Abstract
This paper is intended to serve as a blueprint for the first few chapters of future textbooks on continuum mechanics and continuum thermomechanics. It gives precise intrinsic formulation of the laws of balance of forces and torques, balance of energy, and the concepts of temperature and entropy. They are intrinsic in the sense that they do not involve external frames of reference such as a “physical space”. In the end, an intrinsic reduced dissipation inequality is derived, which plays a crucial role in formulating frame-free constitutive laws.
Highlights
This paper is intended to serve as a model for the first few chapters of future textbooks on continuum mechanics and continuum thermomechanics
The present paper differs from most existing textbooks on the subject in several important respects: 1) It uses the mathematical infrastructure based on sets, mappings, and families, rather than the infrastructure based on variables, constants, and parameters. (For a detailed explanation, see The Conceptual Infrastructure of Mathematics by W.N. [N1].) 2) It is completely coordinate-free and IR n-free when dealing with basic concepts
Not so in continuum mechanics, where it is often appropriate to neglect inertia, for example when analyzing the motion of toothpaste when it is extruded slowly from a tube
Summary
This paper is intended to serve as a model for the first few chapters of future textbooks on continuum mechanics and continuum thermomechanics. For example: 1) It assumes that internal interactions at a distance, both forces and heat transfers, are absent They should be included in a more inclusive analysis because they are important, for example, in applications of continuum thermomechanics to astrophysics. (absolute space is present neither in relativity nor in continuum physics as presented here.) What is needed is some non-relativistic approximation for electromagnetism. In the future, the issues just described will be treated in the same spirit as the present paper, in particular by using the mathematical infrastructure based on sets, mappings, and families and without using a fixed physical space, 1. Ωp := {q ∈ Ω | q ≺ p} is a materially ordered set and the remainder mapping in Ωp is given by remp := (a → arem ∧ p)
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