The elastic properties of FCC -i ,-I J1- exhibit cubic syngony; i.e., it has
three orthogonal planes of elastic symmetry at every point, which is called
orthogonally-anisotropic or, for brevity, orthotropic. Therefore cubic sym-
metry can be described with three independent constants designated as the
elastic modulus, shear modulus, and Poisson's ratio [4] and hence [a1j] can be
expressed as shown in equation (1.3), in the material coordinate system (FCC
crystal axes are parallel to x, y and z coordinate axes) and the coefficient of
deformation is given as
1 1 Vyx 1 xy (
a11 = t, a44 C a12 F- (1.5)
Exx Gyx Exx Eyy
The elastic constants in the generalized Hooke's law of an anisotropic body,
[aij], vary with the direction of the coordinate axes. For orientations other
than the (x, y, z) axes, the [a1j] matrix varies with the crystal orientation. In
the case of an isotropic body the constants are invariant in any orthogonal
coordinate system. Consider a Cartesian coordinate system (x', y', z) that has
rotated about the origin 0 of (x, y, z). The elastic constant matrix [at] in
the (x', y', z') coordinate system that relates {e'} and {oa'} is
{c'} =- '.]{'} (1.6)
where [at] is given by the following transformation
6 6
] [Q][a] [Q] = a mQm (ij = 1,2, ......,6) (1.7)
rrn in 1
where [Q] is the transformation matrix, which is a function of the direction
cosines between the(x, y, z) and (x', y', z') coordinate axes (Fig. 1-5 on
page 11, Table 1-1 on page 8). Here a, /3 and 7 are the direction cosines of
the material coordinate system relative to the universal coordinate system
(specimen coordinate system).