On the integration schemes for Drucker–Prager's elastoplastic models based on exponential maps

Abstract

SUMMARYRate plasticity equations for the case of Drucker-Prager's model in small strain regime are considered. By defining an augmented stress vector, the formulations convert the original problem into a quasi-linear differential equation system. Two new exponential mapping schemes for integrating model equations are proposed. In addition, two traditional schemes for solving the dynamical system in an explicit manner are discussed. The two semi-implicit schemes developed pose higher accuracy and better convergency. Error contours are provided for all four methods to display the accuracy of each scheme. In order to compare the results, these contours for the classical one-step backward Euler integration method are also displayed. Accuracy and efficiency along with the rate of convergency of the existing and the proposed methods are examined by numerical examples.

Integration of nonlinear mixed hardening models

“…To examine the integration accuracy of the elastic-perfectly plastic von-Mises model, the error contour plots, or iso-error maps, were first presented by Krieg and Krieg (1977). Subsequently, the iso-error maps were used by Ortiz and Popov (1985), Ortiz and Simo (1986), Simo and Hughes (1998), Artioli et al (2006Artioli et al ( , 2007 and Rezaiee-Pajand and Nasirai (2008). In this study, by considering a plane stress state, three different positions of the stress point on the yield surface are adopted for starting points.…”

“…Subsequently, the iso-error maps were used by Ortiz and Popov (1985), Ortiz and Simo (1986), Simo and Hughes (1998), Artioli et al (2006Artioli et al ( , 2007 and Rezaiee-Pajand and Nasirai (2008). In this study, by considering a plane stress state, three different positions of the stress point on the yield surface are adopted for starting points.…”

“…Also, Artioli et al (2007) presented an integration procedure based on exponential maps by considering the von-Mises plasticity with a linear isotropic and Armstrong-Frederick kinematic hardening. A numerical integration based on exponential maps for the Drucker-Prager's elastoplastic models was presented by Rezaiee-Pajand and Nasirai (2008). Rezaiee-Pajand et al (2010) proposed an exponential-based scheme in integrating the constitutive equations along with multi-component nonlinear kinematic hardening.…”

Exact integration of the von Mises elastoplasticity model with combined linear isotropic-kinematic hardening

“…For example, Wallin et al (2008) have proposed an integration scheme for a nonlinear isotropic hardening von Mises model coupled with damage evolution; whereas, Carosio et al (2000) have employed the technique mentioned above in order to obtain a reference solution for a perfect J 2 viscoplasticity model. Loret and Prevost (1986) and Rezaiee-Pajand and Nasirai (2008) have employed the method for Drucker-Prager's elastoplastic equations.…”

An alternative approach to integrating plasticity relations

“…Solutions can be obtained for J2-Flow theory (Krieg and Krieg, 1977), kinematic hardening (Krieg and Xu, 1997;Auricchio and Beirão da Veiga, 2003;Arioli et al, 2006), the Drucker-Prager model (Rezaijee-Pajand and Nasirai, 2008;Szabó, 2009), and potentially other plasticity models. These methods have demonstrated accuracy advantages over purely numerical integration algorithms and do not suffer the return mapping direction issues, but the range of models which can be integrated analytically or semi-analytically is limited.…”