Uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory
Von Karman plates are thin structures with linear constitutive relations involving geometric nonlinearities arising from large rotations although strains are small. In this work, a linear viscoelastic constitutive relation is adopted with a given relaxation function. Hence equations describing the motion of the plate becomes nonlinear integro-differential equations because of the hereditary integral due to the viscoelastic term. In addition, nonlinear dissipative boundary conditions are prescribed on some part the sufficiently smooth boundary of the plate. The complementary part of the boundary is clamped. The authors first prove the existence of the solution for such a plate. To this end they employ the well known Galerkin method involving first a finite expansion. Imposing some restriction on the behaviour of the solution and making use of properties of Hilbert and some Sobolev spaces they manage to show uniqueness by a limiting process. Furthermore, they prove that an appropriately defined energy functional decays to zero with time if the relaxation function decreases exponentially.
nonlinear integro-differential equations
scientific article; zbMATH DE number 5709156