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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Lee, E. H., 1969, “Elastic-Plastic Deformations at Finite Strains,” J. Appl. Mech., 36, pp. 1–6. [CrossRef]
Nemat-Nasser, S., 1979, “Decomposition of Strain Measures and Their Rates in Finite Deformation Elastoplasticity,” Int. J. Solids Struct., 15, pp. 155–166. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 2006, “Elastoplasticity Beyond Small Deformations,” Acta Mech., 182, pp. 31–111. [CrossRef]
Belytschko, T., Liu, W. K., and Moran, B., 2000, Nonlinear Finite Elements for Continua and Structures, John Wiley, Chichester, England.
Truesdell, C., 1955, “The Simplest Rate Theory of Pure Elasticity,” Commun. Pure Appl. Math., 8, pp. 123–132. [CrossRef]
Rivlin, R. S., 1955, “Further Remarks on the Stress Deformation Relations for Isotropic Materials,” J. Ration Mech. Anal., 4, pp. 681–702.
Dienes, J. K., 1979, “On the Analysis of Rotation and Stress Rate in Deforming Bodies,” Acta Mech., 32, pp. 217–232. [CrossRef]
Nagtegaal, J. C., and de Jong, J. E., 1981, “Some Aspects of Non-Isotropic Work-Hardening in Finite Strain Plasticity,” Proceedings of the Workshop on Plasticity of Metals at Finite Strain: Theory, Experimental and Computation, Stanford University, Stanford, CA, June 29–July 1, pp. 65–102.
Hill, R., 1978, “Aspects of Invariance in Solid Mechanics,” Adv. Appl. Mech., 18, pp. 1–75. [CrossRef]
Kojic, M., and Bathe, K. J., 1987, “Studies of Finite Element Procedures— Stress Solution of a Closed Elastic Strain Path With Stretching and Shearing Using the Updated Lagrangian Jaumann Formulation,” Comput. Struct., 26, pp. 175–179. [CrossRef]
Meyers, A., Xiao, H., and Bruhns, O. T., 2006, “Choice of Objective Rate in Single Parameter Hypoelastic Deformation Cycles,” Comput. Struct., 84, pp. 1134–1140. [CrossRef]
Truesdell, C., and Noll, W., 2003, The Non-Linear Field Theories of Mechanics, 3rd ed., Springer, Berlin.
Ericksen, J. L., 1958, “Hypo-Elastic Potentials,” Q. J. Mech. Appl. Math., XI, pp. 67–72. [CrossRef]
Bernstein, B., 1960, “Relations Between Hypo-Elasticity and Elasticity,” Trans. Soc. Rheol., IV, pp. 23–28. [CrossRef]
Oldroyd, J. G., 1950, “On the Formulation of Rheological Equations of State,” Proc. R. Soc., London, Ser. A,200, pp. 523–541. [CrossRef]
Cotter, B. A., and Rivlin, R. S., 1955, “Tensors Associated With Time-Dependent Stress,” Q. Appl. Math., 13, pp. 177–182.
Dafalias, Y., 1983, “Corotational Rates for Kinematic Hardening at Large Plastic Deformations,” J. Appl. Mech., 50, pp. 561–565. [CrossRef]
Dubey, R. N., 1987, “Choice of Tensor Rates—A Methodology,” Solid Mech. Arch., 12, pp. 233–244.
Simo, J. C., and Pister, K. S., 1984, “Remarks on Rate Constitutive Equations for Finite Deformation Problems: Computational Implications,” Comput. Methods Appl. Mech. Eng., 46, pp. 201–215. [CrossRef]
Lehmann, T., Guo, Z. H., and Liang, H. Y., 1991, “Conjugacy Between Cauchy Stress and Logarithm of the Left Stretch Tensor,” Eur. J. Mech., A/Solids, 10, pp. 395–404.
Reinhardt, W. D., and Dubey, R. N., 1996, “Application of Objective Rates in Mechanical Modeling of Solids,” J. Appl. Mech., 118, pp. 692–698. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 1997, “Logarithmic Strain, Logarithmic Spin and Logarithmic Rate,” Acta Mech., 124, pp. 89–105. [CrossRef]
Hoger, A., 1986, “The Material Time Derivative of Logarithmic Strain Tensor,” Int. J. Solids Struct., 22, pp. 1019–1032. [CrossRef]
Hoger, A., 1987, “The Stress Conjugate to Logarithmic Strain,” Int. J. Solids Struct., 23, pp. 1645–1656. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 1997, “Hypo-Elasticity Model Based Upon the Logarithmic Stress Rate,” J. Elast., 47, pp. 51–68. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 2007, “The Integrability Criterion in Finite Elastoplasticity and its Constitutive Implications,” Acta Mech., 188, pp. 227–244. [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, pp. 479–520. [CrossRef]
Simo, J. C., and Ortiz, M. A., 1985, “A Unified Approach to Finite Deformation Elastoplastic Analysis Based on the Use of Hyperelastic Constitutive Equations,” Comput. Methods Appl. Mech. Eng., 49, pp. 221–245. [CrossRef]
Simo, J. C., 1988, “A Framework for Finite Strain Elasto-Plasticity Based on Maximum Plastic Dissipation and the Multiplicative Decomposition: Part I. Continuum Formulation,” Comput. Methods Appl. Mech. Eng., 66, pp. 199–219. [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, pp. 173–202. [CrossRef]
Eterovic, A. L., and Bathe, K. J., 1991, “A Note on the Use of the Additive Decomposition of the Strain Tensor in Finite Deformation Inelasticity,” Comput. Methods Appl. Mech. Eng., 93, pp. 31–38. [CrossRef]
Gabriel, G., and Bathe, K. J., 1995, “Some Computational Issues in Large Strain Elastoplastic Analysis,” Comput. Struct., 56, pp. 249–267. [CrossRef]
Lubarda, V. A., 1999, “Duality in Constitutive Formulation of Finite-Strain Elastoplasticity Based on F = FeFp and F = FpFe Decompositions,” Int. J. Plast., 15, pp. 1277–1290. [CrossRef]
Sansour, C., and Wagner, W., 2003, “Viscoplasticity Based on Additive Decomposition of Logarithmic Strain and Unified Constitutive Equations, Theoretical and Computational Considerations With Reference to Shell Applications,” Comput. Struct., 81, pp. 1583–1594. [CrossRef]
Lubarda, V. A., 2004, “Constitutive Theories Based on the Multiplicative Decomposition of Deformation Gradient: Thermoelasticity, Elastoplasticity, and Biomechanics,” Appl. Mech. Rev., 57, pp. 95–108. [CrossRef]
Montans, F. J., and Bathe, K. J., 2005, “Computational Issues in Large Strain Elasto-Plasticity: An Algorithm for Mixed Hardening and Plastic Spin,” Int. J. Numer. Methods Eng., 63, pp. 159–196. [CrossRef]
Sidoroff, F., 1973, “The Geometrical Concept of Intermediate Configuration and Elastic-Plastic Finite Strain,” Arch. Mech., 25, pp. 299–308.
Lee, E. H., 1981, “Some Comments on Elastic-Plastic Analysis,” Int. J. Solids Struct., 17, pp. 859–872. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 1999, “A Natural Generalization of Hypoelasticity and Eulerian Rate Type Formulation of Hyperelasticity,” J. Elast., 56, pp. 59–93. [CrossRef]
Spencer, A. J. M., 1984, Continuum Theory of the Mechanics of Fiber Reinforced Composites, Springer, New York.
Holzapfel, G. A., Gasser, T. C., and Ogden, R. W., 2000, “A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elast., 61, pp. 1–48. [CrossRef]
Panoskaltsis, V. P., Polymenakos, L. C., and Soldatos, D., 2008, “Eulerian Structure of Generalized Plasticity: Theoretical and Computational Aspects,” ASCE J. Eng. Mech., 134(5), pp. 354–361. [CrossRef]
Panoskaltsis, V. P., Polymenakos, L. C., and Soldatos, D., 2008, “On Large Deformation Generalized Plasticity,” J. Mech. Mater. Struct., 3(3), pp. 441–457. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 1998, “On Objective Corotational Rates and Their Defining Spin Tensors,” Int. J. Solids Struct., 35, pp. 4001–4014. [CrossRef]
Xiao, H., Bruhns, O. T., and Meyers, A., 1998, “Strain Rates and Material Spins,” J. Elast., 52, pp. 1–41. [CrossRef]
Green, A. E., and Naghdi, P. M., 1965, “A General Theory of an Elastic-Plastic Continuum,” Arch. Ration. Mech. Anal., 18, pp. 251–281. [CrossRef]
Scheidler, M., 1994, “The Tensor Equation AX+XA= ϕ(A,H) With Applications to Kinematics of Continua,” J. Elast., 36, pp. 117–153. [CrossRef]
Marsden, J., and Hughes, T. J. R., 1994, Mathematical Foundations of Elasticity, Dover, New York.
Ogden, R. W., 1997, Nonlinear Elastic Deformations, Dover, New York.
Norris, A., 2008, “Eulerian Conjugate Stress and Strain,” J. Mech. Mater. Struct., 3(2), pp. 243–260. [CrossRef]
Mooney, M., 1940, “A Theory of Large Elastic Deformation,” J. Appl. Phys., 6, pp. 582–592. [CrossRef]
Rivlin, R. S., 1948, “Large Elastic Deformations of Isotropic Materials I. Fundamental Concepts,” Philos. Trans. R. Soc. London, Ser. A, 240, pp. 459–490. [CrossRef]
Reese, S., and Govindjee, S., 1998, “A Theory of Finite Viscoelasticity and Numerical Aspects,” Int. J. Solids Struct., 35, pp. 3455–3482. [CrossRef]
Blatz, R. J., and Ko, W. L., 1962, “Application of Finite Elastic Theory to the Deformation of Rubbery Materials,” Trans. Soc. Rheol., 6, pp. 223–251. [CrossRef]


Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

Plane strain four-step loading

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

Plane strain elliptical loading

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Fig. 5

Stresses produced by two cycles of plane strain elliptical loading

Grahic Jump Location
Fig. 6

Stresses produced by 50 cycles of plane strain elliptical loading



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