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Research Articles

Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part II: Finite Strain Elastoplasticity

[+] Author and Article Information
Amin Eshraghi

Research Associate
e-mail: maeshrag@uwaterloo.ca

Hamid Jahed

Professor
e-mail: hjahedmo@uwaterloo.ca
Department of Mechanical
and Mechatronics Engineering,
University of Waterloo,
Waterloo, ON, N2L 3G1, Canada

Katerina D. Papoulia

Associate Professor
Department of Applied Mathematics,
University of Waterloo,
Waterloo, ON, N2L 3G1, Canada
e-mail: papoulia@uwaterloo.ca

1Corresponding author.

Manuscript received July 15, 2012; final manuscript received September 8, 2012; accepted manuscript posted September 29, 2012; published online January 30, 2013. Assoc. Editor: Krishna Garikipati.

J. Appl. Mech 80(2), 021028 (Jan 30, 2013) (11 pages) Paper No: JAM-12-1326; doi: 10.1115/1.4007724 History: Received July 15, 2012; Revised September 08, 2012; Accepted September 29, 2012

An Eulerian rate formulation of finite strain elastoplasticity is developed based on a fully integrable rate form of hyperelasticity proposed in Part I of this work. A flow rule is proposed in the Eulerian framework, based on the principle of maximum plastic dissipation in six-dimensional stress space for the case of J2 isotropic plasticity. The proposed flow rule bypasses the need for additional evolution laws and/or simplifying assumptions for the skew-symmetric part of the plastic velocity gradient, known as the material plastic spin. Kinematic hardening is modeled with an evolution equation for the backstress tensor considering Prager’s yielding-stationarity criterion. Nonlinear evolution equations for the backstress and flow stress are proposed for an extension of the model to mixed nonlinear hardening. Furthermore, exact deviatoric/volumetric decoupled forms for kinematic and kinetic variables are obtained. The proposed model is implemented with the Zaremba–Jaumann rate and is used to solve the problem of rectilinear shear for a perfectly plastic and for a linear kinematic hardening material. Neither solution produces oscillatory stress or backstress components. The model is then used to predict the nonlinear hardening behavior of SUS 304 stainless steel under fixed-end finite torsion. Results obtained are in good agreement with reported experimental data. The Swift effect under finite torsion is well predicted by the proposed model.

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Figures

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Fig. 1

Tensor variables defined on different configurations

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Fig. 2

Stress, backstress, shift stress, and yield surface on different configurations

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Fig. 3

Normal components of the Kirchhoff stress, top: whole curve, bottom: stress versus γ-γp

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Fig. 4

Shear components of the Kirchhoff stress, top: whole curve, bottom: stress versus γ-γp

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Fig. 5

Kirchhoff stress, linear kinematic hardening, top: shear component, bottom: normal components

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Fig. 6

Backstress, top: shear component, bottom: normal components

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Fig. 7

Cauchy stress prediction for SUS 304 stainless steel under fixed-end torsion

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Fig. 8

Yield surface size for nonlinear mixed hardening behavior of SUS 304 under fixed-end torsion

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