Dynamics of unsaturated poroelastic solids at finite strain

We derive the governing equations for the dynamic response of unsaturated poroelastic solids at finite strain. We obtain simplified governing equations from the complete coupled formulation by neglecting the material time derivative of the relative velocities and the advection terms of the pore fluids relative to the solid skeleton, leading to a so‐called u^s ? p^w ? p^a formulation. We impose the weak forms of the momentum and mass balance equations at the current configuration and implement the framework numerically using a mixed finite element formulation. We verify the proposed method through comparison with analytical solutions and experiments of quasi‐static processes. We use a neo‐Hookean hyperelastic constitutive model for the solid matrix and demonstrate, through numerical examples, the impact of large deformation on the dynamic response of unsaturated poroelastic solids under a variety of loading conditions. Copyright © 2011 John Wiley & Sons, Ltd.     

A comparison of variational formats for porous media subjected to dynamic loading

This paper presents a comparison of two variational formats for fully saturated porous media subjected to dynamic loading, whereby the general situation of relative fluid acceleration is considered: (1) the classical three‐field ( u , p, w )‐format and (2) a novel two‐field ( u , p)‐format, where the seepage velocity w is a spatially ‘local’ field whose treatment resembles that of internal variables in material models. The limited numerical comparison shows that the ( u , p)‐format competes well with the ( u , p, w )‐format. Indeed, it is consistent with the general acceleration modeling in the full range of permeabilities. Moreover, in the low permeability regime (where the magnitude of w is insignificant), the new format reflects the situation pertinent to ‘added‐mass’ and is more efficient than the classical ( u , p, w )‐format. Finally, the ( u , p)‐formatcan conveniently be implemented in existing FE‐codes based on the ‘added mass’ formulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of fully saturated porous materials

Fully coupled, porous solid–fluid formulation, implementation and related modeling and simulation issues are presented in this work. To this end, coupled dynamic field equations with u?p?U formulation are used to simulate pore fluid and soil skeleton (elastic–plastic porous solid) responses. Present formulation allows, among other features, for water accelerations to be taken into account. This proves to be useful in modeling dynamic interaction of media of different stiffnesses (as in soil–foundation–structure interaction). Fluid compressibility is also explicitly taken into account, thus allowing excursions into modeling of limited cases of non‐saturated porous media. In addition to these features, present formulation and implementation models in a realistic way the physical damping, which dissipates energy. In particular, the velocity proportional damping is appropriately modeled and simulated by taking into account the interaction of pore fluid and solid skeleton. Similarly, the displacement proportional damping is physically modeled through elastic–plastic processes in soil skeleton. An advanced material model for sand is used in present work and is discussed at some length. Also explored in this paper are the verification and validation issues related to fully coupled modeling and simulations of porous media. Illustrative examples describing the dynamical behavior of porous media (saturated soils) are presented. The verified and validated methods and material models are used to predict the behavior of level and sloping grounds subjected to seismic shaking. Copyright © 2008 John Wiley & Sons, Ltd.     

Two formulations for dynamic response of a cylindrical cavity in cross‐anisotropic porous media

Two formulations for calculating dynamic response of a cylindrical cavity in cross‐anisotropic porous media based on complex functions theory are presented. The basis of the method is the solution of Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u–w formulation of Biot's equations are solved. Difference of these two solutions refers to use of two various potential functions. Equations for calculating stress, displacement and pore pressure fields of the medium are mentioned based on each two formulations. Copyright © 2009 John Wiley & Sons, Ltd.     

B‐bar based algorithm applied to meshfree numerical schemes to solve unconfined seepage problems through porous media

The object of this work is to establish a meshfree framework for solving coupled, steady and transient problems for unconfined seepage through porous media. The Biot's equations are formulated in displacements (or u − w) assuming an elastic solid skeleton. The free surface location and its evolution in time are obtained by interpolation of pore water pressures throughout the domain. Shape functions based on the principle of local maximum entropy are chosen for the meshfree approximation schemes. In order to avoid the locking involved in the fluid phase of the porous media, a B‐bar based algorithm is devised to compute the average volumetric strain in a patch composed of various integration points. The efficiency of such an implementation for one phase problems is shown through the Benchmark problem, Cook's membrane loaded by a distributive shear load. The proposed methodology is firstly applied to various classical examples in unconfined steady seepage problems through earth dams, then to the dynamic consolidation of a soil column. The results obtained for both problems are quite satisfactory and demonstrate the feasibility of the proposed method in solving coupled problems in porous media. Copyright © 2015 John Wiley & Sons, Ltd.     

Three‐dimensional nonlinear finite element analysis of pile groups in saturated porous media using a new transmitting boundary

The dynamic behaviour of pile groups subjected to an earthquake base shaking is analysed. An analysis is formulated in the time domain and the effects of material nonlinearity of soil, pile–soil–pile kinematic interaction and the superstructure–foundation inertial interaction on seismic response are investigated. Prediction of response of pile group–soil system during a large earthquake requires consideration of various aspects such as the nonlinear and elasto‐plastic behaviour of soil, pore water pressure generation in soil, radiation of energy away from the pile, etc. A fully explicit dynamic finite element scheme is developed for saturated porous media, based on the extension of the original formulation by Biot having solid displacement (u) and relative fluid displacement (w) as primary variables (u–w formulation). All linear relative fluid acceleration terms are included in this formulation. A new three‐dimensional transmitting boundary that was developed in cartesian co‐ordinate system for dynamic response analysis of fluid‐saturated porous media is implemented to avoid wave reflections towards the structure. In contrast to traditional methods, this boundary is able to absorb surface waves as well as body waves. The pile–soil interaction problem is analysed and it is shown that the results from the fully coupled procedure, using the advanced transmitting boundary, compare reasonably well with centrifuge data. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical theory for one‐dimensional consolidation of clayey soils exhibiting rheological characteristics under time‐dependent loading

This paper presents analytical solutions to the one‐dimensional consolidation problem taking into consideration the rheological properties of clayey soil under variable loadings. A four‐element rheological model is introduced, and different loading types are involved, i.e. constant loading, one‐step loading, triangular loading, rectangular loading, and isosceles–trapezoidal cyclic loading. The differential equations governing consolidation are solved by the Laplace transform. Based on the solutions obtained, the influences of the rheological parameters and loading conditions on the consolidation process are investigated. It has been shown that the consolidation behavior is mainly governed by four dimensionless parameters, a₁, a₂, b, and T_v0. Load shape has a great influence on the rate of consolidation. A decrease either in the modulus of the spring in the Kelvin body or in the viscosity coefficient of independent dashpot will slow down the rate of consolidation. An increase in the viscosity coefficient of the dashpot in the Kelvin body will make the rate of consolidation increase at an early stage but decrease at a later stage. For isosceles–trapezoidal cyclic loading, the consolidation rate in each cycle reaches a maximum at the end of the constant loading phase and the minimum at the end of this cycle. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

Based on the consolidation theory raised by Fredlund, the solutions for the equal-strain consolidation of unsaturated foundation with the prefabricated vertical drain considering smear effect and drain resistance are analytically formulated in this paper. Firstly, governing equations for excess pore pressures (i.e., excess pore-air and pore-water pressures) under the equal-strain hypothesis are derived with the introduction of radial boundary conditions. Afterwards, the obtained coupled equations are solved by applying general integration, decoupling process, and Fourier sine series expansion. The smear coefficients and factors of drain resistance corresponding to air and water phases are both captured explicitly in the final solutions. Furthermore, the degenerated solutions are employed to verify the reliability of the current solutions. Finally, a parametric study is conducted to study the consolidation characteristics of the proposed foundation model against modeling sizes (S and N), smear coefficients (α_a and α_w), and drain resistance factors (G_a and G_w).     

Improved Bell's method for the stability analysis of slopes

A new procedure based on the approximation to the total normal pressure along the slip surface is developed to compute the factor of safety of slopes for slip surfaces of all shapes. By taking the whole sliding body, instead of an individual slice, as the loaded object, all the equilibrium equations are formulated according to the three‐moment equilibrium conditions rather than the two force equilibrium conditions and one‐moment equilibrium condition. The system of nonlinear equations deduced in this way is well‐scaled and enjoys excellent numerical properties such as the existence of solution with a positive factor of safety, a nearly unlimited scope of convergence and a rapid convergence rate associated with the Newton method. In the case of ?_u =0—the situation where no drainage and no consolidation are involved, furthermore, the system has a unique solution and the factor of safety has an explicit expression. Some typical examples are analyzed to demonstrate the numerical properties of the proposed procedure. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear radial consolidation of vertical drains under a general time‐variable loading

By incorporating the nonlinear variation of a soil's compressibility and permeability during the process of consolidation, an analytical solution for the radial consolidation of vertical drains has been developed for a general time‐variable loading. The general solution was verified for the cases of instantaneous loading and ramp loading. Detailed solutions were further derived for two special loading schemes: multistage loading and preloading–unloading–reloading. The nonlinear consolidation behavior of a vertical drain subjected to these two types of loading schemes was then investigated by a parametric study. The results show that the loading rate, the ratio of the compressibility index to the permeability index (C_c/C_k), and the initial stress state have a significant influence on the consolidation rate. A smaller value of C_c/C_k, a larger initial stress, or a fast loading rate always leads to a rapid consolidation rate. During the unloading period, a negative excess pore water pressure may occur, and a slower unloading rate may reduce this negative value. Copyright © 2014 John Wiley & Sons, Ltd.     

Constitutive modelling of fine-grained gassy soil: A composite approach

Fine-grained marine sediments containing large undissolved gas bubbles are widely distributed around the world. Presence of the bubbles could degrade the undrained shear strength (s_u ) of the soil, when the gas pressure u_g is relatively high as compared with the effective stress in the saturated soil matrix. Meanwhile, the addition of bubbles may also increase s_u when the difference between u_g and pore water pressure u_w becomes smaller than the water entry value, causing partial water drainage from the saturated matrix into the bubbles (bubble flooding) during globally undrained shearing. A new constitutive model for describing the two competing effects on the stress-strain relationship of fine-grained gassy soil is proposed within the framework of critical state soil mechanics. The gassy soil is considered as a three-phase composite material with compressible cavities, which allows water entry from the saturated matrix. Bubble flooding is modelled by introducing an additional positive volumetric strain increment of the saturated clay matrix, which is dependent on the difference between pore gas and pore water pressure based on experimental observations. A modified hardening law based on that of the modified Cam clay model is employed, which in conjunction with the expression for bubble flooding, can describe both the detrimental and beneficial effects of gas bubbles on soil strength and plastic hardening in shear. Only two extra parameters in addition to those in the modified Cam clay model are used. It is shown that the key features of the stress-strain relationship of three fine-grained gassy soils can be reproduced satisfactorily.     

Analytical solution for 1D consolidation under haversine cyclic loading

Presented and discussed in this paper is an exact analytical solution of the nonhomogeneous partial differential equation governing the conventional one‐dimensional consolidation under haversine repeated loading. The derived analytical solution to the 1D consolidation equation is compared with the numerical solution of the same consolidation problem via FEM. The series solution takes into account the frequency of repeated loading through a dimensionless time factor T₀. The paper reveals that an increase in the frequency of imposed repeated haversine loading (a decrease in period of repeated loading) causes an increase in the number of cycles required to achieve the steady state, whereas the effect of frequency on the maximum excess pore water pressure at the bottom of a clay layer with permeable top and impermeable bottom for the range of frequencies studied is generally insignificant. The effective stress at the bottom of the clay deposit with permeable top and impermeable bottom increases with time but with some fluctuations without changing the sign. These fluctuations become more pronounced for increasing values of T₀. An increase in T₀ also causes an increase in maximum effective stress. Copyright © 2013 John Wiley & Sons, Ltd.     

Coupled HM analysis using zero‐thickness interface elements with double nodes. Part I: Theoretical model

In recent years, the authors have proposed a new double‐node zero‐thickness interface element for diffusion analysis via the finite element method (FEM) (Int. J. Numer. Anal. Meth. Geomech. 2004; 28 (9): 947–962). In the present paper, that formulation is combined with an existing mechanical formulation in order to obtain a fully coupled hydro‐mechanical (or HM) model applicable to fractured/fracturing geomaterials. Each element (continuum or interface) is formulated in terms of the displacements (u) and the fluid pressure (p) at the nodes. After assembly, a particular expression of the traditional ‘u–p’ system of coupled equations is obtained, which is highly non‐linear due to the strong dependence between the permeability and the aperture of discontinuities. The formulation is valid for both pre‐existing and developing discontinuities by using the appropriate constitutive model that relates effective stresses to relative displacements in the interface. The system of coupled equations is solved following two different numerical approaches: staggered and fully coupled. In the latter, the Newton–Raphson method is used, and it is shown that the Jacobian matrix becomes non‐symmetric due to the dependence of the discontinuity permeability on the aperture. In the part II companion paper (Int. J. Numer. Anal. Meth. Geomech. 2008; DOI: 10.1002/nag.730 ), the formulation proposed is verified and illustrated with some application examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical manifold method for dynamic nonlinear analysis of saturated porous media

It is well known that the Babuska–Brezzi stability criterion or the Zienkiewicz–Taylor patch test precludes the use of the finite elements with the same low order of interpolation for displacement and pore pressure in the nearly incompressible and undrained cases, unless some stabilization techniques are introduced for dynamic analysis of saturated porous medium where coupling occurs between the displacement of solid skeleton and pore pressure. The numerical manifold method (NMM), where the interpolation of displacement and pressure can be determined independently in an element for the solution of u–p formulation, is derived based on triangular mesh for the requirement of high accurate calculations from practical applications in the dynamic analysis of saturated porous materials. The matrices of equilibrium equations for the second‐order displacement and the first‐order pressure manifold method are given in detail for program coding. By close comparison with widely used finite element method, the NMM presents good stability for the coupling problems, particularly in the nearly incompressible and undrained cases. Numerical examples are given to illustrate the validity and stability of the manifold element developed. Copyright © 2006 John Wiley & Sons, Ltd.     

Solution of H2O in diopside melts: a thermodynamic model

Structural similarities between dry diopside melt and superhydrous albite melt (X _w >0.5) — both lack three-dimensional silicate units — suggest that thermodynamic relations may be similar. A model based on that assumption successfully predicts diopside melting relations and H₂O solubilities. For the model, the three partial differential equations describing solution of H₂O in albite melt for X _w >0.5 have been integrated for diopside melt from X _w =0 to X _w at least as large as 0.76, with two exceptions: an alternative partial differential equation for Henrian solution of H₂O in dilute melts was applied for X _w <0.20, and an alternative differential equation for the pressure dependence of a _w at pressures below 2 kbar was developed. The latter alternative equation yields relatively small ¯V_w's at low pressures rather than the large ¯V_w's calculated from the equation from the albite system. Available experimental solubility data are not precise enough to offer a choice between the small-¯V_w and large-¯V_w equations. Integration of all the partial differential equations was constrained solely by the P and T of a single experimentally-determined point on the H₂O-saturated solidus.Solubilities calculated by a Henrian-analogue solution model (a _di=X _di ² ) from the experimental H₂O saturated solidus lie outside experimental solubility constraints for dilute melts. On the other hand, a Henrian model (a _di=X_di) successfully predicts solubilities in dilute melts. The formulation of the Henrian model and magnitudes of model molar entropies of solution are consistent with the hypothesis that H₂O dissolves in diopside melt as an essentially undissociated species with little ordering on melt structural sites. That species could in turn be consistently, if not uniquely, interpreted to be molecular H₂O or a hydroxylation (OH^–) complex formed from nonbridging oxygens.     

Statistical thermodynamic models for ideal oxide and silicate solid solutions,with application to plagioclase

Activity-composition relations are derived for ideal substitutional solid solutions through the Helmholtz free energy expressed in terms of the partition function. For solutions of the type (A, B)_uZ_w involving mixing on one type of atom site, ideal activities of end-member components are expressed by: a_AuZw = (X_AuZw)^u, and a_BuZw = (X_BuZw)^u. With multi-site mixing excluding charge balance restrictions, as in (A, B)^α_u (C, D)^β_vZ_w, the ideal activity of an end-member component such as A_uC_vZ_w is calculated as: a_AuCvZw = (X^α_A)^u (X^β_c)^v. These expressions support the ‘ionic solid solution model for the activities of components in ideal solid solutions. Ideal solution models for coupled substitutions involving charge balance are considered using plagioclase as an example. Ideal activity expressions for solid solution of albite and anorthite are derived with and without adherence to the Al avoidance principle. Mixing models involving local electrostatic balance are contrasted with those involving independent, random mixing of Na-Ca and Al-Si. Of several possible ideal solution models for plagioclase, only that specifying complete Al-Si ordering and local electrostatic neutrality yields activities conforming to Raoult's Law.     

Comparison of numerical formulations for the modeling of tensile loaded suction buckets

The tensile resistance of a suction bucket is investigated using three different numerical formulations. The first formulation utilizes the three-field u-p-U formulation accounting for solid and fluid displacements, u and U, as well as the pore-fluid pressure, p. The two other formulations comprise the simpler u-p formulation in its dynamic and quasi-static form, accounting only for solid displacement and pore-fluid pressure. As basis for comparison, the tensile resistance of a single suction bucket is investigated using a velocity-driven model for a wide range of velocities. It is found, that the quasi-static u-p formulation is sufficient for most relevant velocities.     

Dust emission and transport in Mali, West Africa

Vertical dust fluxes were measured in the Inland Delta region of the Niger River, Mali, West Africa, during April-June, 1989 and 1990. Measurements of dust flux represent, for the most part, non-dust storm conditions or ‘dust haze’ periods. The observed concentration versus height relationships are similar to data presented by other investigators. The relationship between wind shear velocity (u_*, m s^?1) and vertical dust flux (F, μg m^?2 s^?1) can be described by a relationship in which F is proportional to u_*⁴. However, there is considerable scatter within the data set which is attributable to textural controls and surface conditions. The vertical dust fluxes measured in Mali are compared to dust fluxes measured in Texas, USA, and Yukon Territories, Canada. The significantly different values for the constant of proportionality (a) in the F α a u_*⁴ relationship for these geographically diverse areas is a function of surficial controls on the release of sediments to the air stream. Dust concentrations measured in Mali were found to be uniformly high and in general exceed WHO health standards for acceptable total suspended particulate loadings. As a result background dust may be considered a long term stress on health for the people of this region of Mali.     

Application of statistical learning algorithms to ultimate bearing capacity of shallow foundation on cohesionless soil

This study employs two statistical learning algorithms (Support Vector Machine (SVM) and Relevance Vector Machine (RVM)) for the determination of ultimate bearing capacity (q_u) of shallow foundation on cohesionless soil. SVM is firmly based on the theory of statistical learning, uses regression technique by introducing varepsilon‐insensitive loss function. RVM is based on a Bayesian formulation of a linear model with an appropriate prior that results in a sparse representation. It also gives variance of predicted data. The inputs of models are width of footing (B), depth of footing (D), footing geometry (L/B), unit weight of sand (γ) and angle of shearing resistance (?). Equations have been developed for the determination of q_u of shallow foundation on cohesionless soil based on the SVM and RVM models. Sensitivity analysis has also been carried out to determine the effect of each input parameter. This study shows that the developed SVM and RVM are robust models for the prediction of q_u of shallow foundation on cohesionless soil. Copyright © 2010 John Wiley & Sons, Ltd.     

Predicting the initiation of static liquefaction of cross‐anisotropic sands under multiaxial stress conditions

Experimental evidence has shown that the liquefaction instability of sands can be affected by its material density, stress state, and inherent anisotropy. In order to predict the initiation of the static liquefaction of inherent cross‐anisotropic sands under multidimensional stress conditions, a rational constitutive model is needed. An elastoplasticity model able to capture the influences of intermediate principal stress ratio (b  = (σ ₂ ? σ ₃)/(σ ₁ ? σ ₃)) and loading direction on stress–strain relationships and volumetric properties was proposed. The yield function was formulated to be controlled by Lode angle, loading direction, and material state; the stress–dilatancy was a material state‐dependent function. After using the existing drained hollow cylinder tests to validate the proposed model, this model was used to simulate the existing undrained hollow cylinder tests. During this simulation, the second‐order work criterion was used to determine the initiation of static liquefaction. The results showed that an increase in both the intermediate principal stress ratio and the loading angle induces a decrease in the second‐order work. Static liquefaction is initiated more easily at a stress state with a large intermediate principal stress ratio and a large loading angle, and the mobilized friction angle at the instability points decreases with the intermediate principal stress ratio and the loading angle. Copyright © 2017 John Wiley & Sons, Ltd.