18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS SCATTERING OF ANTI-PLANE SH-WAVE BY MULTIPLE CYLINDRICAL CAVITIES AND A LINEAR CRACK H.L. Li 1 * 1 Department of Engineering Mechanics, Harbin Engineering University, Harbin ,150001, China * Corresponding author( leehl@sina.com ) Keywords: multiple cylindrical cavities; crack; Green’s Function; SH-wave scattering; dynamic stress concentration factor (DSCF) using perturbation method Coussy(1982) studied the 1 Introduction In natural medium, engineering materials and problem of SH wave scattering by a cylindrical structures, it can be found that there are cavities inclusion and an interface crack. By using integral everywhere. When structure is impacted by dynamic equation method, Norris and Yang (1991) studied load, the scattering field will be produced because of the influence of Static and dynamic axial load to a the cavities, and it could cause dynamic stress partially bonded fiber. Liu(1999) solved the problem concentration at the edge of the cavities. When the of Scattering of SH-wave by Cracks Originating at structure is overloaded or the load is changed A Circular Hole Edge and Dynamic Stress Intensity regularly, cracks emerge and spread near the cavities. Factor. In this article, authors used Green’s function In theory of elastic wave motion, cavity and crack and the technique of crack-division, and found that are two danger factor. Dynamic stress concentration interaction of a cavity and a crack by SH wave must could greatly decrease the bearing capacity of be considered in some cases. By using the same structure, and reduce the service life of structure. method, Liu and his student(2004) solved the In monograph of Pao(1973), it solved dynamic stress problem of Scattering of SH-wave by an interface concentration problem in an infinite elastic space linear crack and a circular cavity near biomaterial with a cavity by anti-plane SH wave, and it indicated interface. In doctoral dissertation, Li(2004) used that dynamic stress concentration factor is greater Green’s function, crack-division technique and than static concentration factor. Datta(1974), assembly method to solve the problem of interaction Miklowitz(1978) and Moodie(1981) studied some of circular cavity, inclusion with beeline crack at correlative problems by different methods. The arbitrary position by SH-wave. By using the same methods for solving such boundary value problems method, Li(2007) and Yang(2009) solved some included wave function expansion, integral equation, correlative problems. So this method is effective. integral transforms, matched asymptotic expansion. Sometimes, there are some complex engineering To regular shape cavity, wave function expansion problems. For example, there are two or more method is more widely used. By applying the theory underground pipelines in city. Inevitably, there of complex function, Liu(1982) solved irregular would be some crannies near the pipelines. So it is shape cavity problem. On the other hand, dynamic important to study the problem of scattering of stress intensity problems in an infinite elastic space elastic waves by multiple cylindrical cavities and a with cracks were studied by several scientist. linear crack near the cavities. There are lots of Through solving a system of coupled integral materials obtained by theoretical research and equations, Loeber and Sih(1968) studied dynamic earthquake damage investigation. These problems stress intensity problem in an infinite elastic space are complicated, It is hard to obtain analytic with a finite crack by anti-plane shear wave, and solutions except for several simple conditions [1,2]. gave numeric solution of dynamic stress intensity In this paper, the method of Green’s function is used factor. By solving Cauchy singular integral equation, to investigate the problem of dynamic stress Achenbach(1981) did important work to this kind of concentration of multiple cylindrical cavities and a problems. From 1980s, interaction of cavity or linear crack near the cavities for incident SH wave. inclusion and crack in elastic space by SH wave was Multi-polar coordinate system is used too, Which asked to study because of engineering problems. By was used to solve the problem of interaction of
multiple semi-cylindrical canyons by plane SH- ∂ ∂ W W θ − θ τ = μ i + i waves in anisotropic media by Liu(1993). The train ( e e ) r z ∂ ∂ z z of thoughts for this problem is that: Firstly, a ∂ ∂ Green’s function is constructed for the problem, W W e θ − θ τ = μ − i i ( ) e θ z ∂ ∂ which is a fundamental solution of displacement z z field for an elastic space possessing multiple Y cylindrical cavities while bearing out-of-plane harmonic line source force at any point: Secondly, in Y m terms of the solution of SH-wave’s scattering by an T m elastic space with multiple cylindrical cavities, anti- T R s plane stresses which are the same in quantity but m c R m X c opposite in direction to those mentioned before, are s m s loaded at the region where the crack is in existent z 2 actually, this process is called “crack-division”; X O Finally, the expressions of the displacement and z T T 1 stresses are given when multiple cylindrical cavities 1 3 T and a linear crack exist at the same time. Then, by c R 2 R c 1 1 3 3 using the expressions, an example is studied to show c R α the effect of crack on the dynamic stress 2 2 concentration around cylindrical cavities. Fig.1. Model of the problem 2 Model and Green’s function The Green’s function used in this paper is regarded The model is shown as Fig.1, elastic space as the displacement response to the elastic space containing multiple cylindrical cavities and a linear containing multiple cylindrical cavities impacted by crack. In this paper, the anti-plane shear wave anti-plane harmonic linear source force at any point . model is studied. The displacement is expressed The dependence of the displacement function G on as （ ） , and the displacement function , , − ω W x y t i t time t is e . W satisfies the following governing equation: In complex plane, the governing equation of G can be written as: ∂ 2 ∂ 2 W W + + 2 = k W 0 ∂ 2 ∂ 2 x y ∂ 2 1 G + 2 = δ − ( ) (1) k G z z 0 ∂ ∂ 4 z z ω μ = = , ω is the circular where , k C ρ S C The displacement in the elastic space is expressed as S （ ） , z stands for the position of the linear G z z t , , frequency of the displacement （ ） , , , W x y t 0 source force in complex plane. The boundary stands for the shear wave velocity, C s conditions can be expressed as below: ρ and μ are the mass density and the shear τ = δ − = ( ) ( ) z z z z modulus of elasticity respectively. θ z 0 0 (2) τ = − = = Based on the complex function theory, the 0 ( , 1,2,. . , . ) z c R m N r z m m governing equation of W can also be written as The basic solution which satisfies the control ∂ equation (1) and the boundary conditions (2) should 2 1 W z z + 2 = 0 4 k W include two parts of motion: the disturbance of anti- ∂ ∂ plane linear source force and the scattering wave In polar coordinate system, the corresponding incited by multiple cylindrical cavities. The wave stresses are given by: displacement of the complete elastic space due to the
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