Abstract
We consider two quasistatic frictionless contact problems for viscoelastic bodies with long memory. In the first problem the contact is modelled with Signorini's conditions and in the second one is modelled with normal compliance. In both problems the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the mechanical problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and a fixed point theorem. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solutions of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.
Highlights
The adhesive contact between deformable bodies, when a glue is added to prevent relative motion of the surfaces, has received recently increased attention in the mathematical literature
The novelty in all the above papers is the introduction of a surface internal variable, the bonding field, denoted in this paper by β; it describes the pointwise fractional density of active bonds on the contact surface, and sometimes referred to as the intensity of adhesion
There, models for dynamic or quasistatic process of frictionless adhesive contact between a deformable body and a foundation have been analyzed and simulated; the contact was described with normal compliance or was assumed to be bilateral, and the behavior of the material was modelled with a nonlinear Kelvin-Voigt viscoelastic constitutive law; the models included the bonding field as an additional dependent variable, defined and evolving on the contact surface; the existence of a unique weak solution to the models has been obtained by using arguments of evolutionary equations in Banach spaces and a fixed point theorem
Summary
The adhesive contact between deformable bodies, when a glue is added to prevent relative motion of the surfaces, has received recently increased attention in the mathematical literature. There, models for dynamic or quasistatic process of frictionless adhesive contact between a deformable body and a foundation have been analyzed and simulated; the contact was described with normal compliance or was assumed to be bilateral, and the behavior of the material was modelled with a nonlinear Kelvin-Voigt viscoelastic constitutive law; the models included the bonding field as an additional dependent variable, defined and evolving on the contact surface; the existence of a unique weak solution to the models has been obtained by using arguments of evolutionary equations in Banach spaces and a fixed point theorem.
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