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.

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.

A piezoelectric contact problem with slip dependent friction and damage

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abderrezak Kasri
Keyword(s):  
Weak Solution ◽  
Contact Problem ◽  
Dry Friction ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Normal Compliance ◽  
Electrically Conductive ◽  
Coulomb’S Law ◽  
Electrical Condition ◽  
Coulomb's Law

Abstract The aim of this paper is to study a quasistatic contact problem between an electro-elastic viscoplastic body with damage and an electrically conductive foundation. The contact is modelled with an electrical condition, normal compliance and the associated version of Coulomb’s law of dry friction in which slip dependent friction is included. We derive a variational formulation for the model and, under a smallness assumption, we prove the existence and uniqueness of a weak solution.

scholarly journals Analysis of an Antiplane Contact Problem with Adhesion for Electro-Viscoelastic Materials

2008 ◽  
Vol 13 (3) ◽  
pp. 379-395
Author(s):  
M. Sofonea ◽  
L. Chouchane ◽  
L. Selmani
Keyword(s):  
Variational Formulation ◽  
Evolution Equations ◽  
Variational Equation ◽  
Monotone Operators ◽  
Antiplane Shear ◽  
Electrically Conductive ◽  
Variational Equality ◽  
Electric Potential Field ◽  
Contact Surfaces ◽  
System Coupling

We consider a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The adhesion of the contact surfaces, caused by the glue, is taken into account. The material is assumed to be electro-viscoelastic and the foundation is assumed to be electrically conductive. We derive a variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field, a time-dependent variational equation for the electric potential field and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators and fixed point.

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.

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.

scholarly journals Theoretical Study of a Chain Sliding on a Fixed Support

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Jérôme Bastien ◽  
Claude-Henri Lamarque
Keyword(s):  
Dry Friction ◽  
Theoretical Study ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Rheological Models ◽  
Implicit Euler Scheme ◽  
Linear Spring ◽  
A Chain ◽  
Uniqueness Theory

A chain sliding on a fixed support, made out of some elementary rheological models (dry friction element and linear spring) can be covered by the existence and uniqueness theory for maximal monotone operators. Several behavior from quasistatic to dynamical are investigated. Moreover, classical results of numerical analysis allow to use a numerical implicit Euler scheme.

Two-field variational formulations for a class of nonlinear mechanical models

2022 ◽  
pp. 108128652110661
Author(s):  
Andaluzia Matei ◽  
Madalina Osiceanu
Keyword(s):  
Variational Formulation ◽  
Nonlinear Boundary ◽  
Convex Set ◽  
Constitutive Law ◽  
Fenchel Conjugate ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Nonlinear Mechanical ◽  
Mechanical Models ◽  
Variational Formulations

A nonlinear boundary value problem arising from continuum mechanics is considered. The nonlinearity of the model arises from the constitutive law which is described by means of the subdifferential of a convex constitutive map. A bipotential [Formula: see text], related to the constitutive map and its Fenchel conjugate, is considered. Exploring the possibility to rewrite the constitutive law as a law governed by the bipotential [Formula: see text], a two-field variational formulation involving a variable convex set is proposed. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed.

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 Well-posedness of the fractional Zener wave equation for heterogeneous viscoelastic materials

2020 ◽  
Vol 23 (1) ◽  
pp. 126-166 ◽  
Author(s):  
Ljubica Oparnica ◽  
Endre Süli
Keyword(s):  
Weak Solution ◽  
Positive Constant ◽  
Time Derivative ◽  
Strain Tensor ◽  
Constitutive Law ◽  
Time Interval ◽  
Well Posedness ◽  
Unique Weak Solution ◽  
Load Vector ◽  
Bounded Below

AbstractZener’s model for viscoelastic solids replaces Hooke’s law σ = 2με(u) + λ tr(ε(u)) I, relating the stress tensor σ to the strain tensor ε(u), where u is the displacement vector, μ > 0 is the shear modulus, and λ ≥ 0 is the first Lamé coefficient, with the constitutive law (1 + τDt) σ = (1 + ρDt)[2με(u) + λ tr(ε(u)) I], where τ > 0 is the characteristic relaxation time and ρ ≥ τ is the characteristic retardation time. It is the simplest model that predicts creep/recovery and stress relaxation phenomena. We explore the well-posedness of the fractional version of the model, where the first-order time-derivative Dt in the constitutive law is replaced by the Caputo time-derivative $\begin{array}{} D_t^\alpha \end{array} $ with α ∈ (0, 1), μ, λ belong to L∞(Ω), μ is bounded below by a positive constant and λ is nonnegative. We show that, when coupled with the equation of motion ϱü = Div σ + f, considered in a bounded open Lipschitz domain Ω in ℝ3 and over a time interval (0, T], where ϱ ∈ L∞(Ω) is the density of the material, assumed to be bounded below by a positive constant, and f is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions u(0, x) = g(x), u̇(0, x) = h(x), σ(0, x) = S(x), for x ∈ Ω, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of g ∈ [$\begin{array}{} \mathrm{H}^1_0 \end{array} $(Ω)]3, h ∈ [L2(Ω)]3, and S = ST ∈ [L2(Ω)]3×3, and any load vector f ∈ L2(0, T; [L2(Ω)]3), and that this unique weak solution depends continuously on the initial data and the load vector.

scholarly journals On the frictionless unilateral contact of two viscoelastic bodies

2003 ◽  
Vol 2003 (11) ◽  
pp. 575-603 ◽  
Author(s):  
M. Barboteu ◽  
T.-V. Hoarau-Mantel ◽  
M. Sofonea
Keyword(s):  
Evolution Equations ◽  
Maximal Monotone ◽  
Viscoelastic Behavior ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Constitutive Law ◽  
Test Problems ◽  
Fully Discrete ◽  
Deformable Bodies ◽  
Augmented Lagrangian Approach

We consider a mathematical model which describes the quasistatic contact between two deformable bodies. The bodies are assumed to have a viscoelastic behavior that we model with Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the classical Signorini condition with zero-gap function. We derive a variational formulation of the problem and prove the existence of a unique weak solution to the model by using arguments of evolution equations with maximal monotone operators. We also prove that the solution converges to the solution of the corresponding elastic problem, as the viscosity tensors converge to zero. We then consider a fully discrete approximation of the problem, based on the augmented Lagrangian approach, and present numerical results of two-dimensional test problems.

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