AMukherjee, H.S. Rao/ Computational Materials Science 4(1995)249-262 cracking, one dimensional strain analysis etc. )Lo 2. The finite element model ake the problem amenable to solution. Finite element studies of wake toughening mechanisms In this paper, a two dimensional plane are scant [8] and they address only the behavi stress/strain finite element model is developed of a single fibre near the crack tip in a centrally for the analysis of CMCs. Though, these 2D cracked plate under mode I loading at loads less models are approximations of the real 3D com- than the failure load of the fibre. Laird Il and posites, they can still provide a good insight into Kennedy [9] used discrete spring slider elements the toughening mechanisms and their depen to model the interfacial connectivity between the dence on the whisker/matrix interface strength matrix and whisker in the finite element analysis The salient features of the model have been of CMCs. This approach models the connectivity between the whisker and matrix at discrete nodal Developing an adequate Fe model for the points only. As a result, the chances of overlap- analysis of ceramic-matrix-composites, involves ping of the nodes along the interface is not elimi- modelling of: (a) the matrix (b) fibre or whisker lated. Further, this type of idealisation of inter- and (c) the interface between the two materials face considers only the shear failure of the inter- (Fig. elling the mechanics of whisker/matrix face and cannot consider the tensile (normal) failure of the interface which is also possible interface is rather complex. The complexity is due CMCS. Therefore, this approach cannot model to the possible relative displacements at the bi the interface taking into account the interfacial material interface further, if the interface is in mechanics realistically compression, a frictional force between the two In the present work a llicIOnechanical finite Inaterial exists. This frictional grip betweell the element model is developed to analyse the Cmos, matrix and the whisker resists the sliding of the taking the whisker/ matrix interface into effect. whisker against a cracked matrix. If this frictional The matrix, and fibre or whisker are modelled by force is directly considered in the analysis, the eight node isoparametric quadrilateral elements. system becomes non-conservative and the global The whisker/matrix interface is modelled by six stiffness matrix in finite element analysis becomes noded isoparametricinterface elements. Eigh un-symmetric. Such un-symmetric matrices pose noded isoparametric quadrilateral elements are difficulties in the solution routines, resulting ill used for modelling the matrix as well as the numerical instability. On the other hand, if the hiker. As elements of compatible shape func- interface is in tension, the contact between the tions are used to model all the components, i.e., matrix and the whisker ceases to exist and hence whisker, matrix and the interface the interaction there will not be any frictional force. Hence it has between them is modelled realistically. moreover, this model the whisker/matrix interface is modelled all along the interface instead of dis hiker Whisker crete nodal points. The model is capable of con sidering both shear and normal failures of the interface. The possibility of overlapping is also MatrixMatr effectively eliminated. The model is then vali- dated by characterising the non-linear force-dis placement response of a-SiC ceramic whisker cmbcddcd in Al,O3 ceramic matrix and compar- ing it with the simplified analytical solutions. The FE model is then used to demonstrate the effect of whisker/ matrix interface strength on the toughening behaviour of the Al2O,(matrix)/Sic Fig. 1. Longitudinal section of a typical CMC showing the hiker )ceramic-matrix-composite whisker, matrix and interface250 A. Mukherjee, H.S. Rae/Computational Materials Science 4 (1995) 249-262 cracking, one dimensional strain analysis etc.) to make the problem amenable to solution. Finite element studies of wake toughening mechanisms are scant 181 and they address only the behaviour of a single fibre near the crack tip in a centrally cracked plate under mode I loading at loads less than the failure load of the fibre. Laird II and Kennedy [9] used discrete spring slider elements to model the interfacial connectivity between the matrix and whisker in the finite element analysis of CMCs. This approach models the connectivity between the whisker and matrix at discrete nodal points only. As a result, the chances of overlap￾ping of the nodes along the interface is not elimi￾nated. Further, this type of idealisation of inter￾face considers only the shear failure of the inter￾face and cannot consider the tensile (normal) failure of the interface which is also possible in CMCs. Therefore, this approach cannot model the interface taking into account the interfacial mechanics realistically. In the present work a micromechanical finite element model is developed to analyse the CMCs, taking the whisker/matrix interface into effect. The matrix, and fibre or whisker are modelled by eight node isoparametric quadrilateral elements. The whisker/matrix interface is modelled by six noded isoparametricinterface elements. Eight noded isoparameteric quadrilateral elements are used for modelling the matrix as well as the whisker. As elements of compatible shape func￾tions are used to model all the components, i.e., whisker, matrix and the interface, the interaction between them is modelled realistically. Moreover, in this model the whisker/matrix interface is modelled all along the interface instead of dis￾crete nodal points. The model is capable of con￾sidering both shear and normal failures of the interface. The possibility of overlapping is also effectively eliminated. The model is then vali￾dated by characterising the non-linear force-dis￾placement response of a-Sic ceramic whisker embedded in Al,O, ceramic matrix and compar￾ing it with the simplified analytical solutions. The FE model is then used to demonstrate the effect of whisker/matrix interface strength on the toughening behaviour of the Al,O, (matrix)/SiC (whiskerjceramic-matrix-composite. 2. The finite element model In this paper, a two dimensional plane stress/strain finite element model is developed for the analysis of CMCs. Though, these 2D models are approximations of the real 3D com￾posites, they can still provide a good insight into the toughening mechanisms and their depen￾dence on the whisker/matrix interface strength. The salient features of the model have been described below. Developing an adequate FE model for the analysis of ceramic-matrix-composites, involves modelling of: (a) the matrix, (b) fibre or whisker, and (c) the interface between the two materials (Fig. 1). Modelling the mechanics of whisker/matrix interface is rather complex. The complexity is due to the possible relative displacements at the bi￾material interface. Further, if the interface is in compression, a frictional force between the two material exists. This frictional grip between the matrix and the whisker resists the sliding of the whisker against a cracked matrix. If this frictional force is directly considered in the analysis, the system becomes non-conservative and the global stiffness matrix in finite element analysis becomes un-symmetric. Such un-symmetric matrices pose difficulties in the solution routines, resulting in numerical instability. On the other hand, if the interface is in tension, the contact between the matrix and the whisker ceases to exist and hence there will not be any frictional force. Hence it has Whisker Fig. 1. Longitudinal section of a typical CMC whisker, matrix and interface. showing the