Research On The Solutions Of Variational And H-semivariational Inequalities In Viscoelastic Contact Problem

Viscoelastic contact problems are widely used in the analysis of mechanical effects of new composite materials,biomaterials and aerospace materials.Variational inequalities and hemivariational inequalities are important mathematical tools for studying contact problems.In this paper,we mainly study the existence and uniqueness of the solution of the viscoelastic contact problem and the convergence of the solution of the perturbation problem,covering three cases of quasistatic contact with wear,quasistatic contact with impulsive traction on the surface and dynamic contact with temperature.The specific research contents are as follows:First,a long memory viscoelastic quasistatic contact model with wear is studied.Based on the principle of variational method,the contact problem is transformed into the coupling form of differential equation and variational inequality.Using monotone operator theory and fixed point theorem,the existence and uniqueness of the solution of the contact problem are studied.On this basis,the approximation problem of the original problem is constructed,and the existence,uniqueness and boundedness of the solution of the sequence approximation problem are proved by introducing the penalty operator and adopting the(generalized)penalty technique.Furthermore,sufficient conditions are given to ensure that the solution of the sequence approximation problem converges to the solution of the original problem.Secondly,the long memory viscoelastic quasistatic contact problem model of contact surface subjected to pulse traction force is studied.The original contact problem is transformed into the coupling form of differential equation and hemivariational inequality.The system is discretized into a group of evolution operator inclusion problems(Rothe problem)by using the time discrete technique(Rothe method).The existence,uniqueness and boundedness of the solution are proved by using the pseudomonotone operator surjection theorem.Then,using the fixed point theory,the existence and uniqueness theorem of the solution of the contact problem is given.Furthermore,the perturbation problem of the original problem is given,and it is proved that the solution of the original problem can be approximated by the solution of the perturbation problem under appropriate conditions.Thirdly,the viscoelastic dynamic contact model affected by temperature is studied.By using the principle of variational method,the original contact problem model is equivalently transformed into a coupled system of variational-hemivariational inequalities and hemivariational inequalities.Applying the theory of nonsmooth analysis,the nonlinear evolution inclusion problem of the system is derived.Then,the existence and uniqueness theorem of the solution of the inclusion problem is given by using the multi-valued operator surjection theorem.Furthermore,by using the fixed point theorem,a sufficient condition for the existence and uniqueness of the solution of the contact problem is given.On this basis,the approximation relationship between the solution of the original problem and the perturbation problem is established.Finally,the theory of history-dependent variational inequalities is developed to the variational-hemivariational inequalities with fully history-dependent operators.The existence and uniqueness theorem of the solution is obtained by using the theory of nonsmooth analysis and nonlinear functional analysis.On this basis,the solution of the original problem is approximated by the solution sequence of the approximation problem after all parameters of the variational inequality are perturbed.