Response of a solid infinite cylinder embedded in a poroelastic medium and subjected to a lateral load

This paper presents an analytical solution for the response of a poroelastic medium around a laterally loaded rigid cylinder using Biot’s consolidation theory. A plane-strain section of the cylinder-porous medium system is considered and the problem is formulated in polar coordinates. Expressions for the pore fluid pressure, stresses and displacements in the Laplace domain are derived analytically. The inverse of the Laplace transform is evaluated numerically using an efficient scheme. Curves showing decay of the pore fluid pressure with time, the corresponding change in mean effective stress and the variation of displacement, are plotted in non-dimensional form.     

Reconstruction of inhomogeneous characteristics of a transverse inhomogeneous layer in antiplane vibrations

Direct and inverse problems of forced antiplane vibrations of a transverse inhomogeneous elastic layer are considered. The mechanical characteristics of the layer (density and shear modulus) are considered to be functions of the transverse coordinate. A method for solving the direct problem, based on using the integral Fourier transform and solving the boundary problem by the shooting method, is proposed. The inverse problem of determining the distributions of the mechanical parameters based on the known information on the wave field on some part of the upper surface is considered. Iterative sequences of integral equations are constructed. Results of numerical experiments and recommendations on the optimal choice of the vibration frequency and the interval, on which the displacements are determined, are given.     

Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption

A poromechanical model is presented for estimating swelling of nano-porous media fully saturated with a fluid phase. From the Gibbs adsorption isotherm, the effective pore pressure and the volumetric strain are estimated incrementally taking into account the variations of porosity upon swelling and therefore the variations of the poromechanical properties (apparent modulus, Biot coefficient, Biot modulus). Moreover, the interaction between swelling and the adsorption isotherms are examined by proposing a correction to the Gibbs formalism by taking into account the pore volume variation upon swelling. First, comparisons with experimental data found in the literature are performed, and a fair agreement is observed.     

Flexural gravity wave motion over poroelastic bed

Using Biot’s consolidation theory, effect of poroelastic bed on flexural gravity wave motion is analyzed in both the cases of single-layer and two-layer fluids. The model for the flexural gravity waves is developed using linear water wave theory and small amplitude structural response in finite water depth. The effects of permeability and shear modulus of poroelastic bed and time period on flexural gravity wave motion are studied by analyzing the dispersion relation, phase speed, plate deflection, interface elevation and pressure distribution along water depth. Various results for surface gravity waves are analyzed as special cases. The study reveals that bed permeability retards the hydrodynamic pressure distribution along the water depth significantly compared to shear modulus whilst, floating plate deflection decreases significantly with change in shear modulus compared to permeability of the poroelastic bed. The present study can be generalized to analyze various wave–structure interaction problems over poroelastic bed.     

Optimum Young��s Modulus of a Homogeneous Cylinder Energetically Equivalent to a Functionally Graded Cylinder

For a functionally graded (FG) circular cylinder loaded by uniform pressures on the inner and the outer surfaces and Young??s modulus varying in the radial direction, we find lower and upper bounds for Young??s modulus of the energetically equivalent homogeneous cylinder. That is, the strain energies of the FG and the homogeneous cylinders are equal to each other. For a typical power law variation of Young??s modulus in the FG cylinder, it is shown that taking only two series terms, yields good values for bounds of the equivalent modulus. We also study two inverse problems. First, an investigation is made to find the radial variation of Young??s modulus in the FG cylinder, having a constant Poisson??s ratio, that gives the maximum value of the equivalent modulus. Second, the complementary problem of finding the radial variation of Poisson??s ratio in the FG cylinder, having a constant stiffness, that gives the maximum value of the equivalent modulus, is considered. It is found that the spatial variation of the elastic properties, that maximizes the equivalent modulus, depends strongly upon the external loading on the cylinder.     

Hydrodynamic instability in fluid layers with uniform volumetric energy sources

Linear and energy theory stability criteria are presented for fluid layers of infinite horizontal extent heated internally by a uniform volumetric energy source. The thermal coupling between the layer and its environment is modeled by a general mixed boundary condition in both the conduction state and the disturbance temperature. Rigid-rigid, free-free, free-rigid, and rigid-free boundaries are considered in the computations. For a fixed ratio of upper surface Biot number to that at the lower surface, decreasing the Biot number is strictly destabilizing for both linear and energy theory criteria. A region of possible subcritical instability is found; its size is strongly dependent on Biot number and becomes small for small values of lower surface Biot number and large Biot number ratio. For two rigid surfaces and an upper and lower surface Biot number of 47.5, mean energy transport measurementswithin the convecting layer indicate a critical Reyleigh number close to that predicted by linear theory. Subcritical instability is not observed when finite amplitude disturbances are introduced at a Rayleigh number between the critical values predicted by the linear theory and the energy theory.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

State space approach to the generalized thermoelastic problem with temperature-dependent elastic moduli and internal heat sources

A two-dimensional equation of generalized thermoelasticity with one relaxation time in an isotropic elastic medium with the elastic modulus dependent on temperature and with an internal heat source is established using a Laplace transform in time and a Fourier transform in the space variable. The problem for the transforms is solved in the space of states. The problem of heating of the upper and the lower surface of a plate of great thickness by an exponential time law is considered. Expressions for displacements, temperature, and stresses are obtained in the transform domain. The inverse transform is obtained using a numerical method. Results of solving the problem are presented in graphical form. Comparisons are made with the results predicted by the coupled theory and with the case of temperature independence of the elastic modulus.     

解析法求解成层渗透各向异性地基Biot固结轴对称问题

采用解析法研究成层渗透各向异性地基,该法从渗透各向异性Ｂｉｏｔ固结轴对称问题的基本方程（静力平衡方程,物理方程及渗透连续方程）出发,利用Ｌａｐｌａｃｅ～Ｈａｎｋｅｌ变换及有关矩阵理论等,得到Ｂｉｏｔ固结基本量不同深度之间的传递矩阵。利用传递矩阵,边界条件以及Ｌａｐｌａｃｅ～Ｈａｎｋａｌ逆变换技术可求解多层渗透各向异性地基体系,采用更为有效的Ｆ．Ｄｕｒｂｉｎ的方法实现Ｌａｐｌａｃｅ逆变换。编制了计算     

饱和多孔介质中的混合有限元法和有限应变下应变局部化分析

对基于Biot理论的饱和多孔介质中动力-渗流耦合分析提出了一个耦合场混合元．固相位移、应变和有效应力以及流相压力、压力梯度和Darcy速度在单元内均处理为独立变量分别插值．基于胡海昌-Washizu三变量广义变分原理给出的饱和多孔介质动力-渗流耦合问题控制方程的单元弱形式，导出了单元公式．进一步导出了考虑压力相关非关联塑性的非线性单元公式和发展了相应的一致性算法．对几何非线性分析，采用了共旋公式途径．数值结果例题显示所发展耦合场混合元模拟大应变下由应变软化引起以应变局部化为特征的渐进破坏现象的性能．     

Hydrodynamic instability in a porous layer saturated with a heat generating fluid

Critical Rayleigh numbers determined by linear stabiliy theory are presented for porous-fluid layers of infinite horizontal extent heated internally by a uniform volumetric energy source in the fluid. The thermal coupling between the layer and its environment is represented by a general mixed boundary condition for both the conduction state and the disturbance temperature. Rigid-rigid, rigid-constant pressure, and constant pressure-rigid boundaries are considered in the computations. For a fixed ratio of upper surface Biot number to that at the lower surface, decreasing the Biot number is strictly destabilizing for values of this ratio greater than or equal to one. A layer with a rigid upper surface is generally the most stable; however, a layer with a rigid upper surface and a constant pressure lower surface exhibits the largest values of critical Rayleigh numbers for large values of Biot number.     

Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics

The structural dynamics problems, such as structural design, parameter identification and model correction, are considered as a kind of the inverse generalized eigenvalue problems mathematically. The inverse eigenvalue problems are nonlinear. In general, they could be transformed into nonlinear equations to solve. The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations. This method had no requirements for initial value essentially because of the homotopy path to solution. Numerical examples were presented to illustrate the homotopy method.     

Out-of-plane modulus of semi-auxetic laminates

Materials that possess negative Poisson's ratio are termed “auxetic solids”. The out-of-plane modulus of a laminate consisting of alternating positive and negative isotropic laminas (semi-auxetic laminate) is investigated in this paper. It is herein shown that the use of the inverse rule-of-mixture for obtaining the out-of-plane Young's modulus of a laminate is valid only for conventional laminates and fully auxetic laminates. The Young's modulus by inverse rule-of mixture significantly underestimates the out-of-plane Young's modulus of a semi-auxetic laminate. It is also shown that under certain conditions, the out-of-plane Young's modulus of a semi-auxetic laminate exceeds even the direct rule-of-mixture. A correction term is developed herein for incorporation into the inverse rule-of-mixture.     

Various methods of reconstruction of a cavity in an orthotropic layer

Direct and inverse problems of oscillations of an anisotropic layer with a cylindrical cavity of an arbitrary cross-sectional shape under the action of a load applied to the layer surface are considered. An asymptotic approach to solving these problems with cavities of small relative sizes is proposed. Numerical results of solving direct and inverse problems are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 181–189, May–June, 2009.     

Waves in partially saturated porous media

The propagation of compressional waves in a porous medium is investigated in case the pore liquid contains a small volume fraction of gas. The effect of oscillating gas bubbles is taken into account by introducing a frequency-dependent fluid bulk modulus, which is incorporated in the Biot theory. Using a shock tube technique, new experimental data are obtained for a porous column subjected to a pressure step wave. An oscillatory behaviour is observed, consisting of two distinct frequency bands, which is predicted by the theoretical analysis.     

域外奇点法在弹性问题及其物性值反问题中的应用

本文采用域外奇点法中的位移法和应力法求解弹性问题，并对弹性物性值进行反分析，由有限个观测点的位移值。同时反算出材料的纵向弹性模量Ｅ和泊桑比Ｖ。本文方法属于边界型解法，可有效地避免解的奇异性，不需要数值积分，具有输入数据少，精度高，计算时间短和程序编制容易等优点。     

Hydrodynamic methods in the inverse problem of gravitational prospecting

In [1, 2], a dynamical method is proposed for solving stationary inverse problems of potential theory, including the inverse problem of gravitational prospecting. It is based on analogy with the problem of establishing the interface of two immiscible fluids flowing in a porous medium. In the present paper, a system of two functional equations is derived from which one can obtain, as special cases, an equation corresponding to the method of [1, 2], and also a system of equations that enables one to propose a new and different method for solving the inverse problem of gravitational prospecting. Equations are derived in polar coordinates for plane Cauchy problems corresponding to both methods, and the results are also given of the solution of some model problems by these methods. Finally, ways of generating new methods of solution of the inverse problem of gravitational prospecting are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 63–71, July–August, 1980.     

Wave Dynamics of Saturated Porous Media and Evolutionary Equations

Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Biot approach. Mathematical models under research are the Biot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Biot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg–de Vries–Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Biot model and its generalization are shown.