Keywords: multiscale modelling, upscaling, hybrid finite element method, Trefftz stress element, micromechanics.

An accurate simulation technique accounting for the full microstructural detail of heterogeneous materials is an important tool in the quest for highly optimized fabrication. This paper presents a numerically efficient and accurate finite element for heterogeneous materials that fully captures the microscopic mechanical fields without the need for the finite element mesh to explicitly resolve the microscopic geometry.

Classical homogenization techniques, that are built on the premise that either an irregular or periodic representative volume element (RVE) exists, fail when one attempts to reproduce the detailed microscopic response from the overall macroscopic performance [1,2,3]. This paper presents a synergistic, accurate and yet computationally efficient upscaling technique that couples the favourable features of both the hybrid Trefftz stress (HTS) finite element method (FEM), well suited for the structural scale, and analytical micromechanics. The result is a composite HTS element for the analysis of heterogeneous materials.

The HTS finite element formulation [4] utilised in this work is characterized by the approximation of stresses within the domain of the element, chosen to automatically satisfy equilibrium. The stiffness of such an element can be expressed using a boundary integral of the stress approximation fields and makes explicit use of the constitutive law. Therefore, these functions can take into account a variety of physical effects resulting from the material microstructure. In this paper, the perturbation of the stress fields due to the presence of inclusions determined by the micromechanical closed form solution are used to augment the classical Trefftz approximation functions. The response resulting from the mutual interaction of multiple heterogeneities is obtained by superposition of the contribution from each single inclusion using a self-balancing algorithm.

The efficiency and accuracy of the key components of the proposed strategy (micromechanical solution, self-balancing algorithm, etc.) have been explored by means of an academic problem comprising a cube containing three closely placed ellipsoidal inclusions embedded in a matrix.

Although the proposed strategy is limited to simple (elliposidal) inclusion shapes, it still has the potential to be applicable to a wide range of composite materials, for example fibre-reinforced concrete, porous media, functionally graded materials, etc. Furthermore, the method is currently applied to elastic composites but has been designed to be extended to include fracturing, heterogeneous, quasi-brittle materials.

1

J. Zeman, M. Šejnoha, "From random microstructures to representative volume elements", Modelling and Simulation in Materials Science and Engineering, 15, S325, 2007. doi:10.1088/0965-0393/15/4/S01

2

L. Kaczmarczyk, C. Pearce, N. Bicanic, "Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization", Int. J. Numer. Meth. Engng, 74, 506-522, 2008. doi:10.1002/nme.2188

3

T. Mori, K. Tanaka, "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metallurgica, 21(5), 571-574, 1973. doi:10.1016/0001-6160(73)90064-3

4

L. Kaczmarczyk, C.J. Pearce, "A corotational hybrid-Trefftz stress formulation for modelling cohesive cracks", Computer Methods in Applied Mechanics and Engineering, 198(15-16), 1298-1310, 2009. doi:10.1016/j.cma.2008.11.018

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