A large strain finite element formulation for one dimensional membrane elements

A finite element formulation suitable for large strain analysis of one-dimensional membranes is described which has applications to the analysis of problems in soil mechanics involving reinforced earth. The finite element equations, which are derived for a plane strain iso-parametric element of arbitrary order, are cast in the familiar small displacement form, but with a modified material stiffness matrix. In this formulation, full account is taken of element rotations during each load step, an effect which is important when stresses cease to be insignificant when compared with the material modulus. The method is illustrated with reference to a large displacement test problem which has a known closed form solution. An example is included demonstrating the application of the formulation to the analysis of a reinforced unpaved road.     

Localization of deformations in elastic-plastic solids

Localization of deformation in elastic-plastic solids subject to plane strain deformation are investigated numerically. It is shown that the localization may be captured accurately in finite element models by employing (1) the elastic-plastic material stiffness to form the global stiffness, (2) in the case of symmetrical configurations, an imperfection in the form of a weak element, and (3) in the case of incompressible materials, a reduced selective integration scheme which alleviates mesh ‘locking’. Accuracy of the technique is demonstrated by applying it to analyse the classical punch and slope stability problems. Its versatility is illustrated by applying it to analyse finite deformation problems and shear bands formations in associative and non-associative elastic-plastic solids.     

An operator‐split ALE model for large deformation analysis of geomaterials

Analysis of large deformation of geomaterials subjected to time‐varying load poses a very difficult problem for the geotechnical profession. Conventional finite element schemes using the updated Lagrangian formulation may suffer from serious numerical difficulties when the deformation of geomaterials is significantly large such that the discretized elements are severely distorted. In this paper, an operator‐split arbitrary Lagrangian–Eulerian (ALE) finite element model is proposed for large deformation analysis of a soil mass subjected to either static or dynamic loading, where the soil is modelled as a saturated porous material with solid–fluid coupling and strong material non‐linearity. Each time step of the operator‐split ALE algorithm consists of a Lagrangian step and an Eulerian step. In the Lagrangian step, the equilibrium equation and continuity equation of the saturated soil are solved by the updated Lagrangian method. In the Eulerian step, mesh smoothing is performed for the deformed body and the state variables obtained in the updated Lagrangian step are then transferred to the new mesh system. The accuracy and efficiency of the proposed ALE method are verified by comparison of its results with the results produced by an analytical solution for one‐dimensional finite elastic consolidation of a soil column and with the results from the small strain finite element analysis and the updated Lagrangian analysis. Its performance is further illustrated by simulation of a complex problem involving the transient response of an embankment subjected to earthquake loading. Copyright © 2007 John Wiley & Sons, Ltd.     

A multiscale finite element formulation for axisymmetric elastoplasticity with volumetric locking

Similar to plane strain, axisymmetric stress problem is also highly kinematics constrained. Standard displacement‐based finite element exhibits volumetric locking issue in simulating nearly/fully incompressible material or isochoric plasticity under axisymmetric loading conditions, which severely underestimates the deformation and overestimates the bearing capacity for structural/geotechnical engineering problems. The aim of this paper is to apply variational multiscale method to produce a stabilized mixed displacement–pressure formulation, which can effectively alleviate the volumetric locking issue for axisymmetric stress problem. Both nearly incompressible elasticity and isochoric J₂ elastoplasticity are investigated. First‐order 3‐node triangular and 4‐node quadrilateral elements are tested for locking issues. Several representative simulations are provided to demonstrate the performance of the linear elements, which include the convergence study and comparison with closed‐form solutions. A comparative study with pressure Laplacian stabilized formulation is also presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Smeared crack approach: back to the original track

This paper briefly reviews the formulations used over the last 40 years for the solution of problems involving tensile cracking, with both the discrete and the smeared crack approaches. The paper focuses on the smeared approach, identifying as its main drawbacks the observed mesh‐size and mesh‐bias spurious dependence when the method is applied ‘straightly’. A simple isotropic local damage constitutive model is considered, and the (exponential) softening modulus is regularized according to the material fracture energy and the element size. The continuum and discrete mechanical problems corresponding to both the weak discontinuity (smeared cracks) and the strong discontinuity (discrete cracks) approaches are analysed and the question of propagation of the strain localization band (crack) is identified as the main difficulty to be overcome in the numerical procedure. A tracking technique is used to ensure stability of the solution, attaining the necessary convergence properties of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is stable and remarkably robust. As a consequence, the results obtained do not suffer from spurious mesh‐size or mesh‐bias dependence, comparing very favourably with those obtained with other fracture and continuum mechanics approaches. Copyright © 2006 John Wiley & Sons, Ltd.     

Modelling strain localization in granular materials using micropolar theory: mathematical formulations

It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill‐posedness and mesh dependency. Therefore, a non‐local theory with internal length scales is needed to overcome such problems. The micropolar and high‐order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical study of localization in soil systems

A numerical study of the mechanical behavior of heterogeneous soil systems, consisting of a bulk of sand with embedded stiff gravel inclusions or soft clay inclusions, is performed. A solution scheme using parallel computing is employed when analyzing two different categories of problems. First, a homogenization problem is studied, where use of a single representative volume element subjected to plane strain compression offers the possibility to investigate the coupling between the response at a local scale and at a global scale. Second, a plane strain footing problem with different heterogeneous soil systems is analyzed using a traditional finite element formulation. The material model utilized for the soil is a large deformation formulation of non-associated elasto-plasticity with an isotropic hardening law, able to represent dilation. It was found that the shape of the gravel or clay inclusions in the systems had no significant effect on the global responses, whereas the strain localizations in the two different soil systems, sand–gravel and sand–clay, were found to have different character. The effect of the initial density on the response was clearly observed in the localization patterns.     

Investigation of a hybrid polygonal finite element formulation for confined and unconfined seepage

This study presents a formulation for field problems using hybrid polygonal finite elements, taking steady state seepage through a porous material as the focus. We make comparisons with a conventional finite element formulation based on a single primary variable, focussing on the advantages of the hybrid formulation in terms of flux field accuracy and extension to convex polygonal shaped elements. For the unconfined case, we adopt a head dependent hydraulic conductivity that does not require remeshing. The performance of the hybrid polygonal element formulation is demonstrated through a series of numerical examples. The results show a sensitivity of the location of the free surface in unconfined seepage to mesh configuration for hybrid quadrilateral meshes with various aspect ratios, but not for hybrid polygonal meshes with various orientations and irregularity. Examination of the free surface location results for several conforming shape function options shows an insensitivity to choice of interpolation function, provided that it conforms with the assumptions in the formulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical modelling of large deformations associated with driving of open‐ended piles

This paper presents a ‘Eulerian‐like’ finite element technique to simulate the large accumulated displacements of piles subjected to multiple hammer blows. For each hammer blow, results are obtained using a standard small strain finite element model and, at the end of each hammer blow, material flow is taken into account with reference to a fixed finite element mesh. Residual stresses calculated at the Gauss integration points of the deformed finite element mesh are mapped on to the fixed finite element mesh, and these stresses are used as initial stresses for the next hammer blow. At the end of each hammer blow, stiffness and mass matrices are recalculated for the volume of material remaining inside the fixed finite element mesh. Results obtained with and without allowing material to flow through the fixed mesh are compared for several hammer blows. Build up of residual stresses, soil flow and yielded points around the pile are presented for plugged, partially‐plugged and unplugged piles. Using the new finite element technique, the driving of a pile from the soil surface is studied. The ability to analyse this and other large deformation problems is the main advantage of the new finite element technique. Copyright © 2000 John Wiley & Sons, Ltd.     

Axisymmetric compression of a Mohr–Coulomb medium around a circular hole

An analytical solution is presented for the stress and strain fields in a Mohr–Coulomb material in plane strain around a circular hole when it is compressed by an axisymmetric far-field pressure. It is shown that several solutions arise involving one to three plastic zones depending on the values of Poisson's ratio and the friction angle. The solution chosen for presentation was obtained and used to validate the functioning of the Mohr–Coulomb yield condition that was added to the NONSAP finite element code. Stress and strain field comparisons are made.     

Three dimensional FEM of moving coordinates for the analysis of transient vibrations due to moving loads

This paper is concerned with the transient vibration analysis of railway-ground system under fast moving loads. A 3D finite element method in a convected coordinate system moving with the load is formulated, together with viscous-elastic transmitting boundary conditions in order to limit the finite element mesh. A method is proposed to introduce Rayleigh type material damping in the finite element formulation in the moving coordinate system, while measures have also been taken to improve the numerical stability of the solution procedure. The performance of the transmitting boundary and the entire solution procedure are assessed via comparison with the ordinary finite element solution of some relatively simple problems and through a comparison with field measurements. The reasonable agreement found from these comparisons demonstrates the validity of the proposed method.     

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.     

On the well‐posedness of some mechanical variational problems

The paper aims to establish the existence and uniqueness of the solution of some variational problems constituting the basis of finite element modellings encountered in mechanics and civil engineering. And indeed, by expanding to the approximate problems coming from the space discretization, such theoretical results contribute to strengthen the robustness of the modelling softwares and the quality of their numerical results. More particularly, three kinds of mixed variational problems involving rheological non‐linearities are considered here: the evolution problems of incompressible continua (solids or fluids) subjected to quasistatic small transformations, the problems of hydromechanical coupling and those coming from quasistatic large transformations of continua. Copyright © 2005 John Wiley & Sons, Ltd.     

An application of the calculus of variations to rectilinear flow of granular materials

The calculus of variations is an attractive tool for the determination of critical slip surfaces, but there are unresoived difficulties in its application to plane strain problems. Here we consider the simpler problem of rectilinear flow under gravity or other constant body force. In particular we investigate the stability of an infinitely long slope of finite width for material with cohesion and internal friction. From a variational formulation, the differential equation of the critical slip surface is formulated. An explict solution of this equation is obtained, and hence we determine the critical slope angle in terms of the width of the slope and the material properties. The results are compared with those of an earlier limit analysis treatment.     

Variational principles and finite element simulations for thermo-elastic consolidation

In this paper, a general variational principle for the initial boundary value problem of quasi-static thermoelastic consolidation is developed by assuming infinitesimal deformation and an incompressible fluid flowing through a linearly elastic solid. By manipulating the coupling operators, an extended form of the variational pronciple is derved. The associated finite element formulation based on this principle is presented and numerical applications for plane strain thermo-elastic consolidation are revealed.     

Numerical stability of non-local viscoplastic FEM analyses for the study of localisation processes

As is well known, numerically handling, by means of finite element codes, localisation problems involving softening materials is still quite delicate. As soon as strain localisation occurs, mesh dependence and serious problems of convergence take place. This paper deals with this type of problem in the case where localisation occurs in an ideal naturally cemented granular specimen tested under plane strain conditions. Different versions (a local elasto-plastic, a local viscoplastic and a non-local viscoplastic) of the same strain softening model are taken into consideration and the relative numerical results are critically discussed and compared. The snap-back problem is numerically taken into account and it has been demonstrated to be affected not only by the softening parameters but also by the viscous nucleus definition. To highlight the relationship between viscosity and non-locality, the results of a numerical parametric analysis are finally discussed.     

Natural convection of compressible and incompressible gases in undeformable porous media under cold climate conditions

A numerical model for convective heat and mass transport of compressible or incompressible gas flows with soil-water phase change is presented. In general, the gaseous phase is considered as compressible and the model accounts for adiabatic processes of compression heating and expansion cooling. The inherently compressible gaseous phase may nevertheless be considered as incompressible by adopting the Oberbeck–Boussinesq approximations. The numerical method used to solve the equations that describe natural convection is based on a Galerkin finite element formulation with adaptive mesh refinement and dynamic time step control. As most existing numerical studies have focused on the behavior of incompressible fluids, model substantiation examines the influence of fluid compressibility on two-widely used benchmarks of steady-state convective heat and mass transport. The relative importance of the effect of pressure-compressibility cooling is shown to increase as the thermal gradient approaches the magnitude of the adiabatic gradient. From these results, it may be concluded that pore-air compressibility cannot be neglected in medium to large-sized enclosures at small temperature differentials. After demonstrating its ability to solve fairly complex transient problems, the model is used to further our understanding of the thermal behavior of the toe drain at the LA2-BSU dam in the province of Quebec, Canada.     

Semi‐analytical elastostatic analysis of unbounded two‐dimensional domains

Unbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement‐based numerical methods, such as the finite‐element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi‐analytical scaled boundary finite‐element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill‐conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd.     

Coupled hydromechanical‐fracture simulations of nonplanar three‐dimensional hydraulic fracture propagation

This paper presents an algorithm and a fully coupled hydromechanical‐fracture formulation for the simulation of three‐dimensional nonplanar hydraulic fracture propagation. The propagation algorithm automatically estimates the magnitude of time steps such that a regularized form of Irwin's criterion is satisfied along the predicted 3‐D fracture front at every fracture propagation step. A generalized finite element method is used for the discretization of elasticity equations governing the deformation of the rock, and a finite element method is adopted for the solution of the fluid flow equation on the basis of Poiseuille's cubic law. Adaptive mesh refinement is used for discretization error control, leading to significantly fewer degrees of freedom than available nonadaptive methods. An efficient computational scheme to handle nonlinear time‐dependent problems with adaptive mesh refinement is presented. Explicit fracture surface representations are used to avoid mapping of 3‐D solutions between generalized finite element method meshes. Examples demonstrating the accuracy, robustness, and computational efficiency of the proposed formulation, regularized Irwin's criterion, and propagation algorithm are presented.