Keywords: damage, energetic solution, FETI, local state method, quasi-brittle interfaces, sequentially linear analysis, snap-back.

In this paper, an efficient energy- and duality-based approach to the numerical modelling of heterogeneous structures with imperfect interfaces based on sequential linear analysis, which leads to a robust numerical tool for the modelling of brittle and quasi-brittle interfaces is introduced. Brittle and quasi-brittle interfaces are characterised by a rapid dissipation of a reversible stored energy which is commonly accompanied by the so-called snap-back phenomenon. It leads to numerical instability of standard mathematical approaches.

The small-strain setting model, presented, is subservient to quasistatic processes and individual linear elastic domains are joined by (quasi-)brittle friction-less interfaces. The method of local state is adopted and a current interface states is defined by one internal variable, the well-known scalar damage parameter. A constitutive law of interfaces is adopted from cohesive zone models as introduced by Pandolfi and Ortiz [1]. Finally, proportional loading is assumed.

The common discretisation of the problem using a primal variant of the finite element method is substituted by the finite element tearing and interconnecting (FETI) method which was originally developed as a domain decomposition method by Farhat and Roux [2]. In the model, the FETI method is used as a transformation tool for the conversion of the problem to discretised dual variables and discretized surface tractions on the interfaces. It is useful for the reason, that all non-linear phenomena happen only on the interfaces and the satisfaction of the contact condition is more natural by this way. In addition, there are efficient duality based solvers that are possible to use for the modelling of the contact. Not least, the FETI method almost rigid interfaces to be treated without numerical problems connected with matrix conditioning. It should be mentioned, that the first application of the FETI method for a modelling of simple constitutive laws on interfaces was introduced by Kruis and Bittnar [3].

The resulting non-linear discrete problem with (quasi-)brittle interfaces is generally very sensitive to the selection of a load or (pseudo-)time increment. This critical part of analysis is overcome by so-called sequential linear analysis (SLA) introduced by Rots [4]. The SLA, instead of an iterative solution procedure commonly applied to the non-linear problem, divides the problem to a finite number of linear analyses. This is achieved by an incremental increasing of the dissipation instead of the load or pseudo-time. This approach is able to describe the overall response of structures with (quasi-)brittle interfaces, even the snap-backs parts. This conclusion is supported by the numerical experiment of a two-layered laminated beam failing as a result of a mode-II conditions and characterised by snap-back behaviour.

1

M. Ortiz, A. Pandolfi, "Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis", International Journal for Numerical Methods in Engineering, 44(9), 1267-1282, 1999. doi:10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.3.CO;2-Z

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, 1991. doi:10.1002/nme.1620320604

3

J. Kruis, Z. Bittnar, "Reinforcement-Matrix Interaction Modeled by FETI Method", in "Domain Decomposition Methods in Science and Engineering", Springer, XVII, 567-573, 2008. doi:10.1007/978-3-540-75199-1_71

4

J.G. Rots, S. Invernizzi, "Regularized sequentially linear saw-tooth softening model", Int. J. Numer. Anal. Meth. Geomech., 28(7-8), 821-856, 2004. doi:10.1002/nag.371

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