Page 250 - Intro to Tensor Calculus

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Restrictions on Elastic Constants due to Symmetry

The equations (2.4.1) and (2.4.2) can be replaced by an equivalent set of equations which are easier to
analyze. This is accomplished by defining the quantities

e 1 , e 2 , e 3 , e 4 , e 5 , e 6
σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6

where
   
e 1 e 4 e 5 e 11 e 12 e 13
 e 4 e 2 e 6  =  e 21 e 22 e 23 
e 5 e 6 e 3 e 31 e 32 e 33
and
   
σ 1 σ 4 σ 5 σ 11 σ 12 σ 13
 σ 4 σ 2 σ 6  =  σ 21 σ 22 σ 23  .
σ 5 σ 6 σ 3 σ 31 σ 32 σ 33
Then the generalized Hooke’s law from the equations (2.4.1) and (2.4.2) can be represented in either of
the forms
where i, j =1,... , 6 (2.4.4)
σ i = c ij e j or e i = s ij σ j
where c ij are constants related to the elastic stiffness and s ij are constants related to the elastic compliance.
These constants satisfy the relation

where i, m, j =1,... , 6 (2.4.5)
s mi c ij = δ mj
Here
e i , i = j =1, 2, 3

e ij =
e 1+i+j , i 6= j, and i =1, or, 2
and similarly
σ i , i = j =1, 2, 3

σ ij =
σ 1+i+j , i 6= j, and i =1, or, 2.
These relations show that the constants c ij are related to the elastic stiffness coefficients c pqrs by the
relations
c m1 = c ij11 c m4 =2c ij12

c m2 = c ij22 c m5 =2c ij13
c m3 = c ij33 c m6 =2c ij23
where

i, if i = j =1, 2, or3
m =
1+ i + j, if i 6= j and i =1 or 2.
A similar type relation holds for the constants s ij and s pqrs . The above relations can be verified by expanding
the equations (2.4.1) and (2.4.2) and comparing like terms with the expanded form of the equation (2.4.4).