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Assessment of mechanical properties of bone trabeculae as an inverse problem of heterogeneous material modeling

Janc K., Tarasiuk J., Lipinski P., Bonnet A. S., Wronski S.

Archives of Mechanics

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2020

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Vol. 72, nr 5

385--414

EN

In this study, the trabecular bone was treated as a composite material that consists of a bone matrix weakened by ellipsoidal pores. Under the hypothesis that all information concerning the local properties and the microarchitecture are “encrypted” in the apparent properties of a given volume element (VE) of the bone, a method of retrieving these data was proposed. Software based on a genetic algorithm, combined with the incremental scale transition method was developed to this end. To test the approach, μCT measurements of four bone samples were performed providing their real micro-architecture. Tensors of apparent properties of the samples were next computed by numerical (finite element) homogenization method for a large range of the elastic properties of trabeculae. They were considered as the fitness function for the proposed algorithm. Very good agreement was found between the obtained and target values of the apparent elastic properties of the samples and volume fraction of pores. The approach is fast and accurate enough in comparison to the finite element homogenization. As an auxiliary result it was shown that the anisotropy of apparent elastic properties is mainly related to the microarchitecture of the bone, not to the intrinsic properties of trabeculae.

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Response discontinuities in the solution of the incremental Mori-Tanaka scheme for elasto-plastic composites

Sadowski P., Kowalczyk-Gajewska K., Stupkiewicz S.

Archives of Mechanics

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2017

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Vol. 69, nr 1

3--27

EN

The incremental Mori–Tanaka model of elasto-plastic composites is discussed, and the corresponding finite-step formulation is shown to lead to discontinuities in the overall response at the instant of elastic-to-plastic transition in the matrix. Specifically, two situations may be encountered: the incremental equations may have two solutions or no solution. In the former situation, switching between the two solutions is associated with a jump in the overall stress. Response discontinuities are studied in detail for a special case of proportional deviatoric loading. The discontinuities constitute an undesirable feature of the incremental Mori–Tanaka scheme that apparently has not been discussed in the literature so far. Remedies to the related problems are briefly discussed.

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Modelling effective properties of composite materials using the inclusion concept. General considerations

Vieville P., Bonnet A. S., Lipiński P.

Archives of Mechanics

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2006

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Vol. 58, nr 3

207-239

EN

This paper is devoted to some general theoretical considerations concerning the modelling of the effective properties of composite materials based on the inclusion concept. Starting from the kinematical integral equation for inhomogeneous materials, all principal homogenisation methods are reviewed and analysed. Special attention is focused on three approaches, namely the self-consistent scheme, the Mori-Tanaka method and incremental procedure derived from the differential scheme. Mono-site and multi-site versions of these approximate solutions are considered. Limitations of the traditional self-consistent scheme are recognized. Improvements are proposed such as composite or coated inclusions and the incremental method mentioned above. Direct and iterative procedures allowing the determination of strain concentration tensors derived from the integral equation are established. The numerical implementation of all the schemes presented in this article will be considered in the next paper. Extreme configurations will be analysed such as composites with voids or very stiff inclusions with respect to matrix properties.