Time-Dependent Response of an Axially Loaded Elastic Bar in a Multilayered Poroelastic Medium

This paper considers load transfer from an axially loaded long elastic bar into a multilayered poroelastic half-space. The problem is analyzed by decomposing the bar-half-space system into an extended half-space governed by Biot’s theory of poroelasticity and a one-dimensional fictitious bar. The interaction problem is formulated in the Laplace transform domain. Vertical displacement of the bar is approximated by an exponential series with a set of arbitrary functions. The arbitrary functions are determined by using a variational method. The vertical displacement influence function of a multilayered half-space subjected to a buried uniform vertical patch load is required in the variational formulation. The required influence function is obtained by employing a previously developed exact stiffness matrix method. Time domain solutions are computed by using a numerical Laplace inversion scheme. Selected numerical results are presented to portray the influence of the bar length–radius ratio, layer configuration, poroelastic material parameters, and loading time history on the time dependent response of a bar.     

Vertical Vibration of a Flexible Plate with Rigid Core on Saturated Ground

In this paper, the vertical vibration of a flexible plate with rigid core resting on a semi-infinite saturated soil is studied analytically. The behavior of the soil is assumed to follow Biot’s poroelastodynamic theory with compressible soil skeleton and pore water, and the response of the time-harmonic excited plate is governed by the classical thin-plate theory. By virtue of the Hankel transform technique, the fundamental solutions of the skeleton displacements, stresses, and pore pressure are derived, and a set of dual integral equations associated with the relaxed boundary and completely drained condition at the soil-foundation contact interface are also developed. These governing integral equations are further reduced to the standard Fredholm integral equations of the second kind and solved by numerical procedures. Comparison with existing solutions for a rigid permeable plate on saturated soil confirms the accuracy of the present solution. Selected numerical results are presented to show the influence of the permeability, the size of the rigid core, and the plate flexibility on the dynamic interaction between the elastic plate with rigid core and the underlying saturated soil.     

Vertical Vibrations of a Rigid Foundation Embedded in a Poroelastic Half Space

This paper considers the steady-state vertical vibrations of a rigid, cylindrical massive foundation embedded in a poroelastic soil. The foundation is subjected to time-harmonic vertical loading and is perfectly bonded to the surrounding soil. The contact surface between the foundation and the soil is assumed to be smooth and fully permeable. Biot’s poroelastodynamic theory is used in the analysis. The soil underlying the foundation base is assumed to be a homogeneous poroelastic half space while the soil along the side of the foundation is assumed to consist of a series of infinitesimally thin layers. The dynamic interaction problem is solved by a simplified analytical method. The accuracy of the present solution is verified by comparisons with existing solutions for both elastodynamic and poroelastodynamic interaction problems. Selected numerical results for the vertical dynamic impedance and response factor of the rigid foundation are presented to demonstrate the influence of nondimensional frequency of excitation, depth ratio, mass ratio, shear modulus of the backfill, and poroelastic material properties on dynamic interaction between an embedded foundation and a poroelastic half space.     

Dynamic Axial Load Transfer from Elastic Bar to Poroelastic Medium

The time-harmonic response of a cylindrical elastic bar (pile) partially embedded in a homogeneous poroelastic medium and subjected to a vertical load is considered. The bar is modeled using 1D elastic theory valid for long bars in the low-frequency range, and the porous medium using Biot's 3D elastodynamic theory. The bar is bonded to the surrounding medium along the contact surface. The problem is formulated by decomposing the bar∕porous medium system into a fictitious bar and an extended porous medium. A Fredholm's integral equation of the second kind governs the distribution of axial force in the fictitious bar. The integral equation involves kernels that are displacement and strain influence functions of a poroelastic half-space subjected to a buried, uniform vertical patch load. The governing integral equation is solved by applying numerical quadrature. The solutions for axial displacement and axial force of the bar, and the pore pressure are also derived. Selected numerical results for vertical impedance, axial force, and pore pressure profiles are presented to portray the influence of bar stiffness and length∕radius ratio, frequency of excitation, and poroelastic properties.     

Dynamic Analysis of Multilayered Soils to Water Waves and Flow

In nature, a soil profile generally consists of several heterogeneous layers. This study is aimed at discussing the interactive problem of oscillatory water waves and flow passing over multilayered soils. The soil behavior is considered as viscoelastic in the present mathematical model modified from Biot’s poroelastic theory. Employing this model, the dynamic response including the profiles of pore water pressure and effective stress in the multilayered soils is discussed. The results reveal that the perturbed pore pressure is different from that inside a single-layered soil where the thickness of the first soil layer is less than the water wavelength. The discrepancy of the vertical effective stresses between multilayered and single-layered soils is even much more apparent under the same conditions. Moreover, seepage force is examined and is found to be larger near the bed surface and the bottom of the first soil layer where soils are easily disturbed by external disturbance. The locations where soil failure might happen are found near the troughs of surface water waves.     

Effects of Weakly Nonlinear Water Waves on Soft Poroelastic Bed with Finite Thickness

Since porous material is usually of a finite thickness in nature, the effects of periodically nonlinear water waves propagating over a soft poroelastic bed with finite thickness are hence noticed and studied in this work. The water waves are simulated by potential theory while the porous bed is governed by Biot’s theory of poroelasticity herein. The conventional Stokes expansion of water waves based on a one-parameter perturbation expansion fails to solve the soft poroelastic bed problem; therefore, the boundary layer correction approach combined with a two-parameter perturbation expansion is proposed, which enables us to solve the problem of soft poroelastic bed with finite thickness. The results are compared to the similar problem with an infinite-thickness porous bed. The boundary effects of the impervious rock are significant on wave-induced pore water pressure and effective stresses, but are of very little significance on wave profiles at the free surface and the porous bed surface. However, the rigid boundary is insignificant to the pore water pressure and effective stresses when the thickness of porous bed is larger than about one wavelength.     

Relative Periodic Motion of a Rigid Body Pendulum on an Ellipse

In this paper, the properties of the relative periodic motions of a rigid body suspended on an elastic string in a vertical plane are considered. The supported point of this pendulum moves on an elliptic path with the rigid body suspended at the end point of the pendulum. Applying Lagrange’s equation, the equations of motion are obtained in terms of small parameter (ε). These equations represent a quasi-linear system of second order, which can be solved in terms of generalized coordinates Θ, Φ, and β using the method of small parameters. At the end, a discussion of the motion and a conclusion of the results are considered to show the orientation of the body and the geometric interpretation of this motion at any instant of time. Also, computerized data with graphical representations of the solutions are given for describing the behavior of the body in some periods of time.     

Acoustic Measurements of the Anisotropy of Dynamic Elastic and Poromechanics Moduli under Three Stress/Strain Pathways

A series of triaxial compression experiments have been conducted to investigate the effects of induced stress on the anisotropy developed in dynamic elastic and poroelastic parameters in rocks. The measurements were accomplished by utilizing an array of piezoelectric compressional and shear wave sensors mounted around a cylindrical sample of porous Berea sandstone. Three different types of applied states of stress were investigated using hydrostatic, triaxial, and uniaxial strain experiments. During the hydrostatic experiment, where an isotropic state of stress was applied to an isotropic porous rock, the vertical and horizontal acoustic velocities and dynamic elastic moduli increased as pressure was applied and no evidence of stress induced anisotropy was visible. The poroelastic moduli (Biot’s effective stress parameter, α) decreased during the test but also with no evidence of anisotropy. The triaxial compression test involved an axisymmetric application of stress with an axial stress greater than the two constant equal lateral stresses. During this test a marked anisotropy developed in the acoustic velocities, and in the dynamic elastic and poroelastic moduli. As axial stress increased the magnitude of the anisotropy increased as well. The uniaxial strain test involved axisymmetric application of stresses with increasing axial and lateral stresses but while maintaining a zero lateral strain condition. The uniaxial strain test exhibited a quite different behavior from either the triaxial or hydrostatic tests. As both the axial and lateral stresses were increased, an anisotropy developed early in the loading phase but then was effectively “locked in” with little or no change in the magnitude of the values of the acoustic velocities, or the dynamic elastic and poroelastic parameters as stresses were increased. These experimental results show that the application of triaxial states of stress induced significant anisotropy in the elastic and poroelastic parameters in porous rock, while under the uniaxial strain condition the poromechanics, Biot’s effective stress parameter, exhibited the largest variation among the three test conditions.     

Generalized Bending of Shear-Deformable Plate with Elastic Inclusion

The generalized bending of a large plate with a circular elastic inclusion is discussed here. The solution for a hole or a rigid inclusion, which is the limiting case of an elastic inclusion, has been discussed frequently, but there is little work on an inclusion with arbitrary rigidity. Early investigators solved the problem of a large thin plate with a circular elastic inclusion, subjected to uniaxial bending, balanced biaxial bending, pure twisting, and transverse shear. This work was recently generalized to arbitrary bending loading, and explicit formulas were presented for dimensionless maximum circumferential bending moments in the plate and in the inclusion. This paper solves the analogous problem with Reissner's shear-deformable plate theory. Arbitrary bending loading is considered. A general solution is derived for the governing differential equations. Explicit formulas are developed for the maximum circumferential and radial moments in the plate and the inclusion. The results for four typical loadings are illustrated graphically, and two limiting forms of the solution are obtained.     

Porochemothermoelastic Solution for an Inclined Borehole in a Transversely Isotropic Formation

A generalized anisotropic poromechanics formulation for chemically active poroelastic media under nonisothermal conditions, termed as porochemothermoelastic, is presented. The pore fluid is modeled as a two-species constituent comprised of the solute and the solvent. Governing equations are developed and applied to obtain the analytical solution for an inclined borehole in chemically active transversely isotropic formation subjected to a three-dimensional state of stress and nonisothermal conditions. Numerical examples are presented to demonstrate the thermochemical effects on stress and pore pressure distributions in the vicinity of the borehole and their potential impacts on borehole stability.     

Consolidation of Clays for Variable Permeability and Compressibility

The theory of consolidation has its origin in the effective stress concept developed by Terzaghi, which was derived based on several assumptions to arrive at a simplified theory. Considering the limitations involved in Terzaghi’s theory, various attempts are being made by researchers to idealize the problem to represent various field situations. This paper presents a more generalized theory for vertical consolidation of a compressible medium of finite thickness, subjected to suddenly applied loading, assuming small strain and no creep. The theory assumes small deformations, and thus the settlement is governed by vertical strains generated by an increment of loading, neglecting the effect of self-weight of the soil. The analytical solution presented here takes into account e–log?K and e–log?σ′ linear responses under instantaneous loading. With Cc as the slope of the e–log?σ′ line and M the slope of the e–log?K line, a parameter Cc/M is identified which is found to govern the rate of consolidation. In this paper, an analytical closed form solution is obtained for vertical consolidation considering the variation in the compressibility and permeability.     

Dynamic Model of an Astronaut Equipped with a Manned Maneuvering Unit in Virtual Reality

A virtual reality simulation system for a manned maneuvering unit (MMU) and a training system for an astronaut space walk is introduced. The system can simulate an astronaut’s outer space walk by means of manipulating the MMU, and it also can be used as a training environment by taking advantage of virtual reality properties of imagination, interaction, and immersion. As in microgravity space, an astronaut’s moving limbs cause changes in the MMU posture; these changes result in great offsets to correct direction and have a great effect on the MMU’s manipulation performance. A multibody astronaut model is developed by using relative coordinates according to the motion characteristics of an astronaut equipped with MMU. Motion information on the user’s limbs are captured by means of a motion capture subsystem and are mapped to an astronaut in virtual environment to simulate the astronaut’s limb motions while the astronaut is navigating in outer space. It is an open-chain system for an astronaut multibody model. As an astronaut can control his limbs, the whole system dynamics is a hybrid forward and inverse dynamic problem. Recursive dynamic formulations of a previous method are introduced to solve the forward dynamic problem. For the inverse dynamic part, because the system can get motion characteristics of parts of the relative coordinates, unknown generalized driving forces are applied on these known relative coordinates. All the unknown generalized driving forces and unknown relative coordinates can be combined in a set of hybrid dynamic formulations based on a forward dynamic formulation. The motion characteristics of an astronaut multibody model can be obtained by solving the hybrid formulations. Finally, some simulation results for astronaut forward-flying mode with moving limbs are presented; these results show that the movement of an astronaut’s limbs has great effects on astronaut navigation and MMU manipulation performance. The multibody astronaut model has the capability of simulating real space walk with an MMU equipped astronaut.     

Torsional Surface Waves in an Initially Stressed Anisotropic Porous Medium

It is well known that the homogeneous elastic half space does not allow the torsional surface wave to propagate. The present article attempts to study the possibility of propagation of such waves in a liquid-filled initially stressed poroelastic layer over nonhomogeneous half space. The study concludes that torsional surface waves may propagate in the poroelastic layer. The presence of fluid in pores and initial compressive stresses diminishes the velocity. The nonhomogeneity in rigidity and density of the half space also has a certain role to play in initiating, enhancing, or diminishing the velocity of torsional surface waves.     

Time-Dependent Poromechanical Responses of Saturated Cylinders

The uniaxial, hydrostatic, and triaxial tests of saturated cylindrical rock samples are very common in a rock mechanics laboratory. The conventional solution for the mechanical responses of the sample under such testing conditions (an axial load and a confining pressure) is trivial within the elastic range. For saturated samples, however, these elastic solutions can only be applied to the drained or undrained cases, and there has been a lack of transient analyses of cylindrical samples under such tests, taking into account the pore fluid pressure buildup and the coupling effects, especially for samples with low permeability. In this paper, poroelasticity is employed to develop the solution for saturated cylindrical samples subjected to an axial load and a confining pressure. Significant poroelastic effects on the tests were observed through the analyses of a uniaxial test and a triaxial test based on the proposed poroelastic solution. Without consideration of the poroelastic effects, erroneous interpretations of the testing results can be expected.     

Analysis of Lightweight Deflectometer Test Based on In Situ Stress and Strain Response

The lightweight deflectometer (LWD) is gaining acceptance and popularity as an in situ spot-testing device for quality control/quality assurance of earthwork compaction. Little research has been conducted to investigate the stress–strain response within the soil during LWD testing. Similarly, little research has been performed to examine the appropriateness of using homogeneous, isotropic, linear elastic half-space theory to estimate soil modulus (ELWD) from LWD results. With this aim, an array of vertical stress and strain sensors was placed within the soil to measure the stress–strain response during LWD loading. Measured in situ stress values matched well with stresses predicted using homogeneous, isotropic, linear elastic half-space theory. In situ stress data revealed that the contact stress distribution between the soil surface and loading plate is a function of the soil type. Measured in situ strain values did not correspond well with strains predicted using homogeneous, isotropic, linear elastic elasticity. An exponentially increasing modulus function was required to match experimental with theoretical elastic strains. The results indicate that the commonly used form to predict ELWD is inappropriate if the goal is to extract constitutive soil properties. Analysis of strain data suggests the LWD depth of influence (measurement depth) is 0.9–1.1 times the plate diameter.     

Electroelastic Response of a Laminated Composite Plate with Piezoelectric Sensors and Actuators

The paper deals with the fully coupled response characteristics of a multilayered composite plate with piezoelectric layers. The response quantities of the plate are coupled by the mechanical field and the electric field. Based on the three-dimensional linear piezoelectricity and the first-order shear deformation theory, the fundamental unknowns, such as the displacements and the electric potential, are assumed to be expandable through the plate thickness coordinate. The governing equations of motion of the plate are presented in terms of the unknown displacement and electrical potential coefficients. When the boundary conditions and electromechanical inputs are specified, the double Fourier series is used to obtain the response of the simply supported multilayered plates. Numerical results for the static and dynamic response of the laminated composite plates with different lamination schemes and having a PIC-151 piezoelectric material layer are obtained. The effects of the plate thinness ratio, plate aspect ratio, lamination scheme, fiber orientations, and piezoelectric coupling on the static and dynamic response are presented.     

Mixture Theory for a Fluid-Saturated Isotropic Elastic Matrix

The theory for a fluid saturated linearly isotropic elastic matrix is still the basis for many geophysical applications, and commonly adopts Biot’s symmetric stress–strain laws for the matrix stress and fluid pressure. These involve a shear modulus and three elastic moduli governing the mixture and constituent compressions, in contrast to four compression moduli if Biot’s invalid potential energy argument is not applied. We now show that an energy argument applied to undrained loading also leads to three compression moduli, but distinct from those derived by Biot (Biot symmetry). However, there are two distinct solutions of this energy balance, corresponding to the Voigt and Reuss limits of the analogous theory of a linear two-phase elastic composite, whereas a unique undrained modulus not at either limit would be expected. It is proposed that an energy contribution is lost due to the idealised assumptions made for undrained loading, which therefore does not determine a further restriction, so that there are four independent compression moduli. The general and restricted combinations of the total pressure and fluid pressure (effective stress) governing the matrix compression are then presented, together with the alternative forms of the partial differential equations governing the deformation and flow.     

Poroelastic creep response analysis of a lumbar motion segment in compression

The nonlinear three-dimensional poroelastic creep response of a lumbar motion segment under a constant axial compression (400, 1200, or 2000 N) is investigated for a period of 2 h. The role of facet joints, strain-dependent variable permeability, boundary pore pressure, and coupled sagittal rotation on response is studied. Biomechanics of annulus excision, nucleotomy, and facetectomy are also investigated. Both material and geometric nonlinearities are considered. The annulus bulk is modelled as a nonhomogeneous composite of collagenous fibers and annulus bulk. As time progresses, axial displacement increases, pore pressure decreases, annulus bulk undergoes larger compressive stresses, fiber layers become slack, and facets carry larger loads. Surgical alterations markedly soften the temporal response and increase facets forces. In contrast, the strain-dependent variable permeability and boundary pore pressure stiffen the response and decrease forces on the facets. Changes in the nucleus fluid content, facet joints, boundary pore pressure, and disc permeability markedly influence the lumbar biomechanics.     

Micromechanical Approach to Nonlinear Poroelasticity: Application to Cracked Rocks

This paper considers a saturated porous medium in which the matrix is a cracked solid. Progressive crack closure is responsible for an overall nonlinear poroelastic behavior. The state equations of nonlinear poroelasticity are derived in a differential form within a micromechanical framework. When a hydraulic connection exists between the cracks and the pores of the porous space, the tangent drained stiffness tensor as well as the tangent Biot tensor and modulus are shown to depend on Terzaghi effective stress. Estimates for these coefficients as functions of Terzaghi effective stress are then derived with the tools of homogenization for disordered media. They are based on a crack closure criterion giving the condition for a crack to be closed under a given macroscopic stress state, depending on its aspect ratio. In the case of an isotropic orientation of cracks, it is shown that the influence of cracks on the overall poroelastic properties is governed by the crack density parameter which characterizes the distribution of aspect ratios. Conversely, an experimental methodology for the determination of the distribution of aspect ratios from the measurement of the macroscopic compliance is proposed.     

Vertical and Radial Consolidation Analysis of Multilayered Soil Using the Spectral Method

A new, easy to implement, solution to the consolidation of multilayered soil based on the spectral method is presented. Combined vertical and radial drainage under instantaneous or single ramp loading is considered, ignoring well resistance. Flow in the vertical direction is based on the average hydraulic gradient at a particular depth which allows smear effects to be included. The excess pore-water pressure profile across all soil layers is described by a single expression calculated with common matrix operations. Average excess pore pressures within or across any number of layers are easily calculated from the single expression. The new model is verified against other solutions from the current literature indicating that the more general spectral method model can replace the separate solutions developed for specific problems.