Dynamic Analysis of an Infinite Cylindrical Hole in a Saturated Poroelastic Medium

In this paper, the dynamic response of an infinite cylindrical hole embedded in a porous medium and subjected to an axisymmetric ring load is investigated. Two scalar potentials and two vector potentials are introduced to decouple the governing equations of Biot’s theory. By taking a Fourier transform with respect to time and the axial coordinate, we derive general solutions for the potentials, displacements, stresses and pore pressures in the frequency-wave-number domain. Using the general solutions and a set of boundary conditions applied at the hole surface, the frequency-wave-number domain solutions for the proposed problem are determined. Numerical inversion of the Fourier transform with respect to the axial wave number yields the frequency domain solutions, while a double inverse Fourier transform with respect to frequency as well as the axial wave number generates the time-space domain solution. The numerical results of this paper indicate that the dynamic response of a porous medium surrounding an infinite hole is dependant upon many factors including the parameters of the porous media, the location of receivers, the boundary conditions along the hole surface as well as the load characteristics.     

A half-space saturated poro-elastic medium subjected to a moving point load

In this paper, an analytical solution for the dynamic response of a half-space porous medium subjected to a moving point load is derived. In the model, the displacements of the solid skeleton and the pore pressure are expressed in terms of two scalar potentials and one vectorial potential. Based on Biot’s theory, the frequency domain Holmholtz equations for the potentials are derived through the Fourier transformation with respect to time. The general solutions for the potentials are derived through the Fourier transformation with respect to the horizontal coordinates. Numerical results suggest that moving loads have very complicated effects on the dynamic response of the porous medium. Generally speaking, a moving load with a high speed will generate a larger response in the porous medium than a static or a lower speed load.     

Dynamic response of an axially loaded rigid sphere embedded in a saturated poroelastic medium

In this paper, an analytical solution for the response of a rigid sphere embedded in a full space poroelastic medium subjected to a dynamic lateral load is derived. The solution is obtained using Biots theory for acoustic waves. In this solution, the displacements of the solid skeleton and the pore pressure are expressed in terms of three scalar potentials. These potentials correspond to the wave velocities of the slow and fast compressional waves and to the shear wave. The governing equation for the dynamic motion is expressed in the frequency domain using Fourier transformation. Different boundary and load conditions were investigated. Curves showing variation in the fluid pressure and solid displacements with the loads frequency were plotted in non-dimensional forms.     

层状横观各向同性饱和土的非轴对称动力响应

通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的Biot非轴对称波动方 程转化为一组一阶常微分方程组. 然后基于径向Hankel变换,建立问题的状态方程;求解状态方程后,得到传递矩阵. 进而利用传递矩阵,结合饱和层状地基的边界条件、排水条件及层间接触和连续条件,求解 了任意震源力作用下层状横观各向同性饱和地基频域动力响应问题. 时域解可通过频率的Fourier积分得到.     

条形荷载下梯度非均匀非饱和土的动力响应分析

基于多孔介质混合物理论, 建立了梯度非均匀非饱和土地基模型, 研究了条形荷载作用下梯度非均匀非饱和土地基的动力响应问题. 通过傅里叶积分变换和Helmholtz矢量分解原理, 获得频域内非饱和土地基动力响应问题的通解, 结合回传射线矩阵法和边界条件, 求解获得了非均匀非饱和土层中位移、应力以及孔隙压力的计算列式. 假设沿深度方向梯度非均匀非饱和土的物理力学性质按幂函数连续变化, 通过数值傅里叶逆变换得到了非均匀非饱和土地基中的应力、位移以及孔隙压力等物理量的数值解, 分析讨论了土体非均匀性对非饱和土介质动力响应的影响规律. 结果表明: 土体非均匀性显著改变了非饱和土中竖向位移、正应力和孔隙压力在其深度方向上的振动模态, 其中孔隙气压在其深度方向的振动频率随着梯度因子的增加而不断增大, 波峰值不断靠近地表处附近; 竖向位移随着梯度因子的增大不断减小; 正应力和孔隙水压随着梯度因子的增大先增大后减小, 并且土体非均匀程度越高, 正应力与孔隙水压的幅值越大.      

Piezoelectric effect on transversal vibrations and buckling of a beam with varying cross section

In this study, the static and dynamic response of a system composed of an Euler-Bernoulli beam with axially restrained ends and a pair of piezo patches symmetrically bonded at a specified localization is investigated. The system is kinematically loaded as a result of the prescribed displacement of one or both supports. By applying an electric field to the piezo patches a residual in-plane stress is generated in the system. The residual force, depending on the direction of the electric field vector, may diminish or enhance the system buckling capacity as well as affecting its natural vibration frequency. In order to acquire approximate solutions to the non-linear dynamic equilibrium equation, a version of the Lindstedt-Poincare method is utilized. With this in mind, the transversal displacements, vibration frequency and axial dynamic force are expanded into exponential series with respect to the small amplitude parameter. The numerical results show the effect of the structural parameters and induced axial piezoelectric force on the stability of the system and its vibration frequency. The amplitude-frequency relationship of the actuated system is also investigated.     

Backward Transfer-Matrix Method for Elastic Analysis of Layered Solids with Imperfect Bonding

This paper presents a backward transfer-matrix method for the elastic analysis of layered solids with an imperfect bonding at the layer interfaces. Literature review reveals that the conventional transfer-matrix method has an intrinsic fault which leads to ill-conditioned matrices for thick layers and accumulative numerical errors for a large number of layers and that there are a few publications available in the relevant literature regarding analytical analysis of layered solids by taking into account the effects of imperfectly bonded interfaces. The backward transfer-matrix method adopted in this paper completely overcomes the ill-posedness associated with the conventional transfer-matrix method and fully retains the highest efficiency of the classical transfer-matrix concept for analytical formulation of solutions in layered elastic solids with imperfectly bonded interfaces. Numerical results indicate that there is no problem in the numerical evaluation of the solutions with high accuracy and efficiency, and that the interfacial bonding conditions have a significant effect on the elastic response of layered solids due to external loading. This revised version was published online in August 2006 with corrections to the Cover Date.     

DYNAMIC RESPONSE OF A DOUBLE-LAYERED SUBGRADE WITH ROCK SUBSTRATUM TO A MOVING POINT LOAD

In this paper, an analytical solution for the dynamic response of a double-layered subgrade with rock substratum to a moving point load is derived. The subgrade profile is divided into two layers. The upper layer is modeled by an elastic medium and the lower layer by a fully saturated poroelastic medium governed by Biot’s theory. In the meanwhile, the subgrade is resting on the rock substratum. The analytical solutions for stress, displacement and pore pressure are derived by using the Fourier transform. Numerical results obtained by using the inverse fast Fourier transform (IFFT) are used to analyze the influence of the moving load velocity, the thickness of an elastic medium layer and a fully saturated poroelastic medium layer on the dynamic response.     

Numerical study of formation of solitary strain waves in a nonlinear elastic layered composite material

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov et al., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge–Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs–Wilbraham phenomenon is reduced with the help of Padé approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement.     

The dynamic behaviour of a piezoelectric actuator bonded to an anisotropic elastic medium

Surface-bonded piezoelectric actuators can be used to generate elastic waves for monitoring damages of composite materials. This paper provides an analytical and numerical study to simulate wave propagation in an anisotropic medium induced by surface-bonded piezocermic actuators under high-frequency electric loads. Based on a one-dimensional actuator model, the dynamic load transfer between a piezoceramic actuator and an anisotropic elastic medium under in-plane mechanical and electrical loading is obtained. The wave propagation induced by the surface-bonded actuator is also studied in detail by using Fourier transform technique and solving the resulting integral equations in terms of the interfacial shear stress. Typical examples are provided to show effects of the geometry, the material combination, the loading frequency and the material anisotropy of the composite upon the load transfer and the resulting wave propagation.     

Dynamic response of a deepwater riser subjected to combined axial and transverse excitation by the nonlinear coupled model

In offshore engineering long slender risers are simultaneously subjected to both axial and transverse excitations. The axial load is the fluctuating top tension which is induced by the floater’s heave motion, while the transverse excitation comes from environmental loads such as waves. As the time-varying axial load may trigger classical parametric resonance, dynamic analysis of a deepwater riser with combined axial and transverse excitations becomes more complex. In this study, to fully capture the coupling effect between the planar axial and transverse vibrations, the nonlinear coupled equations of a riser’s dynamic motion are formulated and then solved by the central difference method in the time domain. For comparison, numerical simulations are carried out for both linear and nonlinear models. The results show that the transverse displacements predicted by both models are similar to each other when only the random transverse excitation is applied. However, when the combined axial dynamic tension and transverse wave forces are both considered, the linear model underestimates the response because it ignores the coupling effect. Thus the coupled model is more appropriate for deep water. It is also found that the axial excitation can significantly increase the riser’s transverse response and hence the bending stress, especially for cases when the time-varying tension is located at the classical parametric resonance region. Such time-varying effects should be taken into account in fatigue safety assessment.     

Dynamic and static asymmetric response of a layered thick disk

Treated is the asymmetric static and dynamic response of a stack of layered thick disks from external load. Variables of the three-dimensional equations are separated assuming approximate simple supports along the cylindrical perimeter, yielding non-orthogonal eigenfunctions. This also couples the truncated set of radial wave numbers. Applying a radial transform to all variables eliminates radial dependence producing a diagonal eigenproblem in all coupled axial wave numbers. Comparing 3-D and 2-D asymmetric models of industrial glass disks reveals that the 3-D resonances are close to their 2-D counterparts adopting the Mindlin model. A Fourier analysis of a specific asymmetric line-load from pressure or thermal expansion produces a scale factor to static stress from a limited number of asymmetric solutions each with a different circumferential wave number.     

高速荷载下多孔饱和地基的动力响应

研究高速荷载作用下梁与多孔饱和半空间的动力响应。由Fourier变换求解多孔饱和固体的动力基本方程，根据梁与半空间的接触条件得出多孔饱和半空间上梁的垂直位移的表达式。文中的数值算例考虑了荷载移动速度对梁的动力位移的影响，并与相应的弹性半空间问题作了对比。从算例中可以发现荷载移动速度对动力位移有很大的影响，当移动速度与半空间的表面波速相近时，地面会当产生很大的振动，同时还发现当速度大于介质的剪切波速时，多孔饱和半空间上梁的动力响应与弹性半空间上梁的动力响应有很大的差别。     

移动荷载作用下饱和土地基中的波动特性分析

基于Biot波动方程,经过Fourier变换和逆Fourier变换后可获得波数-频率域以及时间-空间域的解析解。通过数值分析的手段研究了移动荷载作用下饱和多孔弹性地基中波的传播特性。重点就弥散曲线、多谱勒效应、波的成分和动力响应频率等几个特性进行了分析,发现饱和土地基由于比弹性地基多了一项流体介质,波动特性明显差异于弹性介质。     

圆柱壳在两种非轴对称同步移动载荷作用下的动力响应

本文根据工程实例计算的需要,研究了有限长弹性圆柱薄壳在两种非轴对称同步移动载荷作用下的动力响应问题。两种非轴对称同步移动载荷作用是指非轴对称移动的集中载荷,以及同步移动且作用范围随移动位置增加的均布载荷的共同作用。建立了在上述两种不同类型载荷作用下的具有对称形式的动力学微分方程组;分别采用Dirac函数与Heaviside函数表示移动的均布载荷与集中载荷,设定位移函数的基础上,应用Galerkin法及Laplace变换,求得了圆柱薄壳应力与位移动态响应的解析解;通过具体算例,将所得到解析解的计算结果与ANSYS数值解进行了对比分析,验证了解析解的可靠性。     

Dynamic Response of a Multilayered Poroelastic Half-Space to Harmonic Surface Tractions

The propagator matrix method is developed to study the dynamic response of a multilayered poroelastic half-space to time-harmonic surface tractions. In a cylindrical coordinate system, a method of displacement potentials is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to those equations are obtained. After that, the propagator matrix method and the vector surface harmonics are employed to derive the solutions for a multilayered poroelastic half-space subjected to surface tractions. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. Finally, a high-order adaptive integration method with continued fraction expansions for accelerating the convergence of the truncated integral is adopted to numerically evaluate the integral solutions expressed in terms of semi-infinite Hankel-type integrals with respect to horizontal wavenumber. Furthermore, to validate the present approach, the response of a uniform poroelastic half-space is examined using the formulation proposed in this article. It is shown that the numerical results computed with this approach agree well with those computed with the analytical solution of a uniform half-space.     

Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect

The authors proposed an analytical method for the analysis of the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. The method is based on the use of simplifying hypotheses and is reduced to the solution of a system of fourth-order differential equations along the axial coordinate with respect to unknown coefficients in the expansion for tangential displacements. An approximate analytical solution, which has a trapezoidal structure and is written in terms of Krylov’s functions, has been obtained. Boundary conditions are formulated in terms of the tangential and longitudinal displacements and axial and shearing stress resultant. For the flexibility factor, analytical solutions are presented in the case where a bend is approximated by a rigid restraint on both ends. To verify the analytical solution and its applicability limits, two numerical procedures were developed, which are based on the finite difference method and the reduction to the Kochi problem by the expansion of the unknowns in the Fourier series over the circumferential coordinate. The authors compare the results obtained with data from the literature, discuss the advantages and disadvantages of the methods, and present recommendations for their practical application.     

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh-Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.     

Multiple scattering of electro-elastic waves from a buried cavity in a functionally graded piezoelectric material layer

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and the dynamic stress around a buried cavity in a functionally graded piezoelectric material layer bonded to a homogeneous piezoelectric material. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the cavity. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. According to the analytical expression of this problem, the numerical solutions of the dynamic stress concentration factor around the cavity are presented. The effects of the piezoelectric property, the position of the cavity in the layer, the incident wave number and the material properties on the dynamic stress around the cavity are analyzed. Analyses show that the piezoelectric property has great effect on the dynamic stress in the region of higher frequencies, and the effect increases with the decrease of the thickness of FGPM layer. If the material properties of the homogeneous piezoelectric material are greater than those at the surface of the structure, the dynamic stress resulting from the piezoelectric property is greater. The effect material properties at the two boundaries of FGPM layer on the distribution of dynamic stress around the cavity is also examined.     

Dynamic interaction of an eccentric multipole cylindrical radiator suspended in a fluid-filled borehole within a poroelastic formation

Acoustic radiation and the dynamic field induced by a cylindrical source of infinite extent, undergoing angularly periodic and axially-dependent harmonic surface vibrations, while eccentrically suspended in a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation, are analyzed in an exact manner. This configuration, which is a realistic idealization of an acoustic logging tool suspended in a fluid-filled borehole within a permeable surrounding formation, is of practical importance with a multitude of possible applications in seismo-acoustics. The formulation utilizes the novel features of Biot dynamic theory of poroelasticity along with the translational addition theorem for cylindrical wave functions to obtain a closed-form series solution. The basic dynamic field quantities such as the resistive and the reactive components of the modal acoustic radiation impedance load on the source in addition to the radial and transverse stresses induced in the surrounding formation by an eccentric pulsating/oscillating cylinder in a water-filled borehole within a water-saturated Ridgefield sandstone medium are evaluated and discussed. Special attention is paid to the effects of source eccentricity, excitation frequency, and mode of surface oscillations on the modal impedance values and the dynamic stresses. Limiting cases are considered and good agreements with available solutions are obtained.