Symmetry‐preserving return mapping algorithms and incrementally extremal paths: A unification of concepts

Abstract

in this work we seek to characterize the conditions under which an elastic-plastic stress update algorithm preserves the symmetries inherent to the material response. From a numerical standpoint, the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality. For the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving. For hardcning plasticity, symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations. We show that these restrictions amount to the existence of a pseudo-internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli. In view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and, hence, in symmetric tangents, we address the question of whether any connections exist between the two approaches. We show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence.

Capturing strain localization in dense sands with random density

“…and therefore, by pushingà forward with Equation (79) and using the fact that˜ 2 aS a = a we obtaiñ c = We observe from Equation (81) that the CTO possesses minor symmetries but lacks major symmetry ifã ep ab =ã ep ba , which is certainly the case for nonassociative plasticity or when the integration algorithm destroys the symmetry of the tangent operator [12,37].…”

Numerical differentiation for local and global tangent operators in computational plasticity

“…For convenience, the terms generalized stress and generalized flow vector are used to refer to (σ σ σ, q) and (m σ σ σ , m q ) respectively [15,23]. For some constitutive models, the generalized flow vector is expressed as the derivative of a generalized flow potential G(σ σ σ, q) [11,20]: m σ σ σ = ∂G/∂σ σ σ and m q = ∂G/∂q.…”

“…The upper-left block-matrix of the inverse of J contains the consistent tangent moduli. In the literature, compact expressions (after inverting J and taking the upper-left block-matrix) of these moduli can be found for particular models [3,15,23]. However, the more general expression given in equation (7) is preferred here because it highlights an important fact in the context of this work: the derivatives of the generalized flow vector needed to solve the local problem are also required in the computation of the consistent tangent matrix.…”

Microstructure-Based Computational Modeling of Mechanical Behavior of Polymer Micro/Nano Composites

“…However, if the rotation is lumped to the plastic portion, the solution needs more mathematical manipulation, which is fully discussed in Section 3.5. It is also crucial to recall that since the flow rule in plasticity is defined in the intermediate configuration (plastic or unloaded configuration) the frame indifference concept has been consistently satisfied (Ortiz and Martin, 1989).…”