A quasistatic contact problem with slip‐dependent coefficient of friction

We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. Copyright © 1999 John Wiley & Sons, Ltd.     

A quasistatic viscoplastic contact problem with normal compliance and friction

We consider a quasistatic contact problem between a viscoplasticbody and an obstacle, the so-called foundation. The contactis modelled with normal compliance and the associated versionof Coulomb's law of dry friction. We derive a variational formulationof the problem and, under a smallness assumption on the normalcompliance functions, we establish the existence of a weak solutionto the model. The proof is carried out in several steps. Itis based on a time-discretization method, arguments of monotonicityand compactness, Banach fixed point theorem and Schauder fixedpoint theorem.     

Analysis of a history-dependent frictional contact problem

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation. The material’s behaviour is modelled with a constitutive law with long memory. The contact is frictional and is modelled with normal compliance and memory term, associated to the Coulomb’s law of dry friction. We present the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. Then we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities. We also study the dependence of the weak solution with respect to the data and prove a convergence result.     

A dynamic frictional contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is dynamic, the material's behavior is modeled with an electro-viscoelastic constitutive law and the contact is described by subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving a second order evolutionary hemivariational inequality for the displacement field coupled with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators.     

Variational analysis of fully coupled electro‐elastic frictional contact problems

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a foundation. The material behavior is described with a nonlinear electro‐elastic constitutive law. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive and both the frictional contact and the conductivity on the contact surface are described with subdifferential boundary conditions which involve a fully coupling between the mechanical and electrical variables. We derive a variational formulation of the problem which is in the form of a system coupling two 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 recent results for inclusions of subdifferential type in Sobolev spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis and numerical solution of a piezoelectric frictional contact problem

We consider a mathematical model which describes the frictional contact between an electro-elastic–visco-plastic body and a conductive foundation. The contact is modelled with normal compliance and a version of Coulomb’s law of dry friction, in which the stiffness and the friction coefficients depend on the electric potential. We derive a variational formulation of the problem and we prove an existence and uniqueness result. The proof is based on a recent existence and uniqueness result on history-dependent quasivariational inequalities obtained in [15]. Then we introduce a fully discrete scheme for solving the problem and, under certain solution regularity assumptions, we derive an optimal order error estimate. Finally, we present some numerical results in the study of a two-dimensional test problem which describes the process of contact in a microelectromechanical switch.     

On well‐posedness for some thermo‐piezoelectric coupling models

There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost‐effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor, the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well‐posedness of such models. There is a need to extend the use of such sensors into high‐temperature environments, and so this paper shows, for a broad class of models, the well‐posedness of the magneto‐electro‐thermo‐elastic problem. Because of its widespread use in the literature, we also show the well‐posedness of the quasi‐electrostatic case. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonsmooth dynamic frictional contact of a thermoviscoelastic body

Abstract

This paper studies a system of two hemivariational inequalities modeling a dynamic thermoviscoelastic contact problem with general nonmonotone and multivalued subdifferential boundary conditions. Thermal effects are included in the Kelvin–Voigt thermoviscoelastic constitutive law and in the boundary conditions, and so in frictional heat generation, which takes place on the boundary and enters the condition for the temperature. The existence of a weak solution to the problem is established using a recent surjectivity result for differential inclusions associated with pseudomonotone operators.     

Numerical analysis of a bilateral frictional contact problem for linearly elastic materials

We consider a mathematical model which describes the contactbetween a linearly elastic body and an obstacle, the so-calledfoundation. The process is quasistatic and the contact is bilateral,i.e. there is no loss of contact during the process. The frictionis modelled with Tresca's law. The variational formulation ofthe problem is a nonlinear evolutionary inequality for the displacementfield which has a unique solution under certain assumptionson the given data. We study spatially semi-discrete and fullydiscrete schemes for the problem with finite-difference discretizationin time and finite-element discretization in space. The numericalschemes have unique solutions. We show the convergence of thescheme under the basic solution regularity. Under appropriateregularity assumptions on the solution, we derive optimal ordererror estimates. Finally, we present numerical results in thestudy of two-dimensional test problems.     

Analysis of a quasistatic contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is quasistatic, the material behavior is modeled with an electro-viscoelastic constitutive law and the contact is described with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving two history-dependent hemivariational inequalities in which the unknowns are the velocity and electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on a recent result on history-dependent hemivariational inequalities obtained in Migórski et al. (submitted for publication) [16].     

Green's function approach to study the propagation of SH‐wave in piezoelectric layer influenced by a point source

Green's function technique serves as a powerful tool to find the particle displacements due to SH‐wave propagation in layer of a shape different from the space between two parallel planes. Therefore, the present paper undertook to study the propagation of SH‐wave in a transversely isotropic piezoelectric layer under the influence of a point source and overlying a heterogeneous substrate using Green's function technique. The coupled electromechanical field equations are solved with the aid of Green's function technique. Expression for displacements in both layer and substrate, scalar potential and finally the dispersion relation is obtained analytically for the case when wave propagates along the direction of layering. Numerical computations are carried out and demonstrated with the aid of graphs for six different piezoelectric materials namely PZT‐5H ceramics, Barium titanate (BaTiO₃) ceramics, Silicon dioxide (SiO₂) glass, Borosilicate glass, Cobalt Iron Oxide (CoFe₂O₄), and Aluminum Nitride (AlN). The effects of heterogeneity, piezoelectric and dielectric constants on the dispersion curve are highlighted. Moreover, comparative study is carried out taking the phase velocity for different piezoelectric materials on one hand and isotropic case on the other. Dispersion relation is reduced to well‐known classical Love wave equation with a view to illuminate the authenticity of problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Positive solutions for a fourth order discrete p‐Laplacian boundary value problem

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p‐Laplacian boundary value problem where φ_p(s) = | s | ^{p ? 2}s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.