Some primitive concepts in continuum mechanics regarded in terms of space-time molecular averaging: the key role played by inertial observers

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Murdoch, A.I. (2006) Some primitive concepts in continuum mechanics regarded in terms of space-time molecular averaging: the key role played by inertial observers. Journal of Elasticity, 84 (1). pp. 69-97. ISSN 0374-3535 (http://dx.doi.org/10.1007/s10659-005-9048-7)

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Abstract

In Continuum Mechanics the notions of body, material point, and motion, are primitive. Here these concepts are derived for any (possibly time-dependent) material system via mass and momentum densities whose values are local spacetime averages of molecular quantities. The averaging procedure necessary to ensure molecular-based densities can be agreed upon by all observers (that is, are objective) has implications for constitutive relations. Specifically, such relations should first be expressed in terms of Galilean-invariant functions of the motion relative to an inertial frame. Thereafter such relations can be re-phrased for general observers, thereby yielding general-frame constitutive relations compatible with material frame-indifference. Two postulates concerning observer agreement (which together constitute a statement of material frame-indifference) are shown to imply that any stress response function which is assumed to depend upon the motion in an inertial (general) frame must be Galilean-invariant (invariant under superposed rigid body motions). Accordingly, invariance under superposed rigid body motions is not a fundamental tenet of continuum physics, but rather a consequence of material frame-indifference whenever constitutive dependence upon motion in a general observer frame is postulated.

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