Elastic-viscoplastic Body Research Articles

VARIATIONAL ANALYSIS OF AN ELASTIC-THERMO-VISCOPLASTIC CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE

We consider a quasistatic frictional contact problem between an elastic-viscoplastic body and an obstacle. The contact is modelled with normal damped response and a local friction law. The material is elastic-viscoplastic with two internal variables which may describe a temperature parameter and the damage of the contacting surface. We provide a variational formulation of the problem and prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities and fixed point.

A QUASISTATIC FRICTIONAL CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE FOR THERMO-ELECTRO-ELASTIC-VISCOPLASTIC BODIES

We consider a mathematical problem for quasistatic contact between a thermo-electro--elastic-viscoplastic body and an obstacle. The contact is modeled by a general normal damped response condition with friction law and heat exchange. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution. The proof is based on the formulation of four intermediate problems for the displacement field, the electric potential field and the temperature field, respectively. We prove the unique solvability of the intermediate problems, then we construct a contraction mapping whose unique fixed point is the solution of the original problem.

Deformation and Loss of Stability Modeling for Elastic-Viscoplastic Mixtures

To describe the behavior of a number of real structures consisting of n components, a generalized model of an elastic-viscoplastic body is used. Within the framework of the proposed model of a multicomponent body in a rigorous linearized three-dimensional formulation, the stability of multicomponent elastic-viscoplastic mixtures is investigated. It is shown that the study of the stability of the subcritical state of the mixture can be reduced to studying the stability of the continuum with complex physical and mechanical parameters. For homogeneous ground states, solutions of static linearized problems are given. An example is considered.

A dynamic Tresca's frictional contact problem with damage for thermo elastic-viscoplastic bodies

We consider a dynamic contact problem between an elastic-viscoplastic body and a rigid obstacle. The contact is frictional and bilateral, the friction is modeled with Tresca's law with heat exchange. We employ the elastic-viscoplastic with damage constitutive law for the material . The evolution of the damage is described b y an inclusion of parabolic type.We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniquness result on parabolic inequalities, differentiel equations and fixed point argument.

The substantiation of the elastic-viscoplastic model of the human spine for modeling the correction process of kyphoscoliotic deformation.

Purpose:The relevance of the problem is caused by an increase in the number of spine-related diseases among children, including scoliosis. Currently, there are no methodologies for the treatment of scoliosis, which ensure an unambiguous positive result. The purpose of the article is to justify the spinal model as an elastic viscoplastic body for further mathematical modeling of the process of spine correction and search for its optimal conditions.Methodology:The leading approach to the study of this problem is the development of techniques for the surgical treatment of deformities of the vertebral column with the aid of an external fixation device for the spine, providing for a rigid connection of the elements of the apparatus with each other and with the spine. The rigid connection between the elements of the external fixation device increases the degree of static indeterminacy of the design, which leads to the occurrence of additional dangerous stresses in the details of the apparatus and in the vertebrae. The control actions in such devices do not provide an adequate result for the process of correction of the vertebral column.Results:The main result is the substantiation of the spine model as an elastic viscoplastic body. This will allow more detailed consideration of the medical and biological features of the spine and the physical and mechanical properties of human bone and soft tissues. The proposed model will allow developing an adaptive design of the device, taking into account specific features of the organism and more effectively managing the correction process.Value:The materials of the article can be useful for scientists, doctors and specialists in conducting scientific research on the problem of spine deformation correction and the development of appropriate technical means.

BILATERAL CONTACT PROBLEM WITH FRICTION AND WEAR FOR AN ELECTRO ELASTIC-VISCOPLASTIC MATERIALS WITH DAMAGE

We consider a mathematical problem for quasistatic contact between anelectro elastic-viscoplastic body and an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. We employ the electro elastic-viscoplastic with damage constitutive law for the material. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a system of an elliptic variational inequality for the displacement, a parabolic variational inequality for the damage and a variational equality for the electric stress. We establish a variational formulation for the model and we give the wear conditions for the existence of a unique weak solution to the problem. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.

A DYNAMIC PROBLEM OF FRICTIONLESS CONTACT FOR ELASTIC-THERMO-VISCOPLASTIC MATERIALS WITH DAMAGE

We consider a dynamic frictionless contact problem between an elastic-viscoplastic body and a reactive foundation. The contact is modelled with normal compliance. The material is elastic-viscoplastic with two internal variables which may describe a temperature parameter and the damage of the system caused by plastic deformations. We derive a weak formulation of the system consisting of a motion equation, an energy equation, and an evolution damage inclusion. We prove existence and uniqueness of the solution, and the positivity of the temperature. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and unique- ness result on parabolic type inequalities, differential equations and fixed-point arguments.

A Dynamic Frictionless Elastic-Viscoplastic Problem with Normal Damped Response and Damage

We consider a mathematical model for the process of a frictionless contact between an elastic-viscoplastic body and a reactive foundation. The material is elastic-viscoplastic with internal state variable which may describe the damage of the system caused by plastic deformations. We establish a variational formulation for the model and prove the existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear equations with monotone operators, on parabolic type inequalities and fixed point.

An Elastic-viscoplastic Quasistatic Contact Problem: Existence and Uniqueness of a Weak Solution

In this paper, the contact between an elastic-viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, is also considered. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is established by using approximate problems, the pseudomonotone operators theory, Schauder fixed-point theorem and a comparison result.

An elastic–viscoplastic quasistatic contact problem with damage

In this paper, the contact between an elastic–viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to tension or compression, is also considered. The variational formulation leads to a coupled system of evolutionary equations for which an existence and uniqueness result is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained and, as a consequence, the convergence of the algorithm is deduced. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm.

Numerical analysis of a contact problem between two elastic–viscoplastic bodies with hardening and nonmatching meshes

We consider a quasistatic frictionless contact problem between two elastic–viscoplastic bodies with strain hardening. The latter is modeled as an internal state variable which verifies an ordinary differential equation. We present a variational formulation of the problem and state an existence and uniqueness result. Then, we study a fully discrete approximation for the two-dimensional case using a nonconforming finite element method, based on the mortar projection operator for the approximation of the spatial variables, and the backward Euler's discretization for the time derivatives. The existence of a unique solution for this scheme is given and error estimates are derived. Finally, four numerical examples are presented.

Numerical Approximation of the Elastic-Viscoplastic Contact Problem with Non-matching Meshes

This work deals with the approximation of a time dependent variational inequality modelling the unilateral contact problem of elastic-viscoplastic bodies in a bidimensional context. The problem is approximated in the space variable with nonconforming finite element methods which allow the handling of nonmatching meshes on the contact zone. Several error estimates are established and the corresponding numerical experiments are achieved.

Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

We consider the quasistatic Signorini′s contact problem with damage for elastic‐viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

A frictionless contact problem for elastic–viscoplastic materials with normal compliance and damage

We study a quasistatic frictionless contact problem with normal compliance and damage for elastic–viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modelled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

A frictionless contact problem for elastic-viscoplastic materials with normal compliance: Numerical analysis and computational experiments

In this paper we consider a frictionless contact problem between an elastic–viscoplastic body and an obstacle. The process is assumed to be quasistatic and the contact is modeled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using strongly monotone operators arguments and Banach's fixed point theorem. We also study the numerical approach to the problem using spatially semi-discrete and fully discrete finite elements schemes with implicit and explicit discretization in time. We show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions. Finally, we present some numerical results involving examples in one, two and three dimensions.

Existence for viscoplastic contact with Coulomb friction problems

We present existence results in the study of nonlinear problem of frictional contact between an elastic‐viscoplastic body and a rigid obstacle. We model the frictional contact both by a Tresca′s friction law and a regularized Coulomb′s law. We assume, in a first part, that the contact is bilateral and that no separation takes place. In a second part, we consider the Signorini unilateral contact conditions. Proofs are based on a time‐discretization method, Banach and Schauder fixed point theorems.

Résolution numérique d'un problème de contact entre corps élasto-viscoplastiques et maillages éléments finis incompatibles

We consider a quasistatic frictionless unilateral contact problem between two elastic-viscoplastic bodies. The corresponding time-dependent variational inequality is approximated in the space variable with nonconforming finite element methods which allow the handling of nonmatching meshes on the contact part. A backward Euler scheme is used for the time discretization. We show the existence and uniqueness of the discrete solution and the convergence towards the continuous solution is obtained. Finally, some corresponding numerical results are shown.

Numerical analysis of a frictionless contact problem for elastic–viscoplastic materials

We consider a mathematical model which describes the unilateral quasistatic contact of two elastic–viscoplastic bodies. The contact is without friction and it is modeled by the classical Signorini boundary conditions. The model consists of an evolution equation coupled with a time-dependent variational inequality. It has been shown that the variational problem of the model has a unique solution. Here we consider numerical approximations of the problem. We use the finite element method to discretize the spatial domain. Spatially semi-discrete and fully discrete schemes are studied. For both schemes, we show the existence of a unique solution, and derive error estimates. Under appropriate regularity assumptions of the solution, we have the optimal order convergence.

Computation for Rheological Phenomena of Elastic-Viscoplastic Body

A computational method is proposed for simulating rheological phenomena of a body composed of a material such as soft soil, powder, and granular material under large excitations. These phenomena are considered to be elastic-viscoplastic (E-VP) flows with moving boundaries. Fundamental equations for the dynamics are constructed from the Eulerian viewpoint. The constitutive laws for pressure volume and deviatoric stress strain are E-VP and are expressed by differential equations. The pressure and deviatoric stress are combined to satisfy yield conditions. The equations construct the first-order partial differential equation system. A domain containing an E-VP body is divided into cells by applying a finite difference method to the partial differential equation system. The locations of markers distributed in the body are calculated to trace the moving free boundaries. The collapse of an E-VP square body is shown as an application example.

On a contact problem for elastic-viscoplastic bodies

On a contact problem for elastic-viscoplastic bodies