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2020 | OriginalPaper | Buchkapitel

8. Fourth Rank Properties

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 extension to fourth rank tensors of the ideas presented in the previous chapter is straightforward. Properties like magnetoresistivity \(\underline{\underline{\underline{\underline{\rho }}}}^{mag}\), the Kerr effect coefficient \(\underline{\underline{\underline{\underline{K}}}}\), the piezo-optic coefficient \(\underline{\underline{\underline{\underline{\pi }}}}^{opt}\), the electrostriction coefficient \(\underline{\underline{\underline{\underline{M}}}}\), and the elastic compliance \(\underline{\underline{\underline{\underline{s}}}}\) and stiffness \(\underline{\underline{\underline{\underline{c}}}}\) are fourth rank properties whose RSs are defined in the same way.

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Vorheriges Kapitel Third Rank Properties

Nächstes Kapitel Transversality and Skew Symmetric Properties

Check explicitly that adding an inversion center to the single fourfold axis of tetragonal class 4 increases the symmetry to that of point group 4/mmm. Hint: take a generic point of coordinates \((x_1,x_2,x_3)\), transform its coordinates by all actions of the 4 axis and of the inversion center \(\bar{1}\). To which point group does the resulting set of points belong?

It is indirectly related because these entities are responsible for the numerical values of the property components.

Why have we not discussed hexagonal classes \(6,\bar{6},6/m\), for which the convention does not define ② ③ axes either?

In the following we will use \(\underline{\underline{\underline{\underline{s}}}}\) only, knowing that all results are valid for \(\underline{\underline{\underline{\underline{c}}}}\) as well, except for the factors of 2 or 4 required by Voigt’s notation.

This shear stress-longitudinal deformation coupling is zero for the isotropic material, but non-zero in the general case.

Threads of the same weight (size) and the same number of ends as picks per unit length [5].

\(\tau _{11}\) is not homogeneous. Can you plot \(\tau _{11}\) as a function of distance from the mid-plane?

So that strain and stress can be considered constant through the thickness.

For tetragonal point group 4/mmm it is also correct to align the axes ① ② along the diagonals of the squares formed by warp and weft threads.

References [6, 7] are excellent references.

E.g. if the conventional ③ axes of the metal grains in a wire have a tendency to be oriented along the drawing direction.

E.g. the orientation of the PVDF molecules in the poling process of Fig. 4.9.

Why are only even exponents used in these examples?

What would be the value of the integral if the integrand were \(\underline{n}\)? If \(\underline{n}\,\underline{n}\,\underline{n}\)? If \(\underline{n}\cdot \underline{n}\)?

Why the third row? Could we have taken any other row or column of \({\underset{{{{\backsim }}}}{L}}\)?

Does the VRH-averaged material satisfy (8.50)?

Or all three, if an average like \(\langle c'_{1223}\rangle \) were to be computed.

Again, maybe three for \(\langle c'_{1223}\rangle \).

Hint: you will need three orthonormal vectors.

Can you show why \(\underline{\underline{\underline{\underline{\underline{s}}}}}\) has 56 independent components?

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

Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. The Foundations of Mechanics and Thermodynamics, pp. 145–156. Springer, Berlin (1974)

Weiner, J.H.: Statistical Mechanics of Elasticity. Wiley, New York (1983)

Jones, R.M.: Mechanics of Composite Materials. Taylor & Francis, Boca Raton (1999)

Collier, A.M.: A Handbook of Textiles. Pergamon Press, Oxford (1970)

Kocks, F., Tomé, C.N., Wenk, H.-R.: Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties. Cambridge University Press, Cambridge (1998)

Bunge, H.-J.: Texture Analysis in Materials Science: Mathematical Methods. Elsevier, Amsterdam (2013)

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

Gurvich, M.R., Skudra, A.M.: Effect of the geometry of the structure on the strength distribution of multilaminate tridirectional reinforced plastics. Mech. Compos. Mater. 24(5), 602–609 (1989)

10.

Bechmann, R., Hearmon, R.F.S.: The third-order elastic constants. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, vol. 2, p. 112. Springer, Berlin (1969)

Titel: Fourth Rank Properties
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_8

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