Weak solvability of a piezoelectric contact problem

2009 ◽  
Vol 20 (2) ◽  
pp. 145-167 ◽  
Author(s):  
STANISŁAW MIGÓRSKI ◽  
ANNA OCHAL ◽  
MIRCEA SOFONEA
Keyword(s):  
Mathematical Model ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Constitutive Law ◽  
Potential Fields ◽  
Material Behaviour ◽  
Abstract Result ◽  
Electrically Conductive ◽  
Piezoelectric Body ◽  
Weak Solvability

We consider a mathematical model which describes the frictional contact between a piezoelectric body and a foundation. The material behaviour is modelled with a non-linear electro-elastic constitutive law, the contact is bilateral, the process is static and the foundation is assumed to be electrically conductive. Both the friction law and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system of two coupled hemi-variational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proof is based on an abstract result on operator inclusions in Banach spaces.

MODELING AND ANALYSIS OF AN ANTIPLANE PIEZOELECTRIC CONTACT PROBLEM

2009 ◽  
Vol 19 (08) ◽  
pp. 1295-1324 ◽  
Author(s):  
STANISŁAW MIGÓRSKI ◽  
ANNA OCHAL ◽  
MIRCEA SOFONEA
Keyword(s):  
Electrical Conductivity ◽  
Frictional Contact ◽  
Material Behavior ◽  
Constitutive Law ◽  
Potential Fields ◽  
Hemivariational Inequalities ◽  
Abstract Result ◽  
Electrically Conductive ◽  
Modeling And Analysis ◽  
Friction Laws

We study a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is static, the material behavior is described with a linearly electro-elastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is in the form of a system of two coupled hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on an abstract result on operator inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our results are valid.

scholarly journals A dynamic electroviscoelastic problem with thermal effects

2021 ◽  
Vol 66 (4) ◽  
pp. 769-781
Author(s):  
Sihem Smata ◽  
◽  
Nemira Lebri ◽  
Keyword(s):  
Thermal Effects ◽  
Dry Friction ◽  
Variational Formulation ◽  
Classical Result ◽  
Monotone Operators ◽  
Constitutive Law ◽  
Unique Weak Solution ◽  
Electrically Conductive ◽  
Piezoelectric Body ◽  
Signorini Condition

We consider a mathematical model which describes the dynamic pro- cess of contact between a piezoelectric body and an electrically conductive foun- dation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law with thermal e ects. Contact is described with the Signorini condition, a version of Coulomb's law of dry friction. A variational formulation of the model is derived, and the existence of a unique weak solution is proved. The proofs are based on the classical result of nonlinear rst order evolution inequali- ties, the equations with monotone operators, and the xed point arguments.

A PIEZOELECTRIC CONTACT PROBLEM WITH SLIP DEPENDENT COEFFICIENT OF FRICTION

2004 ◽  
Vol 9 (3) ◽  
pp. 229-242 ◽  
Author(s):  
M. Sofonea
Keyword(s):  
Mathematical Model ◽  
Weak Solution ◽  
Contact Problem ◽  
Coefficient Of Friction ◽  
Dry Friction ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Coupled System ◽  
Piezoelectric Body ◽  
The Coefficient Of Friction

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and an obstacle. The constitutive relation of the material is assumed to be electroelastic and involves a nonlinear elasticity operator. The contact is modelled with a version of Coulomb's law of dry friction in which the coefficient of friction depends on the slip. We derive a variational formulation for the model which is in form of a coupled system involving as unknowns the displacement field and the electric potential. Then we provide the existence of a weak solution to the model and, under a smallness assumption, we provide its uniqueness. The proof is based on a result obtained in [14] in the study of elliptic quasi‐variational inequalities.

Analysis of a piezoelectric contact problem with subdifferential boundary condition

2014 ◽  
Vol 144 (5) ◽  
pp. 1007-1025 ◽  
Author(s):  
Stanisław Migórski ◽  
Anna Ochal ◽  
Mircea Sofonea
Keyword(s):  
Boundary Condition ◽  
Variational Formulation ◽  
Contact Process ◽  
Constitutive Law ◽  
Time Dependent ◽  
Hemivariational Inequalities ◽  
Material Behaviour ◽  
Nonlinear Integral Equations ◽  
Frictionless Contact ◽  
System Coupling

We consider a mathematical model which describes the frictionless contact between a piezoelectric body and a foundation. The contact process is quasi-static and the foundation is assumed to be insulated. The novelty of the model consists in the fact that the material behaviour is described with an electro-elastic–visco-plastic constitutive law and the contact is modelled with a subdifferential boundary condition. We derive a variational formulation of the problem which is in the form of a system coupling two nonlinear integral equations with a history-dependent hemivariational inequality and a time-dependent linear equation. We prove the existence of a weak solution to the problem and, under additional assumptions, its uniqueness. The proof is based on a recent result on history-dependent hemivariational inequalities obtained by Migórski, Ochal and Sofonea in 2011.

scholarly journals A CLASS OF QUASISTATIC CONTACT PROBLEMS FOR VISCOELASTIC MATERIALS WITH NONLOCAL COULOMB FRICTION AND TIME-DELAY

2014 ◽  
Vol 19 (4) ◽  
pp. 491-508 ◽  
Author(s):  
Si-sheng Yao ◽  
Nan-jing Huang
Keyword(s):  
Mathematical Model ◽  
Variational Inequality ◽  
Time Delay ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Contact Problems ◽  
Existence And Uniqueness Theorem ◽  
Differential Variational Inequality

In this paper, a mathematical model which describes the explicit time dependent quasistatic frictional contact problems is introduced and studied. The material behavior is described with a nonlinear viscoelastic constitutive law with time-delay and the frictional contact is modeled with nonlocal Coulomb boundary conditions. A variational formulation of the mathematical model is given, which is called a quasistatic integro-differential variational inequality. Using the Banach's fixed point theorem, an existence and uniqueness theorem of the solution for the quasistatic integro-differential variational inequality is proved under some suitable assumptions. As an application, an existence and uniqueness theorem of the solution for the dual variational formulation is also given.

Analysis and control of an electro-elastic contact problem

2021 ◽  
pp. 108128652110441
Author(s):  
Tao Chen ◽  
Rong Hu ◽  
Mircea Sofonea
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Strain Field ◽  
Variational Formulation ◽  
Continuous Dependence ◽  
Existence Result ◽  
Frictional Contact ◽  
Elastic Contact ◽  
Tresca’S Friction ◽  
And Control

We consider a mathematical model that describes the frictional contact of an electro-elastic body with a semi-insulator foundation. The process is static; the contact is bilateral and associated to Tresca’s friction law. We list the assumptions on the data and derive a variational formulation of the model, in the form of a system that couples two inclusions in which the unknowns are the strain field and the electric field. Then we prove the unique solvability of the system, as well as the continuous dependence of its solution with respect to the data. We use these results in the study of an associated optimal control problem, for which we prove an existence result. The proofs are based on arguments of monotonicity, compactness, convex analysis, and lower semicontinuity.

History-dependent quasi-variational inequalities arising in contact mechanics

2011 ◽  
Vol 22 (5) ◽  
pp. 471-491 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
ANDALUZIA MATEI
Keyword(s):  
Variational Inequalities ◽  
Frictional Contact ◽  
Deformable Body ◽  
Regularity Result ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Signorini Condition ◽  
Normal Damped Response ◽  
Weak Solvability

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

scholarly journals Quasistatic Adhesive Contact of Piezoelectric Cylinders

2009 ◽  
Vol 14 (1) ◽  
pp. 123-142
Author(s):  
M. Sofonea ◽  
L. Chouchane
Keyword(s):  
Variational Formulation ◽  
Adhesive Contact ◽  
Elastic Problem ◽  
Antiplane Shear ◽  
Potential Fields ◽  
Rigid Foundation ◽  
Viscoelastic Problem ◽  
Electrically Conductive ◽  
Variational Equations ◽  
System Coupling

We consider two mathematical models which describe the antiplane shear deformation of a piezoelectric cylinder in adhesive contact with a rigid foundation. The material is assumed to be electro-viscoelastic in the first model and electro-elastic in the second one. In both models the process is quasistatic, the foundation is electrically conductive and the adhesion is described with a surface variable, the bonding field. We derive a variational formulation of the models which is given by a system coupling two variational equations for the displacement and the electric potential fields, respectively, and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to each model. We also investigate the behavior of the solution of the electro-viscoelastic problem as the viscosity converges to zero and prove that it converges to the solution of the corresponding electro-elastic problem.

scholarly journals VARIATIONAL ANALYSIS OF A FRICTIONAL CONTACT PROBLEM WITH WEAR AND DAMAGE

2021 ◽  
Vol 26 (2) ◽  
pp. 170-187
Author(s):  
Mohammed Salah Mesai Aoun ◽  
Mohamed Selmani ◽  
Abdelaziz Azeb Ahmed
Keyword(s):  
Long Memory ◽  
Dry Friction ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Variational Analysis ◽  
Friction And Wear ◽  
Frictional Contact ◽  
Constitutive Law ◽  
Quasistatic Problem ◽  
Elliptic Variational Inequalities

We study a quasistatic problem describing the contact with friction and wear between a piezoelectric body and a moving foundation. The material is modeled by an electro-viscoelastic constitutive law with long memory and damage. The wear of the contact surface due to friction is taken into account and is described by the differential Archard condition. The contact is modeled with the normal compliance condition and the associated law of dry friction. We present a variational formulation of the problem and establish, under a smallness assumption on the data, the existence and uniqueness of the weak solution. The proof is based on arguments of parabolic evolutionary inequations, elliptic variational inequalities and Banach fixed point.

On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers

2013 ◽  
Vol 143 (5) ◽  
pp. 1047-1059 ◽  
Author(s):  
Andaluzia Matei
Keyword(s):  
Lagrange Multipliers ◽  
Frictional Contact ◽  
Variational Equation ◽  
Deformable Body ◽  
Variational Problems ◽  
Abstract Result ◽  
Continuous Maps ◽  
Mixed Variational Problem ◽  
Fixed Point Technique ◽  
Weak Solvability

We study an abstract mixed variational problem, the set of the Lagrange multipliers being dependent on the solution. The problem consists of a system of a variational equation and a variational inequality. We prove the existence of the solution based on a fixed-point technique for weakly sequentially continuous maps. We then apply the abstract result to the weak solvability of a boundary-value problem that models the frictional contact between a cylindrical deformable body and a rigid foundation.

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