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Lecture

The Czech Society for Mechanics – the expert group for Computational mechanics, the Institute of Thermomechanics AS CR and the Mathematical Institute, Charles University wish to invite you to a lecture, which is a part of the Nečas seminar on continuum mechanics, by

Zdeněk P. Bažant

Northwestern University, Evanston, Illinois, USA

**Abstract**

The objective stress rates used in most commercial finite element (FE) codes for the implicit incremental (updated Lagrangian or Riks) analysis is the Jaumann rate of Cauchy (true) stress. However, if the material is compressible, this rate violates the law of conservation of energy because it is not work-conjugate to any finite strain tensor. For bifurcation load analysis, most programs use the Jaumann rate of Kirchhoff stress. Work-conjugate though it is to the Hencky (or logarithmic) strain tensor, in the case of buckling of highly orthotropic soft-in-shear structures it causes another kind of energy conservation error—incorrect work of initial stresses. For such structures, only the Truesdell stress rate which is work-conjugate to the Green-Lagrangian finite strain tensor is correct (it is used in commercial code ATENA and open-source code OOFEM, both developed in Prague). The Green-Naghdi rate is not work-conjugate to any finite strain tensor and can be associated with a tangential material stiffness matrix that lacks major symmetry although, in commercial explicit programs, this rate in used in a symmetrized form.

These problems with work-conjugacy were, in principle, theoretically demonstrated in 1971. Nonetheless, they have generally been ignored, partly because of some extraneous considerations, partly because in the vast majority of applications, mainly to metals, the errors are negligible. However, with the growing interest in highly compressible materials, soft-in-shear orthotropic materials and quasibrittle materials with softening damage, large discrepancies have been noticed. Often, though, they have been regarded as a matter of choice depending on the type of material or application, rather than as problems with the first law of thermodynamics.

Numerical examples document that the errors can be serious. One is the indentation of a naval-type sandwich plate with a polymeric foam core, in which the error amounts to 29% of the indentation force and 15% in the work of load. Errors from 40% to 100% are demonstrated for the buckling of homogenized sandwich panels planned for a novel type of ribbed ship hulls of super-light and long ships. Errors of similar kind must be expected for all highly compressible materials, such as metallic and ceramic foams, fiber reinforced foam cores, honeycomb, loess, silt, organic soils, pumice, tuff, corral, light wood, osteoporotic bone and various biologic tissues, and for compression damage models in which the material is rendered incrementally highly orthotropic due to dense axial splitting cracks.

A remedy can be achieved if the previously derived equations relating the tangential moduli tensors associated with the Jaumann rates of Cauchy or Kirchhoff stresses and with the Truesdell rate are used in the user's material subroutine of a black-box implicit commercial program. However, these corrections must be delayed, which accumulates additional numerical integration error. A better remedy would be a revision of the black-box commercial programs.

**References**

1. Bažant, Z.P. (1971). “A correlation study of incremental deformations and stability of continuous bodies.” Journal of Applied Mechanics ASME, 38, 919--928.

2. Bažant, Z. P., and Cedolin, L., 1991, Stability of Structures: Elastic, Inelastic,Fracture and Damage Theories, Oxford University Press, New York; and 2nd ed, Dover 2003; 3rd ed., World Scientific 2010.

3. Bažant, Z. P., 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.

4. Bažant, Z. P., and Beghini, A., 2006, “Stability and Finite Strain of Homogenized Structures Soft in Shear: Sandwich or Fiber Composites, and Layered Bodies,” Int. J. Solids Struct., 43 (6), pp. 1571–1593.

5. Ji, W., and Waas, A. M., 2008, “Wrinkling and Edge Buckling in Orthotropic Sandwich Beams,” J. Eng. Mech., 134, 455–461.

6. Ji, Wooseok, Waas, A.M., and Bažant, Z.P., 2010. “Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs.” ASME J. of Applied Mechanics 77 (July), 044504-1—044504-5.

7. 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. Royal Society A, in press (already published online).

More information: Ing. Jiří Plešek, CSc.