Introduction to Nonlinear Thermomechanics of Solids by Michał Kleiber & Piotr Kowalczyk

Author:Michał Kleiber & Piotr Kowalczyk
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Substituting this into Eq. (7.44) we can derive the constitutive relation

Comparing the above equation with Eq. (7.13), one can see the similarity: substituting and we come at nonlinear, strain-dependent expressions on isotropic Lamé constants.

The tangent stiffness components can be determined by differentiation of the constitutive equation with respect to :

Careful readers will surely notice that the form of the resulting components is not symmetric with respect to the pairs kl and mn. This is because the symmetry was not fully taken into account in differentiation of the strain components (to do it consistently, we should have rather assumed that ). Note, however, that the full product of and the symmetric tensor is equivalent to the full product of the symmetric part of (with respect to the indices mn) and . Hence, the term in the above result may be replaced by its symmetric part which yields the final, symmetric form of the tangent stiffness tensor .

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