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, pp. 81-82

Section 1.3.3.2. Elastic constants

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. Elastic constants

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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[link] . Let us take the contracted product of the two sides of each of the two equations of (1.3.3.2)[link] 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)[link] 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[link]. Tables for third-order elastic constants are given in Fumi (1951[link], 1952[link], 1987[link]). The accompanying software to this volume enables these tables to be derived for any point group.

1.3.3.2.2. Matrix notation – reduction of the number of independent components

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It is convenient to write the relations (1.3.3.2)[link] in matrix form by associating with the stress and strain tensors column matrices [1\times 9] and with the tensors of the elastic stiffnesses, c, and of the elastic compliances, s, square matrices [9\times 9] (Section 1.1.4.10.4[link] ); these two [9\times 9] matrices are inverse to one another. The number of independent components of the fourth-rank elastic tensors can be reduced by three types of consideration:

  • (i) Intrinsic symmetry: it was shown in Section 1.1.1.4[link] that tensors representing principal properties are symmetric. This is the case of the elastic tensors and can be shown directly using expression (1.3.2.7)[link] of the energy stored per unit volume in the medium when we allow it to deform from the state [S_{ij}] to the state [S_{ij} + \delta S_{ij}] under the action of the stress [T_{ij}]: [\delta W = T_{ij} \delta S_{ji}. ]

    Applying relation (1.3.3.1)[link], we get [\partial W / \partial S_{ij} = c_{ijkl}S_{kl}. \eqno(1.3.3.3)]Hence, one has by further differentiation [\partial ^{2}W/ (\partial S_{ij}\partial S_{kl}) = c_{ijkl}. ]

    Nothing is changed by interchanging the role of the pairs of dummy indices ij and lk: [\partial ^{2}W/ (\partial S_{kl}\partial S_{ij}) = c_{klij}. ]

    Since the energy is a state function with a perfect differential, one can interchange the order of the differentiations: the members on the left-hand sides of these two equations are therefore equal; one then deduces [c_{ijkl}= c_{klij}. \eqno(1.3.3.4)]

    The tensor of elastic stiffnesses (and also the tensor of elastic compliances) is thus symmetrical. As shown in Section 1.1.4.5.2.2[link] , the number of their independent components is therefore reduced to 45.

  • (ii) Symmetry of the strain and stress tensors: the strain tensor [S_{ij}] is symmetric by definition (Section 1.3.1.3.1[link]) because rotations are not taken into account and the stress tensor [T_{ij}] is symmetric (Section 1.3.2.4[link]) because body torques are neglected. For this reason, summation (1.3.3.2)[link], [S_{ij} = s_{ijkl}T_{kl}], can be factorized [equation (1.1.4.11)[link] ]: [S_{ij} = \textstyle\sum\limits_{l}\displaystyle s_{ijll}T_{ll} + \textstyle\sum\limits_{k \ne l}\displaystyle \left(s_{ijkl} + s_{ijlk}\right)T_{kl}. ]This shows that the number of independent components of tensor [s_{ijkl}] is reduced. This effect of the symmetry of the strain and stress tensors was discussed systematically in Section 1.1.4.10.4[link] . It was shown that [\eqalign{s_{ijkl} &= s_{ijlk} = s_{jikl} = s_{jilk}\cr c_{ijkl} &= c_{ijlk} = c_{jikl} = c_{jilk}\cr} \eqno(1.3.3.5) ]and that the number of independent elastic compliances or stiffnesses is reduced to 21. They are replaced by two-index coefficients constituting [6\times 6] matrices according to Voigt's notation: [\matrix {& \cr i = j\semi\ \ k = l\cr i = j\semi\ \ k \ne l\cr i \ne j\semi\ \ k = l\cr i \ne j\semi\ \ k \ne l\cr }\left|\matrix{{\rm compliances}\hfill\cr s_{\alpha \beta} = s{iil l}\hfill\cr s_{\alpha \beta} = s{iil l} + s{iil k}\hfill\cr s_{\alpha \beta} = s{ijl l} + s{jil l}\hfill\cr s_{\alpha \beta} = s{ijk l} + s{jik l} + s{ijl k} + s{jil k}\hfill\cr}\right\} \matrix{{\rm stiffnesses}\cr\cr c_{\alpha \beta} = cijkl.\cr\cr\cr} \eqno(1.3.3.6) ]Using these notations and Voigt's notations for the strain and stress tensors, equations (1.3.3.2)[link] become: [\eqalign{S_{\alpha \beta} &= s_{\alpha \beta}T_\beta\cr T_{\alpha \beta} &= c_{\alpha \beta}S_{\beta}.\cr} \eqno(1.3.3.7) ]For instance, the first of these two relations is written, once developed, as[\pmatrix{S _{1}\cr S_{2}\cr S_{3}\cr S_{4}\cr S_{5}\cr S_{6}\cr} = \pmatrix{s_{11} &s_{12} &s_{13} &s_{14} &s_{15} &s_{16}\cr s_{12} &s_{22} &s_{23} &s_{24} &s_{25} &s_{26}\cr s_{13} &s_{23} &s_{33} &s_{34} &s_{35} &s_{36}\cr s_{14} &s_{24} &s_{34} &s_{44} &s_{45} &s_{46}\cr s_{15} &s_{25} &s_{35} &s_{45} &s_{55} &s_{56}\cr s_{16} &s_{26} &s_{36} &s_{46} &s_{56} &s_{66}\cr} \pmatrix{T_{1}\cr T_{2}\cr T_{3}\cr T_{4}\cr T_{5}\cr T_{6}\cr}. \eqno(1.3.3.8) ]

    Matrices [s_{\alpha \beta}] and [c_{\alpha \beta}] are the inverse of each another. It is important to note that they are matrices and not tensors. One cannot apply to them the usual rules of transformation under a change of base since they are only valid for the components of a tensor. In cases where the coordinate system is changed, it is necessary to use the components [s_{ijkl}] and [c_{ijkl}] or to establish the rules of transformation for [s_{\alpha \beta}] and [c_{\alpha \beta}].

  • (iii) Symmetry of the crystal: the reduction of the number of independent components of the matrices [s_{\alpha \beta} ] and [c_{\alpha \beta}] was discussed in Section 1.1.4.10.6[link] . As a summary, Table 1.3.3.1[link] gives the number of independent components for each Laue class.

    Table 1.3.3.1 | top | pdf |
    Number of independent components of the elastic compliances and stiffnesses for each Laue class

    Laue class No. of independent components
    [\bar{1}, 1] 21
    [2/m, 2, m] 13
    [mmm, 222, 2mm] 9
    [\bar{3}, 3] 7
    [\bar{3}m, 32, 3m] 6
    [4/m, 4,\bar{4}] 7
    [4/mmm, 422, \bar{4}2m, 42m ] 6
    [6/m, 6, \bar{6}] 5
    [6/mmm, 622, \bar{6}2m, 62m ] 5
    [m\bar{3}, 23] 3
    [m\bar{3}m, 432, \bar{4}32 ] 3
    [\infty A_{\infty}/M C, \infty A_{\infty} ] 2

  • (iv) Cauchy relation: the form that the strain energy of a discrete-particle structure takes when its potential energy depends only on the magnitude of the distance separating pairs of particles is called the central or point-to-point force law. It is shown that, when the cohesive forces have this special form, structures that are such that every atom is situated at a centre of symmetry have elastic constants that are totally symmetric in their four indices. Thus [c_{ijkl} = c_{ikjl} = c_{iljk} ]in addition to the symmetries required by the previous considerations. This additional symmetry implies the following relations between the two-index components: [\matrix{c_{23} = c_{44}\semi \quad\hfil& c_{14} = c_{56} \hfil\cr c_{13} = c_{55}\semi\quad\hfil&c_{25} = c_{46}\hfil\cr c_{12} = c_{66}\semi \quad\hfil& c_{36} = c_{45}.\hfil\cr} ]These are known as the Cauchy relations. The further symmetry implied by the Cauchy relations reduces the maximum number of independent constants from 21 to 15. In crystal structures for which they might be valid, the extent to which these relations are fulfilled is often used to assess the validity of the assumption of a central-force law. It is important to apply such a test only to structures having the necessary symmetry properties for Cauchy relations to hold. For instance, f.c.c and b.c.c. metals have the required symmetry (an inversion centre at each atomic site), while diamond, silicon and germanium do not. Consequently, any apparent fulfilment of the Cauchy relation [c_{12} = c_{66}] for a diamond-type structure offers no ground for conclusions about the nature of the force field.

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.

References

First citation Fumi, F. G. (1951). Third-order elastic coefficients of crystals. Phys. Rev. 83, 1274–1275.Google Scholar
 First citation Fumi, F. G. (1952). Third-order elastic coefficients in trigonal and hexagonal crystals. Phys. Rev. 86, 561.Google Scholar
 First citation Fumi, F. G. (1987). Tables for the third-order elastic tensors in crystals. Acta Cryst. A43, 587–588.Google Scholar








































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