Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results. Received: February 16, 2005     

Variational approach to contact problems in nonlinear elasticity

We use variational methods to study problems in nonlinear 3-dimensional elasticity where the deformation of the elastic body is restricted by a rigid obstacle. For an assigned variational problem we first verify the existence of constrained minimizers whereby we extend previous results. Then we rigorously derive the Euler-Lagrange equation as necessary condition for minimizers, which was possible before only under strong smoothness assumptions on the solution. The Lagrange multiplier corresponding to the obstacle constraint provides structural information about the nature of frictionless contact. In the case of contact with, e.g., a corner of the obstacle, we derive a qualitatively new contact condition taking into account the deformed shape of the elastic body. By our analysis it is shown here for the first time rigorously that energy minimizers really solve the mechanical contact problem. Received: 20 October 2000 / Accepted: 7 June 2001 / Published online: 5 September 2002     

Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field 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 abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.     

Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory

We consider a mathematical model which describes the bilateral contact between a deformable body and an obstacle. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the friction is modeled with Tresca’s law. The problem has a unique weak solution. Here we study spatially semi-discrete and fully discrete schemes using finite differences and finite elements. We show the convergence of the schemes under the basic solution regularity and we derive order error estimates. Finally, we present an algorithm for the numerical realization and simulations for a two-dimensional test problem.     

Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents

We formulate the concentration-compactness principle at infinity for both subcritical and critical case. We show some applications to the existence theory of semilinear elliptic equations involving critical and subcritical Sobolev exponents.     

On the boundary operator of a fourth order differential problem

We study an unilateral elliptic fourth order problem and establish some uniqueness results. The expression of the boundary operation is also given     

Quasistatic crack growth in finite elasticity with non-interpenetration

We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking into account the non-interpenetration condition.     

Modelling of topological derivatives for contact problems

The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization. Partially supported by the grant 4 T11A 01524 of the State Committee for the Scientific Research of the Republic of Poland     

Recovering complex elastic scatterers by a single far-field pattern

We consider the inverse scattering problem of reconstructing multiple impenetrable bodies embedded in an unbounded, homogeneous and isotropic elastic medium. The inverse problem is nonlinear and ill-posed. Our study is conducted in an extremely general and practical setting: the number of scatterers is unknown in advance; and each scatterer could be either a rigid body or a cavity which is not required to be known in advance; and moreover there might be components of multiscale sizes presented simultaneously. We develop several locating schemes by making use of only a single far-field pattern, which is widely known to be challenging in the literature. The inverse scattering schemes are of a totally “direct” nature without any inversion involved. For the recovery of multiple small scatterers, the nonlinear inverse problem is linearized and to that end, we derive sharp asymptotic expansion of the elastic far-field pattern in terms of the relative size of the cavities. The asymptotic expansion is based on the boundary-layer-potential technique and the result obtained is of significant mathematical interest for its own sake. The recovery of regular-size/extended scatterers is based on projecting the measured far-field pattern into an admissible solution space. With a local tuning technique, we can further recover multiple multiscale elastic scatterers.     

Moser iteration for (quasi)minimizers on metric spaces

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

Multiplicity of solutions for a class of fourth elliptic equations

We show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under Navier boundary conditions by linking approaches.     

Nonlinear BVPS with functional parameters

We study the existence, nonexistence and properties of solutions for a certain class of second-order ODEs and their dependence on functional parameters, also in the case when nonlinearities are, in some sense, singular. This approach is based on variational methods and cover both sublinear and superlinear cases. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principle which enables approximation of the solution for our problem. We apply these results to obtain the existence of bounded, radial and positive classical solutions for the BVP of elliptic type. Observe that our method allows us to investigate a certain class of elliptic systems in both bounded annular domain and exterior domain.     

Thermodynamics and analysis of rate-independent adhesive contact at small strains

We address a model for adhesive unilateral frictionless Signorini-type contact between bodies of heat-conductive viscoelastic material, in the linear Kelvin-Voigt rheology, undergoing thermal expansion. The flow rule for debonding the adhesion is considered rate-independent and unidirectional, and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned.     

Solitary waves in a tapered prestressed fluid-filled elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solution with variable wave speed. It is observed that, the wave speed increases with distance for positive tapering while it decreases for negative tapering.     

Variational principles with integrands defined through a minimum

We revisit a systematic approach for the computation and analysis of the convex hull of non-convex integrands defined through the minimum of convex densities. It consists in reformulating the non-convex variational problem as a double minimization and exploiting appropriately the nature of optimality of the inner minimization. This requires gradient Young measures in the vector case, even if the initial problem was scalar, as the full problem is recast through the computation of a certain quasiconvexification. We illustrate this strategy by looking at two typical non-convex scalar problems. We hope to address vector problems in the near future.     

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.