Second-order and higher-order elastic stiffnesses

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. 93

Section 1.3.6.4. Second-order and higher-order elastic stiffnesses

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

1.3.6.4. Second-order and higher-order elastic stiffnesses

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Following Brugger (1964), the strain-energy density, or strain energy per unit volume Φ, is assumed to be a polynomial in the strain: $[\Phi = \Phi_{0} + c_{ij}S_{ij} + {1\over 2!}c_{ijkl}S_{ij}S_{kl} + {1\over 3!}c_{ijklmn}S_{ij}S_{kl}S_{mn}, \eqno(1.3.6.9) ]$ where $[\Phi = \rho_{0} U(X,S_{ij})]$ , $[\Phi_{0} = \rho_{0} U(X,0)]$ , X denotes the configuration of the initial state and the $[S_{ij}]$ 's are the Lagrangian finite strain-tensor components.

If the initial energy and the deformation of the body are both zero, the first two terms in (1.3.6.9) are zero. Note that $[c_{ij}]$ is a stress and not an intrinsic characteristic of the material. In this expression, the elastic stiffnesses $[c_{ijkl}]$ and $[c_{ijklmn}]$ are the second- and third-order stiffnesses , respectively. Since the strain tensor is symmetric, pairs of subscripts can be interchanged [see equation (1.3.3.4)]: $[\eqalign{c_{ijkl} &= c_{jikl} = c_{ijlk} = c_{jilk},\cr c_{ijklmn} &= c_{jiklmn} = c_{ijlkmn} = c_{jilkmn} = c_{ijklnm} \cr&= c_{jiklnm} = c_{ijlknm} = c_{jilknm}.\cr} ]$

More accurately, the isentropic and the isothermal elastic stiffnesses are defined as the nth partial derivatives of the internal energy and the Helmholtz free energy, respectively. For example, the third-order isentropic and isothermal stiffnesses are, respectively, $[\eqalign{c_{ijkl mn}^{\sigma} &= \rho_{0} {\partial^{3} U \over \partial S_{ij}\partial S_{kl}\partial S_{mn}},\cr c_{ijkl mn}^{\Theta} &= \rho_{0} {\partial^{3} F \over \partial S_{ij}\partial S_{kl}\partial S_{mn}},\cr} ]$ where the internal energy, U, is a function of X, $[S_{ij}]$ and σ, and the Helmholtz free energy, F, is a function of X, $[S_{ij}]$ and Θ.

From these definitions, it follows that the Brugger stiffness coefficients depend on the initial state. When no additional information is given, the initial state is the natural state.

The third-order stiffnesses form a sixth-rank tensor containing $[3^{6} = 729]$ components, of which 56 are independent for a triclinic crystal and 3 for isotropic materials (the independent components of a sixth-rank tensor can be obtained for any point group using the accompanying software to this volume). The three independent constants for isotropic materials are often taken as $[c_{123}]$ , $[c_{144}]$ and $[c_{456}]$ and denoted respectively by $[\nu_{1} ]$ , $[\nu_{2}]$ , $[\nu_{3}]$ , the `third-order Lamé constants '.

The `third-order Murnaghan constants' (Murnaghan, 1951), denoted by [l, m, n] , are given in terms of the Brugger constants by the relations $[l = \textstyle{1\over 2} c_{112} ; \ m = c_{155} ; \ n = 4c_{456}. ]$

Similarly, the fourth-order stiffnesses form an eighth-rank tensor containing $[3^{8} = 6561]$ components, 126 of which are independent for a triclinic crystal and 11 for isotropic materials (the independent components of a sixth-rank tensor can be obtained for any point group using the accompanying software to this volume).

For a solid under finite strain conditions, the definition of the elastic compliance tensor has to be reconsidered. In linear elasticity , the second-order elastic compliances $[s_{ijkl}]$ were defined through the relations (1.3.3.2): $[S_{ij} = s_{ijkl} T_{kl} \quad \hbox{or} \quad s_{ijkl} = {\partial S_{ij} \over \partial T_{kl}}, ]$ while, in nonlinear elasticity, one has $[s_{ijkl} = {\partial S_{ij} \over \partial t_{kl}}, ]$ where $[t_{kl} = \rho_0 {\partial U \over \partial S_{kl}}. ]$

References

Brugger, K. (1964). Thermodynamic definition of higher-order elastic coefficients. Phys. Rev. 133, 1611–1612.Google Scholar

Murnaghan, F. D. (1951). Finite deformation in an elastic solid. New York: John Wiley and Sons.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 93