Abstract
In this paper the algebraic structure of the set of fourth-order tensors is examined. Special emphasis is given to the two different tensor products which can be defined over this set, when it is regarded as a vector space and as the set of all linear transformations over a vector space, respectively.
The new formalism introduced here makes it possible to write in a simple form the restrictions imposed on the elasticity tensors, in finite elasticity, by balance of angular momentum and by the principle of material frame-indifference.
A reassessment of the various elasticity tensors used in the literature is presented, and some classical results are re-stated in a simple and natural way.
Riassunto
Questo articolo tratta della struttura algebrica dell' insieme dei tensori del quarto ordine. Particolare rilievo è dato alla definizione di prodotto tensoriale tra tensori del secondo ordine, che è diversa a seconda che essi siano considerati transformazioni lineari su di uno spazio vettoriale, oppure essi stessi elementi di uno spazio vettoriale.
Il nuovo formalismo introdotto rende semplice la determinazione delle restrizioni imposte ai tensori elastici, in Elasticità finita, dal bilancio della quantità di moto e dal principio di indifferenza materiale.
Diventa anche possibile un riordinamento dei tensori elastici piò frequentemente usati nella letteratura. Ciò permette di ritrovare, in modo semplice e naturale, alcuni risultati classici.
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Gurtin, M. E.,The Linear Theory of Elasticity. Handbuch der Physik, Vol. VIa/2, Berlin, Heidelberg, New York: Springer 1972.
Noll, W., A new mathematical theory of simple materials.Arch. Rational Mech. Anal. 48 (1972) 1–50.
Halmos, P. R.,Finite-dimensional vector spaces, New York: Van Nostrand 1958.
Noll, W., Lectures on the foundations of Continuum Mechanics and Thermodynamics.Arch. rational Mech. Anal. 52 (1973) 62–92.
Truesdell, C., & W. Noll,The non-linear field theories of Mechanics. Handbuch der Physik, Vol. III/3. Berlin, Heidelberg, New York: Springer 1965.
Coddington, E. A., & N. Levinson,Theory of ordinary differential equations. New York: Mc. Graw-Hill 1955.
Noll, W., On the continuity of the solid and fluid states.J. Rational Mech. Anal. 4 (1955) 3–81.
Chadwick, P., & R. W. Ogden, “On the definition of elastic moduli”.Arch. Rational Mech. Anal. 44 (1971) 41–53.
Gurtin, M. E.,An introduction to classical Continuum Mechanics (to appear).
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Dedicated to Guido Stampacchia,in mortem
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Del Piero, G. Some properties of the set of fourth-order tensors, with application to elasticity. J Elasticity 9, 245–261 (1979). https://doi.org/10.1007/BF00041097
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DOI: https://doi.org/10.1007/BF00041097