Definition

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 81

Section 1.3.3.2.1. Definition

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.3.2.1. Definition

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Young's modulus is not sufficient to describe the deformation of the bar: its diameter is reduced, in effect, during the elongation. One other coefficient, at least, is therefore necessary. In a general way, let us consider the deformation of a continuous anisotropic medium under the action of a field of applied stresses. We will generalize Hooke's law by writing that at each point there is a linear relation between the components $[T_{ij}]$ of the stress tensor and the components $[S_{ij}]$ of the strain tensor: $[\eqalign{S_{ij} &= s_{ijkl}T_{kl}\cr T_{ij} &= c_{ijkl}S_{kl}.\cr} \eqno(1.3.3.2) ]$ The quantities $[s_{ijkl}]$ and $[c_{ijkl}]$ are characteristic of the elastic properties of the medium if it is homogeneous and are independent of the point under consideration. Their tensorial nature can be shown using the demonstration illustrated in Section 1.1.3.4 . Let us take the contracted product of the two sides of each of the two equations of (1.3.3.2) by the components $[x_{i}]$ and $[y_{j}]$ of any two vectors, x and y: $[\eqalignno{S_{ij}x_{i}y{_j} &= s_{ijkl}T_{kl}x_{i}y_j\cr T_{ij}x_{i}y{_j} &= c_{ijkl}S_{kl}x_{i}y _{j}.\cr} ]$ The left-hand sides are bilinear forms since $[S_{ij}]$ and $[T_{ij}]$ are second-rank tensors and the right-hand sides are quadrilinear forms, which shows that $[s_{ijkl}]$ and $[c_{ijkl}]$ are the components of fourth-rank tensors, the tensor of elastic compliances (or moduli) and the tensor of elastic stiffnesses (or coefficients), respectively. The number of their components is equal to 81.

Equations (1.3.3.2) are Taylor expansions limited to the first term. The higher terms involve sixth-rank tensors, $[s_{ijklmn}]$ and $[c_{ijklmn}]$ , with $[3^{6} = 729 ]$ coefficients, called third-order elastic compliances and stiffnesses and eighth-rank tensors with $[3^{8} = 6561]$ coefficients, called fourth-order elastic compliances and stiffnesses . They will be defined in Section 1.3.6.4. Tables for third-order elastic constants are given in Fumi (1951, 1952, 1987). The accompanying software to this volume enables these tables to be derived for any point group.

References

Fumi, F. G. (1951). Third-order elastic coefficients of crystals. Phys. Rev. 83, 1274–1275.Google Scholar

Fumi, F. G. (1952). Third-order elastic coefficients in trigonal and hexagonal crystals. Phys. Rev. 86, 561.Google Scholar

Fumi, F. G. (1987). Tables for the third-order elastic tensors in crystals. Acta Cryst. A43, 587–588.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 81