Research Papers

How to Realize Volume Conservation During Finite Plastic Deformation

[+] Author and Article Information
Heling Wang

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084 China;
Department of Civil and Environmental
Northwestern University,
Evanston, IL 60208
e-mail: heling.wang@northwestern.edu

Dong-Jie Jiang

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: jiangdj@utexas.edu

Li-Yuan Zhang

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: zhangly@ustb.edu.cn

Bin Liu

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: liubin@tsinghua.edu.cn

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 3, 2017; final manuscript received September 6, 2017; published online September 26, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(11), 111009 (Sep 26, 2017) (18 pages) Paper No: JAM-17-1421; doi: 10.1115/1.4037882 History: Received August 03, 2017; Revised September 06, 2017

Volume conservation during plastic deformation is the most important feature and should be realized in elastoplastic theories. However, it is found in this paper that an elastoplastic theory is not volume conserved if it improperly sets an arbitrary plastic strain rate tensor to be deviatoric. We discuss how to rigorously realize volume conservation in finite strain regime, especially when the unloading stress free configuration is not adopted in the elastoplastic theories. An accurate condition of volume conservation is first clarified and used in this paper that the density of a volume element after the applied loads are completely removed should be identical to that of the initial stress free states. For the elastoplastic theories that adopt the unloading stress free configuration (i.e., the intermediate configuration), the accurate condition of volume conservation is satisfied only if specific definitions of the plastic strain rate are used among many other different definitions. For the elastoplastic theories that do not adopt the unloading stress free configuration, it is even more difficult to realize volume conservation as the information of the stress free configuration lacks. To find a universal approach of realizing volume conservation for elastoplastic theories whether or not adopt the unloading stress free configuration, we propose a single assumption that the density of material only depends on the trace of the Cauchy stress by using their objectivities. Two strategies are further discussed to satisfy the accurate condition of volume conservation: directly and slightly revising the tangential stiffness tensor or using a properly chosen stress/strain measure and elastic compliance tensor. They are implemented into existing elastoplastic theories, and the volume conservation is demonstrated by both theoretical proof and numerical examples. The potential application of the proposed theories is a better simulation of manufacture process such as metal forming.

Copyright © 2017 by ASME
Topics: Deformation , Stress , Tensors
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Fig. 1

Illustration of the accurate volume conservation condition and the definition of plastic volume deformation. The configuration whose plastic volume deformation is to be defined is denoted by a solid square, and its corresponding unloading stress free configuration is denoted by a solid circle. The accurate volume conservation condition is ρ (sf )=ρ (0 ).

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Fig. 2

The definitions of the plastic strain rate suggested by (a) the Rice–Hill theory and (b) the Simo–Ortiz theory

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Fig. 3

An illustration that the strain rate decomposition suggested by the Rice–Hill theory is dependent on the stress/strain measure

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Fig. 4

A uniaxial tension example to illustrate that the strain decomposition suggested by the Rice–Hill theory depends on the stress/strain measure

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Fig. 5

(a) Illustration of the uniaxial loading example and (b) the stress–strain curve used in this example

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Fig. 6

The volume change of the unloading stress free configuration predicted by various elastoplastic theories

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Fig. 7

An illustration of the numerical convergence in the uniaxial loading example

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Fig. 8

The plastic volume deformation predicted by commercial software (a) abaqus; (b) comsol and ANSYS

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Fig. 9

The volume change of the stress free configuration of a material element subjected to subsequent biaxial loading, predicted by the RH-Ini theories (the dashed line) and the theories revised by strategies 1 and 2 (the solid line)

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Fig. 10

The predicted volume change of the stress free configuration under uniaxial loading for elastic-perfectly plastic material with small elastic deformation

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Fig. 11

Illustration of the stress rate decomposition and the plastic stress rate

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Fig. 12

The relative error of the predicted stress by some elastoplastic theories



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