Page 257 - Intro to Tensor Calculus

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The previous discussion has shown that for an isotropic material the generalized Hooke’s law (constitu-
tive equations) have the form

1
e 11 = [σ 11 − ν(σ 22 + σ 33 )]
E
1
e 22 = [σ 22 − ν(σ 33 + σ 11 )]
E
1
e 33 = [σ 33 − ν(σ 11 + σ 22 )]
E , (2.4.20)
1+ ν
e 21 = e 12 = σ 12
E
1+ ν
e 32 = e 23 = σ 23
E
1+ ν
e 31 = e 13 = σ 13
E
where equation (2.4.19) holds. These equations can be expressed in the indicial notation and have the form

1+ ν ν
e ij = σ ij − σ kk δ ij , (2.4.21)
E E
where σ kk = σ 11 + σ 22 + σ 33 is a stress invariant and δ ij is the Kronecker delta. We can solve for the stress
in terms of the strain by performing a contraction on i and j in equation (2.4.21). This gives the dilatation
1+ ν 3ν 1 − 2ν
e ii = σ ii − σ kk = σ kk .
E E E

Note that from the result in equation (2.4.21) we are now able to solve for the stress in terms of the strain.
We find

1+ ν ν
e ij = σ ij − e kk δ ij
E 1 − 2ν
E νE
e ij = σ ij − e kk δ ij (2.4.22)
1+ ν (1 + ν)(1 − 2ν)
E νE
or σ ij = e ij + e kk δ ij .
1+ ν (1 + ν)(1 − 2ν)
The tensor equation (2.4.22) represents the six scalar equations

E E
σ 11 = [(1 − ν)e 11 + ν(e 22 + e 33 )] σ 12 = e 12
(1 + ν)(1 − 2ν) 1+ ν
E E
σ 22 = [(1 − ν)e 22 + ν(e 33 + e 11 )] σ 13 = e 13
(1 + ν)(1 − 2ν) 1+ ν
E E
σ 33 = [(1 − ν)e 33 + ν(e 22 + e 11 )] σ 23 = e 23 .
(1 + ν)(1 − 2ν) 1+ ν