Stress-point Algorithm Research Articles

Comparison of implicit and explicit numerical integration schemes for a bounding surface soil model without elastic range

Standard integration schemes for rate constitutive equations were designed within the classical theory of plasticity. Consequently, they rely on the assumption that a yield criterion defines a range of purely elastic material behaviour. Many constitutive models for non-cohesive soils discard that assumption. They account for inelastic deformations without employing a yield criterion. This simplifies the formulation of the models but raises questions concerning their numerical implementation. In particular, it is not fully clear which algorithmic method is most appropriate for the numerical integration of constitutive models without elastic range. To investigate this, two different stress point algorithms for a critical state bounding surface model for sands were developed and implemented. The explicit method employs substepping to automatically control the local error. The implicit method updates stresses and state variables through a local Newton iteration, the Jacobian of which is computed by numerical differentiation. The two algorithms were compared by means of calculations at integration point level as well as with respect to a boundary value problem. The results show that for a given level of accuracy, the explicit update procedure is significantly more efficient than the implicit one. This holds regardless of initial state parameters and input strain increment magnitudes.

Finite deformation hyperplasticity theory for crushable, cemented granular materials

The work is focused on the formulation of a thermodynamically–based constitutive theory for granular, cemented geomaterials, often characterized by a open structure with high porosity and voids of large diameter. Upon mechanical degradation processes such as bond rupture and grain crushing, these material undergo large volumetric and shear strains, and in some cases the deformations are so large that the usual assumption of linearized kinematics may be not applicable. In the first part of this work, the theory of hyperplasticity is extended to the finite deformation regime by adopting a multiplicative split of the deformation gradient into an elastic and a plastic part, under the assumption of material isotropy. Grain breakage and bond damage processes are accounted for through two micromechanically–inspired internal variables. A specific constitutive model for carbonatic cemented sands and calcarenites is proposed as a relevant example of application. In the second part, an implicit stress–point algorithm has been developed which is amenable to closed form linearization, for the implementation of the model into standard FE platforms. A series of numerical tests have demonstrated the accuracy and efficiency of the proposed algorithm. The simulation of plane strain biaxial tests, modeled as boundary–value problems, has highlighted the role played by geometric non–linearity in determining the evolution of the specimen deformation upon reaching a bifurcation condition.

On the computational applicability of the modified Cam-clay model on the ‘dry’ side

This paper presents a theoretical investigation into the behaviour of the modified Cam-clay model on the ‘dry’ side and shows that: (a) for arbitrary strain-controlled loadings, local uniqueness is violated when the stress ratio η exceeds a certain critical value ηd; (b) for triaxial loadings, local uniqueness is violated when η exceeds another critical value, lower or equal to ηd, and the predicted response to a strain-controlled process becomes plastically inadmissible, while the response to a stress-controlled process presents a ‘snap-back’; (c) stress-point algorithms and displacement-based numerical analysis codes can provide nonsensical results in such cases, which violate plasticity theory.

Three-dimensional concrete impact and penetration simulations using the smoothed particle Galerkin method

In this paper, we model the three-dimensional concrete impact and penetration problems using a stabilized meshfree method. The present method is established using a non-residual penalty term from strain smoothing as a means of stabilizing the meshfree nodal integration method under the Galerkin framework. As a result, the meshfree discretization leads to a dual stress point algorithm with the stabilization parameterized by a measure of a local length scale. An adaptive anisotropic Lagrangian kernel is considered in junction with the stabilized meshfree formulation for the severe deformation analysis. In order to avoid the spurious damage growth and material self-healing in concrete failure analysis, a bond-based failure criterion is introduced. A frictionless self-contact algorithm is also developed to model the interaction between concrete debris in damage. Several impact examples are investigated including the study of scabbing and perforation of concrete under high velocity impact. The numerical results are compared with the experimental data to demonstrate the effectiveness of the present method.

Plasticity with generalized hardening: constitutive modeling and computational aspects

In this work, an extended theory of plasticity with generalized hardening is proposed to describe the response of geomaterials under both mechanical and environmental processes, which include as special cases several elastoplastic constitutive equations proposed in the literature to model such processes as desaturation or suction hardening, thermal softening, chemo-mechanical coupling effects in fine-grained soils, as well as weathering of soft rocks. In the formulation of the theory, the coupling between mechanical and environmental processes takes place at two levels: first, as an additional direct contribution to the constitutive stress changes, taking place in both elastic and elastoplastic processes; and second, as a result of the evolution of the internal state variables induced by changes in the environmental process variables. This last effect is incorporated through a set of generalized hardening rules. As an example of application, the general formulation is specialized to the particular case of weak calcarenite rocks undergoing degradation processes due to plastic deformations, changes in degree of saturation (short-term debonding) and chemical dissolution of the bond material and the solid grains (long-term debonding). The resulting model is implemented in a FE code by means of an implicit generalized backward Euler algorithm, suitably modified to incorporate the full formalism of plasticity with generalized hardening. Results of numerical simulations carried out at the element level show the accuracy and efficiency properties of the proposed stress-point algorithm. The simulation of a representative initial-boundary value problem demonstrates the practical relevance of environmental degradation effects in practical applications, over periods of time comparable with the life cycle of most geotechnical structures.

Implicit integration under mixed controls of a breakage model for unsaturated crushable soils

This paper discusses a series of stress point algorithms for a breakage model for unsaturated granular soils. Such model is characterized by highly nonlinear coupling terms introduced by breakage-dependent hydro-mechanical energy potentials. To integrate accurately and efficiently its constitutive equations, specific algorithms have been formulated using a backward Euler scheme. In particular, because implementation and verification of unsaturated soil models often require the use of mixed controls, the incorporation of various hydro-mechanical conditions has been tackled. First, it is shown that the degree of saturation can be replaced with suction in the constitutive equations through a partial Legendre transformation of the energy potentials, thus changing the thermomechanical state variables and enabling a straightforward implementation of a different control mode. Then, to accommodate more complex control scenarios without redefining the energy potentials, a hybrid strategy has been used, combining the return mapping scheme with linearized constraints. It is shown that this linearization strategy guarantees similar levels of accuracy compared with a conventional strain–suction-controlled implicit integration. In addition, it is shown that the use of linearized constraints offers the possibility to use the same framework to integrate a variety of control conditions (e.g., net stress and/or water-content control). The convergence profiles indicate that both schemes preserve the advantages of implicit integration, that is, asymptotic quadratic convergence and unconditional stability. Finally, the performance of the two implicit schemes has been compared with that of an explicit algorithm with automatic sub-stepping and error control, showing that for the selected breakage model, implicit integration leads to a significant reduction of the computational cost. Such features support the use of the proposed hybrid scheme also in other modeling contexts, especially when strongly nonlinear models have to be implemented and/or validated by using non-standard hydro-mechanical control conditions. Copyright © 2015 John Wiley & Sons, Ltd.

Evaluation of numerical stress-point algorithms on elastic–plastic models for unsaturated soils with hardening dependent on the degree of saturation

Constitutive models of unsaturated soils, and in particular those based on constitutive variables which include both degree of saturation and suction, are characterised by strong non linearities due to hydromechanical coupling. In this paper, a refined Runge–Kutta–Dormand–Prince explicit algorithm and a fully implicit Euler scheme are compared for the integration of the latter class of models. The explicit and implicit procedures have been tested along different hydromechanical paths, involving various hydraulic and mechanical external control conditions. Accuracy and efficiency of the algorithms have been investigated. The results confirm that substepping is mandatory for the explicit algorithm to converge regardless the initial step size and to remain sufficiently accurate. The value of the incremental hydromechanical work per unit volume was calculated during the explicit integration procedure. The numerical results show that the maximum size of the substep which can be adopted to meet a given tolerance depends on the gradient of the incremental work per unit volume. Therefore, the latter appears a good candidate to identify problematic integration steps in terms of convergence. Accuracy of the implicit algorithm also depends on the chosen step size, although the algorithm proved to be convergent in all the paths analysed.

A fully convex reformulation of the original Matsuoka–Nakai failure criterion and its implicit numerically efficient integration algorithm

SUMMARYThis paper presents a reformulation of the original Matsuoka–Nakai criterion for overcoming the limitations which make its use in a stress point algorithm problematic. In fact, its graphical representation in the principal stress space is not convex as it comprises more branches, plotting also in negative octants, and it does not increase monotonically as the distance of the stress point from the failure surface rises. The proposed mathematical reformulation plots as a single, convex surface, which entirely lies in the positive octant of the stress space and evaluates to a quantity which monotonically increases as the stress point moves away from the failure surface. It is an exact reproduction, and not an approximated one, of the only significant branch of the original criterion. It is also suitable for shaping in the deviatoric plane the yield and plastic potential surfaces of complex constitutive models. A very efficient numerical algorithm for the implicit integration of the proposed formulation is presented, which enables the evaluation of the stress at the end of each increment by solving a single scalar equation, both for associated and non‐associated plasticity. The algorithm can be easily adapted for other smooth surfaces with linear meridian section. Finally, a close expression of the consistent Jacobian matrix is given for achieving quadratic convergence in the external structural newton loop. It is shown that all this results in extremely fast solutions of boundary value problems. Copyright © 2013 John Wiley & Sons, Ltd.

A finite element solution scheme for an elastic–viscoplastic soil model

A Newton–Raphson solution scheme with a stress point algorithm is presented for the implementation of an elastic–viscoplastic soil model in a finite element program. Viscoplastic strain rates are calculated using the stress and volumetric states of the soil. Sub-increments of time are defined for each iterative calculation of elastic–viscoplastic stress changes so that their sum adds up to the time increment for the load step. This carefully defined ‘iterative time’ ensures that the correct amount of viscoplastic straining is accumulated over the applied load step. The algorithms and assumptions required to implement the solution scheme are provided. Verification of the solution scheme is achieved by using it to analyze typical boundary value problems.

A stress point algorithm for an elastoplastic model in unsaturated soils

Abstract Two stress fields, combination of total stresses, liquid pressure and gas pressure have to be considered to explain the deformational behaviour of unsaturated media. Elastoplastic models developed for these materials consider generally two yield surfaces, each one associated to a stress field, and whose intersection produces a corner in the space of generalised stress components. In this paper, a stress point algorithm is proposed to cope with the integration of such constitutive laws, which can be seen as non smooth multisurface plastic models in the space of the two stress fields. The basic model developed by Alonso et al. (Alonso, E.E., Gens, A., 1990. A constitutive model for partially saturated soils. Geotechnique 40 (3), 405–430), which will be used to test the algorithm, is first described. Generalised stress and strain variables are then defined. Implementation of the return mapping algorithm, based on an implicit integration scheme, is presented with special attention devoted to the problem of mixed control imposed by the F.E. formulation generally used to analyse the hydromechanical behaviour of unsaturated media. Validation results on distinct generalised stress paths are given at the end.

An evaluation of substepping and implicit stress point algorithms : D. M. Potts & D. Ganendra, Computer Methods in Applied Mechanics & Engineering, 119(3–4), 1994, pp 341–354

An evaluation of substepping and implicit stress point algorithms : D. M. Potts & D. Ganendra, Computer Methods in Applied Mechanics & Engineering, 119(3–4), 1994, pp 341–354

An evaluation of substepping and implicit stress point algorithms

Elasto-plastic finite element analyses of geotechnical boundary value problems involve the use of a non-linear solution strategy. A popular and efficient strategy is the modified Newton Raphson approach of which a key element is the integration of the constitutive laws over a finite strain step. Two algorithms for performing such an integration are the ‘substepping’ and ‘return’ methods. In this paper the basic assumptions behind these methods are described and the associated errors compared. This comparative study is undertaken both at a fundamental level and when applied to a typical boundary value problem. The errors associated with both methods are increment size dependent. It is shown in the problems considered that for a particular increment size, the ‘substepping’ method is the more accurate.

Computational aspects of large elastoplastic deformations in the presence of anisotropy and plastic spin

A constitutive framework for large elastoplastic deformations is briefly presented, along the lines of Mandel's theory, in a form most appropriate for computational convenience. A general purpose stress-point algorithm is adapted to the present large deformation framework. An accuracy analysis is carried out and the performances of several variants are compared, particularly those related to the anisotropic intensity and plastic spin in reference to two specific anisotropic models. The effect of elastic deformations on the overall response is demonstrated for special loading paths.

A note on the accuracy of stress-point algorithms for anisotropic elastic-plastic solids

t is argued that, in order to be meaningful, an error analysis of stress-point algorithms for anisotropic elastic-plastic materials cannot be content to consider only those stress-increments that satisfy the rate plastic loading condition; it must also consider stress-increments that traverse the yield surface because they are observed to lead to the maximum errors, a point that has not been remarked so far. This requirement is illustrated for the Euler-backward and cutting plane algorithms applied to the orthotropic Hill's yield surface. In addition, the closed-form solution used in the error analysis is provided.

Dynamic strain localization in elasto-(visco-)plastic solids, part 2. plane strain examples

Viscoplasticity is introduced as a procedure to regularize the elastic-plastic solid, especially for those situations in which the underlying inviscid material exhibits instabilities which preclude meaningful analysis of the initial-value problem. The procedure is general and therefore has the advantage of allowing the regularization of any inviscid elastic-plastic material. Rate dependence is shown to naturally introduce a length-scale that sets the width of the shear bands in which the deformations localize and high strain gradients prevail. Then, provided that the element size is appropriate for an adequate description of the shear band geometry, the numerical solutions are shown to be pertinent. Stable and convergent solutions with mesh refinements are obtained which are shown to be devoid of spurious mesh length-scale effects. The numerical framework adopted for this study is realistic and relevant to the solution of large scale nonlinear problems. An efficient explicit time stepping algorithm is used to advance the solution in time, and low-order finite elements with only one stress-point are used. An unconditionally stable stress-point algorithm is used to integrate the nonlinear elasto-(visco-) plastic constitutive equations. Therefore, the only numerical restriction of the proposed computational procedure stems from a time step size restriction which emanates from the explicit time integration of the equations of motion. However, since the wave speeds remain elastic, this restriction is trivially dealt with, resulting in a most efficient computational procedure.

Efficient elastic-plastic finite element analysis with higher order stress-point algorithms

The treatment of elastic-plastic problems with finite elements depends essentially on the methods used for state determination and the solution of the nonlinear equations. A systematic formulation of the state determination leads to higher order algorithms, which can better satisfy demands on accuracy and computational costs. The state determination influences highly the solution method for the system of nonlinear equations. In a comparison between Newton and a quasi-Newton method it is shown, that quasi-Newton methods are more suited for an efficient computation in combination with accurate path independent state determination algorithms.