Research Papers

Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation

[+] Author and Article Information
Zdeňek P. Bažant

Hon. Mem. ASME
McCormick Institute Professor and
W. P. Murphy Professor of
Civil and Mechanical
Engineering and Materials Science,
Northwestern University,
Evanston, IL 60208
e-mail: z-bazant@northwestern.edu

Jan Vorel

Assistant Professor
Department of Mechanics,
Faculty of Civil Engineering,
Czech Technical University in Prague,
Prague, Czech Republic
Visiting Scholar
Northwestern University,
Evanston, IL 60208
e-mail: jan.vorel@fsv.cvut.cz

1Corresponding author.

Contributed by the Applied Mechanics of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 21, 2013; final manuscript received March 29, 2013; accepted manuscript posted May 7, 2013; published online September 16, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(2), 021008 (Sep 16, 2013) (5 pages) Paper No: JAM-13-1041; doi: 10.1115/1.4024411 History: Received January 21, 2013; Revised March 29, 2013; Accepted May 07, 2013

The objective stress rates used in most commercial finite element programs are the Jaumann rate of Kirchhoff stress, Jaumann rates of Cauchy stress, or Green–Naghdi rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstrated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.

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Bažant, Z., 1971, “A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies,” ASME J. Appl. Mech., 38(4), pp. 919–928. [CrossRef]
Green, A., and Naghdi, P., 1965, “A General Theory of an Elastic-Plastic Continuum,” Arch. Ration. Mech. Anal., 18(4), pp. 251–281. (Eq. 8.23). [CrossRef]
Červenka, V., and Jendele, L., 2008, ATENA Program Documentation—Part 1: Theory, Cervenka Consulting, www.cervenka.cz
Patzák, B., and Bittnar, Z., 2001, “Design of Object Oriented Finite Element Code,” Adv. Eng. Softw., 32(10–11), pp. 759–767. [CrossRef]
Bažant, Z., Gattu, M., and Vorel, J., 2012, “Work Conjugacy Error in Commercial Finite Element Codes: Its Magnitude and How to Compensate for It,” Proc. Royal Soc. A Math Phys Eng. Sci., 468(2146), pp. 3047–3058. [CrossRef]
Vorel, J., Zant, Z. B., 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., and Beghini, A., 2005, “Which Formulation Allows Using a Constant Shear Modulus for Small-Strain Buckling of Soft-Core Sandwich Structures?,” ASME J. Appl. Mech., 72(5), pp. 785–787. [CrossRef]
Bažant, Z., and Beghini, A., 2006, “Stability and Finite Strain of Homogenized Structures Soft in Shear: Sandwich or Fiber Composites, and Layered Bodies,” Int. J. Solid. Struct., 43(6), pp. 1571–1593. [CrossRef]
Ji, W., Waas, A., and Bažant, Z., 2010, “Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs,” ASME J. Appl. Mech., 77(4), p. 044504. [CrossRef]
Bažant, Z., and Cedolin, L., 1991, Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories, 1st ed., Oxford University Press, New York.
Hibbitt, H., Marcal, P., and Rice, J., 1970, “A Finite Strain Formulation for Problems of Large Strain and Displacement,” Int. J. Solid. Struct., 6, pp. 1069–1086. [CrossRef]
Vural, M., Rittel, D., and Ravichandran, G., 2003, “Large Strain Mechanical Behavior of 1018 Cold-Rolled Steel Over a Wide Range of Strain Rates,” Metal. Mater. Trans. A, 34(12), pp. 2873–2885. [CrossRef]
Dassault Systèmes, 2010, ABAQUS FEA, www.simulia.com
Hughes, T., and Winget, J., 1980, “Finite Rotation Effects in Numerical Integration of Rate Constitutive Equations Arising in Large-Deformation Analysis,” Int. J. Numer. Meth. Eng., 15(12), pp. 1862–1867. [CrossRef]
Fraejis de Veubeke, B., 1965, Displacement Equilibrium Models in the Finite Element Method, John Wiley & Sons Ltd., Chichester, UK.
Jaumann, G., 1911, “Geschlossenes system physikalischer und chemischer differentialgesetze,” Sitzungsberichte Akad. Wiss. Wien, IIa, pp. 385–530.


Grahic Jump Location
Fig. 1

Elevations and plan view skew-notched cylinder analyzed

Grahic Jump Location
Fig. 2

Comparison of computation results for different stress rates: (a) curve of load versus relative displacement w/h (JC = Jaumann rate of Cauchy stress, JK = Jaumann rate of Kirchhoff stress, G–N = Green–Naghdi stress rate); (b) error in energy; (c) average magnitudes of rotation vector (in radians) within the notched part, computed from the rotation increments used in the G–N and JK stress rates, plotted as a function of w/h

Grahic Jump Location
Fig. 3

Axonometric view of (a) undeformed and (b) deformed meshes used in finite element computations; (b) the shown deformation corresponds to Jaumann rate of Cauchy stress, which conserves energy



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