Wave Function Expansion Method Research Articles

Analysis of mode III interface fracture for hole-initiated cracks in magnetic-electro-elastic bimaterials

The analytical solution of two magnetic-electro-elastic materials with hole-initiated cracks are investigated under anti-plane shearing. The modeling of the cracks is achieved by applying a pair of negative shear stresses in the region where the cracks occur. During this investigation, the fundamental solution of the displacement field is expressed by Green’s function method. The expressions of the magneto-electric-elastic wave fields around the interface can be obtained by using the wave function expansion method and Sommerfeld radiation conditions. Then, combined with permeable electromagnetic boundary conditions, the crack problem is reduced to a system of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tip is obtained. Finally, various examples are provided to illustrate the effects of the geometrical parameters, material properties, wave number, and incident angle on the DSIF. Compared with previous traditional numerical and analytical methods, the methods proposed in this article are more applicable in practical engineering and theoretical analysis.

Transient response of bridge piers under eccentric impact of near-fault earthquake

To study the influence of near-fault earthquake on pier failure, establish a double-span continuous girder bridge. The seismic response of the bridge is calculated using the transient wave function expansion method and the indirect mode function superposition method. Solve the dynamic and displacement responses, and the effect of the vertical separation of the bridge on the eccentric compression of the pier is analyzed. The results show that under the near-fault vertical seismic action, the separation can significantly change the horizontal deformation at the top of the pier, and neglecting the separation may underestimate the eccentric compression of the pier. Calculations for different pier heights and girder spans show that separation has a greater effect on the longitudinal deformation of the pier top. Therefore, the reasonable design of longitudinal limit device on pier top in the near-fault area is helpful to reduce the damage of eccentric bridge impact.

Influence of Pier Size on its Failure Mode under Near-Fault Vertical Earthquake

To calculate the influence of pier size on the pier failure mode, a double-span continuous girder bridge model was selected for analysis. The transient wave function expansion method was used to calculate the excitation conditions required for pier separation and the limit solution of pier longitudinal deformation under pier separation, and the indirect mode superposition method was used to calculate the vertical impact force of the pier-beam. The calculation results show that the higher the pier height, the smaller the section size, the larger the vertical seismic acceleration required to separate the pier-beam, the smaller the vertical contact force of the pier-beam, and the larger the most unfavorable value of the longitudinal deformation of the pier caused by separation. The stress of the bridge pier before and after separation is compared and the influence of separation on the failure of the bridge pier is put forward. The study shows that when pier and beam separation occurs, there may be multiple failure modes superimposed on the pier, and the first failure mode will have variation with the change in pier size.

Dynamic analysis of the different types of elliptic cylindrical inclusions subjected to plane SH‐wave scattering

The complex boundary of the elliptical inclusion rendered it difficult to solve the problem of wave scattering. In this study, the steady‐state response was analyzed using the wave function expansion method. Subsequently, the Ricker wavelet was employed as the transient disturbance, and Fourier transform was used to determine the distribution of transient dynamic stress concentration around the elliptical inclusion. The effects of wave number, elliptical axial ratio, and difference in material properties on the distribution of the dynamic stress concentration around the elliptical inclusion were evaluated. The numerical results revealed that the dynamic stress concentration always appeared at both ends of the major axis and minor axis of the elliptical inclusion, and the difference in material properties between the inclusion and medium influenced the variations in the dynamic stress concentration factor with the wave number and elliptical axial ratio.

Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Piezoceramics Containing Regular N-Sided Holes

In this paper, the calculation method of dynamic stress concentration around piezoelectric ceramics containing regular n-sided holes under the action of electroelastic coupling wave was studied, and it was applied to promising barium calcium zirconate titanate material. First, electroelastic governing equations were decomposed by using the auxiliary function method, and the solution forms of the elastic wave field and electric field were obtained by using the wave function expansion method. Then, the triangular boundary was simplified to a circular boundary using the mapping function, and the corresponding modal coefficients were determined according to simplified boundary conditions. Finally, the dynamic stress-concentration factor was calculated to characterize the dynamic stress concentration. We performed numerical simulations with a correlation coefficient of (1 − x)[(Ba0.94Ca0.06) (Ti0.92Sn0.08)]-xSm2O3-0.06 mol% GeO2 (abbreviated as (1 − x)BCTS-xSm-0.06G). The numerical calculation results show that the incident wave number, piezoelectric properties, shape parameters of the hole, and deflection angle have a great influence on the dynamic stress around the defect, and some significant laws are summarized through analysis.

2D dynamic tunnel-soil-aboveground building interaction Ⅰ: Analytical solution for incident plane SH-waves based on rigid tunnel and foundation model

The structure-soil-structure interaction (SSSI) problem between the tunnel and its adjacent aboveground building is investigated by using a 2D model with a rigid circular tunnel and a shear wall supported by a semi-circular rigid foundation embedded in a homogenous half-space. The analytical solution is presented for the model system response under the SH-wave excitation based on the wave function expansion method. The analysis focuses on the influence of the SSSI features between the tunnel and its adjacent aboveground building on the system response. The resonance of the ground structure will have a significant effect on the tunnel response, which is manifested by the sudden change of the tunnel's response transfer function due to structural resonance. The SSSI effect will amplify or reduce the structural relative response at some specific frequencies and tunnel-building distances. The structural relative response amplitude can be amplified by about 40% compared to that of the single structure. The group effect that exists in the interaction of multiple ground structures does not necessarily apply to the tunnel-building interaction. The results in this paper may help to assess the impact of SSSI effects on the seismic response of the tunnel and its aboveground structure, as well as to interpret the observed seismic responses.

Effect of flow-independent viscosity on the propagation behavior of Rayleigh wave in partially saturated soil based on the fractional standard linear solid model

Based on the mixture theory, considering the flow-independent viscosity related to solid skeleton, the present work investigates the propagation characteristics of Rayleigh wave in partially saturated viscoelastic soil. Firstly, the complex history-dependent viscoelastic behavior of solid skeleton is characterized by the fractional standard linear solid constitutive model. The generalized governing equations of motion for the partially saturated viscoelastic soil are established theoretically. Secondly, the analytical solution of Rayleigh wave fields in the partially saturated viscoelastic soil is obtained using the wave function expansion method. Finally, the influences of flow-independent viscosity characterized by fractional viscoelastic parameter on the propagation behaviors of Rayleigh wave are implemented analytically and then discussed in detail. The results show that the fractional index, stress relaxation time and strain relaxation time in the fractional standard linear solid constitutive model have significant influence on the phase velocity and attenuation coefficient of Rayleigh wave. The fractional index, stress relaxation time and strain relaxation time have similar effects on the P1 and S waves included in body waves. Additionally, the influences of the fluid hydraulic conductivity and liquid saturation on the wave propagation cannot be ignored.

Dynamic analysis of piezoelectric materials with an elliptical hole under the action of shear horizontal waves

The dynamic mechanical properties of piezoelectric materials are of interest because of their brittle nature. Previous studies have mainly focused on cylindrical cavities or cracks. However, these are specific forms of elliptical cylindrical cavities. This study aims to develop a semi-analytical approach for analyzing the anti-plane dynamics of piezoelectric materials containing elliptical cavities by exploiting the geometrical generality of ellipses. The Helmholtz and Laplace equations were identified as governing equations by decoupling the displacement and electric potential functions. The wave function expansion method was employed to express the displacement and electric potential functions in the form of a Mathieu function series containing unknown coefficients in the elliptical coordinate system. The unknown coefficients were determined using the traction-free boundary and electric displacement continuous condition. Although the method of intercepting a finite-term series was used, the solutions are reliable because the wave function exhibits convergence. In addition, a parametric study was conducted. The results demonstrated the importance of using large axial ratios, small wave numbers, and large piezoelectric characteristic parameters in the dynamic analysis of piezoelectric materials with elliptical defects.

Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics

In this paper, a method to calculate the dynamic stress concentration around the triangular defect of piezoelectric material under electroelastic coupling is studied and applied to the promising barium calcium zirconate titanate. Firstly, the electroelastic governing equation is decomposed by decoupling technique, and the analytical solutions of elastic wave field and electric field are obtained by wave function expansion method. Then, the conformal transformation is used to simplify the triangle boundary into a circular boundary, and the corresponding modal coefficients are determined according to the simplified boundary conditions. Finally, the analytical solution of the dynamic stress concentration factor can be obtained according to the constitutive equation. Substitute the relevant material parameters of (Ba0.85Ca0.15)(Zr0.1Ti0.9)O3 and set different temperatures, Ce doping amount, and incident wave number for numerical simulation. The numerical results show that the incident wave number, piezoelectric properties, and the shape parameters and deflection angle of the triangular defect have a great influence on the dynamic stress around the defect, and some meaningful laws are summarized through analysis.

Transient responses of deep-buried unlined tunnels subjected to blasting P wave

Dynamic perturbation inevitably induces rock mass damage in deep engineering, especially for high-level in situ stress. The study presented in this paper focuses on the theoretical and numerical responses of deep twin circular tunnels subjected to transient disturbance. Based on wave function expansion method, Graf’s addition theorem, integral transform and its inversion, dynamic responses along the tunnel peripheries are analyzed theoretically. Moreover, with resort to a self-developed code, CASRock, the elasto-plastic responses triggered by coupled static geo-stress and blasting disturbance are investigated numerically. The analytical results indicate that the positions of maximum circumferential stress concentration are approximately perpendicular to the disturbance direction; the most remarkable radial vibrations occur at the incident sidewall and the peak hoop velocities are near the incident side. The numerical results demonstrate that the in situ stress field and characteristics of disturbance load affect the final failure region and damage mechanism around the tunnels. The theoretical and numerical results can provide guideline for the support and design of deep-buried tunnels.

Stress and displacement fields around an arbitrary shape tunnel surrounded by a multilayered elastic medium subjected to harmonic waves under plane strain conditions

This study focuses on the propagation of harmonic mechanical waves around both circular and non-circular tunnels. The excavation disturbed zone is explicitly modeled with a multilayered description of the surroundings of the tunnel. The purpose of this paper is thus to establish closed-form solutions for stress and displacement fields for circular and non-circular geometries taking into account the mechanical heterogeneity of the area adjacent to the tunnel. Particularly, the multilayered non-circular shape configuration is a great step not treated previously. The classical approach based on the wave function expansion method is used to determine the expressions of stresses and displacements around circular geometries. To handle non-circular tunnel shapes, the complex variable method based on the Muskhelishvili approach and conformal mapping functions is introduced. Results indicate that the excavation damaged zone strongly modifies the distribution of stress and displacement fields around the tunnel. Geometrical tunnel shapes also affect stress and displacement polar diagrams.

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.

Dynamic analysis of non‐confocal elliptical lined tunnel under the action of SH wave

The scattering characteristics of non‐confocal elliptical lined tunnels under SH‐wave disturbance are investigated. The problem is essentially a boundary value problem and its theoretical treatment is based on the wave function expansion method, which is an exact treatment of the problem. Multi‐elliptic co‐ordinate systems are employed and wave function translations between multi‐elliptic co‐ordinate systems are performed using the extra‐domain addition theorem of the Mathieu function. The method can be used to fit and handle scattering problems with a wide range of geometries by varying the minor‐and‐major axis ratio, lining thickness and relative position of the tunnel. Calculations have been performed on the truncated infinite linear equation established by considering the boundary conditions. Based on the calculation results a parametric study is carried out to analyze the effect of the dimensionless eccentricity distance, the minor‐and‐major axis ratio and the dimensionless wave number on the dynamic stresses around the tunnel. Two typical application examples of concrete lining in granite and steel lining in granite are presented.

Scattering of Magnetoacoustic Waves and Dynamic Stress Concentration around Double Openings in Piezomagnetic Composites

Based on the magnetoacoustic coupled dynamics theory, the wave function expansion method is used to solve the problem of acoustic wave scattering and dynamic stress concentration around the two openings in e-type piezomagnetic composites. To deal with the multiple scattering between openings, the local coordinate method is introduced. The general analytical solution to the problem and the expression of the dynamic stress concentration are derived. As an example, the numerical results of the dynamic stress distribution around two openings with equal diameters are given. The effects of the parameters, such as the incident wave number and the spacing between the openings, on the dynamic stress concentration factor are analyzed.

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.

Dynamic stress responses of defects in periodic heterogeneous media under SH waves

Due to the complexity of heterogeneous medium models, the corresponding research methods still have great limitations, and hence, there is a dearth of research findings. To study the problem of dynamic stress concentration in heterogeneous media, the scattering of SH waves by circular holes in a full-space periodic heterogeneous continuum medium is investigated based on the propagation characteristics of elastic waves in complex heterogeneous media. The wave equation of SH waves propagating in heterogeneous media with the elastic shear modulus and density varying in the form of trigonometric functions is solved by using the complex function method, the wave function expansion method, and the auxiliary function method. On this basis, the wave field and stress field are derived, and the expression of the dynamic stress concentration factor (DSCF) on the boundary of a circular hole is given. The effects of the reference wavenumber and the variation period of the heterogeneous medium on the distribution of the DSCF on the boundary of a circular hole are analyzed through numerical examples. The results show that when the circular hole defect is within one heterogeneous period of the medium, shortening of the heterogeneous period of the medium reduces the peak of the DSCF due to high-frequency waves on the boundary of the circular hole.

An Analytical Solution for 2D Dynamic Structure-Soil-Structure Interaction for Twin Flexible Tunnels Embedded in a Homogeneous Half-Space

The interaction between subway tunnels is investigated by using a 2D analytic model of a twin tunnels system embedded in a homogenous half-space. The closed-form analytical solution for tunnel displacement response is derived through the wave function expansion method and the mirror method, and the correctness of the solution is verified through residuals convergence and comparison with the published results. The analysis focuses on the effects of tunnel relative stiffness on tunnel–soil–tunnel interaction. Tunnel relative stiffness has a great influence on tunnel displacement response. For small tunnel relative stiffness, tunnel displacement amplitude can be enlarged by 3.3 times that of single rigid tunnel model. The response of the tunnel–soil–tunnel interaction system depends not only on the distances between tunnels but also on the frequency of the incident wave and the incident angle. The strength of the interaction between the tunnels is highly related to the tunnel spacing distance. The smaller the distance between tunnels, the stronger the interaction between them. When the distance between tunnels reaches s/a = 20, the interaction between tunnels can be ignored. It is worth pointing out that the seismic design of underground tunnels should consider the interaction between tunnels when the tunnel distance is small.

On propagation characteristics of Rayleigh wave in saturated porous media based on the strain gradient nonlocal Biot theory

Based on the strain gradient nonlocal Biot theory, the semi-analytical solution of Rayleigh wave fields is obtained using the wave function expansion method. The influence of nonlocal pore size (characterized by nonlocal parameter) and heterogeneity structure effect (characterized by scale factor) on the propagation of Rayleigh wave are numerically implemented and then discussed in detail. The results show that the nonlocal parameter and scale factor have an ignorable influence on the propagation of Rayleigh wave in low frequency region while a significant influence within high frequency range by comparing with the results obtained from the classical Biot theory. A critical ratio of scale factor to nonlocal parameter defining the transition of negative dispersion to positive dispersion is observed. The Rayleigh wave velocity in saturated porous media increases with increasing nonlocal parameter and decreases with the increase of the scale factor. The length of the principal axis and minor axis of the ellipse formed by the displacement fields strongly depends on the ratio of nonlocal parameter to scale factor and the depth of the observation point. Furthermore, the displacement amplitude is intensified by increasing the nonlocal parameter and weakened by increasing the scale factor.

Dynamic Analysis of a Deeply Buried Tunnel Influenced by a Newly-built Adjacent Cavity with a Special Emphasis on the Minimum Seismically Safe Tunnel Distance

Contemporary life streams, more often than ever, impose the necessity for construction of new underground structures in the vicinity of existing tunnels, with an aim to accommodate transportation systems and utility networks. A previously uninvestigated case, in which a newly-constructed tunnel opening is closely positioned behind an existing tunnel, referred to as the tunnel–cavity configuration, has been considered in this study. An exact analytical solution is derived considering a pair of parallel circular cylindrical structures of infinite length, with the horizontal alignment, embedded in a boundless homogeneous, isotropic, elastic medium and excited by time-harmonic plane SV-waves under the plane-strain conditions. The Helmholtz decomposition theorem, the wave functions expansion method, the translational addition theorem for bi-cylindrical coordinates, and the pertinent boundary conditions are jointly employed in order to develop a closed-form solution of the corresponding boundary value problem. The primary goal of the present study is to examine the increase in dynamic stresses at an existing tunnel structure due to the presence of a closely driven unlined cavity, as well as in a localized region around the tunnel (at the position of the cavity in close proximity), under incident SV-waves. A new quantity called dynamic stress alteration factor is introduced and the aspect of the minimum seismically safe distance between the two structures is particularly considered.

An Analytical Solution for Seismic Interaction between Urban River-Canyon Topography and Nearby Buildings

The interaction between urban river-canyon topography and the river-side building is investigated by using a whole analytic model of a semicircle river-canyon and a shear wall supported by a semicircle rigid foundation embedded in a homogenous half-space. The closed-form analytical solution for system response is presented based on the wave function expansion method. The analysis focuses on the effects of the canyon-building interaction on system response. The strength of the interaction between the river-canyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building’s seismic response. The foundation peak response of the building can be amplified by about 10%, and the peak of the building relative response can be amplified by about 40%. The distribution of canyon-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building and the incident angle of the wave. When designing buildings along the river, the building and canyon should be analyzed as a whole model to determine whether the location of the building is in a position with strong interaction with the river-canyon. The model in this paper may be useful for obtaining insight into the effects of canyon-structure interaction and interpreting the observed response in buildings and seismic response estimation in general.