Inclusion Of Parabolic Type Research Articles

CONTACT PROBLEM WITH ADHESION AND WEAR IN ELECTRO-VISCOELASTICITY WITH DAMAGE

We consider a mathematical model which describes the dynamic process of con- tact between a piezoelectric body and two obstacles. The behavior of the material is modeled with a nonlinear electro-viscoelastic constitutive with law long memory and damage. The mechanical damage of the material, caused by excessive stess or strains, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. The contact is modeled with adhesion and wear. The adhesion field, whose evolution is described by a first order differential equation. The evolution of the wear function is described with Ar- chard’s law. For the variational formulation of the contact problem, we present and prove the existence of a unique weak solution to the problem. The proof is based on arguments of time dependent variational inequalities, parabolic inequalities, differential equations and fixed point arguments.

A dynamic problem with wear involving electro-elastic-viscoplastic materials with damage

"A dynamic contact problem is considered in the paper. The material behavior is described by electro-elastic-viscoplastic law with piezoelectric effects. The body is in contact with damage and an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformations. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a coupled system of an elliptic variational inequality for the displacement, variational equation for the electric potential and a parabolic variational inequality for the damage. 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 uniqueness result on parabolic inequalities, differential equations and fixed point arguments."

Optimal control of parabolic differential inclusions in one space dimension

The paper is devoted to the optimisation of parabolic differential inclusions (DFIs) in a bounded rectangular region. For this, a problem with a parabolic discrete inclusion is defined, which is the main inevitable auxiliary problem. With the help of locally adjoint mappings, necessary and sufficient conditions for the optimality of parabolic discrete inclusions are proved. Then, using the method of discretization of parabolic DFIs and the already obtained optimality conditions for discrete inclusions, the necessary and sufficient conditions for the discrete-approximate problem are formulated in the form of the Euler–Lagrange type inclusion. Thus, using a specially proved equivalence theorem, we establish sufficient optimality conditions for a parabolic DFI. To demonstrate the above approach, some linear problems and polyhedral optimisation with inclusions of parabolic type are investigated. In addition, numerical results are also presented.

A Dynamic Contact Problem Between Elasto-Viscoplastic Piezoelectric Bodies with Adhesion

We consider a dynamic frictionless contact problem between two elasto-viscoplastic piezoelectric bodies with damage. The evolution of the damage is described by an inclusion of parabolic type. The contact is modelled with normal compliance condition. The adhesion of the contact surfaces is considered and is modelled with a surface variable, the bonding field whose evolution is described by a first order differential equation. We derive variational formulation for the model and prove an existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed-point arguments.

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 frictional contact problem with wear involving elastic-viscoplastic materials with damage and thermal effects

We consider a mathematical problem for quasistatic contact between a thermo-elastic-viscoplastic body with damage 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 thermo-elastic-viscoplastic with damage constitutive law for the material. The damage of the material caused by elastic deformations. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a coupled system of an elliptic variational inequality for the displacement, a parabolic variational inequality for the damage and the heat equation for the temperature. 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 uniqueness result on parabolic inequalities, differential equations and fixed point arguments.

A frictional contact problem with normal damped response for elastic-visco-plastic materials

We study an evolution problem which describes the dynamic contact of an elastic-visco-plastic body with a foundation. We model the contact with normal damped response and a local friction law. A damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We derive variational formulation for the model which is in the form of a system involving the displacement field, the stress field and the damage field. We prove the existence and uniqueness result of the weak solution. 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 frictional contact problem with wear and damage for electro-viscoelastic materials

We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.

Analysis of two frictional viscoplastic contact problems with damage

In this work we study two quasistatic frictional contact problems arising in viscoplasticity including the mechanical damage of the material, caused by excessive stress or strain and modelled by an inclusion of parabolic type. The variational formulation is provided for both problems and the existence of a unique solution is proved for each of them. Then a fully discrete scheme is introduced using the finite element method to approximate the spatial domain and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the algorithm is deduced. Finally, some numerical examples are presented to show the performance of the method.

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.