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Computational modeling of damage evolution in unidirectional fiber reinforced ceramic matrix composites M. E. Walter, G. Ravichandran, M. Ortiz Abstract A ®nite element model for investigating damage evolution in brittle matrix composites was developed. This model ing is based on an axisy mmetric unit cell composed of a ®ber and its surrounding matrix. The unit cell was discretized into linearly elastic elements for the ®ber and the matrix and cohes ive elements which allow cracking in the matrix, ®ber-matrix interface, and ®ber. The cohesive elements failed according to critical stress and critical energy release rate criteria (in shear and/or in tension). The tension and shear aspects of failure were uncoupled. In order to obtain converged solutions for the axisym- metric composite unit cell problem, inertia and viscous dampi ng were added to the formulati on, and the resulting dynamic problem was solved implicitly using the New- mark Method. Parametric studies of the interface tough- ness and strength and the matrix toughness were performed. Details of the propagation of matrix cracks and the initiation of debonds were also observed. 1 Introduction Tough ening of brittle matrix composite s is achie ved through the energy absorbing processes of damage de- velopment. The toughness of a brittle matrix composite is usually signi®cantly higher than the toughnesses of the individual constituents. For unidirectional ®ber reinforced ceramic matrix composi tes, damage takes the follow ing forms: matrix cracking (with blunting and ®ber bridging), debonding of ®bers, and isolated ®ber failure with pull-out and sliding. Of primary interest in this situation are cracks which propagate through the matrix and interact with strong and stiff ®bers. Thorough understanding of the constitutive response of composites requires knowledge of the formation and propagation of these forms of damage. In addition, further materi al impr oveme nts related to toughening will be achieved by better understanding the sequence of damage propagation. In the present investigation, the overall mechanical re- sponse of a ceramic matrix composite is simulated by a numerical model for a ®ber-matrix unit cell. The model allows for the development of multiple matrix cracks and ®ber debonding. The numerical model developed in the present study is based on a cohesive element formulation similar to that of Needleman (1987). The cohesive law in Needleman's (1987) original paper was modi®ed to include shear failure in the cohesive zone (Tvergaard 1990; Nee- dleman 1990a,b), and more recently a coupled normal and shear response was developed by Xu and Needleman (1993). All of these cohesive element models have been employed to study plastically deforming materials with re- inforcement interfaces. Ortiz and Suresh (1993) employed cohesive elements in their investigation of intergranular fracture in ceramic materials, and Ungsuwarungsri and Knauss (1987) studied fracture of composites and adhe- sives. A cohesive element formulation has also been used in a dynamic setting by Camacho and Ortiz (1995) to study impact damage in brittle materials and by Maurisch and Ortiz (1995) to simulate high speed machining. 2 Finite element formulation Past experience with time-independent simulations re- vealed severe convergence problems when multiple cohe- sive elements began to fail. This is presumably due to large strain energy relea ses and subs equent rapid accele rations of nodes. It is physically realistic that in the presence of rapid cracking, inertial terms are important and internal material damping should be accounted for. Therefore the ®nite element method for the present investigation is applied to the elastodynamic equations. The principle of virtual work with time dependent terms is written at step n 1 as follows: X o P n1 X $ u dV o X o f n1 À qa n1 À jv n1 Á u dV o C o t n1 Á u dS o Y 1 Computational Mechanics 20 (1997) 192±198 Ó Springer-Verlag 1997 192 Communicated by P. E. O'Donoghue, M. D. Gilchrist, K. B. Broberg, 6 January 1997 M. E. Walter I , G. Ravichandran, M. Ortiz Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125, USA Present address: 1 The Ohio State University, Applied Mechanics Program, Columbus, OH 43210, USA Correspondence to: M. E. Walter Supported by a Presidential Young Investigators Award from the National Science Foundation to G. Ravichandran, grant No. MSS-9157846. The autho rs gratefully acknowledge the Caltech Concu rrent Supercomputing Facilities for providing time on the Jet Propulsion Laboratory's CRAY Y-MP2E/232.
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