Nonsmooth Contact Dynamics

The scientific goal of the action is to develop new and non-standard mathematical and numerical tools
directly motivated by the needs of the analysis of various classes of contact problems in mechanics
which are of fundamental importance in technology, industry and real engineering applications. The overall
objective of the project is to conduct the ground-breaking research in the field of the Mathematical Theory
of Contact Mechanics in physical sciences and engineering. The project crosses the borderlines among
mathematics, mechanics, computer science, and material science. It addresses new and emerging areas of
research: theory of contact in quasicrystals, variational inequalities on nonconvex sets, hemivariational
inequalities, shape optimization and optimal shape design, theory of contact in fluid mechanics, and contact
problems in hyperelasticity.

The scientific results and the knowhow of the project are important for society in many aspects.
They are of interest to areas of research and industry. Potential applications include economic profits
due to more accurate prediction of the material stress, displacement, friction and damage under
various contact regimes. Although the project is mainly oriented at mathematics,
and mathematics contributes a lot to a general mass culture, the action has potential applications
in such areas as, for example, automobile industry, building sciences, civil engineering, metal
engineering, or mechanical models in medicine. The interdisciplinary aspects are exemplified
by the fact that the publications written by the project members are cited both in mathematical papers
as well as in engineering ones.
The role of basic science is very important since it has fundamental influence on science, technology
and other disciplines. We agree with the common belief that our culture is only just approaching the
state in which mathematics will be able to begin a wide penetration of social consciousness. The project
also contributes to the promotion of such a statement. European industry will benefit greatly from many
improvements and tools that will come out from the action. There are many European academic and
industrial sites that currently understand and use rigorous methods of Contact Mechanics because
of the economic profits.
On the European level, significant benefits come from promoting science and scientific career as well as
from raising the standards of research. The project contributes to the major objectives of Research Executive
Agency since it will strengthen the process of development of research activities and increase international
and interdisciplinary mobility of researchers in Europe. The main dissemination media of the results
of the project are peer-reviewed publications, lectures, conferences, events promoting the research
aimed at various audiences, and teaching activities of seconded researchers.

The work performed in the project within the two years enters the four workpackages: (1) Novel techniques for inequality and inclusion
problems in Banach spaces, (2) Novel techniques for inequality and inclusion problems in Banach spaces, (3) Control theory for problems
of Nonsmooth Contact Mechanics, and (4) Advanced computational algorithms for nonsmooth problems. The work was performed in
collaboration of all beneficieries and partner institutions.

The main results achieved in this period involve the following mathematical problems: fixed point theorems for multifunctions in theory
of operator and differential inclusions, existence of solutions to systems of operator and differential inclusions, contact with wear diffusion
on contact surfaces, contact problems in hyperelasticity, existence of optimal solutions to differential inclusions and contact problems,
controllability for operator and differential inclusions, active set methods for evolutionary contact problems in viscoelasticity, numerical
codes for frictional unilateral contact problems in hyperelasticity.

The research will go far beyond the current state-of-theart. Within continuum mechanics, we treat contact
problems in solid mechanics (elasticity, viscoelasticity, plasticity, impulsive systems), fracture mechanics
(cracks propagation), fluid mechanics (Newtonian and non-Newtonian fluids), and rheology (which combines
both solid and fluid characteristics). Furthermore, we are able not only to deal with classical smooth mechanics
but also to account for nonsmooth events like stickslip transitions or impacts. In this sense, the project does not
just extend classical mechanics to some nonsmooth special cases but set up a new formulation which incorporates
classical mechanics. The pioneering work on the search for a unified theory for Contact Mechanics will be
accompanied by efficient numerical schemes, which links the mathematical framework with practical applications.

The expected results until thie end of the project are the following.
(1) Find general expressions which describe contact laws (equations, inequalities, inclusions, relations) and
identify the genuine (standard) form for classes of contact models in nonlinear continuum mechanics. This
is an unconventional aspect of the research, such attempts remain unknown in this field. (2) Develop the theory
of contact based on the notion of differential variational-hemivariational inequalities and history-dependent
operators in Banach spaces. (3) Study the intrinsic properties of mechanical processes through the mathematical
analysis of the underlying contact models. The importance for society, industry, economy, etc. remains the same
as it was in the first two years of project implementation.

Periodic Reporting for period 1 - CONMECH (Nonsmooth Contact Dynamics)

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