Xikui Li

An iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics

The discretization of the u–p model for saturated porous media results in the semi-discrete system of mixed type in displacements and pressures. The Babuska–Brezzi condition or the simpler patch test proposed by Zienkiewicz and Taylor precludes the use of elements with the equal low order of interpolation for u and p, unless special stabilization techniques, such as the fractional step algorithms, are used. The purpose of this paper is to present a modified version of the fractional step algorithm which allows much larger time step sizes than those for the existing ones. The method is based on introducing an iteration algorithm. The numerical experiments demonstrate the effectiveness and improved performance of the proposed modified version of the fractional step algorithm.

FINITE ELEMENT METHOD FOR GRADIENT PLASTICITY AT LARGE STRAINS

A finite element method for gradient elasto-plastic continuum in which the yield strength of strain hardening/softening materials not only depends on the effective plastic strain but also on its Laplacian is presented. The consistent integration algorithm to update the stress and the internal state variable at integration points and the consistent compliance matrix for the gradient plasticity are formulated in the non-local sense. The methodology to derive the finite element formulation for the gradient plasticity at large strains presented in this paper is applicable to general finite element analysis; the formulation in the context of the two-dimensional four-noded mixed finite element with one integration point and mean von Mises yield criterion is particularly derived. Numerical examples are tested to demonstrate the capability and performance of the present finite element method at large strain in solving for the strain localization problem.

A discrete particle model and numerical modeling of the failure modes of granular materials

Purpose – To present a discrete particle model for granular materials.Design/methodology/approach – Starting with kinematical analysis of relative movements of two typical circular grains with different radii in contact, both the relative rolling and the relative sliding motion measurements at contact, including translational and angular velocities (displacements) are defined. Both the rolling and sliding friction tangential forces, and the rolling friction resistance moment, which are constitutively related to corresponding relative motion measurements defined, are formulated and integrated into the framework of dynamic model of the discrete element method.Findings – Numerical results demonstrate that the importance of rolling friction resistance, including both rolling friction tangential force and rolling friction resistance moment, in correct simulations of physical behavior in particulate systems; and the capability of the proposed model in simulating the different types of failure modes, such as the...

Finite element method and constitutive modelling and computation for unsaturated soils

Abstract This paper presents a numerical model for the finite element analysis of transient deformation and seepage problems in unsaturated soils. A definition of the generalized effective stress for unsaturated soils is introduced and its rational formula is derived on the basis of the classical definition of the effective stress for saturated soils by Terzaghi and the two stress variables formula accepted for unsaturated soils. The governing equations for unsaturated soils modelled as deforming porous media with immiscible porous water and air are re-formulated.. The passive air assumption is then introduced to result in a simplified form of the governing equations for unsaturated soils. The finite element method is applied to discretize the governing equations in space. The Alonso constitutive model to simulate the non-linear mechanical behaviour of unsaturated soils is integrated into the present numerical model. With introduction of the natural coordinate system in the stress space a backward-Euler integration algorithm and a consistent elastoplastic tangent modulus matrix for the non-smooth multisurface pressure dependent plasticity are derived for the model. Numerical results are illustrated to demonstrate the capability and performance of the present model.

Development of elasto-plastic material models in a natural coordinate system

Abstract By the introduction of a new set of coordinates for the normal stresses a framework is developed which facilitates the implementation and modification of a variety of yield surfaces. An internal variable is proposed which allows the simultaneous description of two strain hardening paths, one in tension, the other in compression. Finally, the numerical aspects of the problem are presented.

CONSIDERATIONS FOR THE APPLICATION AND NUMERICAL IMPLEMENTATION OF STRAIN HARDENING WITH THE HOFFMAN YIELD CRITERION

Abstract This paper extends the work of Schellekens and De Borst [ Comput. Struct. 37 , 1087–1096 (1989)] on the Hoffman yield criterion, which defines the anisotropic yield of nonlinear materials, to include a straightforward representation of strain hardening. Various special cases of the Hoffman criterion are considered with regard to the use of internal state variables to quantify the evolution of the yield surface as strain hardening progresses. One of these forms of strain hardening is then adopted and a fully consistent formulation derived. The formulation consists of an implicit return mapping algorithm at the Gauss point level in which a stress state satisfying the yield criterion is found. In addition, a consistent tangent for use in global structural iterations, in which the displacements are sought as the primary unknowns, is derived. Finally, some numerical examples are presented using the general purpose FE package LUSAS in which the algorithm has been implemented.

Implicit characteristic Galerkin method for convection–diffusion equations

This paper presents a characteristic Galerkin finite element method with an implicit algorithm for solving multidimensional, time-dependent convection–diffusion equations. The method is formulated on the basis of the combination of both the precise and the implicit numerical integration procedures aiming to reference particles. The precise integration procedure with a 2N algorithm is taken as a tool to determine the material (Lagrangian) derivative of the convective function in the operator splitting procedure. The stability analysis of the algorithm and numerical results illustrate good performance of the present method in stability and accuracy. Copyright

A mixed element method in gradient plasticity for pressure dependent materials and modelling of strain localization

This paper presents a mixed strain element method in gradient plasticity for pressure dependent materials at large strains and its application to the modelling of strain localization. The two yield strength parameters of strain hardening/softening materials not only depend on the internal state variable but also on its Laplacian. The evaluation of the Laplacian is based on a least square polynomial approximation of the internal state variable around each integration point. The present non-local approach allows to satisfy exactly the consistency condition in a pointwise fashion but in the non-local sense and at each iteration of a load step. To derive the consistent element formulations in the context of mixed finite element with one point quadrature and hourglass control and the non-local pressure dependent elasto-plasticity, a natural coordinate system in the stress space and a new definition of internal state variable are introduced. The capability of the present approach to preserve ellipticity as strain softening behaviour is incorporated into a computational model is demonstrated by the numerical results, which illustrate that pathological mesh dependence has been overcome.

Micromechanically informed constitutive model and anisotropic damage characterization of Cosserat continuum for granular materials

The microstructures of granular materials are represented by Voronoi cells generated with a Voronoi tessellation of discrete particle assembly. A Voronoi cell model including not only the reference particle laid inside the Voronoi cell but also its intermediate neighboring particles is presented to formulate micromechanically based macroscopic constitutive relations and constitutive modular tensors of effective Cosserat continuum. The anisotropy of effective Cosserat continuum due to intrinsic characters and deformation-induced evolutions of microstructure of granular materials of the Voronoi cell is quantitatively demonstrated. The derived micromechanically informed macroscopic constitutive relation of effective Cosserat continuum reveals that the Cauchy stresses are not only constitutively related to the strains but also to the curvatures defined in Cosserat continuum, likewise, the couple stresses are not only constitutively related to the curvatures but also to the strains. The derived modular tensors are verified by comparisons of them with those given for classical isotropic Cosserat continuum and are used to identify the elastic constitutive parameters of isotropic Cosserat continuum. The micromechanically informed macroscopic damage factor tensor to characterize anisotropic material damage of effective Cosserat continuum is formulated with no need specifying macroscopic phenomenological damage criterion and damage evolution law. The principal directions and values of the derived damage factor tensor along with numerical results reveal the microscopic mechanisms of macroscopic damage phenomenon, i.e. loss of contacts, re-orientation of contacts of the reference particle with its intermediate neighboring particles and concomitant volumetirc dilatation of the Voronoi cell.

Contaminant transport with non‐equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

A non-equilibrium sorption—advection—diffusion model to simulate miscible pollutant transport in saturated–unsaturated soils is presented. The governing phenomena modelled in the present simulation are: convection, molecular diffusion, mechanical dispersion, sorption, immobile water effect and degradation, including both physical and chemical non-equilibrium processes. A finite element procedure, based on the characteristic Galerkin method with an implicit algorithm is developed to numerically solve the model equations. The implicit algorithm is formulated by means of a combination of both the precise and the traditional numerical integration procedures. The stability analysis of the algorithm shows that the unconditional stability of the present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results illustrate good performance of the present algorithm in stability and accuracy, and in simulating the effects of all the mentioned phenomena governing the contaminant transport and the concentration distribution. Copyright