Dynamic Stress Concentration Factor Research Articles

Analysis of the dynamic response and damage characteristic for the tunnel under near-field blasts and far-field earthquakes

The dynamic response and failure characteristics of tunnels vary significantly under various dynamic disturbances. These characteristics are crucial for assessing structural stability and designing effective support for surrounding rock. In this study, the theoretical solution for the dynamic stress concentration factor (DSCF) of a circular tunnel subjected to cylindrical and plane P-waves was derived using the wave function expansion method. The existing equivalent blast stress wave was optimized and the Ricker wavelet was introduced to represent the seismic stress waves. By combining Fourier transform and Duhamel’s integral, the transient response of the underground tunnel under near-field blasts and far-field earthquakes was determined in both the frequency and time domains. The theoretical results were validated by comparing them with those obtained from numerical simulations using ANSYS LS-DYNA software. Numerical simulations were conducted to further investigate the damage characteristics of the underground tunnel and evaluate the effect of initial stress on structural failure under both types of disturbances. The theoretical and numerical simulation results indicated that the differences in the dynamic response and damage characteristics of the underground tunnel were primarily due to the curvature of the stress waves and transient load waveform. The locations of the maximum DSCF values differed between near-field blasts and far-field earthquakes, whereas the minimum DSCF values occurred at the same positions. Without initial stress, the blast stress waves caused spalling damage to the rock mass on the wave-facing side. Shear failure occurred near the areas with maximum DSCF values, and tensile failure occurred near the areas with minimum DSCF values. In contrast, damage occurred only near the areas with maximum DSCF values under seismic stress waves. Furthermore, the initial stress exacerbated spalling and shear damage while suppressing tensile failure. Hence, the blast stress waves no longer induced tensile failure on the tunnel sidewalls under initial stress.

The dynamic performance of semi-infinite piezoelectric strip containing irregular boundaries based on the boundary element method

In this article, the problem of a semi-infinite piezoelectric strip with irregular boundaries is studied by boundary element method, and SH guided wave and line source loads are considered as external loads acting on piezoelectric strip. The superiority of the boundary element method is induced by two different arithmetic examples. Firstly, when the irregular boundary is not subjected to line source loads, the dynamic characteristics are analyzed in the first example, employing the image method and Graf addition theorem. Then, the Green’s identities are introduced and the Green’s function in infinite three-dimensional space is solved. When irregular boundaries are subjected to line source loads, the dynamic characteristics are investigated using the boundary element method in the second example. Finally, results clarified the influence on the dynamic stress concentration factor and electric field intensity 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.

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

Analytical analysis of the interaction between cavities and a crack in bonded isotropic and anisotropic half spaces under SH wave

Interaction between two cavities and a linear crack subjected to elastic shear horizontally (SH) wave is investigated analytically in this work. Two cavities are buried in two bounded half spaces respectively and the cross-section of each cavity has a different shape. The media of two half spaces are isotropic and anisotropic respectively, and the complex function method is applied to first solve the wave equation in each medium. Then, with the help of image principle, Green’s functions and wave field expressions in each separated half space are constructed. Subsequently, utilizing “conjunction” technique, a series of Fredholm integral equations containing undetermined forces at the interface are obtained and reduced in order to be solved. Dynamic stress distribution is mainly considered to discuss the interaction between the cavities and the crack. Finally, a parametric study on dynamic stress concentration factor (DSCF) and dynamic stress intensity factor (DSIF) presents the interaction is affected by several dimensionless factors simultaneously.

Influence of Incident Orientation on the Dynamic Response of Deep U-Shaped Cavern Subjected to Transient Loading

In deep rock engineering, caverns are often disturbed by engineering loads from different directions. To investigate the dynamic response of deep U-shaped caverns under different incident orientations, a theoretical solution of the dynamic stress concentration factor along the cavern boundary was derived based on the wave function expansion and conformal mapping method, and the failure characteristics around the cavern were further investigated by PFC2D (Particle Flow Code in two dimensions). As the incident orientation increases from 0° to 90°, the dynamic compressive stress concentration area transforms from both the roof and the floor to the sidewalls, and the peak dynamic stress concentration factor of the roof decreases from 2.98 to −0.20. The failure of the floor converts from dynamic compression shear failure to dynamic tensile failure. Compared to a stress wave incident from the curved boundary, a stress wave incident from the flat boundary causes severer damage. When the stress wave is incident from the sidewall, the cavern with a larger height-to-width (h/w) ratio exhibits severer damage. Conversely, the cavern with a smaller h/w ratio tends to fail as the stress wave is incident from the floor. This paper provides a basic understanding of dynamic responses of the deep U-shaped cavern.

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.

Anti‐plane stress analysis for V‐notch in a piezomagnetic half space under SH wave

AbstractIn this paper, the anti‐plane stress analysis of a V‐notch with complex boundary conditions in a piezomagnetic half space is studied. Firstly, SH wave is considered as an external load acting on piezomagnetic half space, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the boundary conditions on the boundary of the half space. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V‐notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green's function method is applied, the half space is divided into two parts along the vertical interface, a pair of in‐plane magnetic field and out‐plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity 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.

Dynamic response of semi-cylindrical depression, cylindrical cavity and type-III crack to SH wave in half-space anisotropic media

In this study, the anti-plane dynamic response of an elastic half-space anisotropic medium containing surface semi-cylindrical depressions and internal cylindrical cavity and type-III crack is solved analytically. The wave function expansion method, the complex function method and the Green's function method can be used to effectively construct the free wave field equation and the scattered wave field equation in the case of SH wave incidence, in which the scattered wave field generated by the crack is constructed by the Green's function method and the "crack cut" method. For the scattered wave field generated by a circular cavity is constructed using the mirror method, where the positional offset coefficient in the mirror method of a half-space anisotropic medium is introduced, which well solves the problem of the asymmetry of the scattered wave field. Finally, the effects of anisotropic strength of the medium, SH wave incidence angle and dimensionless incident wave number on the surface displacement amplitude (|W|), dynamic stress concentration factor (DSCF) at the boundary of the circular cavity and dynamic stress intensity factor (DSIF) at the crack tip are investigated in both frequency and time domains under different working conditions, and conclusions different from those of isotropic medium are obtained, which provide some theoretical references for the earthquake-resistant design of the underground structure with canyon topography and the surface building.

Analysis on scattering characteristics of SH guided wave due to V-notch in a piezoelectric bi-material strip

In this paper, the problem of a V-notch with complex boundary conditions in a piezoelectric bi-material strip is studied. First, SH guided wave is considered as an external load acting on the piezoelectric bi-material strip; on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress-free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress-free and electric insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane electric field and out-plane forces is applied on the vertical interface, and the first kind of Fredholm integral equations is set up and solved by applying the orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and electric field intensity 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.

Dynamic Responses of a Shallow Lined Tunnel with Imperfect Interface Under Transient P Wave

Shallow-buried cavities and underground structures are inevitably subjected to transient perturbation from earthquakes or engineering activities during their life circle, leading to the occurrence of dynamic failure. The investigation presented by this paper concentrates on the dynamic responses triggered by an aperiodic transient [Formula: see text] wave around a circular lined tunnel with an imperfect interface in a half-space and the spring model is employed to describe the interface between rock mass and liner. Resorting to the wave function expansion scheme, the large arc assumption, the Graf’s addition theorem and the numerical inversion algorithm of Laplace transformation, the dynamic responses around a circular shallow lined tunnel with an imperfect interface are evaluated theoretically. The good agreement of time–history curves and contour snapshots for stress between the theoretical solutions and the numerical counterparts validates the analytical scheme. The effect of disturbance direction, embedment depth, thickness of liner and interface coefficients on the dynamic responses around the shallow-buried lined tunnel is analyzed. The results indicate that the positions of peak hoop dynamic stress concentration factor (DSCF) are approximately perpendicular to the perturbation direction. The DSCF distributions converge to be stable when the ratio of embedment depth to excavation radius exceeds 10.0. The thickness of the liner mainly affects the dynamic responses around the interface. Increase of elastic coefficient induces a decreasing DSCF in rock mass and an increasing counterpart in the liner. The results may provide the theoretical basis for the transient responses and aseismic design of underground tunnels.

Dynamic analysis for V-notch in a piezomagnetic bi-material strip under SH guided wave

In this paper, the problem of a V-notch with complex boundary conditions in a piezomagnetic bi-material strip is studied. Firstly, SH guided wave is considered as an external load acting on piezomagnetic bi-material strip, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress free and magnetic insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane magnetic field and out-plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity 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.

Dynamic response of water-rich tunnel subjected to plane P wave considering excavation induced damage zone

The stability analysis of a deep buried tunnel subjected to dynamic disturbance is an important issue. In this study, the transient response has been obtained by establishing a water-rich tunnel model considering excavation damage zone (EDZ). Based on Biot’s two-phase dynamic theory and wave function expansion method, the analytical solution of dynamic response around the water-rich tunnel containing EDZ subjected to P wave is derived. Moreover, Fourier transform and Duhamel’s integral technique is introduced to calculate the transient response, and the equivalent blasting curve is adopted to input excitation function. The dimensionless parameters thickness N and shear modulus ratio μ∼ are defined to characterize the degree of damage in the surrounding rock, investigating the influencing factors, such as the parameters and the incident source frequencies. The results indicate that the dynamic stress concentration factor (DSCF) gradually decreases as the dimensionless parameters increase. Additionally, it is observed that the DSCF is more sensitive to changes in the thickness parameter N. Finally, the influence of the waveform parameters has been taken into account in the analysis of transient response, and the stress state and transfer process in each time stage of the EDZ are analyzed. This study establishes a theoretical foundation for comprehending the mechanical behavior and support design considerations associated with a deep-buried water-rich tunnel containing EDZ.

Propagation of spherical wave in a three-layer sphere under cyclic pressure

In this paper, the dynamic characteristics of a three-layer sphere under a spherical wave induced by cyclic pressure are studied. Firstly, on the basis of the Lamb Decomposition and Variable Separation methods, the analytical expression of the spherical wave is conducted, which satisfies the stress equilibrium on the outer and inner surfaces of the sphere. Next, algebraic equations with respective boundary conditions are composed and solved by Legendre Polynomial expansion and effective truncation techniques. The obtained results enable us to reveal the influence on the dynamic stress concentration factor intensity under proper conditions. The conclusions of this article are verified by comparing the analytical solutions to the ones obtained by the finite element method. This paper can provide a theoretical method for the analysis of mechanical properties of multi-layered spherical structures under dynamic loading.

Theoretical analysis of dynamic stress distribution around a circular damaged roadway under transient disturbance

AbstractExcavation damaged zone (EDZ) widely exists in the surrounding rock of underground roadway due to blasting excavation, which can further change the surrounding rock property and deteriorate the stability of the roadway. In the presented research, a damaged roadway model was developed to investigate the dynamic stress concentration factor (DSCF) distribution around the damaged roadway under a transient plane P wave. The influencing factors on DSCF distribution were discussed, including the shear modulus (μe/μd) of the intact surrounding rock to that of the damaged surrounding rock, the thickness of EDZ (Rd/R0), and the wavelength of the incident stress wave. Results indicated that the damaged surrounding rock around the roadway belongs to stress‐relaxed zones, and the existence of EDZ can reduce the overall DSCF of surrounding rock. As the ratio of μe/μd is increasing, both the maximum compressive and tensile stress concentrations in the damaged zones decrease gradually, which is opposite to that in the intact surrounding rock. A larger EDZ can induce a higher DSCF in the damaged surrounding rock and a smaller DSCF in the intact surrounding rock. As the incident stress wave's wavelength is increasing, the peak value of DSCF grows first before declining, finally followed by a constant value. Furthermore, the maximum compressive and tensile stress concentrations are always found near the surrounding rock that is perpendicular to and along the incoming direction of stress waves, respectively. The stress concentration in both intact and damaged surrounding rock near the incoming side of stress waves is higher than that at the transmitted side.

Dynamic 3D effects of single fiber tensile break within unidirectional composites including resin plasticity, residual stress, interfacial debonding and sliding friction

To study the dynamics of a single fiber tensile break within an S-Glass/epoxy composite and the associated effects of stress wave propagation in the fibers, interface and matrix, a 3D micromechanical Finite Element (FE) model with a hexagonal packing of parallel fibers is developed. The effects of a dynamic fiber break on energy dissipation associated with resin plasticity and interface debonding using cohesive traction laws are included. The 3D FE model also incorporates process-induced residual stresses that induces radial compression at the fiber-matrix interface due the mismatch in CTE and Poisson’s ratios between the fiber and matrix. During dynamic fiber failure where interface debonding occurs, radial compressive residual stresses lead to additional frictional energy dissipation in the debond region of the interface. A map of the dynamic stress concentration factor (SCF) values on the surface (along the axial as well as circumferential directions) of the fibers which are neighboring the fiber break is generated. A stability criterion based on an energy balance between elastic strain energy released by the fiber and energy dissipation (resin plasticity, interphase debonding and frictional sliding) is established to predict maximum fiber strength that initiates unstable debonding of the interface characteristic of axial splitting of the unidirectional composite loaded in tension. The FE modeling results also indicate that the dynamic fiber break introduces strain rates on the order of 106/s in the fiber and matrix, and up to 1012/s in the interphase.

Scattering of plane compressional waves by a variable thickness circular lined tunnel

The determination of the dynamic stress concentration factor (DSCF) associated to a circular lined tunnel with variable thickness liner incident upon by compressional P-waves is of particular interest. Due to the lack of symmetry, the treatment of such a problem via the traditional wave-function expansion method is not possible. To circumvent this dilemma it is proposed to employ the dynamic equivalent inclusion method (DEIM). As it will be shown, this method which is based on the notions of the eigenstress and eigenbody-force fields provides an accurate solution for the problem under consideration. In the simple special case where a tunnel with a concentric liner is subjected to plane compressional P-waves the exact wave-function expansion solution given in the literature is recovered. The effect of liner thickness variability on the static and dynamic stress concentration along the interior surface of the liner and along the liner-matrix interface is studied in detail through numerical examples. The maximum value of the DSCF for the concrete and steel liners with positive and negative values of eccentricity is obtained and compared with those of the corresponding concentric cases. It is observed that the eccentricity of the liner results in different behaviors of the stress concentration for the hard (steel) and soft (concrete) liners.

Anti-planar response due to cylindrical cavity with linear crack under semicircular canyon

Surface depressions and subsurface defects have been a trendy topic in ground shaking problems. In the case of canyon tunnels and canyon pipeline projects, surface movement and cavity edge stress concentrations have been a concern in the design, and the effects of cracks need to be considered if the underground structure contains cracks. In this paper, the wave scattering problem in an elastic half-space medium containing semicircular depression(canyon), cylindrical cavity and crack is investigated by using theoretical methods, the wave function expansion method, the complex function method and the mirror image method to solve the control equation in the form of Helmholtz equation satisfying the zero-stress boundary condition, and the corresponding displacement function is solved. The Green function method and the crack “cutting” technique are used to construct cracks. The unknown coefficients in the system of equations to be determined are solved by free boundary conditions combined with Fourier expansion. The displacement field is a superposition of appropriate wave fields. Finally, the effects of the relevant parameters on ground motion |w| (w), dynamic stress intensity factor (DSIF) of the tip of crack and the dynamic stress concentration factor (DSCF)of the cylindrical cavity were investigated by frequency and time domain analysis. This study not only provides a theoretical basis for practical unlined tunnels or pipeline projects, but also provides a basis for the seismic design of underground structures.

Analytical Study of SH Wave Scattering by a Circular Pipeline in an Inhomogeneous Concrete with Density Variation.

In this paper, the shear horizontal (SH) wave scattering by a circular pipeline in an inhomogeneous concrete with density variation is studied. A model of inhomogeneous concrete with density variation in the form of a polynomial-exponential coupling function is established. By using the complex function method and conformal transformation, the incident and scattering wave field of SH wave in concrete are obtained, and the analytic expression of dynamic stress concentration factor (DSCF) around the circular pipeline is given. The results show that the inhomogeneous density parameters, the wave number of the incident wave and the angle of the incident wave in concrete are important factors affecting the distribution of dynamic stress around the circular pipe in concrete with inhomogeneous density. The research results can provide a theoretical reference and a basis for analyzing the influence of circular pipeline on elastic wave propagation in an inhomogeneous concrete with density variation.

Anti-plane dynamics of right-angle rare earth giant magnetostrictive medium with cylindrical conducting inclusion

In this paper, the anti-plane dynamics of rare earth giant magnetostrictive material with a cylindrical magnetically conducting inclusion in right-angle domain under the incidence of SH waves are studied. The governing equation and constitutive equation of magneto-elastic coupling are established. The outward scattered wave and internal standing wave caused by the inclusion are given by combining the complex function method, the wave function expansion method and mirror method. At the same time, the magnetic potential caused by wave is given by solving the Laplace equation. On the basis of boundary conditions, the Fourier series expansion method and effective truncated series are introduced to establish and solve infinite linear algebraic integral equations. Finally, based on numerical calculation, the analysis of dynamic stress concentration factor and magnetic field intensity concentration factor under different parameter combinations is given. The purpose of this paper is to provide an effective analytical method for the research of fracture dynamics of giant magnetostrictive materials under multi-magnetic field coupling.

Dynamic anti-plane response of circular inclusion in bi-material structure with interfacial periodic cracks due to the SH wave

This paper investigates the dynamic anti-plane response of the circular inclusion in the bi-material structure with interfacial periodic cracks due to the SH wave. In our analysis, we use the ‘conjunction’ technology and Green’s function method. The bi-material structure consists of two isotropic elastic right-angle domains consisting of a circular elastic inclusion and a series of periodic cracks at the bi-material interface. The periodic cracks are created to satisfy the free conditions of the stresses by using the interface ‘conjunction’ technology. The Green’s functions are also solved to obtain the displacement solutions of the out-place point source loaded on the vertical boundaries of the right-angle plane. The first Fredholm integral equations with unknown force systems are then established based on the continuity conditions at the bi-material interface. As an example, the distributions of the dynamic stress concentration factor (DSCF) around the circular inclusion are obtained and shown graphically. The numerical results indicate that compared with a single crack having periodic cracks strengthens the situation of the dynamic stress concentration around the inclusion.