Scattering Of SH Waves By Holes/Inclusions In Inhomogeneous Media With Varying Shear Modulus And Density

The dynamic stress concentration caused by the geometric discontinuity of the material medium is one of the main factors leading to damage failure and fatigue instability.New functional gradient materials are widely used in the aerospace,marine engineering and other significant fields.The damage instability of the material medium cannot be ignored.The properties of materials have complex variability due to the inhomogeneity of the medium,which brings unprecedented challenges to the design,production and performance analysis of materials.The functional gradient materials can be designed through different forms of material parameter changes in order to satisfy the practical engineering application,However,the geometric defects such as holes and inclusions are unavoidable in the design and preparation stages of materials in practical engineering applications.Therefore,combined with the designability of gradient materials the dynamic stress response analysis near geometric defects in functional gradient material medium under elastic waves has important practical engineering significance for the design and production of functional gradient materials.In this paper,the variable coefficient wave equation of elastic wave propagating in the medium with continuous inhomogeneous modulus and density is converted and solved.The analytical solutions of elastic shear wave propagation in inhomogeneous medium with continuous modulus and density are given,and the changes of the power,exponential gradient and periodic trigonometric function of modulus and density are analyzed respectively.And the problem of dynamic stress concentration caused by horizontal shear wave scattering by circular(elliptic)holes and inclusions in medium with arbitrary continuous functional forms is discussed.The main content of this paper is as follows:(1)The displacement field of elastic waves is expressed by auxiliary function and modulus change function according to the relation between elastic waves propagating and medium property on the basis of the wave equation problem in homogeneous medium,aiming at the elastic wave propagation problem in continuous inhomogeneous medium of modulus and density without considering the body effects.The auxiliary function and the modulus change function are introduced to construct the density change function considering the correlation between the modulus and density of the medium.The displacement field of the elastic wave propagation in the inhomogeneous medium is related to the changing modulus and density,accordingly,the non-uniform wave equation is simplified.Furthermore,the wave equation with variable coefficients was transformed into the standard Helmholtz equation in the mapping space by coordinate transformation,and the transformation solution of the wave equation with variable coefficients in the medium with inhomogeneous modulus and density was completed.An analytical solution to the SH wave propagation problem in inhomogeneous medium whose modulus and density vary continuously with space coordinates is proposed.(2)The corresponding wave equation solving methods for different problems are given according to the different changing forms of the modulus and density of the inhomogeneous medium based on the wave equation with variable coefficients,and the variable coefficient wave equation is converted and solved.A model of SH wave scattering by holes and inclusion in a medium with infinite inhomogeneous modulus and density was established based on the theory of complex functions.The corresponding analytical expressions of displacement and stress fields are obtained according to the model and the standardized Helmholtz equation.By corresponding the boundary conditions of the problem model,A system of equations is established to solve the unknown coefficients through the corresponding boundary conditions,and then the dynamic stress concentration coefficient(DSCF)around the holes and inclusions in the continuous inhomogeneous medium is obtained by numerical calculation.Finally,specific examples of three types of medium with different inhomogeneous variation forms are given:(a)inhomogeneous medium with power law change of modulus and density;(b)inhomogeneous medium with exponential gradient change of modulus and density;(c)inhomogeneous medium with trigonometric periodic change of modulus and density.The dynamic stress response around the holes and inclusions in the medium is analyzed for each type of changing inhomogeneous medium.The variation of DSCF with the related parameters is discussed,especially the influence of inhomogeneous parameter changes on DSCF which affect the properties of the medium.(3)The free design of wave propagation frequency in functionally graded materials is considered on the basis of solving the wave problems of the above several kinds of inhomogeneous medium,the frequency of the SH wave that can be propagated in the medium is restricted to a certain range through designing the modulus and density to be specific relations.Furthermore,an analytical solution is given for the SH wave propagation problem in a continuous inhomogeneous medium whose shear modulus and density change in arbitrary form considering the generalization of the variation of medium inhomogeneity.Assuming that the medium modulus varies as a natural logarithmic function or an arbitrary power function,and the density is a composite of exponential function and modulus related function.The dynamic stress response around holes and inclusions is analyzed,and the influence of related parameters on DSCF is discussed.