Quasistatic processes for elastic-viscoplastic materials (original) (raw)

An existence and uniqueness theorem in quasi-static viscoelasticity

In some recent works, G. Fichera [1,2] has raised the question of whether unidimensional quasi-static problems are well posed for linearly viscoelastic materials. These problems have the following natural generalization to the three-dimensional CaSC: V • T(x, f) + f(*.

Boundary Value Problems in Thermo Viscoplasticity

In this work, we study two uncoupled quasistatic problems for thermo viscoplastic materials. In the model of the equation of generalised thermo viscoplasticity, both the elastic and the plastic rate of deformation depend on a parameter θ which may be interpreted as the absolute temperature. The boundary conditions considered here as displacement-traction conditions as well as unilateral contact conditions. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem, reducing the isotherm problem to an ordinary differential equation in a Hilbert space.

On the Crack Problem within the Context of Implicity Constituted Quasi-Linear Viscoelasticity

Mathematical Models and Methods in Applied Sciences

A quasi-linear viscoelastic relation that stems from an implicit viscoelastic constitutive body containing a crack is considered. The abstract form of the response function is given first in [Formula: see text], [Formula: see text], due to power-law hardening; second in [Formula: see text] due to limiting small strain. In both the cases, sufficient conditions on admissible response functions are formulated, and corresponding existence theorems are proved rigorously based on the variational theory and using monotonicity methods. Due to the presence of a Volterra convolution operator, an auxiliary-independent variable of velocity type is employed. In the case of limiting small strain, the generalized solution of the problem is provided within the context of bounded measures and expressed by a variational inequality.

Implicit constitutive relations for visco-elastic solids: Part II. Non-homogeneous deformations

International Journal of Non-linear Mechanics, 2020

A constitutive relation was developed in Part I for describing the response of a class of visco-elastic bodies, wherein the left Cauchy-Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are related through an implicit constitutive relation. Here, we study a boundary value problem within the context of the model namely the inhomogeneous deformation of the body, corresponding to the response of an infinitely long slab due to the influence of gravity.

On the dynamics of an elastic-rigid material

We present a model for a class of materials which behave as an elastic solid if the stress exceeds a certain threshold and as a rigid body when the stress is below such a threshold. The constitutive equations for these materials fall in the class of implicit constitutive relations, since the stress is a multivalued function of the strain. The mathematical structure of the constitutive equation is similar to the one of Bingham fluids, the main difference being that above the stress threshold the body behaves like an elastic solid and not as a viscous fluid.

Existence of solution for a nonlinear model of thermo-visco-plasticity

We study a thermodynamically consistent model describing phenomena in a visco-plastic metal subjected to temperature changes. We complete the model with the mixed boundary condition on displacement and stress and Neumann-type condition for temperature. The main result is an existence of solution.

Non-smooth viscoplastic constitutive relations in solid materials

INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021

In this work constitutive relations are presented for non-smooth viscoplasticity problems. Formulations of flow laws and evolutive laws are illustrated which are well suited for non-smooth viscoplasticity. The mathematical formulations of convex analysis are used for handling properly the non-differentiable functions. Extended formulations of evolutive laws are derived within a treatment that proves to be convenient also for the derivation of variational formulations which form the basis for numerical procedures in non-smooth viscoplasticity.

On the nonlinear theory of viscoelasticity of differential type

Mathematics and Mechanics of Solids, 2011

We consider nonlinear viscoelastic materials of differential type and for some special models we derive exact solutions of initial boundary value problems. These exact solutions are used to investigate the reasons for the non-existence of global solutions for such equations.

VISCOELASTIC FLUIDS: FREE ENERGIES, DIFFERENTIAL PROBLEMS AND ASYMPTOTIC BEHAVIOUR

Some expressions for the free energy in the case of incompressible viscoelastic uids are given. These are derived from free energies already in- troduced for other viscoelastic materials, adapted to incompressible uids. A new free energy is given in terms of the minimal state descriptor. The internal dissipations related to these di erent functionals are also derived. Two equiv- alent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible uids. Fi- nally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved.