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. 8085
Section 1.3.3. Linear elasticity^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^{b}Laboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France 
Let us consider a metallic bar of length loaded in pure tension (Fig. 1.3.3.1). Under the action of the uniaxial stress (F applied force, area of the section of the bar), the bar elongates and its length becomes . Fig. 1.3.3.2 relates the variations of Δl and of the applied stress T. The curve representing the traction is very schematic and does not correspond to any real case. The following result, however, is common to all concrete situations:
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 of the stress tensor and the components of the strain tensor: The quantities and 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 . Let us take the contracted product of the two sides of each of the two equations of (1.3.3.2) by the components and of any two vectors, x and y: The lefthand sides are bilinear forms since and are secondrank tensors and the righthand sides are quadrilinear forms, which shows that and are the components of fourthrank 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) are Taylor expansions limited to the first term. The higher terms involve sixthrank tensors, and , with coefficients, called thirdorder elastic compliances and stiffnesses and eighthrank tensors with coefficients, called fourthorder elastic compliances and stiffnesses. They will be defined in Section 1.3.6.4. Tables for thirdorder elastic constants are given in Fumi (1951, 1952, 1987). The accompanying software to this volume enables these tables to be derived for any point group.
It is convenient to write the relations (1.3.3.2) in matrix form by associating with the stress and strain tensors column matrices and with the tensors of the elastic stiffnesses, c, and of the elastic compliances, s, square matrices (Section 1.1.4.10.4 ); these two matrices are inverse to one another. The number of independent components of the fourthrank elastic tensors can be reduced by three types of consideration:

Expression (1.3.2.7) of the strain energy stored per unit volume in a medium for a small deformation can be integrated when the medium is strained under a stress according to linear elasticity. Applying relation (1.3.3.2), one gets for the density of strain energy
Let us apply a hydrostatic pressure (Section 1.3.2.5.2). The medium undergoes a relative variation of volume (the cubic dilatation, Section 1.3.1.3.2). If one replaces in (1.3.3.8) the stress distribution by a hydrostatic pressure, one obtains for the components of the strain tensorFrom this, we deduce the volume compressibility, , which is the inverse of the bulk modulus, κ: This expression reduces for a cubic or isotropic medium to
Under the action of a hydrostatic pressure, each vector assumes a different elongation. This elongation is given by equation (1.3.1.6): where the 's are the direction cosines of r. The coefficient of linear compressibility is, by definition, . Replacing by its value , we obtain for the coefficient of linear compressibility In the case of a cubic or isotropic medium, this expression reduces to
The coefficient of linear compressibility is then equal to one third of the coefficient of volume compressibility. We note that the quadric of elongations is a sphere.
If the applied stress reduces to a uniaxial stress, , the strain tensor is of the form In particular, We deduce from this that Young's modulus (equation 1.3.3.1) is
The elongation of a bar under the action of a uniaxial stress is characterized by and the diminution of the cross section is characterized by and . For a cubic material, the relative diminution of the diameter is One deduces from this that is necessarily of opposite sign to and one calls the ratio Poisson's ratio.
Putting this value into expression (1.3.3.12) for the coefficient of compressibility in cubic or isotropic materials gives
As the coefficient of compressibility, by definition, is always positive, we have
In practice, Poisson's ratio is always close to 0.3. It is a dimensionless number. The quantity represents the departure from isotropy of the material and is the anisotropy factor. It is to be noted that cubic materials are not isotropic for elastic properties. Table 1.3.3.2 gives the values of , , , ν and for a few cubic materials.

It is interesting to calculate Young's modulus in any direction. For this it is sufficient to change the axes of the tensor . If A is the matrix associated with the change of axes, leading to the direction changing to the direction , then Young's modulus in this new direction is with The matrix coefficients are the direction cosines of with respect to the axes , and . In spherical coordinates, they are given by (Fig. 1.3.3.3) where θ is the angle between and , and ϕ is the angle between and . Using the reduction of for the various crystal classes (Section 1.1.4.9.9 ), we find, in terms of the reduced twoindex components, the following.
The representation surface of , the inverse of Young's modulus, is illustrated in Figure 1.3.3.4 for crystals of different symmetries. As predicted by the Neumann principle, the representation surface is invariant with respect to the symmetry elements of the point group of the crystal but, as stated by the Curie laws, its symmetry can be larger. In the examples of Fig. 1.3.3.4, the symmetry of the surface is the same as that of the point group for sodium chloride (Fig. 1.3.3.4a), tungsten (Fig. 1.3.3.4b) and aluminium (Fig. 1.3.3.4c), which have as point group, for tin (Fig. 1.3.3.4e, ) and for calcite (Fig. 1.3.3.4f, ). But in the case of zinc (Fig. 1.3.3.4d, ), the surface is of revolution and has a larger symmetry. It is interesting to compare the differences in shapes of the representation surfaces for the three cubic crystals, depending on the value of the anisotropy factor, which is larger than 1 for sodium chloride, smaller than 1 for aluminium and close to 1 for tungsten (see Table 1.3.3.2). In this latter case, the crystal is pseudoisotropic and the surface is practically a sphere.
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
These relations can equally well be written in the symmetrical form
If one introduces the Lamé constants, the equations may be written in the form often used in mechanics:
Two coefficients suffice to define the elastic properties of an isotropic material, and , and , μ and λ, μ and ν, etc. Table 1.3.3.3 gives the relations between the more common elastic coefficients.

We saw in Section 1.3.2.3 that the condition of equilibrium is
If we use the relations of elasticity, equation (1.3.3.2), this condition can be rewritten as a condition on the components of the strain tensor: Recalling that the condition becomes a condition on the displacement vector, : In an isotropic orthonormal medium, this equation, projected on the axis , can be written with the aid of relations (1.3.3.5) and (1.3.3.9): This equation can finally be rearranged in one of the three following forms with the aid of Table 1.3.3.3.
References
Fumi, F. G. (1951). Thirdorder elastic coefficients of crystals. Phys. Rev. 83, 1274–1275.Fumi, F. G. (1952). Thirdorder elastic coefficients in trigonal and hexagonal crystals. Phys. Rev. 86, 561.
Fumi, F. G. (1987). Tables for the thirdorder elastic tensors in crystals. Acta Cryst. A43, 587–588.