Quadric of elongations

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 73-74

Section 1.3.1.2.5. Quadric of elongations

A. Authier^a ^* and A. Zarembowitch^b

^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

1.3.1.2.5. Quadric of elongations

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Let us project the displacement vector $[{\bf u}({\bf r}) ]$ on the position vector OP (Fig. 1.3.1.2), and let $[u_{r}]$ be this projection. The elongation is the quantity defined by $[{u_{r} \over r} = {{\bf u \cdot r} \over r^2} = {M_{ij}x_ix_j \over r^2}, ]$ where $[x_{1}]$ , $[x_{2}]$ , $[x_{3}]$ 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: $[A = {M - M^T \over 2}; \quad S = {M + M^T \over 2}. ]$

Figure 1.3.1.2 | top | pdf |

Elongation, $[u_{r}/r]$ .

Only the symmetric part of M occurs in the expression of the elongation: $[{u_{r} \over r} = {S_{ij}x_ix_j \over r^2}. \eqno(1.3.1.6) ]$

The geometrical study of the elongation as a function of the direction of r is facilitated by introducing the quadric associated with M: $[S_{ij} y_iy_j = \varepsilon, \eqno(1.3.1.7) ]$ where $[\varepsilon]$ 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, $[\lambda_{1}]$ , $[\lambda_{2}]$ , $[\lambda_{3}]$ . If it is referred to these axes, equation (1.3.1.7) is reduced to $[\lambda_{1}\left(y_{1}\right)^{2}\lambda_{2}\left(y_{2}\right)^{2}\lambda_{3} \left(y_{3}\right)^{2} = \varepsilon. ]$

One can discuss the form of the quadric according to the sign of the eigenvalues $[\lambda_{i}]$ :

(i) $[\lambda_{1}]$ , $[\lambda_{2}]$ , $[\lambda_{3}]$ have the same sign, and the sign of $[\varepsilon ]$ . The quadric is an ellipsoid (Fig. 1.3.1.3a). One chooses $[\varepsilon = +1]$ or $[\varepsilon = - 1 ]$ , depending on the sign of the eigenvalues.

Figure 1.3.1.3 | top | pdf |

Quadric of elongations. The displacement vector, $[{\bf u}({\bf r})]$ , at P in the deformed medium is parallel to the normal to the quadric at the intersection, p, of OP with the quadric. (a) The eigenvalues all have the same sign, the quadric is an ellipsoid. (b) The eigenvalues have mixed signs, the quadric is a hyperboloid with either one sheet (shaded in light grey) or two sheets (shaded in dark grey), depending on the sign of the constant $[\varepsilon]$ [see equation (1.3.1.7)]; the cone asymptote is represented in medium grey. For a practical application, see Fig. 1.4.1.1 .

(ii) $[\lambda_{1}]$ , $[\lambda_{2} ]$ , $[\lambda_{3}]$ are of mixed signs: one of them is of opposite sign to the other two. One takes $[\varepsilon = \pm 1 ]$ . The corresponding quadric is a hyperboloid whose asymptote is the cone $[S_{ij} y_iy_j = 0.]$

According to the sign of $[\varepsilon]$ , the hyperboloid will have one sheet outside the cone or two sheets inside the cone (Fig. 1.3.1.3b). If we wish to be able to consider any direction of the position vector r in space, it is necessary to take into account the two quadrics.

In order to follow the variations of the elongation $[u_{r}/r ]$ with the orientation of the position vector, one associates with r a vector y, which is parallel to it and is defined by $[{\bf y} = {\bf r}/k; \quad {\bf r} = k {\bf y},]$ 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 $[u_{r}/r = \varepsilon/y^{2}. ]$

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):

(i) The eigenvalues all have the same sign; the quadric Q is an ellipsoid: the elongation has the same sign in all directions in space, positive for $[\varepsilon = +1]$ and negative for $[\varepsilon = -1]$ .
(ii) The eigenvalues have different signs; two quadrics are to be taken into account: the hyperboloids corresponding, respectively, to $[\varepsilon = \pm 1]$ . The sign of the elongation is different according to whether the direction under consideration is outside or inside the asymptotic cone and intersects one or the other of the two hyperboloids.

Equally, one can connect the displacement vector $[{\bf u}({\bf r}) ]$ directly with the quadric Q. Using the bilinear form $[f({\bf y}) = M_{ij}y_{i}y_{j},]$ the gradient of $[f({\bf y})]$ , $[\boldnabla (f)]$ , has as components $[\partial f/\partial y^{i} = M_{ij}y_{j} = u_{i}.]$

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 $[\eqalign{&B = \pmatrix{\cos \theta & \sin \theta & 0\cr - \sin \theta & \cos \theta & 0\cr 0 & 0 &1\cr}, \cr\noalign{\vskip5.5pt}\qquad &M = \pmatrix{\cos \theta - 1 & \sin \theta & 0\cr - \sin \theta & \cos \theta - 1 &0\cr 0 & 0 &0\cr}.} ]$ Hence we have $[M_{ij}y_{i}y_{j} = (\cos\theta - 1)\left(y_{1}- y_{2}\right) = \varepsilon. ]$

The quadric Q is a cylinder of revolution having the axis of rotation as axis.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 73-74