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Crack Occurrence in Bodies with Gradient Polyconvex Energies

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Abstract

In a set of infinitely many reference configurations differing from a chosen fit region \({\mathscr {B}}\) in the three-dimensional space and from each other only by possible crack paths, a set parameterized by special measures, namely curvature varifolds, energy minimality selects among possible configurations of a continuous body those that are compatible with assigned boundary conditions of Dirichlet-type. The use of varifolds allows us to consider both “material phase" (cracked or non-cracked) and crack orientation. The energy considered is gradient polyconvex: it accounts for relative variations of second-neighbor surfaces and pressure-confinement effects. We prove existence of minimizers for such an energy. They are pairs of deformations and curvature varifolds. The former ones are taken to be SBV maps satisfying an impenetrability condition. Their jump set is constrained to be in the varifold support.

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Notes

  1. By Lusin’s theorem, measurable functions f into topological spaces with a countable basis can be approximated by continuous functions on arbitrarily large portions of their domain. Also, if \(f:\Omega \rightarrow {{\mathbb {R}}}^N\) is locally summable in Lebesgue’s sense, by the Lebesgue differentiation theorem almost every x in \(\Omega \) is a Lebesgue point of f, i.e., a point such that for some \(\lambda \in {{\mathbb {R}}}^N\)

    $$\begin{aligned} \lim _{r\rightarrow 0^+}\frac{1}{|B(x,r)|}\int _{B(x,r)}|f(z)-\lambda |\;\mathrm {d}x=0 \end{aligned}$$

    with B(x, r) a ball of radius r, centered at x, which Lebesgue measure is |B(x, r)|. The number \(\lambda =f(x)\) is called Lebesgue value of f at x.

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Acknowledgements

This work has been developed within the activities of the research group in “Theoretical Mechanics” of the “Centro di Ricerca Matematica Ennio De Giorgi” of the Scuola Normale Superiore in Pisa. PMM wishes to thank the Czech Academy of Sciences for hosting him in Prague during February 2020 as a visiting professor. We acknowledge also the support of GAČR-FWF project 19-29646L (to MK), GNFM-INDAM (to PMM), and GNAMPA-INDAM (to DM).

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Authors and Affiliations

  1. Czech Academy of Sciences, Institute of Information Theory and Automation, Pod Vodárenskou vĕží 4, CZ-182 00, Prague 8, Czechia

    Martin Kružík

  2. DICEA, Università di Firenze, via Santa Marta 3, I-50139, Firenze, Italy

    Paolo Maria Mariano

  3. DSMFI, Università di Parma, Parco Area delle Scienze 53/A, I-43134, Parma, Italy

    Domenico Mucci

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  1. Martin Kružík
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  2. Paolo Maria Mariano
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  3. Domenico Mucci
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Correspondence to Paolo Maria Mariano.

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Communicated by Arash Yavari.

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Kružík, M., Mariano, P.M. & Mucci, D. Crack Occurrence in Bodies with Gradient Polyconvex Energies. J Nonlinear Sci 32, 16 (2022). https://doi.org/10.1007/s00332-021-09769-3

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