2013
DOI: 10.1002/zamm.201200225
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Symmetry classes for odd‐order tensors

Abstract: We give a complete general answer to the problem, recurrent in continuum mechanics, of determining the number and type of symmetry classes of an odd-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, this problem has been solved for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. In all the aforementioned papers, the results are obtained after some lengthy computations. In a former co… Show more

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Cited by 34 publications
(43 citation statements)
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“…Mindlin's second strain gradient theory must now be extended to the anisotropic case which involves a large number of higher order elastic moduli. The symmetry classes and corresponding independent elastic moduli were recently determined for 6-th order tensors arising in first strain gradient elasticity, see (Auffray et al, 2013b;Olive and Auffray, 2014;Auffray, 2014). The mathematical tools presented in the latter references can be used to extend these representations to the 8th order tensor of elastic moduli arising in the second strain gradient model, together with odd-order coupling tensors.…”
Section: Discussionmentioning
confidence: 98%
See 1 more Smart Citation
“…Mindlin's second strain gradient theory must now be extended to the anisotropic case which involves a large number of higher order elastic moduli. The symmetry classes and corresponding independent elastic moduli were recently determined for 6-th order tensors arising in first strain gradient elasticity, see (Auffray et al, 2013b;Olive and Auffray, 2014;Auffray, 2014). The mathematical tools presented in the latter references can be used to extend these representations to the 8th order tensor of elastic moduli arising in the second strain gradient model, together with odd-order coupling tensors.…”
Section: Discussionmentioning
confidence: 98%
“…The application of symmetry conditions to the constitutive elasticity tensors in gradient continua leads to the definition of proper symmetry classes, as recently done by Olive and Auffray (2013); Auffray et al (2013b); Olive and Auffray (2014). Mindlin (1965) showed in a milestone paper that the linear elastic isotropic first strain gradient theory is insufficient to describe internal strains and stresses that develop close to free surfaces.…”
Section: Introductionmentioning
confidence: 97%
“…At the present time, these questions remain open for the fifth-order tensor spaces M and M ♯ , both in 2D and 3D. Some theoretical results are available concerning the 3D case (Olive and Auffray, 2014;Auffray, 2014), but without explicit construction. In order to have a complete SGE theory to model dispersive media, answering the aforementioned three questions for M and M ♯ is important.…”
Section: Synthesismentioning
confidence: 99%
“…And so Piez is defined by the following {α k } family:{0, 2, 1, 1}. As determined in [13,14,23] the space of piezoelectric tensors can be divided into the following anisotropic systems:…”
Section: Piezoelectricitymentioning
confidence: 99%
“…To avoid any misunderstanding, it is worth noting that the present method does not solve the symmetry classification for a physical tensor, but only provide a way to compute, once the classification has been done, the number of generic coefficients for each anisotropic system. The solution of the classification problem can be found in the following references [25,23]. As an illustration, the obtained formula are applied to the space of third-order piezoelectric tensor [13,14], and to the fifth-order coupling tensor of Mindlin's strain-gradient elasticity [16].…”
Section: Introductionmentioning
confidence: 99%