Implicit integration of elastoplastic constitutive models for frictional materials with highly non-linear hardening functions

Abstract

Constitutive relations in elastoplasticity may be formulated in a variety of ways, and different update algorithms may be employed to solve the resulting equations. Several implicit integration schemes, although some not widely used, have been suggested in the last years. Among them, the closest point projection method (CPPM) has proven to be an effective and robust integration scheme. In order to gain maximum control of the stress projection, a two‐level CPPM iteration scheme is proposed. The hardening variables are fixed during the stress projection onto consequently fixed yield surfaces, and after the stress projection, new values of the hardening variables are calculated defining new yield surfaces. The update of the hardening parameters which, in general, may be highly nonlinear functions, may be determined by a combination of a Picard Iteration (PI) on the hardening variables and an adaptative order inverse interpolation (AOII) on the difference of subsequent iterations of the hardening variables. The integration scheme has been implemented in a general constitutive driver which has been formulated independent of the selected constitutive model and easily linked to finite element codes. A third stress invariant dependent, cone–cap elastoplastic constitutive model, referred to as the MRS–Lade, with a highly non‐linear hardening function has been used to show the applicability of the proposed iteration scheme. Error analyses and accuracy assessment are presented along with some representative test results. © 1997 by John Wiley & Sons, Ltd

Numerical differentiation for non‐trivial consistent tangent matrices: an application to the MRS‐Lade model

“…The MRS-Lade model has been developed at the University of Colorado by Macari, Runesson and Sture [2] and it is a further development of Lade's three-invariant model for cohesionless soils. The model is used to simulate the behaviour of granular materials, such as sand, under both low and high confinement stresses [5,7]. It features 1) a two-surface yield function, comprising a smooth cone surface and a smooth cap surface intersecting in a plane curve (ellipse segment) in the deviatoric plane, 2) hardening and softening variables that depend on dissipated plastic work, and 3) a non-associated flow rule in the meridian plane of the cone region.…”

“…Without these derivatives, it is not possible to compute a full consistent tangent matrix. In the literature there are two techniques to integrate the MRS-Lade model which do not require the computation of the derivatives of the hardening moduli (which are rather more involved to obtain than the derivatives of the stresses): 1) a tangent approach for the stresses and a direct substitution of the internal variable equations [3,6] and 2) a two-level technique with a tangent approach for the stress invariants and a Picard iteration with an adaptive order inverse interpolation for the internal variables [7]. However, in both cases quadratic convergence is never achieved, because these approaches are not based on a consistent linearization of all equations with respect to all unknowns.…”

“…Thus, the improvement of numerical differentiation over other techniques for the MRS-Lade model [6,7] is more clear than for simpler material models: it is the first time that quadratic convergence results are presented for the global and the local problems for this model. As a final introductory remark, it is worth noting that the MRS-Lade model is used here for illustrative purposes.…”

Evaluation of different strategies for the integration of hypoplastic constitutive equations: Application to the CLoE model

“…Thus cJ "1, cJ "cL "cL "0, cL "1, "1/2 and " "0. The relevant coe$cients appearing in Equations (11) and (12) for the RKF-56 method have been evaluated by Dormand and Prince [42], and are given in Reference [39].…”

Numerical study on reinforced concrete beam-column frames in progressive collapse

“…For the integration of constitutive equations, an implicit backward-Euler returnmapping algorithm (Macari et al 1997) is conducted. The suggested algorithm (Cervenka and Papanikolaou 2008) is numerically stable with a fast convergence rate, independent of load step size and does not require differentiation of the failure surface.…”