Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1311–1364 | Cite as

Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

  • Giuliano LazzaroniEmail author
  • Riccarda Rossi
  • Marita Thomas
  • Rodica Toader


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.


Partial damage Rate-independent systems Elastodynamics Phase-field models Heat equation Energetic solutions Local solutions 

Mathematics Subject Classification

35Q74 74H20 74R05 74C05 74F05 



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. 1.DMA, Università degli Studi di Napoli Federico IINaplesItaly
  2. 2.DIMI, Università degli Studi di BresciaBresciaItaly
  3. 3.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  4. 4.DMIF, Università degli Studi di UdineUdineItaly

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