Scattering Of Elastic Waves By A Cavity Or Inclusion In An Inhomogeneous Medium

Based on the concentration of dynamic stress induced by the geometrical discontinuity in medium,scattering of the elastic waves by various scatters has been received consideration attention.It has well-developed theory for the problem of dynamic stress concentration for engineering models constituted by homogeneous medium.With the society development and technology progress,the need of new-style functional material is more and more important in the human production and life.Medium inhomogeneities pose new challenges in the process of materials preparation,manufacturing and production.Furthermore,the problem of dynamic stress concentration around the geometrical discontinuity by elastic waves scattering in inhomogeneous medium is an important research subject.Due to the elastodynamics theory,the present paper researches the mechanism of elastic waves propagation in continuously inhomogeneous medium and solves the inhomogeneous wave equation analytically.The scattering problems of elastic shearing waves are considered for circular(elliptical)cavity and inclusion in inhomogeneous medium with horizontally exponential and radial power variations respectively.Finally,the distribution of dynamic stress around the cavity and inclusion are discussed.The mainwork in the paper can be summarized by three parts as follows:1.Due to the continuum mechanics,the motion equation of elastodynamics is considered.First,with continuously various medium density and no volume force,two-dimensional(2D)elastodynamic governing equation of the elastic waves propagation is decoupled and inhomogeneous wave equations of the horizontal and longitudinal waves are obtained respectively,which are mutual independent.Then,under the time harmonic condition,Helmholtz equation with the variable coefficient is obtained,which is transformed into standard Helmholtz equation by conformal mapping technology with coordinate transformation.Finally,with the variables separation of 2D variable coefficient Helmholtz equation,four transformation relations are outlined to rectangular,parabolic,polar and elliptical coordinates.Helmholtz equation in 2D infinitely inhomogeneous medium is solved and the expressions of the corresponding wave fields and stress by using the complex variable method.2.Based on the complex function theory,the model of the arbitrary cavity embed in infinitely inhomogeneous medium subject to elastic shearing waves with arbitrary incident angle is constructed,and the variable coefficient Helmholtz equation derived from inhomogeneous wave equation is standardized by the space transformation function.The fields of incident and scattering waves of an arbitrary cavity by elastic shearing waves in infinitely inhomogeneous medium,and the analytical expressions of the corresponding hoop and radial stresses are obtained.Under the stress-free boundary condition of arbitrary cavity,the equations are constructed for solving the unknown force system in order to obtain the dynamic stress concentration factor(DSCF)around the arbitrary cavity by elastic shearing waves in continuously inhomogeneous medium.At last,four specific examples are given as follows:(1)exponentially inhomogeneous medium with a circular cavity;(2)radially inhomogeneous medium with a circular cavity;(3)exponentially inhomogeneous medium with an elliptical cavity;(4)radially inhomogeneous medium with an elliptical cavity.The distributions of dynamic stress around the cavity are discussed for every example.The variations of DSCF with the dimensionless parameters are analyzed,especially for the influence of inhomogeneous parameter on the DSCF.3.On the basis of scattering model of arbitrary cavity,a model of arbitrary homogeneous elastic inclusion in infinitely inhomogeneous medium is constructed.Different from scattering wave fields of the cavity,the standing wave field induced by the inclusion should be considered,when the incident and scattering waves fields of elastic shearing waves in infinitely inhomogeneous medium containing an arbitrary inclusion are given.After obtaining the expressions of the corresponding hoop and radial stresses,the equations with the unknown force system are constructed by the continuous boundary conditions of the displacement and stress at the edge of arbitrary inclusion.Then,the dynamic stress concentration factor around the arbitrary inclusion by elastic shearing waves in continuously inhomogeneous medium is obtained.Similarly,four examples of exponentially(radially)inhomogeneous medium with a(an)cavity(elliptical)inclusion are outlined.The variations of dynamic stress concentration factor around the inclusion with dimensionless parameters are discussed for every example.