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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 82

Section 1.3.3.2.3. Passage from elastic compliances [s_{\alpha \beta}] to elastic stiffnesses [c_{\alpha \beta}]

A. Authiera* and A. Zarembowitchb

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

1.3.3.2.3. Passage from elastic compliances [s_{\alpha \beta}] to elastic stiffnesses [c_{\alpha \beta}]

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We have noted already that the matrix [c_{\alpha \beta}] is the inverse of the matrix [s_{\alpha \beta}]. These matrices can be written for cubic and isotropic materials as follows: [\eqalignno{s_{\alpha \beta} &= \pmatrix{s_{11} &s_{12} &s_{12} &0 &0 &0\cr s_{12} &s_{11} &s_{12} &0 &0 &0\cr s_{12} &s_{12} &s_{11} &0 &0 &0\cr 0 &0 &0 &s_{44} &0 &0\cr 0 &0 &0 &0 &s_{44} &0\cr 0 &0 &0 &0 &0 &s_{44}\cr} &\cr c_{\alpha \beta} &= \pmatrix{c _{11} &c_{12} &c_{12} &0 &0 &0\cr c_{12} &c_{11} &c_{12} &0 &0 &0\cr c_{12} &c_{12} &c _{11} &0 &0 &0\cr 0 &0 &0 &c_{44} &0 &0\cr 0 &0 &0 &0 &c_{44} &0\cr 0 &0 &0 &0 &0 &c_{44}\cr}, &\cr} ]where we have, for isotropic materials, [\eqalign{s_{44} &= 2(s_{11} - s_{12})\cr c_{44} &= \textstyle{1 \over 2} (c_{11} - c_{12}).\cr} \eqno(1.3.3.9) ]We easily find that [\eqalign{s_{11} &= {c_{11} + c_{12} \over (c_{11} - c_{12})(c_{11} + 2 c_{12})}\semi \quad c_{11} = {s_{11} + s_{12} \over (s_{11} - s_{12})(s_{11} + 2 s_{12})}\semi \cr s_{12} &= {- c_{12} \over (c_{11} - c_{12})(c_{11} + 2 c_{12})}\semi \quad c_{12} = {- s_{12}\over (s_{11} - s_{12})(s_{11} + 2 s_{12})}\semi \cr s_{44} &= {1 \over c_{44}}\semi \phantom{= {- c_{12} \over (c_{11} - c_{12})(c_{11} c_{12})}\semi} c_{44}= {1 \over s_{44}}. \cr} ]The coefficient [c_{44}] is sometimes called the rigidity modulus.








































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