Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes

Maso, Gianni Dal; Scala, Riccardo

doi:10.1007/s10884-014-9409-7

Cited by 31 publications

(30 citation statements)

References 21 publications

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“…benefitting from having the kinetic energy on the left-hand side, and the by-part integration of the Dirichlet loading term. We stress that the last term in (25) can be rigorously defined as in (40). This way, we can see the estimates…”

Section: Energetics Of the Model And First Estimatesmentioning

confidence: 86%

Dynamic perfect plasticity and damage in viscoelastic solids

2019

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In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto‐plasticity, and unidirectional rate‐dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis relies on the notions of plastic‐strain measures and bounded‐deformation displacements, on sophisticated time‐regularity estimates to establish a duality between acceleration and velocity of the elastic displacement, on the theory of rate‐independent processes for the energy conservation in the dynamical‐plastic part, and on the proof of the strong convergence of the elastic strains. Existence of suitably defined weak solutions (even conserving energy) is proved rather constructively by using a staggered two‐step time discretization scheme.

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Section: Energetics Of the Model And First Estimatesmentioning

confidence: 86%

Dynamic perfect plasticity and damage in viscoelastic solids

2019

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“…We can rephrase the problem, commonly referred as quasistatic limit issue, as follows: is it true that quasistatic evolutions can be approximated by dynamic ones when the external loading becomes slower and slower, or equivalently the speed of internal vibrations becomes faster and faster? Nowadays only partial results on the theme are available; we refer for instance to [19] and [25] for damage models, to [8] in a case of Figure 1. The deformation of the film at time t is represented by the displacement (x 0 , 0) → (x 0 + h(t, x 0 ), u(t, x 0 )).…”

Section: Introductionmentioning

confidence: 99%

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity

Riva¹

2019

J Nonlinear Sci

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In this paper we contribute to studying the issue of quasistatic limit in the context of Griffith's theory by investigating a one-dimensional debonding model. It describes the evolution of a thin film partially glued to a rigid substrate and subjected to a vertical loading. Taking friction into account and under suitable assumptions on the toughness of the glue, we prove that, in contrast to what happens in the undamped case, dynamic solutions converge to the quasistatic one when inertia and viscosity go to zero, except for a possible discontinuity at the initial time. We then characterise the size of the jump by means of an asymptotic analysis of the debonding front.

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“…In fact, our study is akin to the vanishing-viscosity and inertia analysis that has been addressed, in the momentum equation only, for isothermal, rate-independent processes with dynamics in [Rou09,Rou13a], leading to an energetic-type notion of solution. We also refer to [DS14,Sca17] for a combined vanishing-viscosity limit in the momentum equation and in the flow rule, in the cases of perfect plasticity and delamination, respectively…”

Section: Introductionmentioning

confidence: 99%

Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

et al. 2018

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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.2010 MSC: 35Q74, 74H20, 74R05, 74C05, 74F05,

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Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes

Cited by 31 publications

References 21 publications

Dynamic perfect plasticity and damage in viscoelastic solids

Dynamic perfect plasticity and damage in viscoelastic solids

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity

Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

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