Keywords: homogenization, composites with imperfect interfaces, duality-based solvers, sequentially-linear analysis.

Particle-reinforced composites, and the fibrous composites in particular, present a progressive class of materials with a steadily increasing importance in virtually all areas of structural engineering, [1]. It is now generally accepted that one of the key factors governing the mechanical performance of composite materials is the interfacial debonding: the partial separation of reinforcement from the matrix phase when the surface tractions locally exceed the interfacial strength.

To overcome the well-known problems experienced with the primal finite element method with interfacial elements, in the current contribution we investigate the applicability of duality-based solvers to the homogenization of composites with imperfect interfacial bonding. The basic theoretical framework is provided using the first-order strain-based homogenization theories and the finite element tearing and interconnecting (FETI) method proposed by Farhat and Roux [2], extended later by Bittnar and Kruis [3] to accommodate compliant interfaces between individual domains in the linearly elastic regime.

In order to allow for analysis of composites with a general non-linear constitutive law at the interface, the basic principles of the sequentially linear analysis method [4] are adopted, in which the original non-linear problem is converted into a sequence of linear analyses with gradually deteriorated elastic interfaces. A systematic numerical study is performed to identify the most efficient numerical approach to the solution of resulting problems. In particular, we consider the effects of: the modelling of a perfect debonding state (static condensation vs. artificial compliance), a variant of the conjugate gradient solver (basic modified conjugate gradient vs constraint orthogonalization and the orthogonalization of the objective function and the effect of preconditioning (lumped preconditioner and the preconditioner based on the Schur complements). All algorithms incorporate an efficient engineering procedure to ensure the interpenetration of inclusions and the matrix phase. Results of this study clearly demonstrate the superiority of the preconditioned static condensation approach to the alternative solvers, thereby providing a basic step towards efficient and scalable simulation of multi-inclusion unit cells within computational multi-scale strategies.

1

B. Cox, Q. Yang, "In quest of virtual tests for structural composites", Science, 314(5802), 1102-1107, 2006. doi:10.1126/science.1131624

2

C. Farhat, F.X. Roux, "A method of finite element tearing and interconnecting and its parallel solution algorithm", International Journal for Numerical Methods in Engineering, 32(6), 1205-1227, 2005. doi:10.1002/nme.1620320604

3

J. Kruis, Z. Bittnar, "Reinforcement-Matrix Interaction Modeled by FETI Method", in U. Langer, M. Discacciati, D.E. Keyes, O.B. Widlund, W. Zulehner (Editors), Domain Decomposition Methods in Science and Engineering XVII, Volume 60 of Lecture Notes in Computational Science and Engineering, pages 567-573, Springer Verlag, 2008. doi:10.1007/978-3-540-75199-1_71

4

M.J. DeJong, M.A. Hendriks, J.G. Rots, "Sequentially linear analysis of fracture under non-proportional loading", Engineering Fracture Mechanics, 75(18), 5042-5056, 2008. doi:10.1016/j.engfracmech.2008.07.003

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