Page 256 - Intro to Tensor Calculus

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Figure 2.4-2. Element subjected to pure shearing

where α = π . Expanding the transformation equations (2.4.9) we find that
4
σ 1 = σ 11 =cos α sin ασ 12 +sin α cos ασ 21 = σ 12 = σ 4
σ 2 = σ 22 = − sin α cos ασ 12 − sin α cos ασ 21 = −σ 12 = −σ 4 ,

and similarly
e 1 = e 11 = e 4 , e 2 = e 22 = −e 4.

In the barred system, the Hooke’s law becomes

e 1 = s 11 σ 1 + s 12 σ 2 or
e 4 = s 11 σ 4 − s 12 σ 4 = s 44 σ 4 .

Hence, the constants s 11 ,s 12 ,s 44 are related by the relation

1 ν 1
or + = . (2.4.19)
s 11 − s 12 = s 44
E E 2µ
This is an important relation connecting the elastic constants associated with isotropic materials. The
above transformation can also be applied to triclinic, aelotropic, orthotropic, and hexagonal materials to
find relationships between the elastic constants.
Observe also that some texts postulate the existence of a strain energy function U ∗ which has the
property that σ ij = ∂U ∗ . In this case the strain energy function, in the single index notation, is written
∂e ij
∗
U = c ij e i e j where c ij and consequently s ij are symmetric. In this case the previous discussed symmetries
give the following results for the nonzero elastic compliances s ij : 13 nonzero constants instead of 20 for
aelotropic material, 9 nonzero constants instead of 12 for orthotropic material, and 6 nonzero constants
instead of 7 for hexagonal material. This is because of the additional property that s ij = s ji be symmetric.