Page 251 - Intro to Tensor Calculus

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The generalized Hooke’s law can now be expressed in a form where the 36 independent constants can
be examined in more detail under special material symmetries. We will examine the form

     
e 1 s 11 s 12 s 13 s 14 s 15 s 16 σ 1
 s 21 s 22 s 23 s 24 s 25
 e 2  s 26   σ 2 
     
 s 31 s 32 s 33 s 34 s 35
 . (2.4.6)
 e 3  s 36   σ 3 
  =   
 s 41 s 42 s 43 s 44 s 45
 e 4  s 46   σ 4 
     
e 5 s 51 s 52 s 53 s 54 s 55 s 56 σ 5
e 6 s 61 s 62 s 63 s 64 s 65 s 66 σ 6
Alternatively, in the arguments that follow, one can examine the equivalent form
     
σ 1 c 11 c 12 c 13 c 14 c 15 c 16 e 1
 c 21 c 22 c 23 c 24 c 25
 σ 2  c 26   e 2 
     
 c 31 c 32 c 33 c 34 c 35
 .
 σ 3  c 36   e 3 
  =   
 c 41 c 42 c 43 c 44 c 45
 σ 4  c 46   e 4 
     
σ 5 c 51 c 52 c 53 c 54 c 55 c 56 e 5
σ 6 c 61 c 62 c 63 c 64 c 65 c 66 e 6
Material Symmetries
A material (crystal) with one plane of symmetry is called an aelotropic material. If we let the x 1 -
x 2 plane be a plane of symmetry then the equations (2.4.6) must remain invariant under the coordinate
transformation
     
x 1 10 0 x 1
 x 2  =  01 0   x 2  (2.4.7)
x 3 00 −1 x 3
which represents an inversion of the x 3 axis. That is, if the x 1 -x 2 plane is a plane of symmetry we should be
able to replace x 3 by −x 3 and the equations (2.4.6) should remain unchanged. This is equivalent to saying
that a transformation of the type from equation (2.4.7) changes the Hooke’s law to the form e i = s ij σ j where
the s ij remain unaltered because it is the same material. Employing the transformation equations

x 1 = x 1 , x 2 = x 2 , x 3 = −x 3 (2.4.8)

we examine the stress and strain transformation equations

∂x p ∂x q ∂x p ∂x q
σ ij = σ pq and e ij = e pq . (2.4.9)
∂x i ∂x j ∂x i ∂x j
If we expand both of the equations (2.4.9) and substitute in the nonzero derivatives

∂x 1 ∂x 2 ∂x 3
=1, =1, = −1, (2.4.10)
∂x 1 ∂x 2 ∂x 3
we obtain the relations
σ 11 = σ 11 e 11 = e 11
σ 22 = σ 22 e 22 = e 22

σ 33 = σ 33 e 33 = e 33
(2.4.11)
σ 21 = σ 21 e 21 = e 21
σ 31 = −σ 31 e 31 = −e 31
e 23 = −e 23 .
σ 23 = −σ 23