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Finite strain HFGMC analysis of damage evolution in nonlinear periodic composite materials

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Abstract

This work studies the evolution of damage in periodic composites with hyperelastic constituents prone to mechanical degradation under sufficient loading. The micromechanical problem is solved for quasistatic far-field loading for plane-strain conditions, using the finite strain high-fidelity general method of cells (FSHFGMC) approach to discretize the conservation equations. Damage is treated as degradation of material cohesion, modeled by a material conservation law with a stress-dependent damage-source (sink) term. The two-way coupled formulation with the internal variable representing damage is reminiscent of the phase-field approach to gradual cracks growth, albeit with a mechanistically derived governing equation, and with important theoretical differences in consequences. The HFGMC approach consists in enforcing equilibrium in each phase (in the cell-average sense) by stress linearization, using instantaneous tangent moduli, and subsequent iterative enforcement of continuity conditions, a formulation arguably natural for composite materials. The inherent stiffness of the underlying differential equations is treated by use of a predictor–corrector scheme. Various examples are solved, including those of porous material developing cracks close to the cavity, for various sizes and shapes of the cavity, damage in a two-phase composite of both periodic and random structure, etc. The proposed methodology is physically tractable and numerically robust and allows various generalizations.

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Notes

  1. The physical interpretation of the mass flux term can be that crumbs will push each other from the locus of their generation, since there would be no significant attraction forces between them, only repelling forces. Therefore, phenomenologically, eventually there would be a flow of them in the direction opposite to the gradient, the same way it happens with heat (it can be argued that the crumbs will spread due to the entropic principle, or the Liouville dynamical principle, and the spreading would be away from the generation points, or in the direction opposite to the gradient).

  2. The left-hand side includes the smooth part of crumbs transport, which makes no contribution in a short period, and the nonsmooth source term, which makes contribution proportional to the number of rupture events in the short period. The probability for a certain number of rupture events in a continuum-level timescale \(\Delta {t}\) can be assumed to have Gaussian distribution, rather than, say, a power-law-tailed distribution, since the problem is not scale-free, like, say, plasticity, a fact indicated by the divergence operator and \(\bar{l}\). Consequently, the amount of rupture in \(\Delta {t}\) is integrable (an integral over \(\Delta {t}\) of a fixed/typical number of delta functions yields a constant).

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Acknowledgements

KYV gratefully acknowledges the support from the Israel Science Foundation (ISF-394/20).

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Authors and Affiliations

  1. School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, 69978, Tel Aviv, Israel

    Nathan Perchikov & Jacob Aboudi

  2. Faculty of Civil Engineering, Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Konstantin Y. Volokh

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  1. Nathan Perchikov
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  2. Jacob Aboudi
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  3. Konstantin Y. Volokh
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Correspondence to Nathan Perchikov.

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Perchikov, N., Aboudi, J. & Volokh, K.Y. Finite strain HFGMC analysis of damage evolution in nonlinear periodic composite materials. Arch Appl Mech 93, 4361–4386 (2023). https://doi.org/10.1007/s00419-023-02497-y

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