A High-order Multiscale Model And Numerical Algorithm For Mechanical Properties Of Composites With Periodic Configurations

In modern engineering applications,compared with the traditional elastoplastic materials,when viscoelastic materials are subjected to external loads,the material response depends not only on the loading,but also on the loading time,such as concrete,high polymer materials,metal materials under high strain rate,etc.The influence of the rate effect is not considered in elastic mechanical for this type of viscoelastic material with elastic and viscous properties,thus its mechanical properties cannot be accurately described.In recent years,it has been a research hotspot to reasonably discuss the mechanical properties of viscoelastic materials,especially for materials with typical multi-scale characteristics such as concrete.Because of this,this paper mainly studies the multi-scale approach for viscoelastic problems of composite materials,the main contents and achievements are as follows:Firstly,this manuscript introduces the multi-scale asymptotic expansion approach for periodically distributed composite materials,which is based on the homogenization theory and provides the corresponding second-order two-scale algorithm.The error analysis and numerical cases show that the second-order two-scale approximate solution could better approximate the real solution of the original problem,which reveals the necessity of the second-order correction term.When the traditional finite element method is used to solve the corresponding problems,the meshing must be very fine,which requires a huge amount of calculation,even cannot find the solution.The second-order two-scale approach transforms the original problem into a homogenization problem in a macroscopic homogeneous region and several simple cell problems,which greatly promotes the computational efficiency compared with the traditional finite element method.Secondly,based on the second-order two-scale approach,a new high-order three-scale asymptotic expansion approach is systematically put forward.The asymptotic expansion approach is aimed to evaluate the viscoelastic analysis of composite materials with multi-layer and small-scale structures.The heterogeneity of viscoelastic composite materials could be described traverse the periodic arrangement of cells on the micro-scale.Utilizing the homogenization method and Laplace transforms,the high-order three-scale asymptotic expansion formula of the viscoelastic problem is established,and the local cell solutions of micro-scale and mesoscale are defined.In addition,the homogenization coefficients of mesoscale and macro-scale are derived respectively,and the homogenization equation of the whole coarse-scale domain is obtained.Through the microcell solution and homogenization solution,the high-order three-scale approximate solutions of strain field and stress field are constructed.Finally,on the basis of inverse Laplace transform and three-scale asymptotic homogenization,a high-order three-scale finite element method is proposed for viscoelastic problems of periodic composite materials.Some typical numerical examples are used to verify the effectiveness of the high-order three-scale method.The results show that the proposed high-order three-scale approach is effective and accurate for viscoelastic problems of composite materials with multi-layer and small-scale structures.