Georgian Mathematical Journal Volume 16 (2009), Number 4, 629–650 UNILATERAL CONTACT PROBLEMS WITH FRICTION FOR HEMITROPIC ELASTIC SOLIDS AVTANDIL GACHECHILADZE, ROLAND GACHECHILADZE, AND DAVID NATROSHVILI Abstract. We study three-dimensional unilateral contact problems with friction for hemitropic elastic solids. We give their mathematical formu- lation in the form of spatial variational inequalities and show the equivalence to the corresponding minimization problem. Based on our variational in- equality approach, we prove existence and uniqueness theorems. We prove also that solutions continuously depend on the data of the original problem and on the friction coefficient. Our investigation includes the special partic- ular case of only traction-contact boundary conditions without prescribing displacement and microrotation vectors along some part of the boundary of a hemitropic elastic body. Then the problems are not unconditionally solvable and we derive the necessary and sufficient conditions of their solvability. 2000 Mathematics Subject Classification: 35J85, 49J40, 74M10, 74M15. Key words and phrases: Elasticity theory, hemitropic solid, contact prob- lem, unilateral problems, variational inequality. 1. Introduction In recent years, theories of continuum mechanics in which deformation is described not only by a usual displacement vector field, but by scalar, vector or tensor fields as well, have been the object of intensive research. Classical elasticity associates only three translational degrees of freedom with the mate- rial points of a body, and all mechanical characteristics are expressed by the corresponding displacement vector. On the contrary, micropolar theory, by in- cluding intrinsic rotations of particles, provides a rather complex model of an elastic body that can support body forces and body couple vectors as well as force stress vectors and couple stress vectors at the surface. Consequently, in this case all the mechanical quantities are written in terms of the displacement and microrotation vectors. The origin of rational theories of polar continua goes back to brothers E. and F. Cosserat [4], [5], who gave an impetus to the development of mechanics of continuous media in which each material point has six degrees of freedom defined by three displacement components and three microrotation components (for the history of the problem see [7], [17], [22], [31], and the references therein). A micropolar solid which is not isotropic with respect to inversion is called hemitropic, noncentrosymmetric or chiral. Materials may exhibit chirality on an atomic scale, as in quartz and in biological molecules (DNA) and on a large ISSN 1072-947X / $8.00 / c  Heldermann Verlag www.heldermann.de