Page 253 - Intro to Tensor Calculus

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Additional Symmetries

If the material (crystal) is such that there is an additional plane of symmetry, say the x 2 -x 3 plane, then
reversal of the x 1 axis should leave the equations (2.4.14) unaltered. If there are two planes of symmetry
then there will automatically be a third plane of symmetry. Such a material (crystal) is called orthotropic.
Introducing the additional transformation

x 1 = −x 1 , x 2 = x 2 , x 3 = x 3

which represents the reversal of the x 1 axes, the expanded form of equations (2.4.9) are used to calculate the
effect of such a transformation upon the stress and strain tensor. We find σ 1 ,σ 2 ,σ 3 ,σ 6 ,e 1 ,e 2,e 3 ,e 6 remain
unchanged while σ 4 ,σ 5 ,e 4 ,e 5 change sign. The equation (2.4.14) then becomes

     
e 1 s 11 s 12 s 13 s 14 0 0 σ 1
 s 21 s 22 s 23 s 24 0
 e 2  0   σ 2 
     
 s 31 s 32 s 33 s 34 0
 . (2.4.15)
 e 3  0   σ 3 
  =   
 s 41 s 42 s 43 s 44 0
 −e 4  0   −σ 4 
     
−e 5 0 0 0 0 s 55 s 56 −σ 5
e 6 0 0 0 0 s 65 s 66 σ 6
Note that if the constitutive equations (2.4.14) and (2.4.15) are to produce the same results upon reversal
of the x 1 axes, then we require that the following coefficients be equated to zero:
s 14 = s 24 = s 34 =0
s 41 = s 42 = s 43 =0

s 56 = s 65 =0.
This then produces the constitutive equation

   0 0 0   
e 1 s 11 s 12 s 13 σ 1
0 0
 s 21 s 22 s 23
 e 2  0   σ 2 
   0 0   
 s 31 s 32 s 33
 e 3  0   σ 3  (2.4.16)
 0 0 0 0
  =    
 e 4  s 44 0   σ 4 
     
e 5 0 0 0 0 s 55 0 σ 5
e 6 0 0 0 0 0 s 66 σ 6
or its equivalent form
     
σ 1 c 11 c 12 c 13 0 0 0 e 1
 c 21 c 22 c 23 0 0
 σ 2  0   e 2 
     
 c 31 c 32 c 33 0 0
 σ 3  0   e 3 
 0 0 0 c 44 0
  =    
 σ 4  0   e 4 
     
σ 5 0 0 0 0 c 55 0 e 5
σ 6 0 0 0 0 0 c 66 e 6
and the original 36 constants have been reduced to 12 constants. This is the constitutive equation for
orthotropic material (crystals).