On tensor functions whose gradients have some skew-symmetries

Montanaro, Adriano

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 3, pp. 259-268.

Abstract (VO)

Let \( \mathcal{V}_{n} \) be a real inner product space of any dimension; and let \( Q^{\alpha_{1} \dots \alpha_{v}} = \hat{Q}^{\alpha_{1} \dots \alpha_{v}} (X_{\beta{1} \dots \beta_{\tau}}) \) be a \( C^{2} \)-map relating any two tensor spaces on \( \mathcal{V}_{n} \). 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 \( ( \alpha_{\mu},\beta_{\eta} ) \) 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.

Riassunto

Siano \( \mathcal{V}_{n} \) uno spazio lineare di qualunque dimensione, dotato di prodotto interno, e \( Q^{\alpha_{1} \dots \alpha_{v}} = \hat{Q}^{\alpha_{1} \dots \alpha_{v}} (X_{\beta{1} \dots \beta_{\tau}}) \) una funzione di classe \( C^{2} \) tra due arbitrari spazi tensoriali su \( \mathcal{V}_{n} \). Si trova l'espressione di questa funzione dovuta alla condizione di antisimmetria del suo gradiente rispetto a coppie di indici \( ( \alpha_{\mu},\beta_{\eta} ) \). 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.

Bibliographie

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

[2] A. Bressan - A. Montanaro, 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 | Zbl

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

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

[5] A. Montanaro, 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 | Zbl

[6] A. Montanaro, 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] A. Montanaro - D. Pigozzi, 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

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

[9] A. Montanaro - D. Pigozzi, 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

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