed in of friction u=0. 12 terms of tr. Therefore, total unknowns in the slip region are up, u, u and r. There are four boundary 5.2. Extraction of interfacial properties using bour integral equations and four continuity equations dary element method model for each node pair, then the problem can be solved Since BEM is a numerical technique, its results are single data points of maximum pushout force 5. Results for a particular specimen length, friction and residual clamping stress (material 5.1. Extraction of interfacial properties using shear system and fiber volume fraction are fixed in the ig analysis model experiment [13]. Hence, the regression analysis involved is unlike the analytical SLa as discussed By measuring the maximum pushout force, in Section 5.1. An indirect method has to be used Pmax as a function of specimen thickness, L, Eq (4) by assuming a value for the regression parameters, be used to extract the mechanical properties of Oo and u, and calculating the maximum pushout the fiber-matrix interface, Go and H. The experi- force using BEM for the 10 diferent specimen mental data in [13] was used. The material prop lengths in Table 2. Then the sum of the square of erties and geometry are listed in Table I and ar the residuals, S, of the maximum pushout force is used in this paper, unless noted otherwise calculated as follows: Only 10 individual pushout test data points S,=>I[(Pmax)BEMI -[(Pmax)) (33) (Table 2) was used for this study. This was mainly where(Pmax )BEM is the maximum pushout force Table from BEM and(Pmax )EXPT is the maximum push Specimen dimensions and material properties of silicon carbide/ out force from experiment aluminosilicate glass composite sample [131 The regression parameters are changed until Fiber Matrix one obtains the minimum value of the sum of the Diameter, D(um) square of the residuals, S,. The SLA results of oung's modulus, E(GPa) Fo=-75MPa and u=0. 12 were used as a starting Poissons ratio. y 0.24 point. The value of the regression parameters ob Coefficient of thermal expansion, tained via BEM were Go=-74 MPa and u=0.14 x(10-6K-) This shows about 16% difference for the coefficient Table 2 Maximum pushout force(N)vS specimen thickness(um) in a pushout test for specimen given in Table 1[14 Maximum pushout force(M) a30936354313513233415412In the matrix equation, ua z and ub z are solved independently, and their solutions would yield the amount of slip Dus ˆ ua z ÿ ub z : …31† In summary, for Coulomb slip interface condition the following relationship apply: ub r ˆ ua r ; t b r ˆ ÿt a r : …32† So ub r can be expressed in terms of ua r , ua z and ub z can be solved independently and all the unknown tractions in the slip region can be expressed in terms of t a r . Therefore, total unknowns in the slip region are ua r , ua z , ub z and t a r . There are four boundary integral equations and four continuity equations for each node pair, then the problem can be solved. 5. Results 5.1. Extraction of interfacial properties using shear± lag analysis model By measuring the maximum pushout force, Pmax as a function of specimen thickness, L, Eq. (4) can be used to extract the mechanical properties of the ®ber±matrix interface, r0 and l. The experi￾mental data in [13] was used. The material prop￾erties and geometry are listed in Table 1 and are used in this paper, unless noted otherwise. Only 10 individual pushout test data points (Table 2) was used for this study. This was mainly done because the computational time required in BEM is prohibitive not for individual runs but because for the number of runs required to extract the mechanical properties of the interface. To eliminate any bias, every third data point of the total of 30 data points provided in [14] (used in [13] in the range of specimen thickness of up to 2045 lm is used. Using multiple regression analysis [5] of the experimental data from Table 2 to Eq. (4) gave the residual radial stress r0 ˆ ÿ75 MPa and coecient of friction l ˆ 0.12. 5.2. Extraction of interfacial properties using boun￾dary element method model Since BEM is a numerical technique, its results are single data points of maximum pushout force for a particular specimen length, coecient of friction and residual clamping stress (material system and ®ber volume fraction are ®xed in the experiment [13]. Hence, the regression analysis involved is unlike the analytical SLA as discussed in Section 5.1. An indirect method has to be used by assuming a value for the regression parameters, r0 and l, and calculating the maximum pushout force using BEM for the 10 di€erent specimen lengths in Table 2. Then the sum of the square of the residuals, Sr of the maximum pushout force is calculated as follows: Sr ˆ Xf‰…Pmax†BEMŠ i ÿ ‰…Pmax†EXPTŠ i g2 ; …33† where (Pmax)BEM is the maximum pushout force from BEM and (Pmax)EXPT is the maximum push￾out force from experiment. The regression parameters are changed until one obtains the minimum value of the sum of the square of the residuals, Sr. The SLA results of r0 ˆ ÿ75MPa and l ˆ 0.12 were used as a starting point. The value of the regression parameters ob￾tained via BEM were r0 ˆ ÿ74 MPa and l ˆ 0.14. This shows about 16% di€erence for the coecient Table 1 Specimen dimensions and material properties of silicon carbide/ aluminosilicate glass composite sample [13] Fiber Matrix Diameter, D (lm) 137 5500 YoungÕs modulus, E (GPa) 413 87 PoissonÕs ratio, m 0.19 0.24 Coecient of thermal expansion, a (10ÿ6 Kÿ1) 4.3 4.61 Table 2 Maximum pushout force (N) vs. specimen thickness (lm) in a pushout test for specimen given in Table 1 [14] Specimen thickness (lm) 218 374 691 800 1014 1191 1333 1489 1604 2045 Maximum pushout force (N) 0.3100 0.9990 2.745 2.355 4.301 6.185 7.328 5.511 5.921 7.865 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 21