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. 72-74
Section 1.3.1.2. Homogeneous deformation
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 |
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.
References
Salje, E. K. H. (1990). Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press.Google Scholar