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. 85

Section 1.3.3.6. Equilibrium conditions of elasticity for isotropic media

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

1.3.3.6. Equilibrium conditions of elasticity for isotropic media

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We saw in Section 1.3.2.3[link] that the condition of equilibrium is [\partial T_{ij} /\partial x_i + \rho F_{j} =0. ]

If we use the relations of elasticity, equation (1.3.3.2)[link], this condition can be rewritten as a condition on the components of the strain tensor: [c_{ijkl }{\partial S_{kl} \over \partial x_j} + \rho F_{i}= 0. ]Recalling that [S_{kl }= \textstyle{1 \over 2}\displaystyle \left[ {\partial u_k \over \partial x_l } + {\partial u_l\over \partial x_k }\right], ]the condition becomes a condition on the displacement vector, [{\bf u}({\bf r})]: [c_{ijkl} {\partial^2 \over \partial x_l \partial x_{j}} + \rho F_{i}= 0. ]In an isotropic orthonormal medium, this equation, projected on the axis [0x_{1}], can be written with the aid of relations (1.3.3.5)[link] and (1.3.3.9)[link]: [\eqalign{&c_{11}{\partial^2 u_1 \over (\partial x_1){}^2 } + c_{12}\left[{\partial^2 u_2 \over \partial x_1 \partial x_2 } + {\partial^2 u_3 \over \partial x_1 \partial x_3 }\right] \cr&\quad+ \textstyle{1\over 2}\displaystyle (c_{11}-c_{12})\left[{\partial^2 u_1 \over (\partial x_2){}^2 } + {\partial^2 u_3 \over \partial x_1 \partial x_3 } + {\partial^2 u_1 \over (\partial x_3){}^2 }\right]+ \rho F_i \cr&\quad\quad= 0.} ] This equation can finally be rearranged in one of the three following forms with the aid of Table 1.3.3.3[link]. [\eqalign{\textstyle{1 \over 2}\displaystyle (c_{11}-c_{12})\Delta {\bf u} + \textstyle{1 \over 2}\displaystyle (c_{11}+c_{12}) \boldnabla(\boldnabla{\bf u}) + \rho {\bf F}&=0\cr \mu \Delta {\bf u} + (\mu + \lambda) \boldnabla(\boldnabla{\bf u}) + \rho {\bf F}&=0\cr \mu \left[\Delta {\bf u} + {1 \over 1 - 2\nu}\boldnabla(\boldnabla{\bf u}) \right]+ \rho {\bf F}&=0.\cr} \eqno(1.3.3.17) ]








































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