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Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

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Abstract

We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio–Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.

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Acknowledgements

This work has been supported by the Italian Ministry of Education, University, and Research through the PRIN 2010-11 grant for the project Calculus of Variations, by the European Research Council through the two Advanced Grants Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture (290888) and Analysis of Multiscale Systems Driven by Functionals (267802), and by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) through the project Modelli variazionali per la propagazione di fratture, la delaminazione e il danneggiamento. G.L. acknowledges also the support of the University of Würzburg, of the DFG grant SCHL 1706/2-1, of SISSA, of the University of Vienna, and of the FWF project P27052. This paper was submitted on October 24, 2014; the first referee report was received by the authors on December 6, 2017.

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Authors and Affiliations

  1. DMA, Università degli Studi di Napoli Federico II, via Cintia, Monte S. Angelo, 80126, Naples, Italy

    Giuliano Lazzaroni

  2. DIMI, Università degli Studi di Brescia, Via Valotti 9, 25133, Brescia, Italy

    Riccarda Rossi

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Marita Thomas

  4. DMIF, Università degli Studi di Udine, Via delle Scienze 206, 33100, Udine, Italy

    Rodica Toader

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  1. Giuliano Lazzaroni
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Correspondence to Giuliano Lazzaroni.

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Lazzaroni, G., Rossi, R., Thomas, M. et al. Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia. J Dyn Diff Equat 30, 1311–1364 (2018). https://doi.org/10.1007/s10884-018-9666-y

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