Keywords: fracturing, concrete, multi-grid, preconditioner, hybrid-Trefftz, finite elements.

This paper sets out a modelling strategy for simulating fracturing in concrete where the fine-scale heterogeneities are fully resolved. The fine-scale is modelled using an extension to the original hybrid-Trefftz stress element formulation presented by Teixeira De Freitas [1]. This extended element formulation is presented as an efficient framework for modelling propagating cohesive cracking in heterogeneous materials where multiple cracks, crack branching and crack coalescence are the norm and where different material models are required for the various constituents, i.e. aggregate, mortar and interface.

Cohesive cracks are limited to element interfaces and small strains are assumed. Since all element matrices (e.g. stiffness matrix) can be expressed in terms of boundary integrals, an element can have an arbitrary non-convex shape. Furthermore, since the displacement basis is defined independently on each inter-element surface, the overall bandwidth of the stiffness matrix is very small and computationally efficient to solve. The stress approximation is a priori constrained to satisfy the equilibrium condition locally and the boundary displacements are defined independently of the stresses; thus, inconsistencies between the approximation of the stress field within the finite elements and the distribution of cohesive tractions on the interface are avoided. The simplicity and robustness of the presented approach make it an attractive alternative to displacement based finite element approaches.

The very large system of algebraic equations that emerges from the detailed resolution of the fine scale structure requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials. Thus, this paper proposes an extension to the work of Miehe and Bayreuther [2], whereby a preconditioner is constructed using a multi-grid strategy that utilizes scale transition techniques derived for computational homogenization [3-6]. Here, the multi-grid concept is restricted to two scales: the fine-scale heterogeneous structure is discretised using hybrid-Trefftz stress elements and the coarse mesh comprises C¹-continuous elements that are adapted from the plate element presented by Kasparek [7].

The performance of the proposed strategy is demonstrated with a number of numerical examples, with particular emphasis on the tensile loading of concrete dog bone specimens [8]. The computational efficiency and scalability with respect to parallel performance are also demonstrated.

[1]

J.A.T. Teixeira De Freitas, "Formulation of elastostatic hybrid-Trefftz stress elements", Computer Methods in Applied Mechanics and Engineering, 153, 127-151, 1998. doi:10.1016/S0045-7825(97)00042-X

[2]

C. Miehe, C.G. Bayreuther, "On multiscale FE analyses of heterogeneous structures: From homogenization to multigrid solvers", International Journal for Numerical Methods in Engineering, 71, 1135-1180, 2007. doi:10.1002/nme.1972

[3]

V.G. Kouznetsova, "Computational homogenization for the multi-scale analysis of multi-phase materials", Ph.D., TU Eindhoven, The Netherlands, 2002.

[4]

L. Kaczmarczyk, C.J. Pearce, N. Bicanic, "Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization", International Journal for Numerical Methods in Engineering, 74, 506-522, 2008. doi:10.1002/nme.2188

[5]

F. Feyel, J.L. Chaboche, "FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials", Computer Methods in Applied Mechanics and Engineering, 183, 309-330, 2000. doi:10.1016/S0045-7825(99)00224-8

[6]

C. Miehe, "Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials", Computer Methods in Applied Mechanics and Engineering, 171, 387-418, 1999. doi:10.1016/S0045-7825(98)00218-7

[7]

E.M. Kasparek, "An efficient triangular plate element with C-1-continuity", International Journal for Numerical Methods in Engineering, 73, 1010-1026, 2008. doi:10.1002/nme.2108

[8]

M.R.A. van Vliet, "Size effect in tensile fracture of concrete and rock", PhD, Delft University of Technology, 2000.

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