The optimal control problems of hyperbolic H-hemivariational inequalities with the state constraints and nonnomotone multivalued mapping term are considered. The optimal solutions are obtained. In addition, their approximating problems are also studied.

The rule-of-mixture approach has become one of the widely spread ways to investigate the mechanical properties of nano-materials and nano-structures, and it is very important for the simulation results to exactly compute phase volume fractions. The nanocrystalline (NC) materials are treated as three-phase composites consisting of grain core phase, grain boundary (GB) phase and triple junction phase, and a two-dimensional three-phase mixture regular polygon model is established to investigate the scale effect of mechanical properties of NC materials due to the geometrical polyhedron characteristics of crystal grain. For different multi-sided geometrical shapes of grains, the corresponding regular polygon model is adopted to obtain more precise phase volume fractions and exactly predict the mechanical properties of NC materials.

In the present paper, some important characteristics of Fenchel-, Frechet-, Hadamard-. and Gateaux-Subdfferentials are showed up, and properties of functions, especially convexity of functions, are described by subdifferentials.

In this paper the existence of solution to finite elastodynamics constrainted by mixed boundary conditions is derived when the hyperpotential and its gradient (for Green's strain) satisfy adequate conditions.

Under the small deformation assumption th~'s paper shows the existence of sohaion for the system of elastic d.rnamics with the general nonlinear constitutive laws. and the existence of classical solution can be found under weaker conditions.