| With the development of modern society,the underground structure has been increasing,and aseismic research and safety analysis are becoming more and more concerned,the dynamic problem of the underground structure under the action of seismic waves is a very important direction in the related research.In the study of the underground structure,it is usually simplified to the half space problem,but because of the hierarchical structure of the underground soil,the scattering problem of various circular and elliptical defects in the soil layer under the action of SH wave has become an important topic in earthquake engineering research.Most of the previous studies focused on the effect of the presence of soil layer on the dynamic stress concentration,while the research and analysis of scattering of elastic waves in the structure of hole and inclusion in the overburden layer were relatively few.Aiming at the scattering problem of SH wave in the multiple elliptical holes in the half space,according to the image method,it is transformed into the problem of the whole space,and the Helmholtz theorem gives the general form of the solution of the wave function.This paper does not use the traditional conformal mapping method,but through the analysis of the expression of the elliptic equation,we get the relation between the angle of any point on the elliptic boundary and the angle between the vertical line and the coordinate axis of the point two.Thus,the method of using the conformal transformation is avoided and the solution in the original plane is obtained.The effect of different parameters on the dynamic stress concentration coefficient at the edge of an elliptical hole is analyzed by an example.For a plurality of elliptical hole surface layer or a plurality of elliptical inclusion by SH wave dynamic stress concentration problem,this paper uses the method of large circular arc hypothesis,that is to replace the upper and lower boundaries of the cover layer by a circular arc with a great radius,the reflection and refraction waves at the upper and lower boundary of the surface layer are converted into scattered waves,so the problem is simplified.By establishing the relation between the angle between the vertical and the coordinate axes of the elliptical hole or the boundary of the inclusion and the angle between the vertical and the coordinate axes of the point,the stress expression in the original plane of an arbitrary point at the boundary of an elliptical defect is obtained.According to the general expression of Helmholtz theorem of wave function,it can construct representation of the elliptical hole or an elliptical inclusion wave scattering displacement field in the form of Hankel function and the standing wave displacement field in the elliptical inclusion based on the Bessel function.By using the complex function method and the wave function expansion method,we can transform the problem into infinite linear algebraic equations with unknown coefficients in the wave function while meeting the corresponding boundary conditions,and the finite term is truncated to get the coefficient.The influence of different parameters on the dynamic stress concentration coefficient is analyzed by numerical example.In the study,the medium parameters are selected to determine the “soft and hard” relationship between the medium.It is assumed that the medium with large density is “hard”,and the medium with small density is “soft”.The example shows that both elliptical hole and elliptical inclusion,the maximum dynamic stress concentration coefficient in the surrounding area decreases with the increase of the number of incident waves.For the elliptical hole problem,when the surface cover layer is “harder” than half space,its maximum dynamic stress concentration factor is greater than the case when the cover layer is softer than the half space.For the elliptical inclusion,what is the "soft and hard" relationship of the three,no matter the surface layer,the half space and the half space inclusion,we can see that the more the inclusion is "soft",the greater the maximum dynamic stress concentration coefficient is,and the "harder" the inclusion is,the smaller the maximum dynamic stress concentration coefficient is.At present,most of the problems of the surface covering layer adopt the method of big arc assumption,but this method is only an approximate method,and there are many limitations while using it.For the scattering of SH waves of multiple circular holes in the cover layer,In this paper,the holistic conformal mapping method is adopted,and the upper and lower boundaries of the covering layer and the circular holes in the covering layer are mapped to a group of circles with different circles and radius through a transformation equation,and the shape of the problem model is changed by conformal mapping,then the problem is greatly simplified.When the wave field is constructed,the incident wave field is constructed in the original plane and transformed into a new incident wave field by conformal mapping,and the scattered wave field is all constructed in the new plane.According to the connection condition,we can set up an equation set.By using the complex function method and the wave function expansion method,the equation is transformed into an infinite linear algebraic equation set which solves the unknown coefficient in the wave function,and the coefficient is obtained by truncating the finite term.The calculation example shows that when the cover layer is harder than half space,the dynamic stress concentration factor of the edge of the circular hole is greater than the case when the cover layer is softer than the half space. |
