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Second Approach: Stress Strain Equivalence Approach
The same reduction of number of elastic constants can be derived from the stress strain equivalence approach. From Equation (3.12) and Equation (3.16) we have
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(3.21) |
The same can be seen from the stresses on a cube inside such a body with the coordinate systems shown in Figure 3.1. Figure 3.2 (a) shows the stresses on a cube with the coordinate system x1, x2,x3 and Figure 3.2 (b) shows stresses on the same cube with the coordinate system  . Comparing the stresses we get the relation as in Equation (3.21).
Now using the stiffness matrix as given in Equation (3.11), strain term relations as given in Equation (3.17) and comparing the stress terms in Equation (3.21) as follows:
Using the relations from Equation (3.17), the above equations reduce to
Noting that  , this holds true only when 
Similarly,
This gives us the  matrix as in Equation (3.20).
Figure 3.2: State of stress (a) in x1, x2, x3 system
(b) with x1-x3 plane of symmetry |
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