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Jens Frehse - One of the best experts on this subject based on the ideXlab platform.

  • Nonlinear Elliptic Systems Arising from Plasticity Theory
    Regularity Results for Nonlinear Elliptic Systems and Applications, 2020
    Co-Authors: Alain Bensoussan, Jens Frehse
    Abstract:

    In this chapter (which is a continuation of Chapter 9 and will rely heavily on it) we are interested in specific models of Plasticity. We shall need to attach to tensor σ its deviator $${\sigma _D} = \sigma - 1/nI\;tr\;\sigma ,$$ which has trace 0. We will consider two models of Plasticity, the Hencky model, which is a model of Perfect Plasticity where, $$|{\sigma _D}|\mu $$ (µ is a given constant), and the Norton—Hoff model, which is an approximation to the Hencky model, where the constraint of Perfect Plasticity is relaxed with a penalty term. In fact, the Norton—Hoff model will be a particular case of the models considered in Chapter 9, but we shall consider a sequence of these models. Again these models are formulated as variational problems in which the unknown is the stress tensor and the displacement is recovered indirectly. The convergence of the approximation is very natural in the context of variational problems and follows from general penalty methods (see R. Temam [101], G. Duvaut, J.L. Lions [15], J.L. Lions [70], P. Le Tallec [69]).

  • regularity results for two standard models in elasto Perfect Plasticity theory with hardening
    arXiv: Analysis of PDEs, 2020
    Co-Authors: Miroslav Bulicek, Jens Frehse, Maria Specoviusneugebauer
    Abstract:

    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 estimates are uniform and independent of the order of approximation. The main novelty consist of estimates near the boundary. We show that for the stress and the hardening parameter, we control tangential derivative in the Lebesgue space~$L^2$, and for time derivative of the stress and the hardening we control the "half" time derivative and also spatial tangential derivative. Last, for the normal derivative, we show that the stress and the hardening have the $3/5$ derivative with respect to the normal and for the time derivative of the stress and the hardening we show they have the $1/5$ derivative with respect to the normal direction, provided we consider the kinematic hardening or near the Dirichlet boundary. These estimates are independent of dimension. In case, we consider the isotropic hardening near the Neumann boundary we shall obtain $W^{\alpha,2}$ regularity for the stress and the hardening with some $\alpha>1/2$ depending on the dimension and $W^{\beta,2}$ with some $\beta > 1/6$ for the time derivative of the stress and the hardening. Finally, in case of kinematic hardening the same regularity estimate holds true also for the velocity gradient.

  • a revision of results for standard models in elasto Perfect Plasticity theory
    Calculus of Variations and Partial Differential Equations, 2018
    Co-Authors: Miroslav Bulicek, Jens Frehse
    Abstract:

    We consider two most studied standard models in the theory of elasto-Plasticity in arbitrary dimension $$d\ge 2$$ , namely, the Hencky model and the Prandtl–Reuss model subjected to the von Mises condition. There are many available results for these models—from the existence and the regularity theory up to the relatively sharp identification of the plastic strain in the natural function/measure space setting. In this paper we shall proceed further and improve some of known estimates in order to identify sharply the plastic strain. More specifically, we rigorously improve the integrability of the displacement and the velocity (which was known only under a nonnatural assumption that the Cauchy stress is bounded), show the BMO estimates for the stress and finally also the Morrey-like estimates for the plastic strain. In addition, we shall provide the whole theory up to the boundary. As an immediate consequence of such improved estimates, we provide a sharper identification of the plastic strain than that known up to date. In particular, in two dimensional setting, we show that the plastic strain can be point-wisely characterized in terms of the stresses everywhere although the stress is possibly discontinuous and thus the natural duality pairing in the space of measures could be violated.

  • boundary regularity results for models of elasto Perfect Plasticity
    Mathematical Models and Methods in Applied Sciences, 1999
    Co-Authors: Jens Frehse, Josef Malek
    Abstract:

    We study the question of integrability for the gradient of the stress near the boundary for models of elasto-Perfect Plasticity and their approximations.

Alexander Yakhno - One of the best experts on this subject based on the ideXlab platform.

Clement Mifsud - One of the best experts on this subject based on the ideXlab platform.

  • A Numerical Approach of Friedrichs’ Systems Under Constraints in Bounded Domains
    Theory Numerics and Applications of Hyperbolic Problems II, 2018
    Co-Authors: Clement Mifsud, Bruno Després
    Abstract:

    We present here an explicit finite volume scheme on unstructured meshes adapted to first-order hyperbolic systems under constraints in bounded domains. This scheme is based on the work (Coudiere, Vila, Villedieu in C R Acad Sci Paris Ser I Math 331:95–100, 2000, [3]) in the unconstrained case and the splitting strategy of Despres, Lagoutiere, Seguin (Nonlinearity 24:3055–3081, 2011, [4]). We show that this scheme is stable under a Courant–Friedrichs–Lewy condition (and convergent for problems posed in the whole space), and we illustrate the solution constructed by this scheme on the example of the simplified model of Perfect Plasticity. From the theoretical point of view, the interaction between the constraint and the boundary of the domain in the model of Perfect Plasticity is encoded by a nonlinear boundary condition. With this numerical approach, we will show that, even if this scheme uses the underlying linear boundary condition, the results are consistent with the nonlinear model (and in particular with the nonlinear boundary condition).

  • hyperbolic structure for a simplified model of dynamical Perfect Plasticity
    Archive for Rational Mechanics and Analysis, 2017
    Co-Authors: Jeanfrancois Babadjian, Clement Mifsud
    Abstract:

    This paper is devoted to confronting two different approaches to the problem of dynamical Perfect Plasticity. Interpreting this model as a constrained boundary value Friedrichs’ system enables one to derive admissible hyperbolic boundary conditions. Using variational methods, we show the well-posedness of this problem in a suitable weak measure theoretical setting. Thanks to the property of finite speed propagation, we establish a new regularity result for the solution in short time. Finally, we prove that this variational solution is actually a solution of the hyperbolic formulation in a suitable dissipative/entropic sense, and that a partial converse statement holds under an additional time regularity assumption for the dissipative solutions.

  • short time regularity for dynamic evolution problems in Perfect Plasticity
    2017
    Co-Authors: Clement Mifsud
    Abstract:

    In this paper, we study the regularity properties of solutions of dynamic evolution problems in Perfect Plasticity. We prove that for any space dimension and for any closed convex set of constraints containing zero as an interior point, the solutions are regular in space during a short time interval if the data are smooth and compactly supported. The result is based on the hyperbolic structure of the model, namely the finite speed propagation property.

  • Variational and hyperbolic methods applied to constrained mechanical systems
    2016
    Co-Authors: Clement Mifsud
    Abstract:

    In this thesis, we consider constrained hyperbolic partial differential equations and more precisely mechanical problems coming from Perfect Plasticity. The goal of this thesis is to study these problems thanks to different approaches, to analyze the interactions between these different points of view and to confront these various analyzes to get new results. A brief review of the mechanical origin of Perfect Plasticity problems and also of the previous results on these topics are described in Chapter 1. In Chapter 2, we focus our attention on hyperbolic systems with boundary conditions. First, we develop a weak theory for these problems and explain, in a simplified case, why this theory is well-posed. Then, we introduce similarly a notion of weak solutions for constrained hyperbolic systems with boundary conditions. Chapter 3 is devoted to the study of the simplified model of dynamical Perfect Plasticity. We confront the approach introduced in the previous chapter with the one, more standard, coming from calculus of variations that allows us to obtain existence and uniqueness of the solutions for this model. It allows us to bring to light a new interaction between the boundary conditions and the constraints and to get a short-time regularity theorem. Lastly, in Chapter 4, we are interested in the numerical approximation of constrained hyperbolic systems thanks to finite volume schemes. This work allows us to get a convergence result for problems without boundary condition and to show numerically the link between boundary conditions and constraints on the example of the previous chapter.

Jeanfrancois Babadjian - One of the best experts on this subject based on the ideXlab platform.

  • dissipative boundary conditions and entropic solutions in dynamical Perfect Plasticity
    arXiv: Analysis of PDEs, 2019
    Co-Authors: Jeanfrancois Babadjian, Vito Crismale
    Abstract:

    We prove the well--posedness of a dynamical Perfect Plasticity model under general assumptions on the stress constraint set and on the reference configuration. The problem is studied by combining both calculus of variations and hyperbolic methods. The hyperbolic point of view enables one to derive a class of dissipative boundary conditions, somehow intermediate between homogeneous Dirichlet and Neumann ones. By using variational methods, we show the existence and uniqueness of solutions. Then we establish the equivalence between the original variational solutions and generalized entropic--dissipative ones, derived from a weak hyperbolic formulation for initial--boundary value Friedrichs' systems with convex constraints.

  • stress regularity in quasi static Perfect Plasticity with a pressure dependent yield criterion
    Journal of Differential Equations, 2018
    Co-Authors: Jeanfrancois Babadjian, Maria Giovanna Mora
    Abstract:

    Abstract This work is devoted to establishing a regularity result for the stress tensor in quasi-static planar isotropic linearly elastic – Perfectly plastic materials obeying a Drucker–Prager or Mohr–Coulomb yield criterion. Under suitable assumptions on the data, it is proved that the stress tensor has a spatial gradient that is locally squared integrable. As a corollary, the usual measure theoretical flow rule is expressed in a strong form using the quasi-continuous representative of the stress.

  • hyperbolic structure for a simplified model of dynamical Perfect Plasticity
    Archive for Rational Mechanics and Analysis, 2017
    Co-Authors: Jeanfrancois Babadjian, Clement Mifsud
    Abstract:

    This paper is devoted to confronting two different approaches to the problem of dynamical Perfect Plasticity. Interpreting this model as a constrained boundary value Friedrichs’ system enables one to derive admissible hyperbolic boundary conditions. Using variational methods, we show the well-posedness of this problem in a suitable weak measure theoretical setting. Thanks to the property of finite speed propagation, we establish a new regularity result for the solution in short time. Finally, we prove that this variational solution is actually a solution of the hyperbolic formulation in a suitable dissipative/entropic sense, and that a partial converse statement holds under an additional time regularity assumption for the dissipative solutions.

Sergey I Senashov - One of the best experts on this subject based on the ideXlab platform.