Page 268 - Intro to Tensor Calculus

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Differentiate this expression with respect to x k and verify the result
2
∂ u i ∂e ij ∂ω ij
= + . (2.4.52)
∂x j ∂x k ∂x k ∂x k
We further assume that the displacement field is continuous so that the mixed partial derivatives are equal
and
2 2
∂ u i ∂ u i
= . (2.4.53)
∂x j ∂x k ∂x k ∂x j
Interchanging j and k in equation (2.4.52) gives us
2
∂ u i ∂e ik ∂ω ik
= + . (2.4.54)
∂x k ∂x j ∂x j ∂x j
Equating the second derivatives from equations (2.4.54) and (2.4.52) and rearranging terms produces the
result
∂e ij ∂e ik ∂ω ik ∂ω ij
− = − (2.4.55)
∂x k ∂x j ∂x j ∂x k
∂ω ik ∂ω ij ∂ω jk
Making the observation that ω ij satisfies − = , the equation (2.4.55) simplifies to the
∂x j ∂x k ∂x i
form
∂e ij ∂e ik ∂ω jk
− = . (2.4.56)
∂x k ∂x j ∂x i
The term involving ω jk can be eliminated by using the mixed partial derivative relation
2 2
∂ ω jk ∂ ω jk
= . (2.4.57)
∂x i ∂x m ∂x m ∂x i
To derive the compatibility equations we differentiate equation (2.4.56) with respect to x m and then
interchanging the indices i and m and substitute the results into equation (2.4.57). This will produce the
compatibility equations
2 2 2 2
∂ e ij ∂ e mk ∂ e ik ∂ e mj
+ − − =0. (2.4.58)
∂x m ∂x k ∂x i ∂x j ∂x m ∂x j ∂x i ∂x k
This is a set of 81 partial differential equations which must be satisfied by the strain components. Fortunately,
due to symmetry considerations only 6 of these 81 equations are distinct. These 6 distinct equations are
known as the St. Venant’s compatibility equations and can be written as
2 2 2 2
∂ e 11 ∂ e 12 ∂ e 23 ∂ e 31
= − +
∂x 2 ∂x 3 ∂x 1 ∂x 3 ∂x 1 2 ∂x 1 ∂x 2
2 2 2 2
∂ e 22 ∂ e 23 ∂ e 31 ∂ e 12
= − +
∂x 1 ∂x 3 ∂x 2 ∂x 1 ∂x 2 2 ∂x 2 ∂x 3
2 2 2 2
∂ e 33 ∂ e 31 ∂ e 12 ∂ e 23
= − +
∂x 1 ∂x 2 ∂x 3 ∂x 2 ∂x 3 2 ∂x 3 ∂x 1
(2.4.59)
2 2 2
∂ e 12 ∂ e 11 ∂ e 22
2 = 2 + 2
∂x 1 ∂x 2 ∂x 2 ∂x 1
2 2 2
∂ e 23 ∂ e 22 ∂ e 33
2 = 2 + 2
∂x 2 ∂x 3 ∂x 3 ∂x 2
2 2 2
∂ e 31 ∂ e 33 ∂ e 11
2 = + .
2 2
∂x 3 ∂x 1 ∂x 1 ∂x 3
Observe that the fourth compatibility equation is the same as that derived in the example 2.4-3.
These compatibility equations can also be expressed in the indicial form
e ij,km + e mk,ji − e ik,jm − e mj,ki =0. (2.4.60)