On energy consistency of large deformation plasticity models, with application to the design of unconditionally stable time integrators

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Abstract

This paper addresses the concept of energy consistency of plasticity models, with the ultimate goal of designing time integrators that are unconditionally stable for large deformation, nonlinear dynamical systems. In the course of the discussion, some attention is devoted to a widely used plasticity model, which we assert has a fundamental thermodynamical problem if its flow rule is not carefully defined.

Section snippets

Introduction and motivation

It is widely recognized that unconditionally stable algorithms for linear problems are potentially unstable when applied to nonlinear structural and solid mechanics. This has motivated a significant amount of research, beginning in the late 1980s, to develop more robust time integration algorithms for nonlinear systems, with a particular interest in achieving nonlinear stability while maintaining system conservation properties in a discrete sense. Although most of the work focused on nonlinear

Finite deformation plasticity and the dissipation inequality

The free energy in a multiplicative elastoplastic model can be written asEint=ψ(Ce),Ce=FeTFe,FFeFp,where ψ takes the form of a classical hyperelastic stored energy function. It follows then that the time evolution of ψ may be written asψ̇=12SeĊe=12SĊDint,where Dint denotes plastic dissipation, which takes forms as summarized in Table 1 for a few commonly employed finite plasticity models from the literature.

Satisfaction of the thermodynamical requirement Dint⩾0 can be assured by assuming a

A critical review of a multiplicative plasticity model

As can be seen in Table 1, the plastic dissipation is usually written as the double contraction between a stress tensor and a energy conjugate plastic rate of deformation, while the corresponding flow rule describes the evolution of the plastic rate. Departure from this observation (such as is seen for the second flow rule in model 3, which describes a flow rule for a plastic rate not energy conjugate to a stress tensor) may lead to an energy inconsistent plasticity model. In this section we

Design of energy consistent algorithms

As discussed in Section 1, an algorithmic second Piola–Kirchhoff stress tensor needs to be constructed, which satisfies the discrete dissipation inequality (Eq. (4)). Given the fact that the algorithmic dissipation takes the formΔDintYΔγ,an algorithmic second Piola–Kirchhoff stress tensor is proposed asSalgoSn+1/2+2(ψn+1−ψnYΔγ)−Sn+1/2:ΔCΔC:ΔCΔC,where ΔC=Cn+1Cn, Sn+1/2 is the second Piola–Kirchhoff stress tensor evaluated at the mid point, Δγ is the incremental equivalent plastic strain,

Numerical simulations

In this section, two numerical examples are presented to illustrate the importance of energy consistency in algorithmic design.

Conclusion

Unconditional stability for nonlinear dynamical plasticity can be achieved by applying energy consistent algorithms. The basic concepts underlying such algorithms are energy consistent plasticity models and accompanying energy–momentum time discretization schemes. Failure to pay proper attention to either concept may cause instabilities even in the presence of physical dissipation, as illustrated by the numerical examples presented.

Acknowledgements

The authors’ work on this project was supported by the Office of Naval Research under the Young Investigator Program, grant no N00014-97-1-0529, and by the National Science Foundation CAREER program, award CMS-9703356. This support is gratefully acknowledged.

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