3.6 Constitutive Relations
139
From Eq. (3.69), the required 2D strain invariants are (i, j = 1, 2)
I 1 = 3 + 2E ii = 3 + 2(E 11 + E 22 )
I 3 = det[δ ij + 2E ij ] =
1 + 2E 11 2E 12
2E 21 1 + 2E 22
= 1 + ,
where
= 2(E 11 + E 22 ) + 4(E 11 E 22 − E 12 E 21 ).
In Hooke’s law, stress depends linearly on strain. Since S ij = dW/dE ij , this
implies that W should be a quadratic function of the strain components. In the limit
of small strain, therefore, we will drop terms of order higher than E 2
ij .
For || << 1, expanding I
−β
3 in a Taylor series and omitting higher-order terms
yields
I
−β
3
= (1 + )
−β
= 1 − ββ +
1
2 β(β + 1))
2
= 1 − 2β(E 11 + E 22 ) + 2β(β + 1)(E
2
11 + E
2
22 ) + 4β
2 E 11 E 22 + 4βE 12 E 21 .
Note that three terms have been retained in the series expansion involving . This
is very important. Otherwise, some of the quadratic terms would be lost.
Substituting the above expressions for I 1 and I
−β
3 into Eq. (3.225) and simplifying give
W = μ
(β + 1)
E
2
11 + E
2
22
+ 2βE 11 E 22 + 2E 12 E 21
.
(3.226)
For small deformation, all the definitions for stress are essentially equal, and
Eq. (3.208) gives
S ij ≈ σ ij =
∂W
∂E ij
,
which provides the 2D stress components
σ 11 =
∂W
∂E 11
= 2μ
E 11 +
ν
1 − 2ν
(E 11 + E 22 )
σ 22 =
∂W
∂E 22
= 2μ
E 22 +
ν
1 − 2ν
(E 11 + E 22 )
σ 12 =
∂W
∂E 12
= 2μE 21
σ 21 =
∂W
∂E 21
= 2μE 12 .