We consider a dynamical elasto-plasticity system with Kelvin--Voigt viscosity and linear kinematic hardening of Melan--Prager type. The model is formulated in a variational framework in which a constraint set for the stress evolves in time and is translated by an internal (backstress) variable. As a consequence, the flow rule is coupled with an equation of motion through a quasi-variational structure, since the constraint set depends on the unknown internal variable. To construct solutions, we employ Rothe's method and introduce a projection-based time discretization. Each time step consists of solving a linear viscous-elastic subproblem to obtain a trial stress, followed by a projection onto the translated constraint set. We establish stability of the resulting discrete solutions under suitable norms. By compactness and passage to the limit as the time step tends to zero, we prove existence of a weak solution in the variational sense, and uniqueness follows from an energy argument. The results cover time-dependent yield bounds without assuming spatial continuity or a strictly positive lower bound, and the discretization provides a constructive basis for numerical approximation.
We prove a homogenization result for Hencky plasticity functionals with non-convex potentials. We also investigate the influence of a small hardening parameter and show that homogenization and taking the vanishing hardening limit commute.
We present a variational principle governing the quasistatic evolution of a linearized elastoplastic material. In case of linear hardening, the novel characterization allows to recover and partly extend some known results and proves itself to be especially well-suited for discussing general approximation and convergence issues. In particular, the variational principle is exploited in order to prove in a novel setting the convergence of time and space-time discretizations as well as to provide some possible a posteriori error control.
The paper studies the asymptotic analysis of a model coupling elastoplasticity and damage depending on three parameters -- governing viscosity, plastic hardening, and convergence rate of plastic strain and displacement to equilibrium -- as they vanish in different orders. The notion of limit evolution obtained is proven to coincide in any case with a notion introduced by Crismale and Rossi in 2019; moreover, such solutions are closely related to those obtained in the vanishing-viscosity limit by Crismale and Lazzaroni in 2016, for the analogous model where only the viscosity parameter was pres…
We consider a dislocation-based rate-independent model of single crystal gradient plasticity with isotropic or linear kinematic hardening. The model is weakly formulated through the so-called primal form of the flow rule as a variational inequality for which a result of existence and uniqueness is obtained using the functional analytical framework developed by Han-Reddy.
The purpose of this article is to prove the Hölder continuity up to the boundary of the displacement vector and the microrotation matrix for the quasistatic, rate-independent Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects. This model is of non-monotone and non-associated type. In the case of two space dimensions we use the hole-filling technique of Widman and the Morrey's Dirichlet growth theorem.
In this paper we use convex analysis and variational inequality methods to establish an existence result for a model of infinitesimal rate-independent gradient plasticity with kinematic hardening and plastic spin, in which the local backstress tensor remains symmetric. The model features a defect energy contribution which is quadratic in the dislocation density tensor Curl p, giving rise to nonlocal non-symmetric kinematic hardening. Use is made of a recently established Korn's type inequality for incompatible tensor fields. The solution space for the non-symmetric plastic distortion is natura…
We propose an extension of the cyclic hardening plasticity model formulated by Armstrong and Frederick which includes micropolar effects. Our micropolar extension establishes coercivity of the model which is otherwise not present. We study then existence of solutions to the quasistatic, rate-independent Armstrong-Frederick model with Cosserat effects which is, however, still of non-monotone, non-associated type. In order to do this, we need to relax the pointwise definition of the flow rule into a suitable weak energy-type inequality. It is shown that the limit in the Yosida approximation proc…
We prove the lower semicontinuity of functionals of the form \[ \int \limits_Ω\! V(α) \, \mathrm{d} |\mathrm{E} u| \, , \] with respect to the weak converge of $α$ in $W^{1,γ}(Ω)$, $γ> 1$, and the weak* convergence of $u$ in $BD(Ω)$, where $Ω\subset \mathbb{R}^n$. These functional arise in the variational modelling of linearised elasto-plasticity coupled with damage and their lower semicontinuity is crucial in the proof of existence of quasi-static evolutions. This is the first result achieved for subcritical exponents $γ< n$.
We consider two most studied standard models in the theory of elasto-plasticity with hardening in arbitrary dimension $d\ge 2$, namely, the kinematic hardening and the isotropic hardening problem. While the existence and uniqueness of the solution is very well known, the optimal regularity up to the boundary remains an open problem. Here, we show that in the interior we have Sobolev regularity for the stress and hardening while for their time derivatives we have the "half" derivative with the spatial and time variable. This was well known for the limiting problem but we show that these estimat…
In this paper we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor Curl(p) giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and sat…
In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures $α_k\,\mathrm{E}u_k$ with respect to the weak convergence $α_k\rightharpoonup α$ in $W^{1,n}(Ω)$ and the weak$^*$ convergence $u_k\stackrel{*}\rightharpoonup u$ in $BD(Ω)$, $\mathrm{E}$ denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form $α\,\mathrm{E}u+η$, with $η$ supported on an at most countable set.
Consideration is given to a non-convex variational model for a shear experiment in the framework of single-crystal linearised plasticity with infinite cross-hardening. The rectangular shear sample is clamped at each end, and is subjected to a prescribed horizontal or diagonal shear, modelled by an appropriate hard Dirichlet condition. We ask: how much energy is required to impose such a shear, and how does it depend on the aspect ratio? Assuming that just two slip systems are active, we show that there is a critical aspect ratio, above which the energy is strictly positive, and below which it …
In this work we establish the well-posedness for infinitesimal dislocation based gradient viscoplasticity with isotropic hardening for general gradient monotone plastic flows. We assume an additive split of the displacement gradient into non-symmetric elastic distortion and non-symmetric plastic distortion. The thermodynamic potential is augmented with a term taking the dislocation density tensor $\operatorname{Curl} p$ into account. The constitutive equations in the models we study are assumed to be of self-controlling type. Based on the generalized version of Korn's inequality for incompatib…
A rate-independent model coupling small strain associative elasto-plasticity and damage is studied via a 'vanishing-viscosity' analysis with respect to all the variables describing the system. This extends the analysis performed for the same system in [Crismale-Lazzaroni 2016], where a vanishing-viscosity regularization involving only the damage variable was set forth. In the present work, an additional approximation featuring vanishing plastic hardening is introduced in order to deal with the vanishing viscosity in the plastic variable. Different regimes are considered, leading to different n…
We prove existence of globally stable quasistatic evolutions, referred to as energetic solutions, for a model proposed by Marigo and Kazymyrenko in 2019. The behaviour of geomaterials under compression is studied through the coupling of Drucker-Prager plasticity model with a damage term tuning kinematical hardening. This provides a new approach to the modelling of geomaterials, for which non associative plasticity is usually employed. % The present coupling is such that The kinematical hardening is null where the damage is complete, so there the behaviour is perfectly plastic. We analyse the m…
In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter $\eps>0$ denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit $\eps\to 0$. The homogenization limit is based on the needle-problem approach: We verify that the stochastic coefficients "allow averaging": …
We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with $L^p$-hardening penalty. In these relaxed models, where only one slip-plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower-semicontinuous along bounded-energy sequences which satisfy the single-plane condition, meaning precisely that this side condition should be preserved in the weak $L^p$-limit. This is done with the aid of an 'e…
We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.
