Scattering Of Anti-plane Shear Waves And Stress Concentration On Elliptical Cylindrical Surfaces In Elastic Space

In recent decades,elastic wave scattering and dynamic stress concentration have received a lot of attention from the engineering and mathematical communities due to their important applications in different scientific fields such as geophysics,seismology,materials science and non-destructive testing.In order to reveal the mechanism of elastic wave scattering,significant work has been done in numerical simulations(domain methods,boundary methods and hybrid methods)and analytical studies(wave function expansion methods).Although numerical methods are more suitable for handling scattering problems with complex models in today’s world of evolving computer technology.However,the analytical(or semi-analytical)solutions obtained by the wave function expansion method under relatively simple models not only provide more physical insight,but also provide an essential benchmark when various numerical methods are used to test and verify the approximate solutions.Currently,the majority of results for solving elastic wave scattering problems using wave function expansions have focused on cylindrical surfaces,while elliptical cylindrical surfaces have rarely been studied.Due to limitations in the calculation of elliptic wave functions and Mathieu functions,transformation of elliptic wave functions between coordinates and waveform conversion of elliptic wave functions on boundaries,existing studies are mainly limited within the scattering of plane SH waves by simple models in a single elliptic coordinate system.Compared to cylindrical surfaces,elliptic cylindrical surfaces are more geometrically general and can be effectively fitted to more complex scattering surfaces in nature.In addition,considering that Mathieu functions,which are commonly used to deal with elliptic boundary value problems,are becoming more and more computationally mature,Mathieu function addition theorems,which are used to establish connections between elliptic coordinate systems,are becoming more and more widely used.Therefore,this paper is concerned with solving the positive problem of plane SH-wave scattering by elliptic cylindrical surfaces using the wave function expansion method.The aim is to break through the limitations of the inter-coordinate transformation of elliptic wave functions by means of the Mathieu function and the Mathieu function addition theorem,and to extend the exact analytical method applicable to solving the scattering problem by elliptic cylindrical surfaces on the basis of the previous work.First,the out-of-plane wave equation and its separated variable solutions,elliptic coordinate systems and Mathieu functions are introduced in detail.The governing equations for SH wave propagation in the elliptic coordinate system are given,as well as expressions for the plane wave function and the cylindrical wave function in product form of the Mathieu function and the modified Mathieu function.The development of Mathieu function addition theorems is systematically reviewed,and specific expressions for inner-domain and outer-domain Mathieu function addition theorems and their scope of application are given.A reasonable choice of a computationally convenient and highly accurate calculation method for the Mathieu function,the modified Mathieu function and the derivative of the Mathieu function is given.Second,considering the interaction between topography and substructure,the scattering problem of a confocal elliptical cylindrical cavity in semi-elliptical bulge for plane SH waves is modeled;considering the amplification effect of local topography,the scattering problems of an elliptical arc depression and an oblique semi-elliptical depression local topography for plane SH waves are modeled;considering the soil-structure interaction,the scattering problems of a non-confocal elliptical lined tunnel and a shallow buried unlined elliptical tunnel for plane SH waves are modeled.All the emergent materials are isotropic,homogeneous,and linearly elastic.To facilitate utilization of elliptic boundary conditions,these models are built on one or more elliptic coordinate systems.However,this also brings the problem of transforming Mathieu function series form wave functions between multiple elliptic coordinates.By introducing Mathieu function addition theorems,a connection between elliptic wave functions with different parameters and in different elliptic coordinate systems is established.According to the scope of application of Mathieu function addition theorems in the inner and outer domains,the addition theorems applicable to each model are reasonably chosen for the characteristics of the above models.Finally,these problems are summarized as the solution of a linear system of equations with infinite series.The solution of the infinite equation system presupposes that it must be truncated to a finite number in terms,and at the same time,this number of terms must be sufficient for the equation system to converge.The equation system,although complex,can be solved by standard matrix techniques due to its linearity.Semi-analytic solutions for the scattering of incident plane SH-waves by confocal elliptical cylinder cavity in in semi-elliptical bulge,elliptical arc-shaped depression,oblique semi-elliptical depression,non-confocal elliptical lined tunnel and shallow buried un-lined elliptical tunnel are obtained,respectively.Using the traction-free boundary conditions on the elliptical cylindrical surface,the accuracy of the numerical results is ensured by calculating the stress residuals.Using the geometrical generality of the ellipse,the results obtained for the elliptical cylindrical surface tending to the cylindrical surface are compared with the previous results for the cylindrical surface to verify the validity of the method.Various factors affecting the displacement amplitude and dynamic stress concentration are investigated,including the incidence angle and the incidence frequency of the plane SH wave,multiple scattering,the minor-to-major axis ratio of the elliptical cylindrical surface,the relative dimensions and the relative positions.Numerical results for the various parameters in the problem are given,and based on the numerical results,the displacement and stress responses are discussed.The significant effect of the presence of an elliptical cylindrical surface under the action of plane SH waves on the nearby dynamic stress concentration factors and displacement amplitudes are observed,evaluated and analysed.