An equilibrium problem for a thermoelectroconductive body with the Signorini condition on the boundary

We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators

A thermoelastic contact problem with a phase transition

We investigate a thermomechanical contact problem with phase transitions. The system of equations consists of a quasistatic momentum balance and a semilinear energy balance. The phase transition is described by an ordinary differential equation. Different mechanical properties of the respective phases are taken care of by a mixture ansatz. We prove the existence of a weak solution and a uniqueness result, the latter only being valid in one space dimension

On safe crack shapes in elastic bodies

According to the Griffith criterion, a crack propagation occurs provided that the derivative of the energy functional with respect to the crack length reaches some critical value. We consider a generalization of this criterion to the case of nonlinear cracks satisfying a non-penetration condition and investigate the dependence of the shape derivative of the energy functional on the crack shape. In the paper, we find the crack shape which gives the maximal deviation of the energy functional derivative from a given critical value and, in particular, prove that this optimality problem admits a solution

On an equilibrium problem for a cracked body with electrothermoconductivity

We consider a problem related to resistance spot welding. The mathematical model describes the equilibrium state of an elastic, cracked body subjected to heat transfer and electroconductivity and can be viewed as an extension to the classical thermistor problem. We prove existence of a solution in Sobolev spaces

A mathematical model for impulse resistance welding

We present a mathematical model of impulse resistance welding. It accounts for electrical, thermal and mechanical effects, which are nonlinearly coupled by the balance laws, constitutive equations and boundary conditions. The electrical effects of the weld machine are incorporated by a discrete oscillator circuit which is coupled to the field equations by a boundary condition. We prove the existence of weak solutions for a slightly simplified model which however still covers most of its essential features, e.g. the quadratic Joule heat term and a quadratic term due to non-elastic energy dissipation. We discuss the numerical implementation in a 2D setting, present some numerical results and conclude with some remarks on future research

Griffith Formulae for Elasticity Systems with unilateral Conditions

In the paper we consider the elasticity equations in nonsmooth domains in $R^n, n=2,3$. The domains have a crack whose length may change. At the crack faces, inequality type boundary conditions describing a mutual nonpenetration of the crack faces are prescribed. The derivative of the energy functional with respect to the crack length is obtained. The Griffith formulae are derived in 2D and 3D cases and the other properties of the solutions are established. In two-dimensional case the Rice--Cherepanov's integral over a closed curve is constructed. The path independence of the Rice--Cherepanov's integral is shown

Shape and topological sensitivity analysis in domains with cracks

Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method \cite{Fulman} can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics

Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions

We consider a 3D elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that non-penetration conditions are imposed at the crack faces which do not allow the opposite crack faces to penetrate each other. We analyze the variational formulation of the problem and provide shape and topology sensitivity analysis of the solution

Thin inclusion in elastic body: identification of damage parameter

In the paper, we consider an equilibrium problem for a 2D elastic body with a thin elastic inclusion crossing an external boundary. The elastic body has a defect which is characterized by a positive damage parameter. The presence of a defect means that the problem is formulated in a non-smooth domain. Non-linear boundary conditions at the defect faces are imposed to prevent the mutual penetration between the faces. Both variational and differential problem formulations are proposed, and existence of solutions is established. We study an asymptotics of solutions with respect to the damage parameter as well as with respect to a rigidity parameter of the inclusion. Identification problems for finding the damage parameter are investigated. To this end, existence of solutions of optimal control problems is proven

On the Crossing Bridge between Two Kirchhoff–Love Plates

The paper is concerned with equilibrium problems for two elastic plates connected by a crossing elastic bridge. It is assumed that an inequality-type condition is imposed, providing a mutual non-penetration between the plates and the bridge. The existence of solutions is proved, and passages to limits are justified as the rigidity parameter of the bridge tends to infinity and to zero. Limit models are analyzed. The inverse problem is investigated when both the displacement field and the elasticity tensor of the plate are unknown. In this case, additional information concerning a displacement of a given point of the plate is assumed be given. A solution existence of the inverse problem is proved