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. 7276
Section 1.3.1. Strain 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 medium that undergoes a deformation. This means that the various points of the medium are displaced with respect to one another. Geometrical transformations of the medium that reduce to a translation of the medium as a whole will therefore not be considered. We may then suppose that there is an invariant point, O, whose position one can always return to by a suitable translation. A point P, with position vector , is displaced to the neighbouring point P′ by the deformation defined by The displacement vector constitutes a vector field. It is not a uniform field, unless the deformation reduces to a translation of the whole body, which is incompatible with the hypothesis that the medium undergoes a deformation. Let Q be a point that is near P before the deformation (Fig. 1.3.1.1). Then one can write
After the deformation, Q is displaced to Q′ defined by
In a deformation, it is more interesting in general to analyse the local, or relative, deformation than the absolute displacement. The relative displacement is given by comparing the vectors and PQ. Thus, one has Let us set Replacing by its expansion up to the first term gives
If we assume the Einstein convention (see Section 1.1.2.1 ), there is summation over j in (1.3.1.2) and (1.3.1.3). We shall further assume orthonormal coordinates throughout Chapter 1.3; variance is therefore not apparent and the positions of the indices have no meaning; the Einstein convention then only assumes repetition of a dummy index. The elements and are the components of dr and , respectively. Let us put where represents the Kronecker symbol; the 's are the components of matrix unity, I. The expressions (1.3.1.2) can also be written using matrices M and B: The components of the tensor are nonzero, unless, as mentioned earlier, the deformation reduces to a simple translation. Two cases in particular are of interest and will be discussed in turn:
If the components are constants, equations (1.3.1.3) can be integrated directly. They become, to a translation,
The fundamental property of the homogeneous deformation results from the fact that equations (1.3.1.4) are linear: a plane before the deformation remains a plane afterwards, a crystal lattice remains a lattice. Thermal expansion is a homogeneous deformation (see Chapter 1.4 ).
Some crystals present a twin microstructure that is seen to change when the crystals are gently squeezed. At rest, the domains can have one of two different possible orientations and the influence of an applied stress is to switch them from one orientation to the other. If one measures the shape of the crystal lattice (the strain of the lattice) as a function of the applied stress, one obtains an elastic hysteresis loop analogous to the magnetic or electric hysteresis loops observed in ferromagnetic or ferroelectric crystals. For this reason, these materials are called ferroelastic (see Chapters 3.1 to 3.3 and Salje, 1990). The strain associated with one of the two possible shapes of the crystal when no stress is applied is called the macroscopic spontaneous strain.
Let be the basis vectors before deformation. On account of the deformation, they are transformed into the three vectors The parallelepiped formed by these three vectors has a volume V′ given by where is the determinant associated with matrix B, V is the volume before deformation and represents a triple scalar product.
The relative variation of the volume is It is what one calls the cubic dilatation. gives directly the volume of the parallelepiped that is formed from the three vectors obtained in the deformation when starting from vectors forming an orthonormal base.
1.3.1.2.4. Expression of any homogeneous deformation as the product of a pure rotation and a pure deformation
Let us project the displacement vector on the position vector OP (Fig. 1.3.1.2), and let be this projection. The elongation is the quantity defined by where , , are the components of r. The elongation is the relative variation of the length of the vector r in the deformation. Let A and S be the antisymmetric and symmetric parts of M, respectively:
Only the symmetric part of M occurs in the expression of the elongation:
The geometrical study of the elongation as a function of the direction of r is facilitated by introducing the quadric associated with M: where is a constant. This quadric is called the quadric of elongations, Q. S is a symmetric matrix with three real orthogonal eigenvectors and three real eigenvalues, , , . If it is referred to these axes, equation (1.3.1.7) is reduced to
One can discuss the form of the quadric according to the sign of the eigenvalues :
In order to follow the variations of the elongation with the orientation of the position vector, one associates with r a vector y, which is parallel to it and is defined by where k is a constant. It can be seen that, in accordance with (1.3.1.6) and (1.3.1.7), the expression of the elongation in terms of y is
Thus, the elongation is inversely proportional to the square of the radius vector of the quadric of elongations parallel to OP. In practice, it is necessary to look for the intersection p of the parallel to OP drawn from the centre O of the quadric of elongations (Fig. 1.3.1.3a):

Equally, one can connect the displacement vector directly with the quadric Q. Using the bilinear form the gradient of , , has as components
One recognizes the components of the displacement vector u, which is therefore parallel to the normal to the quadric Q at the extremity of the radius vector Op parallel to r.
The directions of the principal axes of Q correspond to the extremal values of y, i.e. to the stationary values (maximal or minimal) of the elongation. These values are the principal elongations.
If the deformation is a pure rotation Hence we have
The quadric Q is a cylinder of revolution having the axis of rotation as axis.
If the deformation is small but arbitrary, i.e. if the products of two or more components of can be neglected with respect to unity, one can describe the deformation locally as a homogeneous asymptotic deformation. As was shown in Section 1.3.1.2.4, it can be put in the form of the product of a pure deformation corresponding to the symmetric part of , , and a pure rotation corresponding to the asymmetric part, : Matrix B can be written where I is the matrix identity. As the coefficients of are small, one can neglect the product and one has is a symmetric matrix that represents a pure deformation. is an antisymmetric unitary matrix and, since A is small, Thus, represents a rotation. The axis of rotation is parallel to the vector with coordinates which is an eigenvector of . The magnitude of the rotation is equal to the modulus of this vector.
In general, one is only interested in the pure deformation, i.e. in the form of the deformed object. Thus, one only wishes to know the quantities and the symmetric part of M. It is this symmetric part that is called the deformation tensor or the strain tensor. It is very convenient for applications to use the simplified notation due to Voigt:One may note that The Voigt strain matrix S is of the form
Let us consider an orthonormal system of axes with centre P. We remove nothing from the generality of the following by limiting ourselves to a planar problem and assuming that point P′ to which P goes in the deformation lies in the plane (Fig. 1.3.1.4). Let us consider two neighbouring points, Q and R, lying on axes and , respectively (, ). In the deformation, they go to points Q′ and R′ defined by

Geometrical interpretation of the components of the strain tensor. , , : axes before deformation; , , : axes after deformation. 
As the coefficients are small, the lengths of P′Q′ and P′R′ are hardly different from PQ and PR, respectively, and the elongations in the directions and are
The components , , of the principal diagonal of the Voigt matrix can then be interpreted as the elongations in the three directions , and . The angles α and β between PQ and P′Q′, and PR and P′R′, respectively, are given in the same way by One sees that the coefficient of Voigt's matrix is therefore The angle is equal to the difference between angles before deformation and after deformation. The nondiagonal terms of the Voigt matrix therefore represent the shears in the planes parallel to , and , respectively.
To summarize, if one considers a small cube before deformation, it becomes after deformation an arbitrary parallelepiped; the relative elongations of the three sides are given by the diagonal terms of the strain tensor and the variation of the angles by its nondiagonal terms.
The cubic dilatation (1.3.1.5) is (taking into account the fact that the coefficients are small).
Matrix M has only one coefficient, , and reduces to (Fig. 1.3.1.5a) The quadric of elongations is reduced to two parallel planes, perpendicular to , with the equation .
References
Salje, E. K. H. (1990). Phase transitions in ferroelastic and coelastic crystals. Cambridge University Press.