Longtime Dynamic Behavior For Several Viscoelastic Models

With the widespread use of viscoelastic materials in national defence and modern industry,viscoelastic mechanics has received a great deal of attention from scholars at home and abroad.Nowadays,viscoelastic mechanics has become a fundamental element of solid mechanics and an important part of modern mechanics of continuous media.The so-called viscoelastic material is a class of material that has both a transient elastic response and a continuous internal friction effect under the action of a load,which is mathematically known as memory properties.With the rapid development of modern technology and industry,machinery is moving towards high speed7 light weight and sophistication,and the vibration of its structure is becoming increasingly prominent.In practice,the most economical and common vibration damping measure is to attach a layer of viscoelastic material to the external surface of the equipment structure.This method does not change the form of the structural elements themselves,but also converts the mechanical vibration energy into thermal energy through the viscoelastic material on the surface of the structural elements,thus achieving the effect of vibration damping.As important structural elements in engineering,beams and rods are widely used in aerospace,aviation,armaments and mechanical engineering.As structural elements of the foundation of mechanical equipment,beams and slabs are bound to be in long-term service.Therefore,this paper takes the vibration model of beams and rods made of viscoelastic materials as the object of study,and discusses the long time dynamic behaviour of the corresponding equation solutions,also known as integral attractors,to provide a theoretical basis for the wide application of viscoelastic materials.This paper provides a theoretical basis for the widespread use of viscoelastic materials and enables a more accurate and profound description and understanding of the motion of objects and physical laws.The details are as follows.Firstly,we considered the initial boundary value problem for the following rod equation of viscoelastic with nonlinear strong damping the existence and uniqueness of the global weak solution is proved by using the FaetoGalerkin approximation method and the Lions-Aubin compactness theorem.Then,the existence of the bounded absorption set of the semigroup is proved by some inequalities,and the existence of the global attractor of the system is obtained by proving the compactness of the semigroup.The Galerkin truncation method is used to transform the existing vibration models of viscoelastic rods with different damping,which have been proved to have global attractors,into ordinary differential equations respectively,and the fourth order Runge-Kutta method is used to solve them numerically.Finally,MATLAB software was used to conduct numerical simulation analysis on the dynamic behavior of each Galerkin truncation system,and the existing theoretical analysis results were presented more intuitively.Secondly,we studied the initial boundary value problem for the following rod equation of viscoelastic with nonlinear strong damping We take M(s)=a+bs and M(s)=1+sm/2,respectively.Under the framework of past history,the existence of the global attractor of the corresponding dynamical system is proved by verifying the asymptotic compactness of the semigroup.Nextly,we discussed the initial boundary value problem for the following nonlinear viscoelastic Kirchhoff rod equation firstly,the existence and uniqueness of global weak solutions and their continuous dependence on initial values are obtained by using the Faedo-Galerkin approximation method and Lions-Aubin compactness theorem.Then,a stability inequality is used to verify the compactness of semigroups,and the existence of global attractors for corresponding dynamical systems is proved.Finally,we studied the initial boundary value problem for the following nonlinear viscoelastic Kirchhoff beam equation the research of this kind of equation mainly focuses on the existence of solution and energy decay estimation,but the research on global attractor is very few.We first obtain the uniqueness of the solution by using the Faedo-Galerkin approximation method,and prove the existence of the global attractor by using a stability inequality.