用波函数展开法求解界面圆孔的SH波散射问题

研究了ＳＨ波绕界面圆孔散射的波函数展开理论。由入射,反射和透射波组成的自由波场与孔的散射场叠加成总波场。将定义于两个半平面的散射场按一定方式延拓到全平面。通过Ｈａｎｋｅｌ－Ｆｏｕｒｉｅｒ型积分来满足界面与孔缘处的边界条件,得到了确定特定系数的封闭代数方程组。给出了具体算例,计算了孔缘处的动应力系数分布。     

Scattering of shear waves by an elliptical cavity in a radially inhomogeneous isotropic medium

Complex function and general conformal mapping methods are used to investigate the scattering of elastic shear waves by an elliptical cylindrical cavity in a radially inhomogeneous medium. The conformal mappings are introduced to solve scattering by an arbitrary cavity for the Helmholtz equation with variable coefficient through the transformed standard Helmholtz equation with a circular cavity. The medium density depends on the distance from the origin with a power-law variation and the shear elastic modulus is constant. The complex-value displacements and stresses of the inhomogeneous medium are explicitly obtained and the distributions of the dynamic stress for the case of an elliptical cavity are discussed. The accuracy of the present approach is verified by comparing the present solution results with the available published data. Numerical results demonstrate that the wave number, inhomogeneous parameters and different values of aspect ratio have significant influence on the dynamic stress concentration factors around the elliptical cavity.     

半无限空间中圆形孔洞周围SH波的散射

建立了求解在含有圆形孔洞的弹性半空间中SH波散射与圆形孔洞附近动应力集中问题的解析方法。利用SH波散射的对称性和多极坐标的方法，构造了一个可以预先满足半空间自由表面上应力自由边界条件的圆形孔洞对SH波散射的波函数。利用这一波函数，则可将该问题转化成对一个圆形孔洞散射的求解问题。该问题的解答最终又可归结为对一组无究代数方程组的求解问题，并可利用截断有限项的方法对其进行计算，最后给出了有关圆形也洞附近动应力集中问题的算例和数值结果，并讨论了波数与圆孔至自由边界距离变化对动应力集中的影响。     

SH波冲击下浅埋任意形孔洞的动力分析

求解了稳态SH波垂直弹性半空间水平表面入射时,浅埋任意形孔洞的动力响应。采用复变函数和多极坐标方法构造了一个能够自动满足水平表面上应力自由边界条件的散射波函数。应用这一波函数,将半空间中的问题转化为求解一个全空间中任意形孔洞的散射问题,最终将问题归结为对一组无穷代数方程组的求解。作为对抗爆问题的研究,给出了浅埋椭圆孔和方孔附近的动应力集中系数的数值结果,并对算例进行了讨论。     

Scattering of SH waves from a partially debonded rigid elliptic cylinder

Scattering of SH waves by an embedded rigid elliptic cylinder of finite length, which is partially debonded from elastic soil is studied. The debonding regions are modeled as multiple elliptic arc-shaped interface cracks with non-contacting faces. The scattered wave field is expressed as a Mathieu function expansion with unknown coefficients. The mixed boundary conditions of the problem lead to a set of singular integral equations of the first type in terms of the dislocation density functions of the cracks. A quadrature method is used to solve these integral equations numerically. The results for dynamic stress intensity factors, far-field pattern of the displacement and scattering cross sections are presented. We particularly discuss the effects of the ratio of the short radius to long radius of the cylinder.     

P波入射时含圆柱形空腔板弹性波散射与动应力集中

采用波函数展开法给出了板的横截面处孔洞在平面P波入射下动应力集中问题的解。本文使用大圆弧假定法来满足板中的自由表面的边界条件。数值结果表明,上表面和孔洞间的距离与下表面和孔洞间的距离都对结果有影响。如果孔洞半径相对板的厚度很小,解近似等于孔洞全空间解。所给出的解是半解析解。因此能被用来验证诸如BEM,FEM,FDM等数值法。     

Scattering of SH-wave by multiple circular cavities in half space

In this paper, an analytic method is developed to address steady SH-wave scattering and perform dynamic analysis of multiple circular cavities in half space. The scattered wave function used for scattering of SH-waves by multiple circular cavities, which automatically satisfies the stress-free condition at the horizontal surface, is constructed by applying the symmetry of the SH-wave scattering and the method of multi-polar coordinates system. Applying this scattered wave function and method of moving coordinates, the original problem can be transformed to the problem of SH-wave scattering by multiple circular cavities in the full space. Finally, the solution of the problem can be reduced to a series of algebraic equations and solved numerically by truncating the infinite algebraic equations to the finite ones. Numerical examples are provided for case with two cavities to show the effect of wave number, and the distances between the centers of the cavities and from the centers to the ground surface on the dynamic stress concentration around the cavity impacted by incident steady SH-wave.     

Diffraction of anti-plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular tunnel

A closed-form analytic solution of two-dimensional scattering and diffraction of plane SH waves by a semicylindrical hill with a semi-cylindrical concentric tunnel inside an elastic half-space is presented using the cylindrical wave functions expansion method. The solution is reduced to solving a set of infinite linear algebraic equations. Fourier expansion theorem with the form of complex exponential function and cosine function is used. Numerical solutions are obtained by truncation of the infinite equations. The accuracy of the presented numerical results is carefully verified.     

柱面波在半空间中洞室周围的散射

采用一种间接边界积分方程法求解了柱面波在半空间中洞室周围的散射问题。通过边界条件的验算以及退化解答与精确解的比较,验证了本文方法的计算精度。文中主要分析了入射波频率、波源与洞室距离等参数对洞室附近地表位移响应和洞室动应力集中的影响,得出了一些有益的结论。     

任意圆弧形凸起地形对平面SH波的散射

本文采用波函数展开方法提出弹性半空间表面一任意圆弧形凸起边界对平面ＳＨ波二维散射的封闭级数解答，利用引入的辅助函数和推广的外域型Ｇｒａｆ加法公式将解答归结为一代无穷代数方程组的求解，解答的数值结果可由无穷方程的截断计算得出，文中从级数项数增加时计算结果的收敛以及边界条件的收敛满足两方面检验了截断计算的精度，并指出了位移解答的收敛速度与主要参数之间的关系，同时，对凸起角点的应力奇异性及其对解答精度的     

Dynamic stress concentration and scattering of SH-wave by interface elliptic cylindrical cavity

In this paper, the dynamic stress concentration and scattering of SH-waves by bi-naterial structures that possess an interface elliptic cavity are investigated. First, by using the complex function method, the Green‘s function is constructed. This yields the solution of the displacement field for an elastic half space with a semi-elliptic canyon impacted by an anti-plane harmonic line source loading on the horizontal surface. Then, the problem is divided into an upper and lower half space along the horizontal interface, regarded as a harmony model. In order to satisfythe integral continuity condition,the unknown anti-plane forces are applied to the interface. The integral equations with unknown forces can be established through the continuity condition, and after transformation, the algebraic equations are solved numerically. Finally, the distribution of the dynamic stress concentration factor (DSCF) around the elliptic cavity is given and the effect of different parameters on DSCF is discussed.     

Scattering of anti-plane (SH) waves by a semi-elliptical hill: II—deep hill

The ground motion and wave propagation resulting from the scattering of anti-plane (SH) waves by a deep semi-elliptical hill on an elastic half-space are investigated. The solution is the closed-form one based on the method of wave-function expansion, and the expansion of elliptical sine and cosine over the half-range [−π/2, π/2]. The analysis shows that the interference of ground motion depends on several factors such as frequency of the incident waves, incident angle, and the aspect ratio of elliptical hills. Similar to part I: Shallow Hill, numerical results when semi-elliptical tends to semi-circular are compared with the work done by Lee et al. [17].     

SH波入射时多个半圆形沉积谷地附近浅埋圆形孔洞的动力分析

本文采用辅助函数的思想,利用复变函数和多级坐标的方法给出了SH波入射条件下多个半圆形沉积谷地附近浅埋圆形孔洞动力分析问题的解答。将整个求解区域分割成两部分来处理,区域I为多个半圆形沉积谷地,区域II为浅埋圆形孔洞附近带半圆形凹陷的半无限弹性空间。在区域I和II中分别构造位移解,并在二个区域的“公共边界”上实施位移应力的连续条件,建立求解该问题的无穷代数方程组。最后,本文给出了算例和数值结果,并对其进行了讨论。     

Surface motion of a cylindrical hill of circular-arc cross-section for incident plane SH waves

A closed-form solution of two-dimensional scattering of plane SH waves by a cylindrical hill of circular-arc cross-section in a half-space is presented using the wave functions expansion method. The solution is reduced to solving a set of infinite linear algebraic equations, using the auxiliary functions and the exterior region form of the Graf's addition theorem. Numerical solutions are obtained by truncation of the infinite equations and their accuracies are demonstrated by convergence of the numerical results and by the extent to which the numerical results fit the exact boundary conditions with increasing the truncation order. The numerical results for some typical cases are then presented for checking accuracies of various numerical methods. The effects of the height-to-width ratio of the hill on surface ground motion are finally illustrated.     

Scattering of plane SH waves by a circular- arc hill with a circular tunnel

IntroductionEarthquakedamageinvestigationsandtheoreticalanalysesshowthatahillamplifiesincidentwavestremendouslyduetomulti-reflectionofthewaveswithinthehill;notabledynamicstressconcentrationisobservedonwallofatunnel(Pao,Mow,1973;Lee,Trifunac,1979).Therefore,itmaybeassumedthattheexistenceofatunnelinahillmighthavegreateramplificationeffectonthegroundmotionnearby,andthedynamicstressconcentrationofthetunnelinthehillmightbemorenotable.Itiscommonthattherailwaysorhighwaysarethroughahilloramountainwi…     

任意圆弧形凸起地形中隧洞对入射平面SH波的影响

借助辅助函数的思想， 采用波函数展开法给出了圆弧形凸起地形中隧洞对入射平面SH波影响问题的一个解析解．数值结果表明，凸起地形中隧洞的存在以及隧洞大小，对凸起地形表面运动，以及凸起地形中的隧洞的动应力集中均具有显著影响．      

Dynamic response of a partially debonded pipeline embedded in a saturated poroelastic medium to harmonic plane waves

A practical model of partially debonded pipeline embedded in a saturated poroelastic medium is proposed, and the dynamic response of this model to harmonic plane waves is theoretical investigated. Biot׳s poroelastic theory is introduced to describe the dynamic equations of the saturated poroelastic medium, and the potentials obtained from Helmholtz decomposition theorem are expressed by wave function expansion method. The debonding areas around the pipeline are assumed to be filled with water. The disturbed solutions of basic field equations in these areas are expressed in terms of a scalar velocity potential. Different boundary conditions of bonded and debonded areas are adopted, and the expanded coefficients are obtained. An example of one partially debonded area is presented and analyzed. It is found that the stresses in the perfectly bonded and debonded areas show great difference, and the jump of dynamic stress at the connection points between these two areas is great in the case of low frequency. The effect of debonded areas on the dynamic stress under different thicknesses of lining is also examined.     

半空间异质隆起的出平面动力响应:闭合级数解答

采用分区思想和辅助函数方法,利用波函数展开技术,提出了水平地表圆形非匀质隆起出平面动力响应的闭合级数解答。将解答归结为一组无穷代数方程组的求解,解答的数值结果可由无穷方程的截断计算得出。计算结果表明,给出的闭合解答在宽频带内能很快收敛。     

周期径向荷载作用下圆弧格构拱动力稳定性分析

基于结构弹性稳定理论,根据能量法推导出以位移为基本未知量的圆弧格构拱总势能,从Hamilton原理出发,建立了考虑剪切变形的圆弧格构拱的动力稳定微分方程.利用Galerkin方法将其转化为二阶常微分Mathieu型参数共振方程,求得周期解所包围的动力不稳定区域,探讨了圆弧格构拱发生参数共振的动力稳定性问题,分析了缀条面...     

SH波入射时半圆形凸起地形附近浅埋圆形衬砌结构的动应力分析

本文给出了地下圆形衬砌结构与地面上的半圆形凸起地形对垂直于地面入射的SH波散射问题的解答。方法是将求解区域分割成两部分。其一为包含半圆形凸起地形在内的圆形区域Ⅰ，其二为带有一个半圆形凹陷和一个圆形衬砌结构的弹性半空间Ⅱ，半圆形凹陷部分为其公共边界，在区域Ⅰ和Ⅱ中分别构造其位移解，然后再通过移动坐标，使其满足“公共边界”上的条件和地下衬砌的边界条件，建立起求解该问题的无穷代数方程组。最后，本文给出了算例，并讨论了数值结果，给出了圆形衬砌结构周边上的动应力集中系数变化规律。