On Volume Average Relations in Continuum Mechanics, Part II

Författare

Per Lidström

Summary, in English

This paper continues the analysis of volume average relations that was initiated in [1]. In this paper, as well as in the previous one, results from various investigations, presented in the literature, have been reviewed and extended and they have, in some cases, been more precisely stated and rigorously proven. Some of the results in the present paper are believed to be entirely new. The key objective is the investigation of the concept of average consistence for continuum mechanical relations and the derivation of various integral representations for volume averaged quantities. This paper is restricted to energy and entropy. Results on kinematics, mass, momentum and moment of momentum are presented in [1]. The role of the so-called Hill-Mandel condition, in connection with average consistence of the net power, has been elucidated. The significance of volume averages of stress moments for the calculation of the average net power is demonstrated. Applications to micromechanics will sometimes require the possibility of introducing different types of singularities, such as singular surfaces and cracks, into the thermo-mechanical description. In this paper it is assumed that cracks are absent, but singular surfaces giving rise to jump discontinuities in the thermo-mechanical variables are allowed for. The Appendices contain some mathematical theorems used in the main text. For instance, a non-standard theorem on the inner product of multi-linear mappings is presented with a proof.

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