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. 7680
Section 1.3.2. Stress tensor^{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 solid C, in movement or not, with a mass distribution defined by a specific mass ρ at each point. There are two types of force that are manifested in the interior of this solid.
Now consider a volume V within the solid C and the surface S which surrounds it (Fig. 1.3.2.2). Among the influences that are exterior to V, we distinguish those that are external to the solid C and those that are internal. The first are translated by the body forces, eventually by volume couples. The second are translated by the local contact forces of the part external to V on the internal part; they are represented by a surface density of forces, i.e. by the stresses that depend only on the point Q of the surface S where they are applied and on the orientation of the normal n of this surface at this point. If two surfaces S and S′ are tangents at the same point Q, the same stress acts at the point of contact between them. The equilibrium of the volume V requires:
The equilibrium of the solid C requires that:
Using the condition on the resultant of forces, it is possible to show that the components of the stress can be determined from the knowledge of the orientation of the normal n and of the components of a ranktwo tensor. Let P be a point situated inside volume V, , and three orthonormal axes, and consider a plane of arbitrary orientation that cuts the three axes at Q, R and S, respectively (Fig. 1.3.2.3). The small volume element PQRS is limited by four surfaces to which stresses are applied. The normals to the surfaces PRS, PSQ and PQR will be assumed to be directed towards the interior of the small volume. By contrast, for reasons that will become apparent later, the normal n applied to the surface QRS will be oriented towards the exterior. The corresponding applied forces are thus given in Table 1.3.2.1. The volume PQRS is subjected to five forces: the forces applied to each surface and the resultant of the volume forces and the inertial forces. The equilibrium of the small volume requires that the resultant of these forces be equal to zero and one can write (including the inertial forces in the volume forces).

As long as the surface element dσ is finite, however small, it is possible to divide both terms of the equation by it. If one introduces the direction cosines, , the equation becomes When dσ tends to zero, the ratio tends towards zero at the same time and may be neglected. The relation then becomes This relation is called the Cauchy relation, which allows the stress to be expressed as a function of the stresses , and that are applied to the three faces perpendicular to the axes, , and . Let us project this relation onto these three axes: The nine components are, by definition, the components of the stress tensor. In order to check that they are indeed the components of a tensor, it suffices to make the contracted product of each side of (1.3.2.4) by any vector : the lefthand side is a scalar product and the righthand side a bilinear form. The 's are therefore the components of a tensor. The index to the far left indicates the face to which the stress is applied (normal to the , or axis), while the second one indicates on which axis the stress is projected.
Let us return to equation (1.3.2.1) expressing the equilibrium condition for the resultant of the forces. By replacing by the expression (1.3.2.4), we get, after projection on the three axes, where and the inertial forces are included in the volume forces. Applying Green's theorem to the first integral, we have
The equilibrium condition now becomes In order that this relation applies to any volume V, the expression under the integral must be equal to zero, or, if one includes explicitly the inertial forces, This is the condition of continuity or of conservation. It expresses how constraints propagate throughout the solid. This is how the cohesion of the solid is ensured. The resolution of any elastic problem requires solving this equation in terms of the particular boundary conditions of that problem.
Let us now consider the equilibrium condition (1.3.2.2) relative to the resultant moment. After projection on the three axes, and using the Cartesian expression (1.1.3.4) of the vectorial products, we obtain (including the inertial forces in the volume forces). is the permutation tensor. Applying Green's theorem to the first integral and putting the two terms together gives
In order that this relation applies to any volume V within the solid C, we must have or
Taking into account the continuity condition (1.3.2.5), this equation reduces to
A volume couple can occur for instance in the case of a magnetic or an electric field acting on a body that locally possesses magnetic or electric moments. In general, apart from very rare cases, one can ignore these volume couples. One can then deduce that the stress tensor is symmetrical:
This result can be recovered by applying the relation (1.3.2.2) to a small volume in the form of an elementary parallelepiped, thus illustrating the demonstration using Green's theorem but giving insight into the action of the constraints. Consider a rectangular parallelepiped, of sides , and , with centre P at the origin of an orthonormal system whose axes , and are normal to the sides of the parallelepiped (Fig. 1.3.2.4). In order that the resultant moment with respect to a point be zero, it is necessary that the resultant moments with respect to three axes concurrent in this point are zero. Let us write for instance that the resultant moment with respect to the axis is zero. We note that the constraints applied to the faces perpendicular to do not give rise to a moment and neither do the components , , and of the constraints applied to the faces normal to and (Fig. 1.3.2.4). The components and alone have a nonzero moment.

Symmetry of the stress tensor: the moments of the couples applied to a parallelepiped compensate each other. 
For face 1, the constraint is if is the magnitude of the constraint at P. The force applied at face 1 is and its moment is
Similarly, the moments of the force on the other faces are
Noting further that the moments applied to the faces 1 and 1′ are of the same sense, and that those applied to faces 2 and 2′ are of the opposite sense, we can state that the resultant moment is where is the volume of the small parallelepiped. The resultant moment per unit volume, taking into account the couples in volume, is therefore It must equal zero and the relation given above is thus recovered.
We shall use frequently the notation due to Voigt (1910) in order to express the components of the stress tensor: It should be noted that the conventions are different for the Voigt matrices associated with the stress tensor and with the strain tensor (Section 1.3.1.3.1).
The Voigt matrix associated with the stress tensor is therefore of the form
1.3.2.5.2. Interpretation of the components of the stress tensor – special forms of the stress tensor
If the surface of the solid C is free from all exterior action and is in equilibrium, the stress field inside C is zero at the surface. If C is subjected from the outside to a distribution of stresses (apart from the volume forces mentioned earlier), the stress field inside the solid is such that at each point of the surface where the 's are the direction cosines of the normal to the surface at the point under consideration.

Consider a medium that is subjected to a stress field . It has sustained a deformation indicated by the deformation tensor S. During this deformation, the forces of contact have performed work and the medium has accumulated a certain elastic energy W. The knowledge of the energy density thus acquired is useful for studying the properties of the elastic constants Let the medium deform from the deformation to the deformation under the influence of the stress field and let us evaluate the work of each component of the effort. Consider a small elementary rectangular parallelepiped of sides , , (Fig. 1.3.2.8). We shall limit our calculation to the components and , which are applied to the faces 1 and 1′, respectively.

Determination of the energy density in a deformed medium. PP′ represents the displacement of the small parallelepiped during the deformation. The thick arrows represent the forces applied to the faces 1 and 1′. 
In the deformation , the point P goes to the point P′, defined by A neighbouring point Q goes to Q′ such that (Fig. 1.3.1.1) The coordinates of are given by
Of sole importance is the relative displacement of Q with respect to P and the displacement that must be taken into account in calculating the forces applied at Q. The coordinates of the relative displacement are
We shall take as the position of Q the point of application of the forces at face 1, i.e. its centre with coordinates (Fig. 1.3.2.8). The area of face 1 is and the forces arising from the stresses and are equal to and , respectively. The relative displacement of Q parallel to the line of action of is and the corresponding displacement along the line of action of is . The work of the corresponding forces is therefore
The work of the forces applied to the face 1′ is the same (, and change sign simultaneously). The works corresponding to the faces 1 and 1′ are thus and for the two stresses, respectively. One finds an analogous result for each of the other components of the stress tensor and the total work per unit volume is
References
Voigt, W. (1910). Lehrbuch der Kristallphysik. 2nd ed. (1929). Leipzig: Teubner. Photoreproduction (1966). New York: Johnson Reprint Corp.