[aij] is the matrix of 36 elastic coefficients, of which only 21 are independent,
since [aij] = [agj]. Depending on the the material structure, the crystalline
material di pl'i different forms of geometric symmetry. There are 32 forms
of geometric symmetry of crystals, which can be further divided into seven
crystal systems called syngony: 1) triclinic, 2) monoclinic, 3) rhombic, 4)
tetragonal, 5) trigonal, 6) hexagonal, and 7) cubic [6]. For an isotropic
material the mechanical properties (E, v, G) are the same at each point of the
material because it can have only two independent elastic constants as
all = a22 a= 33 a12 a= 13 a= 23 a44 a= 55 a= 66 = 2(all a12) (1.2)
and the rest of the coefficients of deformations are zero.
all a12 a12 0 0 0
a12 all a12 0 0 0
] a12 a12 all 0 0 0
[a,1 (1.3)
0 0 0 a44 0 0
0 0 0 0 a44 0
0 0 0 0 0 a44
An isotropic material subjected to multiaxial lti:.- under mechanical
equilibrium, has three principal stresses. These principal stresses act on
orthogonal planes, which are free of shear stresses. The Von Mises and Tresca
criterions are two of the most widely used yield criteria for ductile isotropic
materials.
Anisotropic Elastic Deformation of FCC Single Crystal: The generalized
Hooke's law for a homogeneous anisotropic elastic body in a cartesian
coordinate system (x, y, z with origin at point 0, Fig. 1-5 on page 11) is
given by equation (1.4) [6]
{} = [ai]{} (.
(1.4)