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Minimization of maximal von Mises stress in porous composite microstructures using shape and topology optimization

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Abstract

Strength-oriented optimization of porous periodic microstructures impacts on efficient design of load-bearing lightweight structures avoiding mechanical failure. In this work, the maximal von-Mises stress, predicted by homogenization theory on a planar representative unit-cell domain, is minimized using either shape or topology design changes. Plane stress and linear behaviour are assumed. Two benchmarks problems are revisited, bulk and shear loads. Firstly, a fully stressed design is sought on extremal materials, rank-2 laminates, for comparative purposes. The lamination factors are handled analytically to find a relationship between stress and material volume fraction. Secondly, one numerically minimizes the peak von-Mises stress of single-material unit-cell varying shape or topology. In shear, the optimal topology design tends to approximate its rank-2 counterpart. Under bulk load, the peak stresses are further decreased by allowing an inhomogeneous solid phase. An extra material discrete phase is included in the shape problem while functionally graded material solutions are allowed in the topology problem. The single-material optimal results are consistent with the theoretical ones. This validates not only the proposed shape parameterization problem, based on supershapes, but also the proposed stress-based formulation for microstructural topology optimization, not yet extensively addressed in the literature. The multi-material approaches, to the extent of ideally allowing each spatial point to have a different material property, show that by increasing the material design freedom one achieves lower peak stresses.

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Acknowledgements

The authors wish to thank Professor Krister Svanberg (Royal Institute of Technology, Stockholm, Sweden) for the MMA optimization code.

Funding

This study received financial support from Fundação para a Ciência e a Tecnologia (FCT–MCTES) through the projects UIDB/00667/2020 (UNIDEMI) and UIDB/50022/2020 (IDMEC/LAETA) and PhD scholarship SFRH/BD/136744/2018.

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Authors and Affiliations

  1. UNIDEMI, Faculty of Sciences and Technology, Universidade Nova de Lisboa, FCT, 2829-516, Caparica, Portugal

    Pedro G. Coelho & Fábio M. Conde

  2. Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300, 3001, Leuven, Belgium

    Bruno C. Barroca

  3. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

    Fábio M. Conde & José M. Guedes

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  1. Pedro G. Coelho
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  4. José M. Guedes
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Correspondence to Pedro G. Coelho.

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Replication of results

Our manuscript details the input and output data by means of figures and tables such that results can be reproduced and confirmed. In addition to that, supershape parameters are provided in Appendix 1 Tables 5, 6, 7, and 8. Equations are presented in such way that readers can implement or solve them. Actually, we also use in this paper implementations that are already available in the literature as the homogenization and MMA codes.

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Appendix 1 Values for the supershape parameters

Appendix 1 Values for the supershape parameters

Tables 5, 6, 7 and 8 present the values found for the supershape parameters in the SO problems solved.

Table 5 Supershape parameters for SMSO minimizing maximal stress and bulk loading
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Table 6 Supershape parameters for SMSO minimizing compliance and shear loading
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Table 7 Supershape parameters for SMSO minimizing maximal stress and shear loading
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Table 8 Supershape parameters for BMSO minimizing maximal stress and bulk loading
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Coelho, P.G., Barroca, B.C., Conde, F.M. et al. Minimization of maximal von Mises stress in porous composite microstructures using shape and topology optimization. Struct Multidisc Optim 64, 1781–1799 (2021). https://doi.org/10.1007/s00158-021-02942-y

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