Dynamic Stress Concentration Research Articles

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.

Seismic analysis of a lined tunnel in a multi-layered TI saturated half-space due to qP1-and qSV-waves

A special indirect boundary element method (IBEM) is presented to study the in-plane scattering by a lined tunnel embedded in a multi-layered TI saturated half-space. By integrating the full-space and half-space Green’s functions for the inclined distributed load and pore pressure, these singular-free kernels enables the method to treat an arbitrarily shaped tunnel with a superior flexibility and high efficiency in comparison to the traditional IBEM. For the total system, the substructure process is implemented to decompose the model into a layered TI saturated half-space region and a lining region filled with isotropic elastic medium. The dynamic stiffness matrix method developed by the authors is adopted to solve the free field of the exterior half-space, and the scattered fields existed in the exterior and interior lining regions are simulated by the half-space and full-space Green’s functions in a single-layer potential representation, respectively. Then the total wave-fields in the half-space region can be expressed as the superposition of free-field and scattered field, and the tunnel region merely contains the scattered field. The compatibility conditions on inner and outer interfaces enclosing the lining are satisfied to couple these sub-regions to the original one. The present method is calibrated by comparison with available limited cases. Displacement and dynamic hoop stress concentration at the inner surface of the lining are calculated in the frequency domain to portray the effects of anisotropy degree, drainage condition, incident angle, frequency and layering on the seismic response. Results show that the decrease of anisotropy degree generally leads to larger amplitudes of surface displacement and the stress. Prominent amplification of displacement and stress can be observed for the two-phase medium model in comparison to the one-phase model although the vertical displacement is hardly affected by the drainage condition at the lining-rock interface. When taking the layering into account, the vertical displacement maximum is increased by about 32%, while the DHSC amplitudes at different cases tend to become larger on the whole compared to those for the homogeneous half-space.

Dynamic stress analysis of scattering by a circular cavity in a radially inhomogeneous unbounded space under SH waves

Abstract The density of a radially inhomogeneous unbounded space is derived as a function form. Harmonic dynamics stress of the radially inhomogeneous medium with a circular cavity is investigated by the complex variable function method. The governing equation under incident SH waves in the radially inhomogeneous unbounded medium is expressed as a Helmholtz equation with a variable coefficient. It is equivalently transformed into a standard Helmholtz equation by the conformal transformation method. Then, the stress fields in the radially inhomogeneous medium can be proposed. The results indicate that the changes in density parameter of the medium and wave number further affect the dynamic stress concentration factor around the circular cavity.

Scale Dependence of Earthquake Rupture Prestress in Models With Enhanced Weakening: Implications for Event Statistics and Inferences of Fault Stress

AbstractDetermining conditions for earthquake slip on faults is a key goal of fault mechanics highly relevant to seismic hazard. Previous studies have demonstrated that enhanced dynamic weakening (EDW) can lead to dynamic rupture of faults with much lower shear stress than required for rupture nucleation. We study the stress conditions before earthquake ruptures of different sizes that spontaneously evolve in numerical simulations of earthquake sequences on rate‐and‐state faults with EDW due to thermal pressurization of pore fluids. We find that average shear stress right before dynamic rupture (aka shear prestress) systematically varies with the rupture size. The smallest ruptures have prestress comparable to the local shear stress required for nucleation. Larger ruptures weaken the fault more, propagate over increasingly under‐stressed areas due to dynamic stress concentration, and result in progressively lower average prestress over the entire rupture. The effect is more significant in fault models with more efficient EDW. We find that, as a result, fault models with more efficient weakening produce fewer small events and result in systematically lower b‐values of the frequency‐magnitude event distributions. The findings (a) illustrate that large earthquakes can occur on faults that appear not to be critically stressed compared to stresses required for slip nucleation; (b) highlight the importance of finite‐fault modeling in relating the local friction behavior determined in the lab to the field scale; and (c) suggest that paucity of small events or seismic quiescence may be the observational indication of mature faults that operate under low shear stress due to EDW.

Theoretical responses of shallow-buried circular cavity subjected to transient P wave

Shallow-buried structures are inevitably subjected to time-varying disturbances such as drilling and blasting, earthquake and tectonic activities. The work presented in this paper focuses on the transient responses of a shallow-buried cavity. Based on the wave function expansion method, the large arc approximation and the integral transform, the time-domain responses around a cylindrical cavity subjected to transient P wave in an elastic half space are investigated analytically. The analytical results reveal that the wavelength-to-diameter ratio, the direction of disturbance and the embedment depth have a significant effect on the velocity vibrations and dynamic stress concentration around the tunnel. The theoretical investigations can provide practical value for the support of shallow tunnels.

Transient Invisibility and Cloak of Impact Elastic Wave in Piezoelectric/Piezomagnetic Metamaterials

Based on the scattering cancellation technique, a piezoelectric cloaking is designed, which can make the piezomagnetic cylinder invisible to transient elastic waves. Following the Laplace transform, the piezoelectric cloaking mechanism is presented and dynamic stress concentration factor (DSCF) is derived by the plane wave expansion method. The numerical method of the inverse Laplace transformation is used to show the cloaking behavior that besides the invisible phenomenon, the DSCF is also reduced by the cloak. The piezoelectric parameter plays a positive role on enhancing the object invisibility and improving the stress distribution. These results can offer the fundamental design and application of elastic wave metamaterials.

A Novel Exact Plate Theory for Bending Vibrations Based on the Partial Differential Operator Theory

Thick wall structures are usually applied at a highly reduced frequency. It is crucial to study the refined dynamic modeling of a thick plate, as it is directly related to the dynamic mechanical characteristics of an engineering structure or device, elastic wave scattering and dynamic stress concentration, and motion stability and dynamic control of a distributed parameter system. In this paper, based on the partial differential operator theory, an exact elasto-dynamics theory without assumptions for bending vibrations is presented by using the formal solution proposed by Boussinesq–Galerkin, and its dynamic equations are obtained under appropriate gauge conditions. The exact plate theory is then compared with other theories of plates. Since the derivation of the dynamic equation is conducted without any prior assumption, the proposed dynamic equation of plates is more exact and can be applied to a wider frequency range and greater thickness.

Scattering of SH-guided wave by an elliptic inclusion in an infinite strip region

Dynamic incident anti-plane shearing (SH-guided wave) is considered to calculate dynamic stress in an infinite strip region with an elliptic inclusion. By the means of the conformal mapping method, the field of an ellipse can be transformed into a unit circle. With the aid of series expansion method, the expression of SH-guided wave is established, which satisfies the stress free condition on the upper and lower horizontal boundaries of the strip. The expression of scattering wave is conducted based on repeated image superpositions. Boundary conditions are solved by applying the orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Evolution and control of deformation mechanisms in micro-grooving of Zr-based metallic glass

The evolution of deformation mechanisms with increasing depth of cut (DOC) in micro-grooving of Zr-based metallic glass was deeply investigated in this study for the first time. The micro-grooving characteristics of metallic glass were initially studied by nanoscratch test, which verified that the material removal mode abruptly changes from extrusion mode to cutting mode with increase of DOC. Based on the free-volume theory, a dynamic model coupling stress, temperature and free volume concentration was used to simulate the change of deformation characteristics with DOC. The results showed that the deformation mechanism of metallic glass is closely related to free volume flow, which is significantly affected by average strain rate and free volume flow coefficient in deformation zone and can be tuned with different DOC. Considering that free volume flow is determined by its flow coefficient and concentration, we proposed to control the deformation characteristics of metallic glass in micro-grooving by heat treatment and verified its effectiveness experimentally. • Materials removal mode of metallic glass changes with the increase of depth of cut. • The evolution of deformation mechanisms was studied using a dynamic model. • Free volume flow affects the evolution of deformation mechanisms significantly. • Deformation characteristics of metallic glass were experimentally modulated.

Dynamic response of a functionally graded cylindrical tube with power-law varying properties due to SH-waves

An analytical solution to the dynamic response of a functionally graded cylindrical tube embedded in a full homogeneous space subjected to elastic SH waves is presented using the wavefunction expansion method. The shear modulus of the functionally graded tube is assumed to have a power-law variation with radius. The wave fields in the tube and unbounded surrounding medium are derived in a form of wavefunction series with unknown coefficients. The wave field coefficients are obtained by satisfying the boundary conditions at the inner and outer tube surfaces. The influences of the inhomogeneity exponent, shear modulus and thickness of the functionally graded tube, and the frequency of the incident SH waves on the dynamic stress concentration factor (DSCF) of the tube are systematically discussed. It is concluded that the application of functionally graded materials (FGMs) with a positive inhomogeneity exponent tends to weaken the DSCF along the inner surface of the tube significantly but intensify that along the outer surface slightly, which is beneficial to the long-term use of the tube.

Dynamic anti-plane analysis for a semi-circular salient in an infinite elastic strip region

Dynamic incident anti-plane shearing (SH guided wave) is considered to calculate dynamic stress in an infinite strip region with a semi-salient. With the aid of series expansion method, the expression of SH guided wave is established, which satisfies the stress free condition on the upper and lower horizontal boundaries of the strip. The expression of standing wave is constructed by the means of Fourier series expansion. The expression of scattering wave is conducted based on repeated image superposition. Boundary conditions are solved by applying orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor under proper conditions.

Dynamic anti-plane behavior of rare earth giant magnetostrictive medium with a circular cavity defect in semi-space

An analytical solution to the anti-plane dynamics problem of semi-space rare earth giant magnetostrictive media with circular cavity defects near the horizontal boundary under the action of SH wave is studied. Based on the Helmholtz theorem and the theory of complex function, the elastic-magnetic dynamic equation of magnetostrictive medium is established, and the semi-space incident wave field is written. In addition, based on the theory of complex function and the method of wave function expansion, the expression of the wave function of the scattered displacement field and the corresponding magnetic potential of the scattered wave under the condition of no stress and magnetic insulation of the horizontal boundary are obtained. Then, based on the conditions of free boundary stress, continuous magnetic induction intensity and continuous magnetic potential around the circular cavity, the infinite linear algebraic equations are established. Finally, the analytical expressions of dynamic stress concentration factor and magnetic field intensity concentration factor around circular cavity in semi-space rare earth giant magnetostrictive medium are obtained. Numerical examples show that the analysis results depend on the following parameters: permeability, dimensional-piezomagnetic coefficient, frequency of the incident wave, incident angle, distance between the circular cavity and horizontal boundary. These results have certain reference value for the study of non-destructive testing and failure analysis of rare earth giant magnetostrictive materials.

Scattering of cylindrical inclusions in half space with inhomogeneous shear modulus due to SH wave

Dynamic responses around cylindrical inclusion in inhomogeneous medium are discussed. A mathematical model of inhomogeneous half space is established. The shear modulus of the medium is assumed to change in two dimensions. Based on complex function theory, the governing equations are derived. Meanwhile, the auxiliary function is introduced. By solving the governing equation, the analytical expressions of the displacement field and stress field formed by Bessel function and Hankel function are obtained. The unknown coefficients can be obtained by boundary conditions. According to numerical examples, the results of this paper are compared with published results to verify the validity of the method. Meanwhile, the effects of inhomogeneous parameters, reference wave number and burial location on the dynamic stress concentration factor (DSCF) around a cylindrical inclusion are discussed.

The method of fundamental solutions for the elastic wave scattering in a double-porosity dual-permeability medium

For materials that have a matrix porosity at the microscopic scale, and an interconnecting fracture system at the mesoscopic scale, the double-porosity dual-permeability model provides a better prediction of seismic wave attenuation than the single porosity model, due to the flow exchange mechanism between these different size pores. The current work offers for the first time a numerical solution tool based on the method of fundamental solutions for problems of 3-D wave scattering and dynamic stress concentration around scatterers, such as cavities and embedded inhomogeneous inclusions, in an infinite double-porosity dual-permeability medium. The method of fundamental solutions is based on the spherical wave potentials representing three compressional waves and one shear wave. The accuracy and stability of the method are tested against other available results, and are demonstrated to be excellent. Wave scattering and dynamic stress concentration problems due to spherical cavity and poroelastic inclusion are then investigated for incident plane P1 and SV waves in the frequency domain. The research shows that the seismic responses are sensitive to the incident frequency, the boundary drainage condition, and material properties such as porosity.

Influence of excavation damaged zone on the dynamic response of circular cavity subjected to transient stress wave

In this study, the Hoek Brown criteria is primarily employed to calculate the distribution of excavation damaged zone (EDZ) around the cavity buried at 500 m, 1000 m, and 2000 m based on the variation of the in-situ stress and the mechanical properties of the rockmass. Then, the simulation scheme Spectral Element Method (SEM) is applied to investigate the dynamic responses of the excavation-damaged cavity that subjected to the dynamic disturbance, including peak particle velocity (PPV), particle vibration frequency (PVF), particle displacement, and dynamic stress concentration factors (DSCF). Three two-dimensional (2D) models embedding a circular hole with different shapes of EDZ are prepared to numerically study the dynamic responses of the underground cavity, finding the influencing factors, such as the shapes of EDZ, the incident direction of the dynamic disturbance, and the dynamic source frequency. The cavity that is surrounded by EDZ has resembled as the one with a series of low-velocity areas. The results indicate that the underground cavity and EDZ become a scatter to the incoming stress waves that correspondingly produce high amplitude scattered waves. The influencing factors to the dynamic response of the cavity are summarized to be the shape of EDZ, the angle of stress disturbances, and the wavelength of incoming stress waves. We also find that the amplitude of stress disturbance around the cavity is proportional to the areas of EDZ and the frequency of the incident waves. In particular, only the tensile stress concentrates around the cavity that is surrounded by EDZ.

Dynamic stress concentration of an elliptical cavity in a semi-elliptical hill under SH-waves

The method of wave-function expansion in elliptical coordinates, elliptical cosine half-range expansion and Mathieu function were applied to obtain an exact analytical solution of the dynamic stress concentration factor (DSCF) around an elliptical cavity in a shallow, semi-elliptical hill. An infinite system of simultaneous linear equations for solving this problem was established by substituting the wave expression obtained by the Mathieu function including the standing wave expression of elliptical lining given herein into the boundary condition obtained by the region-matching method. The finite equations system with unknown coefficients obtained by truncation were solved numerically, and the results in the case of an ellipse degenerating into a circle were compared with previous results to verify the accuracy of the method. The effects of different aspect ratios, incident wave angles and aperture ratios on the dynamic stress concentration around the elliptical cavity were described. Some numerical results, when the elliptical hill was changed into a circular one, were analyzed and compared in detail. In engineering, this model can be regarded as a semi-cylindrical hill with an elliptical cylindrical unlined tunnel under the action of SH waves, and the results are significant in aseismic design.

Effects of coating on dynamic stress concentration in fiber reinforced composites

Fiber reinforced materials consisting of aligned fibers within a matrix are effective for applications in nature and engineering in which strength and stiffness are required in a dominant direction. The fibers are typically coated. Optimization of these materials is typically based upon choosing elastic properties of phases to reduce static stresses. However, materials with well-matched moduli can have mismatched acoustic impedance, and thus a fiber reinforced material that is strong in quasistatic loading may be weak in dynamic loading. To explore this trade-off, we modeled perfectly bonded, isotropic, linear elastic coated fibers in an infinite, isotropic, linear elastic matrix and calculated dynamic stresses and interfacial stress concentrations induced by continuous and transient waves using the wave function expansion method. Results revealed ways that the physical properties and geometrical dimensions of a coating around a fiber can be tailored to reduce dynamic stress concentration, and point to a pathway for improving the shock resistance of fiber reinforced materials.

Theoretical study on dynamic responses of an unlined circular tunnel subjected to blasting P-waves

Abstract In conventional studies, the blasting seismic wave is always treated as a time-harmonic wave, which is not suitable in some situations due to its short duration. In this paper, the blasting P-wave is simplified as a triangular impulse. The distribution functions of dynamic stress concentration factor (DSCF) and the radial and hoop vibration velocity scaling factors (RVSF and HVSF) around the circular tunnel are derived based on the Fourier–Bessel expansion method. Then, the effects of the rising duration, the total duration and Poisson's ratio on DSCF, RVSF and HVSF are discussed. Results show that (1) the maximum RVSF and HVSF are located at the incident side, and the location of the maximum DSCF will move from the incident side to the shadow side when the total duration increases; (2) the maximum DSCF increases with the increasing total duration, decreases with the increasing Poisson's ratio, but is immune to the ratio of the rising to total duration; (3) the maximum RVSF decreases with the increase of the total duration and the ratio of the rising to total duration, but increases with the increasing Poisson's ratio; (4) the maximum HVSF decreases with the increase of the total duration and the ratio of the rising to total duration, but is immune to Poisson's ratio; (5) the RVSF and HVSF under triangular impulses gradually change from less than those under time-harmonic waves to greater than those under time-harmonic waves with the increasing total duration, but the DSCF shows the opposite trend.

Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.