252 A Mukherjee, H.S. Rao /Computational Materials Science 4(1995)249-262 (o)of a point exceeds the interface normal strength, the normal stiffness(K)and the shear stiffness(K, of that point is set to zero to allow debonding at the interface. Thus the algorithm of the interface element takes the value of interface stiffness as NO SLIP 1)K,=0, if o> interface shear strength(slip condition): )Kn=10 Em, if o <0(overlapping con dition) 3)K=0, and K.=0 if o> interface normal strength(debonding condition) The iterative finite element analysis has to be carried out to account for shear failure or separa tion at the interface. Though this FE solution is computationally expensive it is still a powerful ool for performing numerical experiments and Fig. 3. Model of whiskers embedded in a matrix with an sensitivity studies on CMCs. However, the pre sent author gaged in the devclopment or where of the whisker, T is the interface shear an artificial neural network approach which gen cralises the limited responses obtained from the strength finite element analysis, by adaptively learning the Fig. 3 shows such a case with slip occurring constitutive relation. This will provide a fast and over a distance L along the matrix/whisker inte computationally economical tool for conducting face causing a displacement u of the atrix due sensitivity studies on CMCs. to some applied stress. Marshall and Cox [6] developed a simple analytical model to calculate this displacement, u, for infinitely long fibres 3. Validation of the Fe model based on the premise that a region of strain continuity or no-slip exists above the slipped in The FE model has been validated by charac terface. Their expression is given below sponse of the a-siC ceramic long and short p=2LuTVTEr(1+n)/R whiskers embedded in Al,o, ceramic matrix with wher EVI/Emm, p is the force per unit the present finite element model. These force- length, T is the interface shear stress, Er is displacement responses have been compared with Young s modulus of the fibre, em is the Youngs a simplified analytical solution, and also with the solution presented by Laird II and Kennedy [9]. whiskers, Vm is the volume fraction of the matrix, R is the radius of the fibre and v is the poisson's 3. Analytical model Based on the above expression Laird Il and For the case of whiskers bridging a crack, slip Kennedy [9] derived the following relation be ill occur for some distance along the tween the force on the whisker (F)and displace whisker/matrix interface, provided the whisker ment u for plane strain has minimum length required for load transfer to W EH(1-)2+1, the whisker by interfacial shear. The critical F=2Erm-v2)IEm m(I-m) length required for such load transfer can be calculated from the following equation. L=01r/7, (1) where252 A. Mukherjee, H.S. Rao /Computational Materials Science 4 (1995) 249-262 (a,) of a point exceeds the interface normal strength, the normal stiffness (K,) and the shear stiffness (K,) of that point is set to zero to allow debonding at the interface. Thus the algorithm of the interface element takes the value of interface stiffness as: 1) K, = 0, if a, > interface shear strength (slip condition); 2) K, = 10’ X E,, if a, < 0 (overlapping con￾dition); 3) K, = 0, and K, = 0 if a,, > interface normal strength (debonding condition). The iterative finite element analysis has to be carried out to account for shear failure or separa￾tion at the interface. Though this FE solution is computationally expensive, it is still a powerful tool for performing numerical experiments and sensitivity studies on CMCs. However, the pre￾sent authors are engaged in the development of an artificial neural network approach which gen￾eralises the limited responses obtained from the finite element analysis, by adaptively learning the constitutive relation. This will provide a fast and computationally economical tool for conducting sensitivity studies on CMCs. 3. Validation of the FE model The FE model has been validated by charac￾terising the non linear force-displacement re￾sponse of the a-Sic ceramic long and short whiskers embedded in Al,O, ceramic matrix with the present finite element model. These force￾displacement responses have been compared with a simplified analytical solution, and also with the solution presented by Laird II and Kennedy [9]. 3.1. Analytical model For the case of whiskers bridging a crack, slip will occur for some distance along the whisker/matrix interface, provided the whisker has minimum length required for load transfer to the whisker by interfacial shear. The critical length required for such load transfer can be calculated from the following equation. L, = u,r/r, (1) f \ F F Fig. 3. Model of whiskers embedded in a matrix with an applied stress at a distance far from the crack plane. where a, is the whisker fracture stress, r is the radius of the whisker, r is the interface shear strength. Fig. 3 shows such a case with slip occurring over a distance L along the matrix/whisker inter￾face causing a displacement u of the matrix due to some applied stress. Marshall and Cox [6] developed a simple analytical model to calculate this displacement, U, for infinitely long fibres based on the premise that a region of strain continuity or no-slip exists above the slipped in￾terface. Their expression is given below. p = 2[ U7V;2Ef(1 + 7))/R]“*, (2) where 77 = E,V,/E,V,, p is the force per unit length, r is the interface shear stress, E, is Young’s modulus of the fibre, E, is the Young’s modulus of matrix, V, is the volume fraction of whiskers, V, is the volume fraction of the matrix, R is the radius of the fibre, and v is the Poisson’s ratio. Based on the above expression Laird II and Kennedy [9] derived the following relation be￾tween the force on the whisker (F) and displace￾ment cf for plane strain. where