Acta Mech (2008) DOI 10.1007/s00707-013-0973-4 I-Shih Liu · Rubens Sampaio Remarks on material frame-indifference controversy Received: 10 February 2013 / Revised: 5 July 2013 © Springer-Verlag Wien 2013 Abstract Regarding a frame of reference as an observer, the notion of frame-indifference (also referred to as objectivity) concerns the transformation properties of physical quantities under change of observer. For kine- matic quantities, they can usually be derived from the deformation/motion, while for non-kinematic quantities, such as force and stress, A frame-indifference property 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 confusions in the interpretation of “material” frame-indifference in the literature. Material frame-indifference deals with the constitutive functions, which characterize intrinsic properties of the material under different observers. We shall carefully render these con- cepts mathematically and deduce the well-known condition of material objectivity as a consequence of the frame-indifference postulate and the principle of material frame-indifference. We shall also emphasize and remark on some persistent controversy and some misleading statements found in recent literature. 1 Introduction The foundation of Modern Continuum Mechanics is based on the fundamental ideas, set forth in the biblical treatise, The Nonlinear Field Theories of Mechanics, by Truesdell and Noll [17], among which one of the most important and controversial ones is the principle of material frame-indifference (MFI). Due to the original somewhat loose statements, attempts for better interpretation of MFI appeared again and again in the literature and some controversy persisted to these days [1, 2, 4, 11, 13, 16, 18]. As we understand, the essential meaning of MFI is the simple idea that material properties are independent of observers. In order to explain this, one has to know what a frame of reference (regarded as an observer) is, so as to define what frame-indifference means. However, frame-indifference concerns only transformation prop- erties of physical quantities under change of frame, and hence has nothing to do with the material properties. On the other hand, material frame-indifference, which deals with constitutive functions characterizing intrinsic properties of the material under different observers, is postulated as a guiding principle for the formulation of constitutive models. Therefore, frame-indifference and material frame-indifference are two different concepts not to be confused. We shall make these different concepts mathematically clear in this paper. A similar version of this paper was included in a conference proceedings [8]. Nevertheless, the present one has been more carefully refined, and it is one of our objectives to make additional remarks and comments on some persistent controversy and misleading statements found in recent literature. I-Shih Liu (B ) Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil E-mail: liu@im.ufrj.br R. Sampaio Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil