A novel ‘optimal’ exponential‐based integration algorithm for von‐Mises plasticity with linear hardening: Theoretical analysis on yield consistency, accuracy, convergence and numerical investigations

Abstract

SUMMARYIn this communication we propose a new exponential-based integration algorithm for associative von-Mises plasticity with linear isotropic and kinematic hardening, which follows the ones presented by the authors in previous papers. In the first part of the work we develop a theoretical analysis on the numerical properties of the developed exponential-based schemes and, in particular, we address the yield consistency, exactness under proportional loading, accuracy and stability of the methods. In the second part of the contribution, we show a detailed numerical comparison between the new exponential-based method and two classical radial return map methods, based on backward Euler and midpoint integration rules, respectively. The developed tests include pointwise stress-strain loading histories, iso-error maps and global boundary value problems. The theoretical and numerical results reveal the optimal properties of the proposed scheme.

An alternative approach to integrating plasticity relations

“…(6) and from subsequent equations, but they are retained as an explicit reminder that the expression are intended to be applied over time increments and are not alternative rate equations. The exponentials are reminiscent of those in analytic solutions for J2-Flow theory (e.g., Arioli et al, 2006) and the Drucker-Pragar model (e.g., Szabó, 2009), but the anisotropy renders these fourth order tensors rather than scalars. A quick check can be made for small and large strain increments to verify that Eq.…”

“…(5b) in closed form for one time step, Dt, given the average rate of deformation tensor over the time step. The anisotropy creates a rate form that is not amenable to the analytic techniques cited above (e.g., Arioli et al, 2006;Szabó, 2009). The equations are instead integrated by summation over infinitesimal parts of the time step.…”

“…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.…”

Integration of nonlinear mixed hardening models

“…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.…”

“…Auricchio and Beirão da Veiga (2003) proposed the exponential maps for solving the differential equation system in the augmented stress space and presented a first-order integration algorithm for the constitutive model with a linear mixed hardening mechanism. Subsequently, this technique was extended to a second-order accuracy scheme (Artioli et al, 2006;Rezaiee-Pajand and Nasirai, 2007). 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.…”