Technical Briefs

Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs

[+] Author and Article Information
Wooseok Ji

Computational Science and Engineering, University of Illinois, Urbana, IL 61801

Anthony M. Waas1

Department of Aerospace Engineering, Composite Structures Laboratory, University of Michigan, Ann Arbor, MI 48109dcw@umich.edu

Zdeněk P. Bažant

Civil Engineering and Materials Science, Northwestern University, Evanston, IL 60208


Corresponding author.

J. Appl. Mech 77(4), 044504 (Apr 12, 2010) (5 pages) doi:10.1115/1.4000916 History: Received May 19, 2009; Revised September 18, 2009; Published April 12, 2010; Online April 12, 2010

Many finite element programs including standard commercial software such as ABAQUS use an incremental finite strain formulation that is not fully work-conjugate, i.e., the work of stress increments on the strain increments does not give a second-order accurate expression for work. In particular, the stress increments based on the Jaumann rate of Kirchhoff stress are work-conjugate with the increments of the Hencky (logarithmic) strain tensor but are paired in many finite element programs with the increments of Green’s Lagrangian strain tensor. Although this problem was pointed out as early 1971, a demonstration of its significance in realistic situations has been lacking. Here it is shown that, in buckling of compressed highly orthotropic columns or sandwich columns that are very “soft” in shear, the use of such nonconjugate stress and strain increments can cause large errors, as high as 100% of the critical load, even if the strains are small. A similar situation may arise when severe damage such as distributed cracking leads to a highly anisotropic tangential stiffness matrix, or when axial cracks between fibers severely weaken a uniaxial fiber composite or wood. A revision of these finite element programs is advisable, and will in fact be easy—it will suffice to replace the Jaumann rate with the Truesdell rate. Alternatively, the Green’s Lagrangian strain could be replaced with the Hencky strain.

Copyright © 2010 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

(a) Finite element model of a orthotropic column and (b) typical deformed shape of the column in the eigenbuckling problem

Grahic Jump Location
Figure 2

Comparison of buckling loads from different formulations

Grahic Jump Location
Figure 3

Relative error of the buckling loads



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