Page 254 - Intro to Tensor Calculus

248

Axis of Symmetry

If in addition to three planes of symmetry there is an axis of symmetry then the material (crystal) is
1
termed hexagonal. Assume that the x axis is an axis of symmetry and consider the effect of the transfor-
mation
1
1
3
2
x = x , x = x 3 x = −x 2
upon the constitutive equations. It is left as an exercise to verify that the constitutive equations reduce to
the form where there are 7 independent constants having either of the forms
     
e 1 s 11 s 12 s 12 0 0 0 σ 1
 s 21 s 22 s 23 0 0
 e 2  0   σ 2 
     
 s 21 s 23 s 22 0 0
 e 3  0   σ 3 
 0 0 0 s 44 0
  =    
 e 4  0   σ 4 
     
e 5 0 0 0 0 s 44 0 σ 5
e 6 0 0 0 0 0 s 66 σ 6
or
     
σ 1 c 11 c 12 c 12 0 0 0 e 1
 c 21 c 22 c 23 0 0
 σ 2  0   e 2 
    

 c 21 c 23 c 22 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 44 0 e 5
σ 6 0 0 0 0 0 c 66 e 6
2
Finally, if the material is completely symmetric, the x axis is also an axis of symmetry and we can
consider the effect of the transformation
2
2
3
3
1
x = −x , x = x , x = x 1
upon the constitutive equations. It can be verified that these transformations reduce the Hooke’s law
constitutive equation to the form
     
e 1 s 11 s 12 s 12 0 0 0 σ 1
 s 12 s 11 s 12 0 0
 e 2  0   σ 2 
     
 s 12 s 12 s 11 0 0
 . (2.4.17)
 e 3  0   σ 3 
 0 0 0 s 44 0
  =   
 e 4  0   σ 4 
     
e 5 0 0 0 0 s 44 0 σ 5
e 6 0 0 0 0 0 s 44 σ 6
Materials (crystals) with atomic arrangements that exhibit the above symmetries are called isotropic
materials. An equivalent form of (2.4.17) is the relation
     
σ 1 c 11 c 12 c 12 0 0 0 e 1
 c 12 c 11 c 12 0 0
 σ 2  0   e 2 
     
 c 12 c 12 c 11 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 44 0 e 5
σ 6 0 0 0 0 0 c 44 e 6
The figure 2.4-1 lists values for the elastic stiffness associated with some metals which are isotropic 1
1 Additional constants are given in “International Tables of Selected Constants”, Metals: Thermal and
Mechanical Data, Vol. 16, Edited by S. Allard, Pergamon Press, 1969.