Anti-plane Steady State Motion Of A Cylinder In An Infinite Elastic Strip

Anti-plane elastodynamics is presented in this paper.Anti-plane steady state motion of a cylinder in an infinite elastic strip,which satisfies stress-free conditions on the upper and lower planar boundaries,is analyzed and solved to obtain displacement and stress fields by elastodynamic theories and methods.This paper can be divided into two parts on the structure and content.The first part is theoretical formulations and analyses,and the second part is two numerical calculating examples.In the first part,systematic analyses of the anti-plane motion within the linear elastic range and a detailed derivation of the basic theories and methods are presented.Expressions of SH wave in the elastic strip is completely discussed.Importantly,scattering guided wave and the definite equations of the cylinder in the strip is formulated.Normalized parameters of near field problem and far field problem are respectively defined,and criteria for the error analysis are proposed.Starting from the particle motion,governing equations for anti-plane motion of the elastic continuum is obtained.Wave function expansion methods of steady state motion are formed by separating variables.The image method of wave function series is proven for stress-free planar bounds.Graf addition formulae are presented for wave functions in different polar coordinate systems.Fourier expansion method is applied to the treatment of the cylindrical boundary conditions.Formal expressions of each order of SH guided wave and the corresponding frequency equations are established by separating variables method with combined stress-free conditions on the upper and lower bounds of the infinite elastic strip.Guided wave expansion method is derived that all consistent shear horizontal displacement waves which satisfy these bound conditions in the elastic strip can be expanded as the series of this guided waves.The formal series of Green function,steady state response of an anti-plane point source in the elastic strip,is formed by using repeated image method.An approximate solution of Green function is obtained by discussing convergence property and applicability of this formal series.SH wave scattering expression of cylinder is expanded by Fourier-Bessel wave functions and Fourier-Hankel wave functions,and effects of the cylinder shape to SH wave scattering are discussed.The formal series of scattering guided wave of the cylinder in the elastic strip is formulated by using repeated image method,and the convergence property of this formal series is discussed.Definite equations of wave function coefficients are given by employing Fourier expansion method on the cylindrical bound conditions with derived stress expressions of the scattering guided wave.Normalized displacement amplitude(NDA)and dynamic stress concentrations factor(DSCF)for near field problem are defined,and displacement mode(DM)and scattering cross section(SCS)for far field problem are defined,respectively.In the second part,steady state scattering of SH guided waves by a circular cavity in the elastic strip and steady state scattering of anti-plane point source load by a circular inclusion in the elastic strip are specifically solved in the calculating example 1 and 2,respectively.The affiliated numerical examples are calculated to display the DSCF around the cylinder.These results have some values of reference and application for theories and engineerings.For example 1:Firstly,compatible guided wave is formulated in the elastic strip,which is satisfied to stress free conditions in upper and lower bounds.Secondly,scattering wave of circular cylindrical cavity is expressed as series form by employed wave function expansion method,compatible scattering guided wave resulted by reflected waves of bounds of the elastic strip is constructed by using repeated images method.Lastly,coefficients of wave function expansion are determined by stress free condition of circular cylindrical bound with pre-given incident guided wave.Numerical calculating examples are presented to solve scattering of circular cylindrical cavity with given SH guided wave,to describe dynamic stress distribution of edge of cavity,to discuss influences of guided orders,frequencies and positions of cavity.For example 2:Wave function expansion method is employed to form series of scattering wave of cylinder.Image method is employed to construct Green function of point source loading and cylindrical wave of cylinder in the elastic strip,which is satisfied stress free condition on bounds of the elastic strip,respectively.Displacement and stress continuity condition on bound of cylinder are employed to determine specific expression of steady state responses of cylinder impacted by point source loading.Numerical calculations are presented to analyze influences of loading frequency and shear modulus and bounds of medium.