This Special Issue of Applicable Analysis reports recent developments in the Mathematical Theory of Contact Mechanics (MTCM) dealing with various mathematical aspects of contact problems. It has been in the making for over a year and a half, and is a follow-up of the special session on ‘Mathematical Analysis of Unilateral Contact Problems’ in the International Conference Emerging Trends in Applied Mathematics and Mechanics (ETAMM 2016) that was held in Perpignan during the week of May 30–June 3, 2016. Some of the results reported here were announced there, others are original.
The peculiarity of MTCM lies in the fact that the most interesting aspects of the problems enter via the boundary conditions, although the equations themselves may be complex, too. Indeed, the various contact conditions and conditions of friction or adhesion are specified on a part of the boundary of the domain occupied by a solid body, and these lead to very interesting and unusual mathematical problems.
As can be seen in the following pages, MTCM has made considerable progress and matured into a rather comprehensive theory that encompasses a wide range of topics: models in the form of Variational Inequalities (VIs) and Hemivariational Inequalities (HVIs), as well as more general differential inclusions, and their mathematical analysis. It also made substantial progress in numerical methods and computations, which are outside of the scope of this issue. The models and the related mathematical problems that can be found here range from very general and very theoretical, to more specific problems that have direct applications. They involve static contact, leading to essentially elliptic mathematical problems; quasistatic or evolution problems that lead to VIs or HVIs; linearly elastic, viscoelastic or nonlinear materials, that may also include material damage. The contact conditions may be local or nonlocal, monotone or nonmonotone, unilateral or with deformable foundations, with friction or frictionless, or with adhesion. Moreover, other processes, such as temperature and electrical currents may be present either on the boundary or in the interior. Each one of the articles below describes a novel combination of these characteristics and provides new mathematical results for the studied problems. Mostly, the results deal with the existence and possible uniqueness of weak solutions. However, some of the papers also provide continuity and additional regularity of the solutions or more detailed structure of the solutions.
We turn now to a short description of the results the reader will find in the following pages.
A dynamic frictional contact problem in a thermomechanical system that, in addition, has an electrical heating term and the associated equation for the electric potential can be found in Bartosz et al. The interest in the problem lies, in addition to a nonmonotone velocity dependent friction condition, in the fact that the coupled nonlinear system consists of equations of motion that are, essentially, hyperbolic, the heat equation that is parabolic and the elliptic equation for the electric potential.
A quasistatic contact problem with adhesion is studied in Bonetti et al. The theoretical interest in problems with adhesion lies in the fact that the adhesive provides tensile resistance to separation of the contacting surfaces. The paper allows for nonlocal adhesive tractions on the contacting boundary and establishes the existence of a weak solution for the problem.
A quasistatic problem for an elastic material with frictional contact is considered by Cocou, where the existence of a solution is obtained by passing to a limit in a series of incremental problems, that have interest in and of themselves. Then, the abstract result is applied to problems involving linear and nonlinear materials of the Hencky type. The paper establishes the existence of a unique weak solution for the problem.
Matei et al. study a class of mixed variational contact problems involving viscoelastic materials. The interest here is in the inclusion of contact tractions in the problem formulation. The existence of weak solutions is established.
Another article that takes thermal effects into account is Migorski et al. where the dynamic problem of contact between a thermoviscoelastic body and a reactive foundation is studied. The existence of a weak solution is established using various tools from the theory of HVIs.
The structure of the solutions to a unilateral contact of a beam that vibrates between two rigid stops is reported in Paoli and Shillor. It is seen that the contact traction, which is a measure, has a regular part and an atomic part related to the impact. It seems that similar ideas may be applied to solutions of other unilateral contact problems.
Another mixed variational quasistatic contact problem that includes adhesion, which uses Lagrange multipliers for the contact tractions, is studied in Patrulescu. Some of the memory effects in the model lead to integral terms.
The solvability of a pseudodifferential linear complementarity problem that arises in the dynamic contact of a viscoeslatic body with a rigid foundation is reported in Petrov.
Rodriguez-Arós derives a contact problem for elliptic membrane shells by applying asymptotic methods to the 3D problem. Obtaining the necessary estimates and passing to the limit leads to a problem for a thin structure.
Next, an interesting frictionless contact problem for a new 2D model for beams, which allows for easy inclusion of transversal and normal contact tractions and the related optimal control problem, is studied in Sofonea et al.
Finally, we mention three articles that are on the boundary of MTCM.
First, a quasistatic problem in which the evolution of material damage is included is studied in Andrews et al. where some of the system coefficients are allowed to be random, so that the measurability of the solutions is of importance. The latter provides a way to deal with uncertainty in the system parameters that often are either not known or have considerable measurements error.
Then, Boukrouche and Paoli deal with a 3D non-stationary Stokes flow with friction boundary conditions that is also history-dependent, and establish the global existence of the solutions.
The last paper by Liu and Motreanu, deals with inclusion problems and HVIs. The paper uses sub-and super-solutions for a (p, q)-Laplacian with multivalued convection term, to prove the existence and location of solutions.
It is clear from the very short description above that MTCM is growing in many interesting directions. It seems to us that issues of optimal control of contact problems are likely to, and should, in view of their applied importance, become more extensively studied and more papers are to be expected. Issues concerning the structure and regularity of the solutions for contact problems are of importance, since they provide both theoretical information and may lead to possibly better numerical methods for such problems. Derivation is needed of the various contact conditions that involve adhesion as a limiting process in which a 3D region that the adhesive occupies is collapsed into a 2D patch on the boundary. In particular, this will provide some insight into the structure of debonding source functions. We expect further mathematical results for increasingly more complex models in the near future, as well as extensions into new directions. It may be of interest to study problems of fluid–solid interactions that involve nonmonotone boundary conditions. Finally, we note that the appropriate contact conditions for the frictional contact of piezoelectric materials are still unknown.
We note that the very recent special issue of the journal Mathematics and Mechanics of Solids, vol. 23(3) (2018), edited by M. Sofonea, and which is also a result of ETAMM2016, is dedicated to more computational and applied aspects of contact processes.
Let us conclude this introduction with the expression of our deepest thanks to the Editor-in-Chief, Professor Steve Xu Yongzhi, for inviting us to be guest editors of this special issue of Applicable Analysis, to all the authors for their very interesting contributions and to the reviewers for their valuable work and comments, and finally to the production team for their help.