Research Articles

Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part I: Rate-Form Hyperelasticity

[+] Author and Article Information
Amin Eshraghi

Research Associate
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada
e-mail: maeshrag@uwaterloo.ca

Katerina D. Papoulia

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

Hamid Jahed

Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada
e-mail: hjahedmo@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), 021027 (Jan 30, 2013) (11 pages) Paper No: JAM-12-1325; doi: 10.1115/1.4007723 History: Received July 15, 2012; Revised September 08, 2012; Accepted September 29, 2012

An integrable Eulerian rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational rate of the Kirchhoff stress with material spin to the same rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integrated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann rate of stress is employed.

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Fig. 1

Stresses produced by plane strain simple shear deformation; lines: proposed model results; symbols: hyperelasticity results

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Fig. 2

Plane strain four-step loading

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Fig. 4

Plane strain elliptical loading

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Fig. 5

Stresses produced by two cycles of plane strain elliptical loading

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Fig. 6

Stresses produced by 50 cycles of plane strain elliptical loading

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Fig. 3

Deviatoric Cauchy stress components produced by plane strain four-step loading




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