Dynamic nonsmooth frictional contact problems with damage in thermoviscoelasticity

Abstract

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem describing contact problem between the body and foundation. The process is dynamic, the material behaviour is described by nonlinear viscoelastic law, strongly coupled with the thermal effects. The contact is modelled by nonmonotone subdifferential boundary conditions. The mechanical damage of the material is described by a parabolic equation. We use recent results from the theory of hemivariational inequailities and fixed point theorems.

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.…”

“…This is described by an interval variable, which is modelled by a parabolic differential inclusion. Moreover, for contact problems involving damage phenomena, we refer to [13,15,16,24] and the references therein.…”

“…Then, we shall deal with a thermo-electro-visco-elastic materials for which the constitutive laws are given, without indicating explicitly the dependence of various functions on the independent variables x ∈ Ω ∪ Γ, as follows: σ(t) = Aε( u(t)) + B(ε(u(t)), ζ(t)) − P T E(ϕ(t)) − Cθ(t), (1.1) D(t) = Pε(u(t)) + βE(ϕ(t)) + Gθ(t), (1.2) θ(t) − div K(∇θ(t)) = Mε(u)(t) − N E(ϕ(t)) + h 0 (t), (1.3) in which σ denotes the stress tensor, u is the displacement field, ζ is the damage field, ϕ is the electric potential field and θ is the temperature. The extension of the thermo-viscoelastic constitutive laws with damage employed in [24] is represented by a constitutive equation of the type (1.1)- (1.3). They generalize the thermo-piezoelectric and thermoelectro-visco-elastic constitutive equations employed in [2,25].…”

On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage

“…As the evolution of the displacement and damage may influence each other, the governing relations for both quantities are mutually coupled. The evolution variational inequality for damage that we use here, was introduced by Frémond (see for example [9,10]) and has been recently extensively studied for springs (see [2,5]), beams (see [1,14]) and for three-dimensional deformable solids (see [4,11,15,16,20,22]).…”

On dynamic contact problem with generalized Coulomb friction, normal compliance and damage

“…According to this model, the damage is represented by a function β : Ω × [0, T ] → [0, 1], where the value 0 means that the body is fully damaged, and the value 1 means that it is not damaged at all. Such approach has been recently extensively studied for springs (see [2,6]), beams (see [1,18]) and for three-dimensional deformable solids (see [4,12,14,19,20,25,27]).…”