On Objectivity and the Principle of Material Frame-Indifference

Abstract

In the nineteen sixties and the seventies, rapid development of Modern Continuum Mechanics was based on the fundamental ideas set forth in the biblical treatise, The Non-Linear Field Theories of Mechanics, by Truesdell and Noll. Of them, one of the most important ideas is the principle of material
frame-indifference (MFI). Unfortunately, due to the original somewhat loose statements, attempts for better interpretation of MFI appeared again and again throughout the following decades even until these days. Some involving serious misunderstandings and misinterpretations, hidden behind some seemingly plausible physical arguments or some impressive yet over-sophisticated mathematics.
As we understand, the essential meaning of MFI is the simple idea that material properties are independent of observers. In order to explain this, we shall describe what a frame of reference (regarded as an observer) is, and one shall never forget that any configuration/motion implies a previous choice of frame.
Transformation properties for kinematic quantities can usually be derived from the deformation/motion under change of frame. For a non-kinematic quantity, such as force and stress, frame-indifference property (also known as objectivity) can not be derived and hence must be postulated. Frame-indifference postulate for the stress, sometimes unsuitably called the principle of frame-indifference, is a universal assumption which has nothing to do with material properties. This has caused some great confusions in the interpretation of “material” frame-indifference in the literature.
Mathematically, the principle of material frame-indifference can be stated as invariance of constitutive function under change of frame. However, great care must be observed of what such constitutive functions are. They should not simply be the constitutive functions relative to some reference configuration in two different frames, because a choice of reference configuration may change material properties.
We shall carefully state the domain of constitutive functions and with clear and simple mathematical reasoning deduce the well-known condition of material objectivity as a consequence of objectivity postulate and the principle of material frame-indifference.
In this paper we show some misunderstandings and misstatements found in the very recent literature and we show how to correct them.