Isotropic materials

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, pp. 84-85

Section 1.3.3.5. Isotropic materials

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

1.3.3.5. Isotropic materials

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The isotropy relation between elastic compliances and elastic stiffnesses is given in Section 1.3.3.2.3. For reasons of symmetry, the directions of the eigenvectors of the stress and strain tensors are necessarily the same in an isotropic medium. If we take these directions as axes, the two tensors are automatically diagonalized and the second relation (1.3.3.7) becomes $[\eqalign{T_{1} &= c_{11}S_{1}+ c_{12}(S_{2}+ S_{3})\cr T_{2} &= c_{12}S_{1}+ c_{11}S_{2}+ c_{12}S_{3}\cr T_{3} &= c_{12}(S_{1}+ S_{2}) + c_{11}S_{3}.\cr} ]$

These relations can equally well be written in the symmetrical form $[\eqalign{T_{1}&= (c_{11}- c_{12})S_{1}+ c_{12}(S_{1}+ S_{2}+ S_{3})\cr T_{2}&= (c_{11}- c_{12})S_{2}+ c_{12}(S_{1}+ S_{2}+ S_{3})\cr T_{3}&= (c_{11}- c_{12})S_{3}+ c_{12}(S_{1}+ S_{2}+ S_{3}).\cr} ]$

If one introduces the Lamé constants , $[\eqalign{\mu & = (1/2) (c_{11}- c_{12}) = c_{44}\cr \lambda &= c_{12},\cr} ]$ the equations may be written in the form often used in mechanics: $[\eqalign{T_{1} &= 2\mu S_{1} + \lambda (S_{1} +S_{2} + S_{3})\cr T_{2} &= 2\mu S_{2} + \lambda (S_{1} +S_{2} + S_{3})\cr T_{3} &= 2\mu S_{3} + \lambda (S_{1} +S_{2} + S_{3}).\cr} \eqno(1.3.3.16) ]$

Two coefficients suffice to define the elastic properties of an isotropic material, $[s_{11}]$ and $[s_{12}]$ , $[c_{11}]$ and $[c_{12}]$ , μ and λ, μ and ν, etc. Table 1.3.3.3 gives the relations between the more common elastic coefficients.

Table 1.3.3.3 | top | pdf |
Relations between elastic coefficients in isotropic media

Coefficient	In terms of μ and λ	In terms of μ and ν	In terms of $[c_{11} \hbox{ and } c_{12} ]$
$[c_{11}]$	$[2\mu + \lambda]$	$[2\mu(1-\nu)(1-2\nu)]$	$[c_{11}]$
$[c_{12}]$	λ	$[2\mu \nu(1-2\nu)]$	$[c_{12}]$
$[c_{44} = 1/s_{44}]$	μ	μ	$[(c_{11}-c_{12})/2]$
$[E = 1/s_{11}]$	$[\mu(2\mu+3\lambda)/(\mu+\lambda) ]$	$[2\mu(1+\nu)]$	See Section 1.3.3.2.3
$[s_{12}]$	$[-\lambda/\left[2\mu(2\mu+3\lambda)\right] ]$	$[-\nu/\left[2\mu(1+\nu)\right] ]$	See Section 1.3.3.2.3
κ	$[3/(2\mu+3\lambda)]$	$[3(1-2\nu)/\left[2\mu(1+\nu)\right] ]$	$[3/(c_{11}+2c_{12})]$
$[\nu = - s_{12}/s_{11}]$	$[\lambda/\left[2(2\mu+3\lambda)\right] ]$	ν	$[c_{11}/(c_{11}+c_{12})]$