On tensor functions whose gradients have some skew-symmetries

Adriano Montanaro

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1991)

  • Volume: 2, Issue: 3, page 259-268
  • ISSN: 1120-6330

Abstract

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Let V n be a real inner product space of any dimension; and let Q α 1 ⁢ … ⁢ α v = Q ⌃ α 1 ⁢ … ⁢ α v ⁢ X β ⁢ 1 ⁢ … ⁢ 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.

How to cite

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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 2.3 (1991): 259-268. <http://eudml.org/doc/244135>.

@article{Montanaro1991,
abstract = {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.},
author = {Montanaro, Adriano},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Tensor calculus; Partial differential equations; Continuous media; tensor gradient; homogeneous partial differential equations},
language = {eng},
month = {9},
number = {3},
pages = {259-268},
publisher = {Accademia Nazionale dei Lincei},
title = {On tensor functions whose gradients have some skew-symmetries},
url = {http://eudml.org/doc/244135},
volume = {2},
year = {1991},
}

TY - JOUR
AU - Montanaro, Adriano
TI - On tensor functions whose gradients have some skew-symmetries
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1991/9//
PB - Accademia Nazionale dei Lincei
VL - 2
IS - 3
SP - 259
EP - 268
AB - 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.
LA - eng
KW - Tensor calculus; Partial differential equations; Continuous media; tensor gradient; homogeneous partial differential equations
UR - http://eudml.org/doc/244135
ER -

References

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  1. BOWEN, R. M. - WANG, C.-C., Introduction to vectors and tensors. Plenum Press, New York-London1976. Zbl0329.53008MR460004
  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. Zbl0725.73004MR1082620
  3. EDELEN, D. G. B., The null set of the Euler-Lagrange operator. Arch. Rational Mech. Anal., vol. 11, 1962, 117-121. Zbl0125.33002MR150623
  4. MONTANARO, A., On heat flux in simple media. Journal of Elasticity, to appear. Zbl0768.73008MR1207018DOI10.1007/BF00041772
  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. Zbl0725.73005MR1088046
  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. Zbl0796.35020
  8. MONTANARO, A. - PIGOZZI, D., On the response function for the heat flux in bodies of the differential type. To appear. Zbl0805.73009MR1292090
  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. Zbl0815.73005
  10. TOUPIN, R. A., Theories of Elasticity with Couple-stress. Arch. Rational Mech. Anal., vol. 17, 1964, 7-112. Zbl0131.22001MR169425

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