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Research Papers

How to Realize Volume Conservation During Finite Plastic Deformation

[+] Author and Article Information
Heling Wang

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

Dong-Jie Jiang

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

Li-Yuan Zhang

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

Bin Liu

AML, CNMM,
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.

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Topics: Deformation , Stress , Tensors
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References

Mandel, J. , 1974, “Director Vectors and Constitutive Equations for Plastic and Visco-Plastic Media,” Problems of Plasticity, A. Sawczwk , ed., Springer, Dordrecht, The Netherlands, pp. 135–143. [CrossRef]
Rice, J. R. , 1971, “Inelastic Constitutive Relations for Solids: An Internal-Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19(6), pp. 433–455. [CrossRef]
Rice, J. R. , 1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanics,” Constitutive Equations in Plasticity, A. S. Argon , ed., MIT Press, Cambridge, UK, pp. 23–79.
Hill, R. , and Rice, J. R. , 1972, “Constitutive Analysis of Elastic-Plastic Crystals at Arbitrary Strain,” J. Mech. Phys. Solids, 20(6), pp. 401–413. [CrossRef]
Hill, R. , and Rice, J. R. , 1973, “Elastic Potential and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25(3), pp. 448–461. [CrossRef]
Moran, B. , Ortiz, M. , and Shih, C. F. , 1990, “Formulation of Implicit Finite Element Methods for Multiplicative Finite Deformation Plasticity,” Int. J. Numer. Methods Eng., 29(3), pp. 483–514. [CrossRef]
Besseling, J. F. , and Van der Giessen, E. , 1994, Mathematical Modelling of Inelastic Deformation, Chapman & Hall, London. [CrossRef]
Yang, Q. , Mota, A. , and Ortiz, M. , 2006, “A Finite-Deformation Constitutive Model of Bulk Metallic Glass Plasticity,” Comput. Mech., 37(2), pp. 194–204. [CrossRef]
Lele, S. P. , and Anand, L. , 2009, “A Large-Deformation Strain-Gradient Theory for Isotropic Viscoplastic Materials,” Int. J. Plast., 25(3), pp. 420–453. [CrossRef]
Rubin, M. B. , and Ichihara, M. , 2010, “Rheological Models for Large Deformations of Elastic-Viscoplastic Materials,” Int. J. Eng. Sci., 48(11), pp. 1534–1543. [CrossRef]
Vladimirov, I. N. , Pietryga, M. P. , and Reese, S. , 2010, “Anisotropic Finite Elastoplasticity With Nonlinear Kinematic and Isotropic Hardening and Application to Sheet Metal Forming,” Int. J. Plast., 26(5), pp. 659–687. [CrossRef]
Volokh, K. Y. , 2013, “An Approach to Elastoplasticity at Large Deformations,” Eur. J. Mech. A-Solids, 39, pp. 153–162. [CrossRef]
Altenbach, H. , and Eremeyev, V. A. , 2014, “Strain Rate Tensors and Constitutive Equations of Inelastic Micropolar Materials,” Int. J. Plast., 63, pp. 3–7. [CrossRef]
Shutov, A. V. , and Ihlemann, J. , 2014, “Analysis of Some Basic Approaches to Finite Strain Elasto-Plasticity in View of Reference Change,” Int. J. Plast., 63, pp. 183–197. [CrossRef]
Rajagopal, K. R. , and Srinivasa, A. R. , 2015, “Inelastic Response of Solids Described by Implicit Constitutive Relations With Nonlinear Small Strain Elastic Response,” Int. J. Plast., 71, pp. 1–9. [CrossRef]
Chaouki, H. , Picard, D. , Ziegler, D. , Azari, K. , Alamdari, H. , and Fafard, M. , 2016, “Viscoplastic Modeling of the Green Anode Paste Compaction Process,” ASME J. Appl. Mech., 83(2), p. 021002. [CrossRef]
Zhu, Y. , Kang, G. , Kan, Q. , Bruhns, O. T. , and Liu, Y. , 2016, “ Thermo-Mechanically Coupled Cyclic Elasto-Viscoplastic Constitutive Model of Metals: Theory and Application,” Int. J. Plast., 79, pp. 111–152. [CrossRef]
Chowdhury, S. R. , Kar, G. , Roy, D. , and Reddy, J. N. , 2017, “ Two-Temperature Thermodynamics for Metal Viscoplasticity: Continuum Modeling and Numerical Experiments,” ASME J. Appl. Mech., 84(1), p. 011002. [CrossRef]
Bertram, A. , 2005, Elasticity and Plasticity of Large Deformations: An Introduction, Springer, Berlin.
Dunne, F. , and Petrinic, N. , 2005, Introduction to Computational Plasticity, Oxford University Press, New York.
Hashiguchi, K. , 2009, Elastoplasticity Theory, Springer, Berlin. [CrossRef]
Lubarda, V. A. , 2001, Elastoplasticity Theory, CRC Press, Boca Raton, FL. [CrossRef]
Naghdi, P. M. , 1990, “A Critical-Review of the State of Finite Plasticity,” J. Appl. Math. Phys., 41(3), pp. 315–394. [CrossRef]
Xiao, H. , Bruhns, O. T. , and Meyers, A. , 2006, “Elastoplasticity Beyond Small Deformations,” Acta Mech., 182(1–2), pp. 31–111. [CrossRef]
Chen, H. S. , Ma, J. , Pei, Y. , and Fang, D. N. , 2013, “ Anti-Plane Yoffe-Type Crack in Ferroelectric Materials,” Int. J. Fract., 179(1–2), pp. 35–43. [CrossRef]
Chen, H. S. , Pei, Y. , Liu, J. , and Fang, D. N. , 2013, “Moving Polarization Saturation Crack in Ferroelectric Solids,” Eur. J. Mech.-A/Solids, 41, pp. 43–49. [CrossRef]
Chen, H. S. , Pei, Y. M. , Liu, B. , and Fang, D. N. , 2013, “Rate Dependant Heat Generation in Single Cycle of Domain Switching of Lead Zirconate Titanate Via In-Situ Spontaneous Temperature Measurement,” Appl. Phys. Lett., 102(24), p. 242912. [CrossRef]
Chen, H. S. , Wang, H. L. , Pei, Y. M. , Wei, Y. J. , Liu, B. , and Fang, D. N. , 2015, “Crack Instability of Ferroelectric Solids Under Alternative Electric Loading,” J. Mech. Phys. Solids, 81, pp. 75–90. [CrossRef]
Wojnar, C. S. , and Kochmann, D. M. , 2017, “Linking Internal Dissipation Mechanisms to the Effective Complex Viscoelastic Moduli of Ferroelectrics,” ASME J. Appl. Mech., 84(2), p. 021006. [CrossRef]
Zhang, H. , Yu, Z. , Pei, Y. , and Fang, D. , 2017, “Butterfly Change in Electric Field-Dependent Young's Modulus: Bulge Test and Phase Field Model,” ASME J. Appl. Mech., 84(5), p. 051009. [CrossRef]
Ziolkowski, A. , 2007, “ Three-Dimensional Phenomenological Thermodynamic Model of Pseudoelasticity of Shape Memory Alloys at Finite Strains,” Continuum Mech. Thermodyn., 19(6), pp. 379–398. [CrossRef]
Thamburaja, P. , 2010, “A Finite-Deformation-Based Phenomenological Theory for Shape-Memory Alloys,” Int. J. Plast., 26(8), pp. 1195–1219. [CrossRef]
Arghavani, J. , Auricchio, F. , and Naghdabadi, R. , 2011, “A Finite Strain Kinematic Hardening Constitutive Model Based on Hencky Strain: General Framework, Solution Algorithm and Application to Shape Memory Alloys,” Int. J. Plast., 27(6), pp. 940–961. [CrossRef]
Kelly, A. , Stebner, A. P. , and Bhattacharya, K. , 2016, “A Micromechanics-Inspired Constitutive Model for Shape-Memory Alloys That Accounts for Initiation and Saturation of Phase Transformation,” J. Mech. Phys. Solids, 97, pp. 197–224. [CrossRef]
Moon, S. , Rao, I. J. , and Chester, S. A. , 2016, “Triple Shape Memory Polymers: Constitutive Modeling and Numerical Simulation,” ASME J. Appl. Mech., 83(7), p. 071008. [CrossRef]
LaMaster, D. H. , Feigenbaum, H. P. , Nelson, I. D. , and Ciocanel, C. , 2014, “A Full Two-Dimensional Thermodynamic-Based Model for Magnetic Shape Memory Alloys,” ASME J. Appl. Mech., 81(6), p. 061003. [CrossRef]
Tang, H. , Barthelat, F. , and Espinosa, H. D. , 2007, “An Elasto-Viscoplastic Interface Model for Investigating the Constitutive Behavior of Nacre,” J. Mech. Phys. Solids, 55(7), pp. 1410–1438. [CrossRef]
Chen, K. , Kang, G. , Yu, C. , Lu, F. , and Jiang, H. , 2016, “ Time-Dependent Uniaxial Ratchetting of Ultrahigh Molecular Weight Polyethylene Polymer: Viscoelastic–Viscoplastic Constitutive Model,” ASME J. Appl. Mech., 83(10), p. 101003. [CrossRef]
Yu, C. , Kang, G. , Lu, F. , Zhu, Y. , and Chen, K. , 2016, “Viscoelastic–Viscoplastic Cyclic Deformation of Polycarbonate Polymer: Experiment and Constitutive Model,” ASME J. Appl. Mech., 83(4), p. 041002. [CrossRef]
Liu, Y. , Zhang, H. , and Zheng, Y. , 2016, “A Micromechanically Based Constitutive Model for the Inelastic and Swelling Behaviors in Double Network Hydrogels,” ASME J. Appl. Mech., 83(2), p. 021008. [CrossRef]
Wang, H. , and Shen, S. , 2016, “A Chemomechanical Model for Stress Evolution and Distribution in the Viscoplastic Oxide Scale During Oxidation,” ASME J. Appl. Mech., 83(5), p. 051008. [CrossRef]
Miehe, C. , Apel, N. , and Lambrecht, M. , 2002, “Anisotropic Additive Plasticity in the Logarithmic Strain Space: Modular Kinematic Formulation and Implementation Based on Incremental Minimization Principles for Standard Materials,” Comput. Methods Appl. Mech. Eng., 191(47–48), pp. 5383–5425. [CrossRef]
Weber, G. , and Anand, L. , 1990, “Finite Deformation Constitutive Equations and a Time Integration Procedure for Isotropic, Hyperelastic–Viscoplastic Solids,” Comput. Methods Appl. Mech. Eng., 79(2), pp. 173–202. [CrossRef]
Simo, J. C. , 1992, “Algorithms for Static and Dynamic Multiplicative Plasticity That Preserve the Classical Return Mapping Schemes of the Infinitesimal Theory,” Comput. Methods Appl. Mech. Eng., 99(1), pp. 61–112. [CrossRef]
Reese, S. , and Govindjee, S. , 1998, “A Theory of Finite Viscoelasticity and Numerical Aspects,” Int. J. Solids Struct., 35(26–27), pp. 3455–3482. [CrossRef]
Dettmer, W. , and Reese, S. , 2004, “On the Theoretical and Numerical Modelling of Armstrong–Frederick Kinematic Hardening in the Finite Strain Regime,” Comput. Methods Appl. Mech. Eng., 193(1–2), pp. 87–116. [CrossRef]
Reese, S. , and Christ, D. , 2008, “Finite Deformation Pseudo-Elasticity of Shape Memory Alloys–Constitutive Modelling and Finite Element Implementation,” Int. J. Plast., 24(3), pp. 455–482. [CrossRef]
Vladimirov, I. N. , Pietryga, M. P. , and Reese, S. , 2008, “On the Modelling of Non-Linear Kinematic Hardening at Finite Strains With Application to Springback–Comparison of Time Integration Algorithms,” Int. J. Numer. Methods Eng., 75(1), pp. 1–28. [CrossRef]
Green, A. E. , and Naghdi, P. M. , 1971, “Some Remarks on Elastic-Plastic Deformation at Finite Strain,” Int. J. Eng. Sci., 9(12), pp. 1219–1229. [CrossRef]
Bruhns, O. T. , 2015, “The Multiplicative Decomposition of the Deformation Gradient in Plasticity—Origin and Limitations,” From Creep Damage Mechanics to Homogenization Methods, H. Altenbach, T. Matsuda, and D. Okumura, eds., Springer, Cham, Switzerland, pp. 37–66. [CrossRef]
Hill, R. , 1978, “Aspects of Invariance in Solids Mechanics,” Advances in Applied Mechanics, Vol. 18, C. S. Yih, ed., Academic Press, New York, pp. 1–75. [CrossRef]
Green, A. E. , and Naghdi, P. M. , 1965, “A General Theory of an Elastic-Plastic Continuum,” Arch. Ration. Mech. Anal., 18(4), pp. 251–281. [CrossRef]
Simo, J. C. , and Ortiz, M. , 1985, “A Unified Approach to Finite Deformation Elastoplastic Analysis Based on the Use of Hyperelastic Constitutive Equations,” Comput. Methods Appl. Mech. Eng., 49(2), pp. 221–245. [CrossRef]
Gross, A. J. , and Ravi-Chandar, K. , 2015, “On the Extraction of Elastic–Plastic Constitutive Properties From Three-Dimensional Deformation Measurements,” ASME J. Appl. Mech., 82(7), p. 071013. [CrossRef]
Bažant, Z. P. , 1971, “A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies,” ASME J. Appl. Mech., 38(4), pp. 919–928. [CrossRef]
Lee, E. H. , Mallett, R. L. , and Wertheimer, T. B. , 1983, “Stress Analysis for Anisotropic Hardening in Finite-Deformation Plasticity,” ASME J. Appl. Mech., 50(3), pp. 554–560. [CrossRef]
Atluri, S. N. , 1984, “On Constitutive Relations at Finite Strain: Hypo-Elasticity and Elasto-Plasticity With Isotropic or Kinematic Hardening,” Comput. Methods Appl. Mech. Eng., 43(2), pp. 137–171. [CrossRef]
Xiao, H. , Bruhns, O. T. , and Meyers, A. , 1997, “Logarithmic Strain, Logarithmic Spin and Logarithmic Rate,” Acta Mech., 124(1–4), pp. 89–105. [CrossRef]
Xiao, H. , Bruhns, O. T. , and Meyers, A. , 1997, “Hypoelasticity Model Based Upon the Logarithmic Stress Rate,” J. Elast., 47(1), pp. 51–68. [CrossRef]
Xiao, H. , Bruhns, O. T. , and Meyers, A. , 1998, “On Objective Corotational Rates and Their Defining Spin Tensors,” Int. J. Solids Struct., 35(30), pp. 4001–4014. [CrossRef]
Bažant, Z. P. , Gattu, M. , and Vorel, J. , 2012, “Work Conjugacy Error in Commercial Finite-Element Codes: Its Magnitude and How to Compensate for It,” Proc. R. Soc. A, 468(2146), pp. 3047–3058. [CrossRef]
Ji, W. , Waas, A. M. , and Bazant, Z. P. , 2013, “On the Importance of Work-Conjugacy and Objective Stress Rates in Finite Deformation Incremental Finite Element Analysis,” ASME J. Appl. Mech., 80(4), p. 041024. [CrossRef]
Vorel, J. , Bažant, Z. P. , and Gattu, M. , 2013, “Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate,” ASME J. Appl. Mech., 80(4), p. 041034. [CrossRef]
Bažant, Z. P. , and Vorel, J. , 2014, “ Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation,” ASME J. Appl. Mech., 81(2), p. 021008. [CrossRef]
Vorel, J. , and Bažant, Z. P. , 2014, “Review of Energy Conservation Errors in Finite Element Softwares Caused by Using Energy-Inconsistent Objective Stress Rates,” Adv. Eng. Software, 72, pp. 3–7. [CrossRef]
Petric, D. , Owen, D. R. J. , and Honnor, M. E. , 1992, “A Model for Finite Strain Elasto-Plasticity Based on Logarithmic Strains: Computational Issues,” Comput. Methods Appl. Mech. Eng., 94(1), pp. 35–61. [CrossRef]
Bruhns, O. T. , Xiao, H. , and Meyers, A. , 1999, “ Self-Consistent Eulerian Rate Type Elasto-Plasticity Models Based upon the Logarithmic Stress Rate,” Int. J. Plast., 15(5), pp. 479–520. [CrossRef]
Xiao, H. , Bruhns, O. T. , and Meyers, A. , 2000, “A Consistent Finite Elastoplasticity Theory Combining Additive and Multiplicative Decomposition of the Stretching and the Deformation Gradient,” Int. J. Plast., 16(2), pp. 143–177. [CrossRef]
Xiao, H. , Bruhns, O. T. , and Meyers, A. , 2001, “Large Strain Responses of Elastic-Perfect Plasticity and Kinematic Hardening Plasticity With the Logarithmic Rate: Swift Effect in Torsion,” Int. J. Plast., 17(2), pp. 211–235. [CrossRef]

Figures

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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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