On tensor functions whose gradients have some skew-symmetries

Montanaro, Adriano

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 2 (1991), p. 259-268 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

Let $V_{n}$ be a real inner product space of any dimension; and let $Q^{α_{1} \dots α_{v}} = {\overset{⌃}{Q}}^{α_{1} \dots α_{v}} (X_{β 1 \dots β_{τ}})$ be a $C^{2}$ -map relating any two tensor spaces on $V_{n}$ . We study the consequences imposed on the form of this function by the condition that its gradient should be skew-symmetric with respect to some pairs $(α_{μ}, β_{η})$ of indexes. Any such a condition is written as a system of linear partial differential equations, with constant coefficients, which is symmetric with respect to certain couples of independent variables. The solutions of these systems appear useful to characterize the possible indéterminations in the admissible systems of constitutive equations for various continuous media.

Siano $V_{n}$ uno spazio lineare di qualunque dimensione, dotato di prodotto interno, e $Q^{α_{1} \dots α_{v}} = {\overset{⌃}{Q}}^{α_{1} \dots α_{v}} (X_{β 1 \dots β_{τ}})$ una funzione di classe $C^{2}$ tra due arbitrari spazi tensoriali su $V_{n}$ . Si trova l'espressione di questa funzione dovuta alla condizione di antisimmetria del suo gradiente rispetto a coppie di indici $(α_{μ}, β_{η})$ . Una tale condizione equivale a un certo sistema di equazioni lineari a coefficienti costanti, alle derivate parziali, simmetrico rispetto a una coppia di variabili indipendenti. La conoscenza delle soluzioni di tale sistema è utile per caratterizzare le possibili indeterminazioni degli ammissibili sistemi di equazioni costitutive per vari mezzi continui.

Publié le : 1991-09-01

MR 1135430 | Zbl 0759.53006

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     author = {Adriano Montanaro},
     title = {On tensor functions whose gradients have some skew-symmetries},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {2},
     year = {1991},
     pages = {259-268},
     zbl = {0759.53006},
     mrnumber = {1135430},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1991_9_2_3_259_0}
}

Montanaro, Adriano. On tensor functions whose gradients have some skew-symmetries. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 2 (1991) pp. 259-268. http://gdmltest.u-ga.fr/item/RLIN_1991_9_2_3_259_0/

Bibliographie

[1] Bowen, R. M. - Wang, C.-C., Introduction to vectors and tensors. Plenum Press, New York-London 1976. | MR 460004 | Zbl 0329.53008

[2] Bressan, A. - Montanaro, A., On the uniqueness of response stress-functionals for purely mechanical continuous media, from the Mach-Painlevé point of view. Mem. Mat. Acc. Lincei, s. 9, vol. 1, fase. 3, 1990, 59-94. | MR 1082620 | Zbl 0725.73004

[3] Edelen, D. G. B., The null set of the Euler-Lagrange operator. Arch. Rational Mech. Anal., vol. 11, 1962, 117-121. | MR 150623 | Zbl 0125.33002

[4] Montanaro, A., On heat flux in simple media. Journal of Elasticity, to appear. | MR 1207018 | Zbl 0768.73008

[5] Montanaro, A., On the response functions of a thermo-elastic body, from the Mach-Painlevé point of view. Mem. Mat. Acc. Lincei, s. 9, vol. 1, fase. 5, 1990, 123-146. | MR 1088046 | Zbl 0725.73005

[6] Montanaro, A., On the response functions for the stress and couple-stress in nonsimple elastic materials of grade two without internal constraints. Part 1 - General smooth solution of certain threefold-symmetric systems of linear PDEs, for tensor functions, usefuld to study physically equivalent response stressfunctions. Part 2 - On the indéterminations in the skew-symmetric parts of stress and couple-stress in nonsimple elastic materials of grade two. To appear.

[7] Montanaro, A. - Pigozzi, D., On a large class of symmetric systems of linear PDEs for tensor functions, useful in mathematical physics. Annali di Matematica Pura e Applicata. To appear. | Zbl 0796.35020

[8] Montanaro, A. - Pigozzi, D., On the response function for the heat flux in bodies of the differential type. To appear. | MR 1292090 | Zbl 0805.73009

[9] Montanaro, A. - Pigozzi, D., On the physical general differential indétermination bodies. Part 1 - Physical solutions of the response functions to some symmetric for systems of linear PDEs in tensor functions. Part 2 - On uniqueness properties of the response functions for the stress and internal energy in differential bodies. To appear. | Zbl 0815.73005

[10] Toupin, R. A., Theories of Elasticity with Couple-stress. Arch. Rational Mech. Anal., vol. 17, 1964, 7-112. | MR 169425 | Zbl 0131.22001