Ronaldo I. Borja

Cam-Clay plasticity, part I: implicit integration of elastoplastic constitutive relations

Abstract Two stress integration algorithms based on implicit calculation of plastic strain are implemented and tested for the modified Cam-Clay plasticity model. The integration rules fall under the category of return mapping algorithms in which the return directions are computed by closest point projection for associative flow rule and by central return mapping for non-associative flow rule applied to the Cam-Clay ellipsoids. Stress updates take place at the Gauss points upon enforcement of the consistency condition in which the appropriate consistency parameters are determined iteratively on the scalar level. Numerical examples with geotechnical applications, which include an analysis of foundation bearing capacity and an investigation of deformations in vertical cuts, are discussed to demonstrate the global accuracy and stability of the numerical solution. The relationships among various return mapping schemes are discussed in the context of both associative and non-associative flow rule formulations.

A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation

This paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard Galerkin approximation. Strain enhancements via jumps in the displacement field are captured and condensed on the material level, leading to a formulation that does not require static condensation to be performed on the element level. The mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band, which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band. Precise conditions for the appearance of slip lines, including their initiation and evolution, are outlined for a rateindependent, strain-softening Drucker‐Prager model, and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting. At post-localization the stress-point integration algorithm along the band is exact and amenable to consistent linearization. Numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow, as well as plane strain compression of dilatant cohesive/frictional materials, are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions. ” 2000 Elsevier Science S.A. All rights reserved.

Cam-Clay plasticity part III: Extension of the infinitesimal model to include finite strains

The infinitesimal version of the modified Cam-Clay model of critical state soil mechanics is reformulated to include finite deformation effects. Central to the formulation are the choice of a hardening law that is appropriate for cases involving large plastic volumetric strains, and the use of a class of two-invariant stored energy functions appropriate for Cam-Clay-type models that includes, as special cases, the constant elastic shear modulus approximation as well as the pressure-dependent shear modulus elastic model. The analytical model is cast within the framework of finite deformation theory based on a multiplicative decomposition of the deformation gradient. For the fully elasto-plastic case, return mapping is done implicitly in the space defined by the invariants of the elastic logarithmic principal stretches, which requires the solution at each stress point of no more than three simultaneous non-linear equations. A finite element analysis of a strip footing problem is presented to illustrate a prototype example where finite deformation effects significantly impact the predicted response.

On the numerical integration of three-invariant elastoplastic constitutive models

We investigate the performance of a numerical algorithm for the integration of isotropically hardening three-invariant elastoplastic constitutive models with convex yield surfaces. The algorithm is based on a spectral representation of stresses and strains for infinitesimal and finite deformation plasticity, and a return mapping in principal stress directions. Smooth three-invariant representations of the Mohr–Coulombmodel, such as the Lade–Duncan and Matsuoka–Nakai models, are implemented within the framework of the proposed algorithm. Among the specific features incorporated into the formulation are the hardening/softening responses and the tapering of the yield surfaces toward the hydrostatic axis with increasing confining pressure. Isoerror maps are generated to study the local accuracy of the numerical integration algorithm. Finally, a boundary-value problem involving loading of a strip foundation on a soil is analyzed with and without finite deformation effects to investigate the performance of the integration algorithm in a full-scale non-linear finite element simulation. 2002 Elsevier Science B.V. All rights reserved.

Strain localization in frictional materials exhibiting displacement jumps

Abstract This paper presents a mathematical model for analyzing strain localization in frictional solids exhibiting displacement jumps. Precise conditions for the appearance of slip lines, including their initiation and evolution, are outlined for a rate-independent, strain-softening Drucker–Prager model, and explicit analytical expressions are used to describe the orientation of the slip line. A stress–displacement relation obtained through the consistency condition is also formulated to describe the quasi-static response in the post-localization regime. The mathematical model, which is cast within the framework of finite element analysis employing the assumed enhanced strain method, circumvents mesh-dependence issues often associated with rate-independent plasticity models. It is shown that the enhancement equation is nothing else but the consistency condition imposed on the band. Numerical examples involving plane strain compression are described to demonstrate objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions.

A mathematical framework for finite strain elastoplastic consolidation. Part 1: Balance laws, variational formulation, and linearization

A mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented. Strong and weak forms of the boundary-value problem are derived using both the material and spatial descriptions. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. Balance laws are written for the soil-water mixture following the motion of the soil matrix alone. It is shown that the motion of the fluid phase only affects the Jacobian of the solid phase motion, and therefore can be characterized completely by the motion of the soil matrix. Furthermore, it is shown from energy balance consideration that the effective, or intergranular, stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil-water mixture. Finally, it is shown that the mathematical model is amenable to consistent linearization, and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method.

Plane strain finite element analysis of pressure sensitive plasticity with strong discontinuity

Numerical simulations of localized deformation in solids should capture the structural phenomenon of localization and associated loss of material body strength in a manner independent of spatial discretization. Many regularization techniques have been proposed to address the ill posedness associated with rate-independent softening plasticity that leads to mesh-dependent numerical simulations. One approach to alleviating mesh dependence is the strong discontinuity approach, which represents localized deformation as a slip surface within a plasticity model; a strong discontinuity is a discontinuous displacement field. This approach is used in this paper to formulate a plane strain, pressure sensitive, nonassociative plasticity model with strong discontinuity and to implement the model, along with an enhanced bilinear quadrilateral element, via an assumed enhanced strain method. Numerical examples demonstrate mesh independence of the method for pressure sensitive materials.

Bifurcation of elastoplastic solids to shear band mode at finite strain

This paper formulates and implements a finite deformation theory of bifurcation of elastoplastic solids to planar bands within the framework of multiplicative plasticity. Conditions for the onset of strain localization are based on the requirement of continuity of the nominal traction vector, and are described both in the reference and deformed configurations by the vanishing of the determinant of either the Lagrangian or Eulerian acoustic tensor. The relevant acoustic tensors are derived in closed form to examine the localization properties of a class of elastoplastic constitutive models with smooth yield surfaces appropriate for pressure-sensitive dilatant/frictional materials. A link between the development of regularized strong discontinuity and its unregularized counterpart at the onset of localization is also discussed. The model is implemented numerically to study shear band mode bifurcation of dilatant frictional materials in plane strain compression. Results of the analysis show that finite deformation effects do enhance strain localization, and that with geometric nonlinearities bifurcation to shear band mode is possible even in the hardening regime of an associative elastoplastic constitutive model.

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach

Abstract A finite element model of localized deformation in frictional materials taking a strong discontinuity approach is presented. A rate-independent, non-associated, strain-softening Drucker–Prager plasticity model is formulated in the context of strong discontinuities and implemented along with an enhanced quadrilateral element within the framework of an assumed enhanced strain finite element method. For simple model problems such as uniform compression, the strong discontinuity approach has been shown to lead to mesh-independent finite element solutions when localized deformation is present. In this paper, a finite element analysis of localized deformation occurring in a more complex model problem of slope stability is conducted in a nearly mesh-independent manner. The effect of dilatancy on the orientation of slip lines is demonstrated for the slope stability problem.

One the solution of elliptic free-boundary problems via Newton's method

Abstract The performance of a numerical technique for locating the free surface without mesh iteration is investigated in the context of steady state pore fluid diffusion through porous and deformable media. The free boundary is located from a fixed mesh solution in which the required constraints are imposed by penalization of the variational equation. Central to the efficiency of the algorithm is the use of a ‘relaxed penalty function’ which makes it possible to linearize the penalized equation consistently. The solution is shown to converge orders of magnitude faster than some of the most widely used algorithms for solving free-boundary problems associated with flow of fluids through porous media.