Scattering Of SH-waves By Vertical Interface Crack And Circular Inclusions In Half Space

Generally, it is hard to avoid various complicated defects when working with the manualmaterials and structures, such as circular holes, inclusions and cracks. Under the externalforce, the material which contains defects will take place the phenomenon of dynamic stressconcentration near the defects because of the geometrical discontinuity. What’s more, itdecides the level of damage. So it is very significant to investigate dynamic stress state nearthe defects to satisfy the theoretical and the engineering needs. As the simplest one among thecalculative models of scattering problems of elastic waves, SH waves scattering problem hasrelative mature theories. However, there are still many boundary value problems unsolved. Inthis paper, the analysis solutions of the scattering of SH waves by the single circular elasticinclusion, the multiple circular elstic inclusions (holes) and the composite defects constitutedof the interfacial crack and circular elastic inclusion are considered respectively based on thelinear elastic theory. Meanwhile, some examples for dynamic stress concentration factor ofthe defects and the amplitude of ground surface displacement in half space are given. Themainwork in present paper can be summarized into three parts as follows:（1） In part one, complex function and Green’s function methods are used toinvestigate the analysis solution of the scattering of SH-wave by the bi-material elastic halfspace which contains the vertical interface and a circualr elastic inclusion. Firstly, Green’sfunction is constructed to meet the needs of the problems, which is an essential solution ofdisplacement field for an elastic quarter plane containing a elastic cylindrical inclusion whilebearing out-of-plane harmonic line source load at any point of its vertical boundary. In thispaper, the method of fictitious line source force is used to constructe the expression ofincident waves which satisfies the stress free condition at the horizontal boundary in quarterplane, and the expression of scattering waves which satisfies the stress free conditions at thetwo vertical boundaries can be obtained with the aid of image method. Then, the unknowncoefficients can be determined by the continuous conditions of the stresses and displacementsaound the circular ealstic inclusion edge. Secondly, the expressions of the incident waves, thereflected waves and the refracted waves, which satisfy the boundary cinditions, can beconstructed using the image method. Then, the bi-material media is divided into two partsalong the vertical interface using the idea of interface “conjunction”, and the undetermined anti-plane forces are loaded at the linking sections respectively to satisfy continuity conditions.So a series of Fredholm integral equations of first kind for determining the unknown forcescan be set up through continuity conditions on interface. In the light of attenuationcharacteristic of the scattering waves, the unknown forces can be obtained by the method ofdirect discrete. Finally, some examples for dynamic stress concentration factor around thecircular elastic inclusion edge and the amplitude of ground surface displacement are given.Numerical results discuss the distribution of dynamic stress concentration factor andamplitude of ground surface with the changes of the nondimensional parameters.（2）In part two, complex method and multi-polar coordinates technology are used toinvestigate the analysis solution for the multiple circular inclusions near the vertical interfacedisturbed by SH waves in bi-material half space. Firstly, the Green’s function should beconstructed in this problem, which is an essential solution to the displacement field for anelastic quarter plane with multiple circular inclusions disturbed by out-plane harmonic linesource loading at vertical surface. Secondly, the bi-material media is divided into two partsalong the bi-material interface based on the idea of interface “conjunction”, and the verticalsurfaces of the quarter space are loaded with undetermined anti-plane forces in order to satisfydisplacement continuity and stress continuity conditions at linking section. Then, the integralequations for determining the unknown forces can be set up through continuity conditions andthe Green’s function. Finally, some examples for dynamic stress concentration factor aroundthe circular elastic inclusion (hole) edge and the amplitude of ground surface displacement aregiven. Numerical results discuss the distribution of dynamic stress concentration factor withthe changes of the nondimensional parameters.（3）In part three, complex method and Green’s function method are used toinvestigate the analysis solution for the circular inclusions and the vertical interfacial crackdisturbed by SH waves in bi-material half space. Firstly, the Green’s function should beconstructed in this problem, which is the same as the part one. Secondly, the interfacial crackis constructed with the aid of the crack-division technique. The detail is as follows: the bi-material media is divided into two parts along the vertical interface, and a pair of oppositeforces which are equal to the original stresses disturbed SH waves. Meanwhile, a series of theunknown forces must be loaded at the linking sections except the region of the crack to satisfycontinuity conditions. Then, the integral equations for determining the unknown forces can be set up through continuity conditions. Finally, some examples for dynamic stress concentrationfactor around the circular elastic inclusion edge and the dynamic stress intensity factors formode III at the crack tip are given. Numerical results discuss the distribution of dynamicstress concentration factor and the dynamic stress intensity factors with the changes of thenondimensional parameters.