Elastic Properties and Anisotropy of Elastic Behavior. Figure 7-1 Anisotropic materials: (a) rolled material, (b) wood, (c) glass-fiber cloth in an epoxy matrix, and (d) a crystal with cubic unit cell. .
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Figure 7-1 Anisotropic materials: (a) rolled material, (b) wood, (c) glass-fiber cloth in an epoxy matrix, and (d) a crystal with cubic unit cell.
We can write:
We can also write:
can be realized through simplifying the matrix notation
for stresses and strains.
We can replace the indices as follows:
11 12 13 1 6 5
22 23 2 4
The foregoing transformation is easy to remember: In other to obtain notation II, one must proceed first along the diagonal ( ) and then back ( ).
and , but
36 (6 x 6) independent components of stiffness can be
Or in short notation, we can write:
Of the 36 constants, there are six constants where i = j, leaving 30 constants where i j.
But only one-half of these are independent constants since Cij = Cji
Therefore, for the general anisotropic linear elastic solid there are:
independent elastic constant.
Triclinic None (or center of symmetry)
Monoclinic 1 twofold rotation
Orthorhombic 2 perpendicular twofold rotation
Tetragonal 1 fourfold rotation around 
Rhombohedral 1 threefold rotation around 
Hexagonal 1sixfold rotation around 
Cubic 4 threefold rotations around <111>
 - c axis; after 60 degrees, the structure superimposes
In terms of a matrix, we have the following:
The number of independent elastic constants in a cubic system
is three (3).
For isotropic materials ( most polycrystalline aggregates can
be treated as such) there are two (2) independent constants, b/c :
define an anisotropy ratio (also called the Zener anisotropy
ratio, in honor of the scientist who introduced it):
Similarly, the 81 components of elastic compliance for the cubic system have been reduced to three (3) independent ones while for the isotropic case, only two (2) independent elastic constants are needed.
Rigidity or Shear modulus
In a cubic material, the elastic moduli can be determined along any orientation, from the elastic constants, by the application of the following equations:
and are the Young’s and shear modulus, respectively, in
the [ijk] direction; are the direction cosines of the
Aluminum 10.82 6.13 2.85 1.57 -0.57 3.15
Copper 16.84 12.14 7.54 1.49 -0.62 1.33
Iron 23.70 14.10 11.60 0.80 -0.28 0.86
Tungsten 50.10 19.80 15.14 0.26 -0.07 0.66
Stiffness constants in units of 10-10 Pa.
Compliance in units of 10-11 Pa
the equation for determining the Elastic Moduli along any direction
is given by:
Typical values of elastic constants for cubic metals are given in
All the relations described in Eqs. 7-12 to 7-20 for obtaining Elastic
constants are applicable. This include:
A hydrostatic compressive stress applied to a material with cubic symmetry results in a dilation of -10-5. The three independent elastic constants of the material are
C11 = 50 GPa, C12 = 40 GPa and C44 = 32 GPa. Write an expression for the generalized Hooke’s law for the material, and compute the applied hydrostatic stress.
Dilation is the sum of the principal strain components:
= 1 + 2 + 3 = -10-5
Cubic symmetry implies that 1 = 2 = 3 = -3.33 x10-5
4 = 5 = 6 = 0
From Hooke’s law,
i = Cijj
the applied hydrostatic stress is:
p = 1 = (50 + 40 + 40)(-3.33) 103 Pa
= -130 x 3.33 x 103 = -433 kPa
in the <111> and <100> directions. What conclusions can be drawn
about their elastic anisotropy? From Table 7.1
Fe: 0.80 -0.28 0.86
W: 0.26 -0.07 0.66
The direction cosines for the chief directions in a cubic lattice are:
<100> 1 0 0
is elastically anisotropic.