Research On Multi-physics Coupling Smoothed Finite Element Method Based On Asymptotic Homogenization

Piezoelectric composite materials have the excellent characteristics such as low acoustic impedance,low dielectric constant,low mechanical quality factor,good flexibility and high sensitivity,which are widely used in the preparation of smart components including energy harvesters,sensors and transducers.The actual service process is always accompanied by multi-physics coupling phenomena,such as piezoelectric effect,pyroelectric effect and hygroelectric effect,etc.In order to further improve the application breadth and depth of piezoelectric composites,it is urgent to solve the basic mechanical properties,multi-physics performance characterization and coupling behavior mechanism of piezoelectric composites.With the rapid development of science and technology,computer software and hardware,numerical calculation methods have become a necessary and effective method for solving multi-physics coupling problems.Finite element method（FEM）is currently the most effective numerical simulation method for solving multi-physics coupling problems.However,FEM is based on discrete elements,it has inherent defects such as overly-stiff,small displacement solution,weak ability to deal with distorted meshes and high requirements for mesh quality,especially when solving multi-physics coupling problems and inhomogeneous composites problems,the complexity of the actual structures often results in low mesh quality,low solution accuracy or even errors.In order to overcome the difficulties above,developing an accurate and efficient numerical calculation method for the performance evaluation of piezoelectric composites,guiding the design,development and directional preparation of new piezoelectric composites with certain specific functions and shortening the research and development cycle of materials are the strategic demand for the long-term development of national economy,which have important scientific significance and engineering application value for the design and development of piezoelectric composites.In order to further improve the mechanical properties of piezoelectric composites,the gradient smoothing technique is introduced into the multi-physics coupling FEM,the electro-mechanical coupling smooth FEM（CS-FEM）based on asymptotic homogenization,and the hygro-electro-mechanical coupling smooth FEM,the thermo-electro-mechanical CS-FEM and the hygro-thermo-electro-mechanical CS-FEM are proposed.Combining with the generalized solution method of multi-physics coupling dynamic equation,the dynamic responses of typical piezoelectric composite structures are analyzed.First,based on the basic theories of piezoelectric composites and asymptotic homogenization method,the piezoelectric equations of the piezoelectric composite are equivalent to two types of local problems:plane problem ₁₁L and ₁₂L,elastic body anti-plane strain and electric potential system coupling problem ₁₃L.The general expressions of the effective performance coefficients of piezoelectric composites are deduced,the equivalent performance parameters of typical piezoelectric composites are solved to verify the validity and correctness of the calculation results.Second,based on the basic equations of FEM and piezoelectric composites,combined with asymptotic homogenization method,the gradient smoothing technique is introduced into the electro-mechanical coupling FEM,the electro-mechanical coupling FEM based on asymptotic homogenization is proposed.The equations of electro-mechanical coupling CS-FEM are deduced for piezoelectric composite structures,the dynamic characteristics of piezoelectric composites are also analyzed.Through solving the dynamic response problems of two-layer piezoelectric composite driver and piezoelectric composite energy harvester,the validity and correctness of the method are verified.Third,starting from the Gibbs free energy function,based on the hygro-electro-mechanical coupling effect,piezoelectric equation,generalized geometric equation,generalized equilibrium equation and generalized boundary conditions of the piezoelectric composite and the effective performance coefficients predicted by the asymptotic homogenization method,the hygro-electro-mechanical multi-physics coupling CS-FEM is proposed.Using the modal analysis method and the generalized Newmark method to solve the inherent frequency and transient responses of typical structures,the inherent frequency and dynamic behavior of piezoelectric composites in moisture environment are analyzed,the results are compared with those of FEM to verify the correctness and effectiveness of the method.Then,starting from the intrinsic characteristics of piezoelectric composites,based on the generalized geometric equation,generalized equilibrium equation,generalized boundary conditions and hygro-thermo-electro-mechanical coupling effect,combined with the asymptotic homogenization method to predict the effective performance coefficients of piezoelectric composites,the thermo-electro-mechanical coupling CS-FEM based on asymptotic homogenization is developed.The dynamic control equation of the thermo-electro-mechanical coupling FEM is deduced and rewritten into a unified form.The Wilson-θmethod is used to solve the dynamic problems of piezoelectric composite structures,and the influence of temperature variation on the structure inherent frequency and dynamic behavior.The results are compared with those of FEM to validate the correctness and effectiveness of the method.Finally,based on the piezoelectric equation,generalized geometric equation,generalized equilibrium equation,generalized boundary conditions and hygro-thermo-electro-mechanical coupling effect and using the Hamilton principle to deduce the dynamic equations of the multi-physics coupling system of piezoelectric composite.Combined with the effective performance coefficients of piezoelectric composites predicted by the asymptotic homogenization method,the hygro-thermo-electro-mechanical multi-physics coupling CS-FEM based on asymptotic homogenization is developed.The Wilson-θmethod is used to analyze the transient responses of piezoelectric composite structures.And the influence of the continuous variation of moisture concentration and temperature on the dynamic response of the piezoelectric composite energy harvester is analyzed.The results are compared with those of FEM to validate the correctness and effectiveness of the method.The work above shows that the multi-physics coupling CS-FEM based on asymptotic homogenization has the advantages of simple solution format,high solution accuracy and low mesh quality requirement when solving the mechanical problems of piezoelectric composites,which has important scientific value and engineering application prospects for the design and development of piezoelectric composite materials.