Dynamic perfect plasticity and damage in viscoelastic solids

Abstract

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.

Models of Dynamic Damage and Phase-field Fracture, and their Various Time Discretisations

“…For the strong convergence of e el,τ , instead of the strategies (18) or (21), we now rely rather on the test of (59a,b) respectively by . u τ − .…”

“…A combination of damage with the perfect plasticity (i.e. G NH = H = 0 and κ 1 = 0) has been analysed in [18] by using the strategy (60) except that .. u was approximated by the backward second time difference.…”

“…After selection of weakly* convergent subsequences, we prove the strong convergence (15b). We use a slightly different estimation comparing to (18) based on a test by . u k − .…”

Fatigue and phase transition in an oscillating elastoplastic beam

“…We remark that our viewpoint is different from the contributions already present in the literature: on one hand we focus on the dynamics (compared, for instance, with [18,26,30] that go into the direction of rateindependent damage processes in nonlinear elasticity and/or thermo-elasto-plasticity, see also [8,28,29]) and also because the approach used in other papers, that interpret damage processes as a kind of phase transition in the material (see for instance [1][2][3]27]), is based on the idea that damage processes are driven by large deformations. Moreover, the multiyield character emerging from our bending problem does not seem to have been taken into consideration before, neither in the multidimensional setting: see for instance [5,7,21,24].…”

Energetic solutions for the coupling of associative plasticity with damage in geomaterials

“…Here C is independent of α but the relevant mixed term including the damage variable degenerates. Restricting to elasticity-damage models, in [8,45,41,33] some cases of complete damage are addressed, and also in the dynamic plasticity-damage model [22] Hooke's tensor is only positive semidefinite. Moreover, in plasticity-damage models the L 2 regularisation in the damage gradient is at the moment not enough, unless having the presence of strain gradient [13] or further assumptions on elastic strain [16]: an L p regularisation is required for p > n [12,22] or for p = n [15].…”