Work conjugacy error in commercial finite-element codes: Its magnitude and how to compensate for it

Zdeněk P. Bažant*, Mahendra Gattu, Jan Vorel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


Most commercial finite-element programs use the Jaumann (or co-rotational) rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligibleerrors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 percent in the load and 15.3 percent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann rates of Cauchy and Kirchh off stresses can be used in the user's material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann rate of Kirchh off stress, which is work-conjugate, were used.

Original languageEnglish (US)
Pages (from-to)3047-3058
Number of pages12
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2146
StatePublished - Oct 8 2012


  • Compressible materials
  • Energy error
  • Finite strain
  • Incremental analysis
  • Objective stress rate
  • Updated Lagrangian method

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)


Dive into the research topics of 'Work conjugacy error in commercial finite-element codes: Its magnitude and how to compensate for it'. Together they form a unique fingerprint.

Cite this