Scattering Of SH Waves Research Articles

Scattering of SH wave in an elastic half-space by a semi-elliptical crater with surface elasticity

Cratering occurs inevitably on the surface of materials at all scales due to comprehensive factors. The presence of craters may have a significant influence on the mechanical performance of the surface of an elastic body. In this paper, we study the SH wave propagation in an elastic semi-infinite medium whose surface has a semi-elliptical crater. We explore the scattering of the SH wave by the crater when surface elasticity is incorporated in the model of deformation to account for possible small-scale effects of the crater. We obtain a series solution for the scattered wave function by employing the Mathieu functions in the corresponding elliptical coordinate system. We discuss in some numerical examples the dynamic stress concentration around the crater relative to the wave frequency and few geometric parameters.

Scattering of SH waves by elliptical cavity and type-III crack in deep anisotropic geology

Most of the underground rock formations are anisotropic, and the study of the dynamic response of anisotropic geology containing internal defects is necessary for the development of underground tunnels or caverns. In this paper, the wave function expansion method, the complex function method, and the Green's function method are used to solve the problem of scattering of SH waves in anisotropic rock in the presence of elliptical cavity and type-III crack. Boundary conditions for solving the problem are given using the conformal transformation method, and crack in the medium is constructed using the "crack cutting" technique. The scattered wave field generated by the crack is constructed based on the Green function method, and the displacement and stress fields in the model are obtained by the wave field superposition principle in the case of SH wave incidence. The dynamic stress concentration factor (DSCF) on the boundary of the elliptical cavity and the dynamic stress intensity factor (DSIF) at the crack tip are derived. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, the effects of the anisotropic strength of the rock mass, the dimensions of the cavern, and the incidence angle and wave number of the SH wave on the DSCF and DSIF are analyzed by numerical calculations in both the frequency and time domains. The results show that the anisotropy of the rock mass, the shape of the cavern and the defects of the surrounding medium have a great influence on the dynamic response of the underground structures, which should be given enough attention in the development and design of deep underground caverns or tunnels.

Scattering of plane SH waves by a concentric semi-cylindrical discontinuity and canyon

Scattering of plane SH waves by a concentric semi-cylindrical discontinuity and canyon

Scattering of SH waves by lined tunnel in inhomogeneous right-angle space with variable shear modulus

Scattering of SH waves by lined tunnel in inhomogeneous right-angle space with variable shear modulus

Scattering of plane SH waves by a circular tunnel in nonlocal fractional-order viscoelastic half-space

This paper proposed a complex function solution to the scattering of plane SH waves by circular tunnels in nonlocal fractional-order viscoelastic half-space. The nonlocal theory and the fractional order viscoelastic model are used to describe the viscoelastic properties and nonlocal effects of the soil, respectively. The basic equations are expressed in the complex domain based on the theory of complex functions, and then the scattering displacement potential function of plane SH waves is obtained using the wave function expansion method. The correctness of the presented method is verified by degrading it to the classical local elastic solution. Finally, the effects of non-local parameters, soil viscosity and fractional order on the distribution of dynamic stress concentration factors inside the tunnel were analyzed, and it was found that they had significant effects on the dynamic stress concentration factors under high-frequency wave incidence.

Scattering of SH waves by cavity with arbitrary shapes and crack at arbitrary positions in anisotropic elastic half-space media

In this paper, the complex function method, Green's function method and coordinate transformation method are used to solve the scattering problem of cavity with arbitrary shape and crack at arbitrary position in the orthotropic elastic half-space under the incidence of SH waves. Firstly, the displacement field and stress field caused by cavity scattering in the orthotropic elastic half-space under the action of incident SH wave are obtained. Secondly, the solution of Green's function is derived when the plane line source load acts on the orthotropic elastic half-space with cavity. Then, the crack is constructed based on the method of crack ‘cutting.’ The total displacement field and stress field can be further obtained. Then, the analytical solutions of the dynamic stress concentration factor (DSCF) around the cavity, the surface displacement amplitude |W| and the dynamic stress intensity factor (DSIF) at the crack tip are obtained. Finally, taking the existence of elliptical cavity and crack in the orthotropic elastic half-space as an example, a large number of numerical calculations were carried out. Numerous numerical examples show that different parameters have a great influence on the DSCF around the cavity, the surface displacement amplitude |W| and the DSIF at the crack tip.

Scattering of SH Waves by a Semi-cylindrical Bump in an Inhomogeneous Half-space

The wave propagation in an inhomogeneous half space with a semi-cylindrical surface bump is investigated by the means of complex function method. With the origin of the coordinate system as the center, the density of the medium changes radially. To solve the Helmholtz equation with variable coefficients caused by the inhomogeneous of the medium, a conformal transformation technique based on the theory of complex variable functions is adopted. Then, by adjusting the inhomogeneous parameters and comparing with the published analytical results, the correctness of the method in this paper is verified. Finally, the displacement amplitudes of typical observation points on the surface and inside are given, and the effects of incident wave angle, wave number and inhomogeneous parameters on wave energy distribution are analyzed.

Scattering of SH wave by an elliptic hole: surface effect and dynamic stress concentration

Scattering of SH wave by an elliptic hole: surface effect and dynamic stress concentration

Scattering of SH-wave by an impedance strip on the rigid wall of the plane elastic waveguide

The problem of SH-wave scattering from an impedance strip on the rigid wall of an elastic waveguide is considered. The opposite waveguide surface is free from tensions. This structure is illuminated by one of the normal SH-waves that propagate along the waveguide without attenuation. The displacement of the particles in this wave is perpendicular to the direction of wave propagation and has the harmonic dependence on time. The problem is two-dimensional and is reduced to the mixed boundary value problem for Helmholtz equation with respect to the unknown diffracted displacement field. The separation of variables and the Fourier integral trans-formation techniques are applied for the solution. Using these techniques, the problem is reduced to the functional equation with respect to the Fourier transforms of the unknown tensions and displacement at the particular intervals of integration. It is shown that this equation is valid in the strip of the complex plane which encompasses the real axis. It is proved that this equation is the equation of the Wiener–Hopf type. Using the factorization and decomposition techniques this equation is reduced to solving an infinite system of linear algebraic equations of the second kind. Its solution is applied to elucidate the behavior of the displacement field on the tensions-free surface of the elastic layer.

Scattering of plane SH waves by a semi-circular discontinuity in an elastic half-plane

A curved discontinuity near the ground can cause the scattering of seismic waves and significantly affect the ground and underground motions, posing some challenges in seismic design. In this paper, a series solution for the scattering and diffraction of plane SH waves from a semi-circular discontinuity in an elastic half-plane is first presented using the wave function expansion method, aiming to reveal the effects of the discontinuity properties on wave scattering and seismic motion. Based on the correspondence principle of viscoelasticity, the influence of the damping ratio of the medium on the seismic motion is also discussed. Finally, by combining the series solution and the inverse Fourier transform, the ground and underground motions induced by a transient plane SH wave are analyzed by exhibiting the wave field snapshots at different times and the waveform curves of different measurement points on the ground. The results indicate that both the stiffness of the discontinuity and the damping ratio of the medium have a great influence on the propagation of seismic waves and play an important role in seismic calculations.

Scattering of SH wave by a crack in a functionally graded one dimensional hexagonal piezoelectric quasicrystals

AbstractIn this paper, the dynamic interaction between SH wave and a crack in functionally graded one‐dimensional hexagonal piezoelectric quasicrystals (PQCs) is studied by using integral transform technique. To make the analysis executable, the material properties are assumed to vary partially in the direction perpendicular to the crack face following an exponential function. The problem was transformed through Fourier transform into three pairs of dual integral equations, which are solved numerically by the Copson method. The explicit expressions of phonon field, phason field and electric field on crack surface are determined. The dynamic field intensity factors of crack tip are given and some special cases are discussed. Numerical examples are then conducted to reveal the influences of amplitude, incident angle and wave number of SH wave and crack length, gradient parameter on the normalized dynamic stress intensity factors (DSIFs). The results of this paper lay a theoretical foundation for the design and nondestructive testing of QCs structures.

Scattering of SH waves by a scalene triangular hill with a shallow cavity in half-space

Scattering of SH waves by a scalene triangular hill with a shallow cavity in half-space

Scattering of SH waves in a bi-material half space with a circular hole and periodic type III interfacial cracks

The dynamic anti-plane response of a bi-material half space including a circular hole and periodic interfacial cracks is investigated. An analytical model is established for periodic type III cracks that adopt the idea of “conjunction” modeling. The definite integral equations suitable for modeling the existence of multiple interfacial cracks are derived that satisfy the continuity conditions at the interface. Numerical calculations are carried out to find the dynamic stress concentration factor (DSCF), whose results indicate that the number of interfacial cracks has a significant effect on the dynamic stress concentration around the hole edge, especially for low-frequency incident SH waves.

Scattering of SH wave in inhomogeneous media based on functional materials

Based on the complex function method, SH wave scattering in continuously inhomogeneous media has been studied in this work. Firstly, an auxiliary function was introduced, and the displacement field was represented by the product of modulus and auxiliary functions. Then, using functional gradient material theory, the continuous variable density function was set to limit wave frequency in the material. Finally, a new coordinate variable was introduced, and a conformal mapping method was applied to solve the wave motion equation under variable velocity. Taking SH scattering by a circular hole as an example, the influences of inhomogeneous parameters were analyzed on the distribution laws and peak values of dynamic stress concentration around circular holes.

Scattering of SH waves by a circular cavity in an inhomogeneous medium: shear modulus and density are a functional form

Aiming at wave problems in continuous inhomogeneous media, this work studies the scattering of SH waves by a circular cavity in an infinite inhomogeneous medium and provides an analytical solution. The shear modulus and density of this inhomogeneous medium are given by a function. Through the construction of the elastic wave displacement field using an auxiliary function and a modulus function, the inhomogeneous governing equation is solved by assuming that both the elastic shear modulus and the medium density vary in the form of an exponential function to obtain the standard form of the Helmholtz equation. Using scattering by a cylindrical cavity as an example, the distribution of the dynamic stress concentration factor (DSCF) on the boundary of the circular hole is calculated and discussed, and the influence of each parameter on the distribution of the DSCF on the boundary of the circular hole is obtained. The results show that the DSCF at the boundary of a circular hole in an inhomogeneous medium is affected by the nonuniform parameters. When the medium density and modulus increase at a fast rate, the dynamic stress concentration coefficient changes significantly, the number of wave peaks increases, and the maximum value increases rapidly.

Scattering of SH waves by orthotropic lining groups in half space

Scattering of SH waves by orthotropic lining groups in half space

The Scattering of SH Wave in a Nano Hole Embedded the Infinite Inhomogeneous Medium

Scattering of the shear waves by a nano-sized cylindrical hole embedded the inhomogeneous is investigated in this study. The Helmholtz equation with a variable coefficient is transformed the standard Helmholtz equation by the complex function method and the conformal mapping method. By wave function expanding method, the analytical expressions of the displacement field and stress field in the inhomogeneous medium are obtained. Considering the surface effect and using the generalized Young-Laplace equation, we obtain the boundary conditions at nano arbitrary-shaped hole, then the field equations satisfying boundary conditions are attributed to solving a set of infinite algebraic equations. Numerical results show that when the radius of the cylindrical cavity shrinks to nanometers, surface energy becomes a dominant factor that affects the dynamic stress concentration factor (DSCF) around the cylindrical cavity. The influence the density variation of the inhomogeneity on the DSCF is discussed at the same time.

Scattering of SH waves by an isosceles flat-topped hill with multistage slopes

This paper presents a flexible mechanical model, which can be used to study the dynamic response of irregular surfaces with complex shapes under the action of seismic waves. Take flat-topped hill as an example, an isosceles model with complex boundaries is established, and an appropriate multi-region-matching technique (MRMT) is proposed to simplify the complex boundary value problem and solve it analytically. Based on the wave function expansion method and the complex function method, the wave field expressions of each subregion, including the singular shape subregion, are constructed. By introducing multipolar coordinate transformation and combining the continuous conditions at three auxiliary boundaries, the infinite algebraic equations are established. To solve the unknown coefficients in the wave field expressions, the Fourier series expansion method in the complex domain is adopted. Finally, some typical numerical examples are calculated to analyze the influence of the shape parameters of the flat-topped hill, the incident angle and the wave number of seismic waves on surface displacement amplitudes.

A Series Solution for 2D Scattering of Cylindrical SH-Waves by Surrounding Loose Rock Zone of Underground Tunnel Lining

This paper presents a closed-form series solution of cylindrical SH-wave scattering by the surrounding loose rock zone of underground tunnel lining in a uniform half-space based on the wave function expansion method and the mirror image method. The correctness of the series solution is verified through residual convergence and comparison with the published results. The influence of the frequency of the incident cylindrical SH-wave, the distance between the wave source and the lining, the lining buried depth, and the properties of the surrounding loose rock zone on the dynamic stress concentration of the tunnel lining is investigated. The results show that the incident wave with high frequency always makes the dynamic stress concentration of the tunnel lining obvious. With the increase of the distance between the wave source and the tunnel lining, the stress around the tunnel lining decreases, but the dynamic stress concentration factor around the tunnel lining does not decrease significantly but occasionally increases. The ground surface has a great influence on the stress concentration of the tunnel lining. The amplitude of the stress concentration factor of tunnel lining is highly related to the shear wave velocity of the surrounding loose rock zone. When the property of the surrounding rock (shear wave velocity) changes more, the amplitude of the stress concentration factor is larger, that is, the stress concentration is more significant.

Scattering of plane SH waves on an irregular piezomagnetic stratum-substrate structure

The present paper investigates the scattering phenomenon of horizontally polarized Shear wave on an irregular surface of the piezomagnetic stratum lying over a piezomagnetic substrate. The expressions of frequency relation and group velocity of scattered Shear wave on the considered piezomagnetic stratum-substrate structure are deduced for magnetically open and magnetically short cases by inducing appropriate boundary conditions. A closed form expressions of reflected mechanical displacement and reflected magnetic potential function related to stratum and substrate are also determined for three different shaped irregularities (rectangular shaped, parabolic shaped and triangular shaped) by employing perturbation technique and residue theorem for both magnetically open and short cases. The efficacy of piezomagnetic coupling parameter, vertical irregular parameter due to presence of surface irregularities is portrayed on the phase velocity, group velocity, reflected mechanical displacement and reflected magnetic potential function related to the stratum for both magnetically open and short conditions graphically. A comparative study is also made to examine the impact of different shaped irregularities on the reflected mechanical displacement and reflected magnetic potential function of the stratum.