Viscoelasticity of fractional type and shape optimization in a theory of rods

The two main focuses of this research are the viscoelastic materials of the fractional type and the problem of the optimization of the shape of elastic rod, subject to certain constraints. We shall formulate of the constitutive equations for the viscoelastic materials, containing either arbitrary number of the fractional derivatives, or the distributed-order fractional derivative, so that the Second Law of Thermodynamics formulated in the terms of the Calusius-Duhamel inequality is satisfied. Namely, we shall determine the restrictions on the constitutive parameters and functions. Constitutive equations obtained as described previously will be coupled with the equations of deformable body and energy balance equations. Initial-boundary value problems will be analyzed, i.e. we shall solve and prove the existence and regularity of the solutions in the appropriate function and distribution spaces. Obtained results will be applied on description of the composites, materials widely used in the dentistry. By the use of the same models, we shall treat the problem of the marginal adaptation. We shall optimize the shape of the elastic rods by the use of the variation principles and the Pontryagin Principle of Maximum. The restrictions on the cross-section area (maximal or minimal value is prescribed) will receive due attention. Post-critical behaviour of the rods will be analyzed, especially the stability of the equilibrium configurations that bifurcates from the initial configuration.