Is the Jaumann Stress Rate Applicable? — Discussions on Choosing Proper Rate Expressions of a Finite Deformation Constitutive Model

Abstract

The rate form constitutive expression plays an important role in finite deformation theory and is widely used in the finite element method (FEM) software. However, the choice of objective stress and strain rates in constitutive expressions has caused many controversies. This paper aims to clarify these controversies by distinguishing the related concepts: constitutive behavior, constitutive model and constitutive expression, and discuss the influence of constitutive model choices and constitutive expression choices separately. As a certain constitutive model may correspond to many different rate form constitutive expressions, the conversion relationships among them are derived, which can help clarify many controversies. First, it is proven that three different stress–strain curves in the controversial simple shear example actually correspond to three constitutive models. For each constitutive model, the corresponding different rate form constitutive expressions yield consistent calculation results. We also demonstrate that the applicability of a constitutive model is completely independent of the choice of objective stress rates in the constitutive expressions. Second, we find that for FEM simulation, the element tangent stiffness matrix can serve as an indicator to evaluate the validity of different constitutive expressions and their corresponding FEM formulations. Moreover, even though non-work-conjugate stress-strain rates are adopted, such as the Jaumann stress rate/deformation rate, different rate form constitutive expressions can correctly reflect the constitutive model with the proper tangent modulus tensors. All these conclusions have been verified by examples. For convenience, we recommend using the constitutive expressions in terms of the second Piola–Kirchhoff stress versus Green’s strain tensor and Truesdell stress rate versus deformation rate for experimental measurements and FEM simulations. Of course, other objective stress rates are all applicable and have no effect on the calculation results as long as the correct conversion relationships are used.