Arbitrary but small deformations
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.3,
p.
[ doi:10.1107/97809553602060000902 ]
properties
1.3.1.3.1. Definition of the strain tensor
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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 ...
Homogeneous deformation
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.2,
p.
[ doi:10.1107/97809553602060000902 ]
properties
If the components are constants, equations (
1.3.1.3) can be integrated directly. They become, to a translation,
1.3.1.2.1. Fundamental property of the homogeneous deformation
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Particular components of the deformation
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.4,
p.
[ doi:10.1107/97809553602060000902 ]
properties
1.3.1.4.1. Simple elongation
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Matrix M has only one coefficient,, and reduces to (Fig.
1.3.1.5 a) The quadric of elongations is reduced to two parallel planes ...
Definition of the strain tensor
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.3.1,
p.
[ doi:10.1107/97809553602060000902 ]
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 ...
Quadric of elongations
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.2.5,
p.
[ doi:10.1107/97809553602060000902 ]
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 ...
Geometrical interpretation of the coefficients of the strain tensor
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.3.2,
p.
[ doi:10.1107/97809553602060000902 ]
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 ...
Simple shear
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.4.3,
p.
[ doi:10.1107/97809553602060000902 ]
properties
Matrix has one coefficient only, a shear (Fig.
1.3.1.5 c): The matrix is not symmetrical, as it contains a component of rotation. Thus we have One can show that the deformation is a pure shear associated ...
Pure shear
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.4.2,
p.
[ doi:10.1107/97809553602060000902 ]
properties
This is a pure deformation (without rotation) consisting of the superposition of two simple elongations along two perpendicular directions (Fig.
1.3.1.5 b) and such that there is no change of volume (the cubic ...
Introduction, the notion of strain field
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.1,
p.
[ doi:10.1107/97809553602060000902 ]
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 ...
Simple elongation
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.1.4.1,
p.
[ doi:10.1107/97809553602060000902 ]
properties
Matrix M has only one coefficient,, and reduces to (Fig.
1.3.1.5 a) The quadric of elongations is reduced to two parallel planes, perpendicular to, with the equation .
Figure ...
