Semi-discrete Finite Element Method for a Class of Visco-elastic Problems with Long Memory under Condition of Friction

Viscoelasticity means that the material under consideration exhibits both creep and relaxation phenomena, i.e., for a fixed stress, the strain in a sample grows in time, whereas for a fixed strain, the stress decreases monotonously. Visco-elatic materials have properties between those of purely elastic materials and viscous fluids. A famfiy of such materials are obtained by allowing the dependence of stress on the current values of both the strain and the velocity gradient. Such materials have memory in the sense that the stress depends not only on the present values of the strain and /or velocity gradient, but on the entire temporal history of the motion. Typically the memory fades with time, i.e., disturbances which occurred in the distant past have less influence on the present stress than those which occurred in the more recent past. In addition, the visco-elastic materials are called linear, if the constituent law establishes a linear relation between the stress tensor σ and the strain tensor∈.