On the Passage from Nonlinear to Linearized Viscoelasticity

Abstract

We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Our main tools are the theory of gradient flows in metric spaces and Γ-convergence.2010 Mathematics Subject Classification. 74D05, 74D10, 35A15, 35Q74, 49J45.

Thermoviscoelasticity in Kelvin–Voigt Rheology at Large Strains

“…A similar regularization approach to isothermal large-strain viscoelasticity was considered in [20], where the H(y) is multiplied with a small parameter that vanishes slower than the loading. Hence, the authors are able to show that their solutions are sufficiently close to the identity which allows them to exploit a simpler Korn's inequality obtained by a perturbation argument.…”

“…After the precise definition of our notion of solution, Theorem 2.2 provides the main existence result for global-in-time solutions for the large-strain thermoviscoelastic system, while Corollary 2.3 gives the corresponding existence result for viscoelasticity at large strain and at constant temperature, which, to the knowledge of the authors, is also new. A related result for isothermal large-strain viscoelasticity is derived in [20], but there the limit of small strains is treated.…”

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

“…Thus, to ensure that our anisotropic penalization does not perturb the qualitative small-strain behavior of (1.3), it is essential that η ≪ ε −1 . Let us mention that related problems in the framework of nonsimple materials have recently been studied in the settings of viscoelasticity [33] and multiwell energies [1]. There, under strong penalization of the full second gradient, rigorous counterparts of the formal linearization (1.6) are performed by Γ-convergence.…”

Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

“…In their seminal work [33], Dal Maso, Negri, and Perivale performed a nonlinear-to-linear analysis in terms of suitably rescaled displacement fields and proved the convergence of minimizers for corresponding boundary value problems. This study has been extended in various directions, including different growth assumptions on the stored energy densities [1], the passage from atomistic-to-continuum models [13,55], multiwell energies [2,54], plasticity [51], and viscoelasticity [43].…”