Variational and Numerical Analysis for Frictional Contact Problem with Normal Compliance in Thermo-Electro-Viscoelasticity

Abstract

In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.

Analysis and Approximation of Hemivariational Inequality for a Frictional Thermo-electro-visco-elastic Contact Problem with Damage

“…Here, conditions (2.1)-(2.4) represent the thermo-electro-visco-elastic constitutive laws with damage, see [2,9,13,24] for more details, where A ∈ L ∞ (Ω) and B ∈ L ∞ (Ω) are the viscous and the elastic tensors, P = (e ijk ) ∈ L ∞ (Ω) is the piezoelectric tensor, β = (β ij ) is the symmetric and coercive electric permittivity tensors, G is the pyroelectric tensor, M = (m ij ) is the thermal expansion tensor, N = (n i ) is the pyroelectric tensor, K = (k ij ) is the thermal conductivity tensor and φ is the mechanical source of damage growth. In addition, ε(u) = (∇u+(∇u) T )/2 is the linearized strain tensor, E(ϕ) = −∇ϕ is the electric field, P T = (P kij ) is the transpose tensor of P, I [0,1] is the indicator function of the interval [0, 1] and ∂I [0,1] denotes its subdifferential.…”

“…In the study of a fully discrete scheme for the numerical solutions, a finite element approach is used to approximate the spatial variable and finite differences are used for the time derivatives, and as result, we obtain some error estimates on the approximate solutions. For further information on numerical aspects of elastic and electro-elastic contact problems, we refer [4,5,8,9,27].…”

Numerical Analysis for a piezoelectric contact problem with long-memory and wear

“…Because of all of these real-world applications, we've seen an increase in the number of researchers and scholars interested in frictional mechanics. Frictional contact problems involving piezoelectric materials have been extensively studied to construct and analyze different electro-elastic models with piezoelectric effect, the reader is welcome to consult the references [1][2][3][4][5][6][7] and the references therein. The literature here is very abundant: for piezo-viscoelastic frictional contact problems with different contact conditions, see e.g., [8][9][10][11][12], for a slip-dependent electro-elastic frictional contact problem, see among others [2] and for electro-viscoelastic problems with adhesion, see [4,13] and more recently [5,14,15].…”

On the dynamic frictionless contact problem with normal compliance in thermo-piezoelectricity

“…In the literature, there are some mathematical results concerning the variational analysis for this kind of problems; see, for instance, [1,3,4] for static models that consider the effects of mechanical, electrical, and thermal interactions in frictional contacts. The mathematical models that describe quasistatic frictional contact with thermo-piezoelectric effects are already addressed in [5,6,7,11,12], and more recently in [8]. Numerical schemes and their error estimates for the aforementioned models were also discussed for both static and quasistatic cases in [1,5,6,8].…”

“…The mathematical models that describe quasistatic frictional contact with thermo-piezoelectric effects are already addressed in [5,6,7,11,12], and more recently in [8]. Numerical schemes and their error estimates for the aforementioned models were also discussed for both static and quasistatic cases in [1,5,6,8].…”