Dynamic Stress Concentration Factor Research Articles

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.

Fuzzy response to SH guided-wave scattering by semicircular depressions on the boundary of a ribbon-shaped elastic plate

In this paper, the fuzzy scattering problem with semicircular depressions on the boundary of a band-shaped elastic plate with steady SH guided wave incident is studied and an analytical solution is given. First, the SH guided wave is constructed by the guided wave expansion method, and then the scattered wave satisfying the free condition of the boundary stress of the strip domain is constructed by the cumulative mirror method. Finally, a definite solution equation is obtained based on the fact that the shear stress at the edge of the semi-circular recessed hole is zero. In this paper, the ambiguity of the number of waves and the width of the bands is taken into account. In order to avoid the irreversibility of interval algorithm and the difficulty of solving non-linear equations, the membership function of fuzzy quantity is segmented to make the membership degree and fuzzy quantity correspond respectively. A deterministic problem that translates into piecewise processing. Two numerical examples are given to examine the changes in fuzzy response of different numbers of fuzzy waves and fuzzy thicknesses to the dynamic stress concentration factor of the hoop at the collapse limit.

Dynamic stress analysis of elliptical inclusions in an inhomogeneous medium with modulus and density variations under the action of shear horizontal (SH) waves

In this paper, in connection with the problem of elastic wave propagation in a continuously inhomogeneous medium, the dynamic stress concentration phenomenon caused by the elliptical inclusions in a complex, continuously inhomogeneous medium under the action of shear horizontal (SH) waves is studied. First, the auxiliary function and the shear modulus function are introduced to construct the wave field function, and the modulus is reduced in order and simplified using a high-order fluctuating wave equation with variable coefficients that has power function variations; then, the auxiliary function and the modulus function are introduced once again to construct the density function; finally, the conformal mapping method is used to carry out coordinate transformation and to solve and obtain the standard form of the Helmholtz equation. Based on this combined problem model, the wave field and the stress field are derived, and the expression for the dynamic stress concentration factor (DSCF) is given. The influence of the inhomogeneous parameters, the ratio of the long and short axes of the elliptical inclusions, and the relative properties of the elliptical inclusions and the matrix on the DSCF around the elliptical inclusions is analyzed through calculation examples.

Analytical model for a circular lined tunnel embedded in the layered soil under seismic waves

AbstractIn this study, based on the wave function expansion method and exact stiffness method (ESM), an analytical model for a circular lined tunnel embedded in the layered soil under seismic waves is established. For convenience, the wave field in the layered‐soil‐tunnel (LST) system is divided into three parts, that is, the free wave field and scattering wave field in the layered soil, as well as the transmitted wave field in the lining. The free wave field in the layered soil is determined by the ESM for the layered soil. The scattering wave field in the layered soil is further decomposed into three parts, namely, the principal and complementary wave fields in the soil layer with the tunnel (the tunnel layer) as well as the modified wave field occurring in the other soil layers. These scattering wave fields are represented by the corresponding principal, complementary, and modified wave functions, respectively. The principal wave functions associated with the direct cylindrical waves emanated from the tunnel are represented by the Hankel functions. The complementary and modified wave functions correspond to the additional responses of the tunnel layer and the other soil layers due to the direct cylindrical waves from the tunnel, and can be obtained by expanding direct cylindrical waves into plane waves together with the application of the ESM. The transmitted wave field in the lining is simply represented by two kinds of Bessel functions. With the aforementioned wave field representations, the boundary value problem for the LST system under seismic waves is formulated. Based on the proposed analytical model, some numerical results and corresponding analyses are presented. Presented numerical results indicate that the dynamic stress concentration factor (DSCF) associated with the stiffer tunnel layer is usually larger than that associated with the softer tunnel layer for the incidence of plane P waves with not very low frequencies, while the circumstance for the incidence of plane SV waves shows an opposite tendency.

Dynamic performance of bimaterial composite sphere under spherical wave

In this article, the dynamic characteristics of bimaterial composite spheres under spherical waves induced by cyclic pressure are studied. First, on the basis of the Lamb Decomposition and Variable Separation method, the analytical expression of 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. Obtained results enable 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 article can provide a theoretical method for the analysis of mechanical properties of multilayered spherical structure under dynamic loading.

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.

Analysis and Prediction of the Dynamic Antiplane Characteristics of an Elastic Wedge-Shaped Quarter-Space Containing a Circular Hole

Based on the wave function expansion method, the dynamic antiplane characteristics of a wedge-shaped quarter-space containing a circular hole are studied in a complex coordinate system. The wedge-shaped medium is decomposed into two subregions along the virtual boundary using the virtual region decomposition method. The scattering wave field in subregion I is constructed by the mirror method, and the standing wave field in region II is constructed by the fractional Bessel function. According to the continuity conditions at the virtual boundary and the stress-free boundary of the circular hole, the unknown coefficients of the wave fields are obtained by the Fourier integral transform, and the analytical solution of the dynamic stress concentration factor (DSCF) of the circular hole is then obtained. Through parametric analysis, the effects of incident wave frequency, geometry of the wedge, and corner slope on the DSCF of the circular hole are discussed. The results show that when the SH-wave is horizontally incidence at high frequencies, the DSCF of the circular hole can be significantly changed by introducing the corner slope. Moreover, when the corner slope is high, the maximum DSCF can be amplified about 1.2 times. Finally, the back propagation (BP) neural network prediction model of DSCF is established, and the coefficient of regression is found to reach more than 0.99.