Numerical integration of elasto-plastic constitutive models using the extrapolation method (original) (raw)
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Computers and Geotechnics, 37(1-2): 68-81, 2010
The paper compares the accuracy and efficiency of explicit stress integration schemes for elasto-plastic unsaturated soil models with automatic error control. Numerical tests are performed with reference to the Barcelona Basic Model (BBM), one of the most popular elasto-plastic models for unsaturated soils, by using eight explicit Runge–Kutta algorithms of various order as well as a novel application of the extrapolation method described in the companion paper. Initially, the results obtained from the lowest order Runge–Kutta scheme (i.e. Modified Euler) as well as the extrapolation method are checked against accurate solutions of a number of BBM paths involving changes of strains and suction. Subsequently, the efficiency and accuracy of all algorithms are assessed for generic increments of strains and suction, while the difference between two alternative error control methods is also analysed. The results presented, although strictly valid for the Barcelona Basic Model, are expected to be general and relevant to other similar unsaturated elasto-plastic models formulated in terms of two independent stress variables such as net stress and suction.
Computers and Geotechnics, 37(1-2): 59-67, 2010
The numerical integration of the stress–strain relationship is an important part of many finite element code used in geotechnical engineering. The integration of elasto-plastic models for unsaturated soils poses additional challenges associated to the presence of suction as an extra constitutive variable with respect to traditional saturated soil models. In this contribution, a range of explicit stress integration schemes are derived with specific reference to the Barcelona Basic Model (BBM), which is one of the best known elasto-plastic constitutive models for unsaturated soils. These schemes, however, do not address possible non-convexity of the loading collapse (LC) curve and neglect yielding on the suction increase (SI) line. The paper describes eight Runge–Kutta methods of various orders with adaptive substepping as well as a novel integration scheme based on Richardson extrapolation. The algorithms presented also incorporate two alternative error control methods to ensure accuracy of the numerical integration. Extensive validation and comparison of different schemes are presented in a companion paper. Although the algorithms presented were coded for the Barcelona Basic Model, they can be easily adapted to other unsaturated elasto-plastic models formulated in terms of two independent stress variables such as net stress and suction.
Implicit numerical integration for a kinematic hardening soil plasticity model
International Journal for Numerical and Analytical Methods in Geomechanics, 2001
Soil models based on kinematic hardening together with elements of bounding surface plasticity, provide a means of introducing some memory of recent history and sti!ness variation in the predicted response of soils. Such models provide an improvement on simple elasto-plastic models in describing soil behaviour under non-monotonic loading. Routine use of such models requires robust numerical integration schemes. Explicit integration of highly non-linear models requires extremely small steps in order to guarantee convergence. Here, a fully implicit scheme is presented for a simple kinematic hardening extension of the Cam clay soil model. The algorithm is based on the operator split methodology and the implicit Euler backward integration scheme is proposed to integrate the rate form of the constitutive relations. This algorithm maintains a quadratic rate of asymptotic convergence when used with a Newton}Raphson iterative procedure. Various strain-driven axisymmetric triaxial paths are simulated in order to demonstrate the e$ciency and good performance of the proposed algorithm.
An interior - point algorithm for hyper-plastic models for soils
In numerical analysis of geotechnical problems it is often necessary to use sophisticated elastoplastic constitutive models. Particularly, the presence of friction as the fundamental mechanism to dissipate energy provides most of the models used in this field with a non-associated character. This characteristic introduces theoretical and numerical complexity at time of deriving the models from thermo-mechanical principles and implementing them. The paper presents the formulation and implementation for a family of non-associated models based on the hyper-plasticity approach. The paper starts with the hyper-plastic framework, that proves to be a powerful technique to derive evolution equations using standard thermo-mechanical procedures based on the differentiation of the energy expressions (free Helmholtz energy fs, Gibbs energy gs, and dissipation d), even for non-associated laws. The characterization of the variational structure behind the discrete equations of the Interior-Point p...
Normally, suction-controlled triaxial tests are used to characterize soil behavior in constitutive modeling of unsaturated soils. However, this type of tests requires sophisticated equipment and is time-consuming. This has been one of the major obstacles to the implementation and dissemination of unsaturated soil mechanics beyond the research context. In contrast to suction-controlled triaxial tests, the suction-controlled oedometer test requires simpler equipment and a shorter testing period. Oedometer tests represent the at-rest earth pressure (K0) condition, which is an important stress state in any simulation. The major disadvantage of the oedometer test is that its lateral stress is controlled by the condition of zero lateral strain and remains unknown during the testing process. At present, no well-established, simple, and objective methods are available that take advantage of oedometer test results for constitutive modeling purposes. This paper derives an explicit formulation of the at-rest coefficient for unsaturated soils and develops an optimization approach for simple and objective identification of material parameters in elasto-plastic models for unsaturated soils using the results from suction-controlled oedometer tests. This is achieved by combining a modified state surface approach (MSSA), recently proposed to model the elasto-plastic behavior of unsaturated soils, with the quasi-Newton method to simultaneously calibrate all parameters governing virgin behavior in elasto-plastic models. The Barcelona Basic Model (BBM) is used to demonstrate the application of the proposed explicit formulation and calibration method. Results predicted using obtained parameters are compared with laboratory test results for the same stress paths in order to evaluate the simplicity and objectivity of the proposed method.
An Implicit Numerical Integration Algorithm for Bai & Wierzbicki (2007) Elasto-Plastic Model
This contribution describes an implicit algorithm for numerical integration of a recently proposed model for metal plasticity and fracture [2]. The constitutive equations of the material model critically include both the effect of pressure through the triaxiality ratio and the effect of third deviatoric stress invariant through the lode angle in the description of material. These effects are directly introduced on the hardening rule of the material. The theoretical basis of the material model is presented in the first part of the paper. Then, the necessary steps required to implement the model within an implicit quasi-static finite element environment are discussed. In particular, the stress update procedure, which is based on the so-called operator split concept resulting in the standard elastic predictor/return mapping algorithm, and the computation of tangent matrix consistent with the stress update are described. Finally, the simulation of a flat grooved specimen subjected to tension [1] is presented to illustrate the robustness and efficiency of the proposed algorithm.
Calibration of elasto-plastic models for unsaturated soils under isotropic stresses
Engineering Geology, 165: 64-72, 2013
This paper presents a new method for selecting parameter values in elasto-plastic models for unsaturated soils under isotropic stress states. The proposed method is an improvement over more traditional calibration approaches as it explicitly takes into account experimental yielding behaviour when defining the unsaturated normal compression surface. This is achieved by “enriching” measured yield stresses with corresponding values of specific volume (estimated from the observed elastic and virgin responses), which generates experimental points that can be interpolated together with virgin compression data. The proposed methodology is general and independent of the chosen mathematical forms of normal compression surface, elastic law and constitutive variables. The validity of the method is demonstrated by calibrating a reference model from a set of constant suction isotropic compression tests. Both improved and traditional calibration approaches are employed to select model parameter values which are then used to predict yield curves in the mean net stress–suction plane. Because of the incorporation of yielding data in the interpolation of the normal compression surface, the improved calibration method produces a more accurate prediction of yield stresses at different suctions compared with the standard method. Due to its generality and limited subjectivity, the proposed method provides an unbiased tool to compare strengths and weaknesses of different constitutive models.