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2020 | OriginalPaper | Buchkapitel

3. Representation of Material Properties by Means of Cartesian Tensors

verfasst von : Manuel Laso, Nieves Jimeno

Erschienen in: Representation Surfaces for Physical Properties of Materials

Verlag: Springer International Publishing

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Abstract

The goal of this chapter is to familiarize the reader with the basic ideas and the few manipulation rules for Cartesian tensors [1‐6] that will be required in the rest of the book. If you have previously been exposed to and perhaps intimidated by expressions like “covariant” and “contravariant”, you are about to make a quantum leap in your study of tensor material properties: when working with Cartesian tensors the concepts of “covariant” and “contravariant” are not necessary. The manipulation rules for Cartesian tensors are but a minor extension of the familiar rules for vector and matrices. In the rest of the book the adjective “Cartesian” will often be dropped for brevity.

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Vorheriges Kapitel Geometric Symmetry

Nächstes Kapitel Representation Surfaces

Although in specific cases such as \(\underline{a}\cdot \underline{b}=\underline{b}\cdot \underline{a}\) the order of the factors can be irrelevant.

The permutation or Levi-Civita symbol provides the componentes of the alternating or totally antisymmetric tensor \({\underline{\underline{\underline{\epsilon }}}}\), which is a third rank, axial, antisymmetric tensor. The values of \(\epsilon _{ijk}\) are the components of \({\underline{\underline{\underline{\epsilon }}}}\), just as the values of Kronecker’s \(\delta _{ij}\) are the components of the second rank unit or isotropic tensor \({\underline{\underline{\delta }}}\).

It is not unusual to find notations like \(\mathbf {T}\,_{\hat{\mathrm{e}}}\), where the subscript \(\underline{\hat{\mathrm{e}}}\) explicitly indicates the frame to which the tensor \(\mathbf {T}\) is referred.

All rank zero tensors (scalars) are isotropic; there is no rank one isotropic tensor. The number of isotropic tensors for ranks \(n=0,1,2,3,4,5,6,\dots \) is \(1,0,1,1,3,6,15\dots \) but we will not need them for \(n>4\).

An infinite number of invariants can be defined from a given one: if I is an invariant, any function containing only I and constants will be invariant as well, although only one of them will be independent.

We strictly follow [6] left-to-right rule: indices are assigned to the terms appearing in an expression in the same order in which they appear when the expression is read from left to right; see (3.13) and (3.14). If \(\underline{v}\) in (3.13) is a velocity field, the left-to-right rule leads to a velocity gradient which is the transpose of that defined by other authors: \(\underline{\delta }_i\underline{\delta }_j\frac{\partial v_i}{\partial x_j }\).

It is also common practice to call \(x_i\) the components of \(\underline{r}\), so that \(\underline{r}=\underline{\delta }_ix_i\).

The use of a “prime” to tag the new frame in this particular case does not imply that the old frame will always be unprimed and the new frame always be primed.

Again, the nine angles \(\theta _{ij}\) (or their nine cosines \(l_{ij}\)) are not independent. Only three are. This is a consequence of the fact that in order to orient a frame of reference with respect to another in two dimensions, a single angle suffices, whereas three are required in three dimensions (for example three Euler angles). However, to orient a single axis or vector with respect to a frame of reference, one angle is enough in two dimensions, and two in three dimensions. We will come back to this point in later chapters.

The fact that \(\underline{\underline{\sigma }}\) is a symmetrical tensor did not play any role in this derivation. Equation 3.24 is valid for asymmetric tensors as well.

The following applies to tensors without any special symmetry. Symmetry reduces the number of independent components.

Why double?

This is the reason for naming the stress tensor \(\underline{\underline{\tau }}\) instead of \(\underline{\underline{\sigma }}\), (that is reserved for the electric conductivity) and the dielectric permittivity \(\underline{\underline{\kappa }}\) instead of \(\underline{\underline{\epsilon }}\) (which is reserved for the deformation or displacement gradient tensor).

What is the second action of the plane?

The extensive compilation [14] goes up to rank seven.

Jeffreys, H.: Cartesian Tensors. Cambridge University Press, Cambridge (1969)

Halmos, P.R.: Finite Dimensional Vector Spaces. Van Nostrand-Reinhold, New York (1958)

Gel’fand, I.M.: Lectures on Linear Algebra. Dover Publications Inc., Mineola (1989)

Knowles, J.K.: Linear Vector Spaces and Cartesian Tensors. Oxford University Press, Oxford (1998)

Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists. Harcourt Academic Press, San Diego (2001)

Bird, B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, Hoboken (2002)

Dodson, C.T.J., Poston, T.: Tensor Geometry: The Geometric Viewpoint and Its Uses. Springer, Berlin (2013)

Das, A.J.: Tensors: The Mathematics of Relativity Theory and Continuum Mechanics. Springer, Berlin (2007)CrossRef

Jeevanjee, N.: An Introduction to Tensors and Group Theory for Physicists. Springer, Berlin (2011)CrossRef

10.

Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer, Berlin (2012)

11.

Newnham, R.E.: Properties of Materials: Anisotropy, Symmetry. Structure. Oxford University Press, Oxford (2005)

12.

Tinder, R.F.: Tensor Properties of Solids: Phenomenological Development of the Tensor Properties of Crystals. Morgan & Claypool Publishers, San Rafael (2008)

13.

Voigt, W.: Lehrbuch der Kristallphysik. Teubner (1928)

14.

Popov, S.V., Svirko, Y.P., Zheludev, N.I.: CD Encyclopedia of Material Tensors. Wiley, Hoboken (1999)

15.

Nye, J.F.: Physical Properties of Crystals. Oxford University Press, Oxford (1985)

Titel: Representation of Material Properties by Means of Cartesian Tensors
verfasst von: Manuel Laso
Nieves Jimeno
Verlag: Springer International Publishing
Buch: Representation Surfaces for Physical Properties of Materials
Print ISBN: 978-3-030-40869-5

Electronic ISBN: 978-3-030-40870-1

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-40870-1_3

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