Page 59 - Physical Principles of Sedimentary Basin Analysis

3.4 Strain 41

du 1

φ 2
dx 2
φ

φ 1 du 2
dx 1

Figure 3.3. The deformation of a right angle.

and an antisymmetric part
1 ∂u i ∂u j
R ij = − (3.10)
2 ∂x j ∂x i
(ε ij is symmetric because ε ij = ε ji , and R ij is antisymmetric because R ij =−R ji .) The
reason for expressing ∂u i /∂x j as these two parts is that the symmetric part ε ij expresses
volume and shape changes, while the antisymmetric part R ij expresses rotations. We
will later see that we have to distinguish rotations from volume and shape changes when
deformation (strain) is related to forces (stress).
We have already observed that a diagonal element of the symmetric part ε ij , for instance
ε 11 , represents relative length change in the x-direction. (If two points along the x-axis, x 1
and x 1 + dx 1 , become displaced by u 1 (x 1 ) and u 1 (x 1 + dx 1 ) respectively, then the relative
length change between the points is du 1 /dx 1 ≈ ∂u 1 /∂x 1 = ε 11 .)
The off-diagonal elements of ε ij are the amount a right-angle becomes deformed. See
Figure 3.3, where we have that φ 1 ≈ du 2 /dx 1 and φ 2 ≈ du 1 /dx 2 , because we can make
the approximation tan φ ≈ φ for small angles (resulting from small deformations). The
right-angle then becomes deformed by an amount

du 1 du 2
φ 1 + φ 2 = + = 2 ε 12 . (3.11)
dx 2 dx 1
The off-diagonal term ε 12 is thus one-half the amount a right-angle becomes deformed.
The interpretation of the two other off-diagonal elements ε 13 and ε 23 is similar.
It is the sum of the diagonal elements of ε ij that expresses local volume changes. This
is seen by considering a rectangular box with sides dx i (i = 1, 2, 3) before it is deformed.
The sides of the box are dx i + du i after the deformation, where du i is the difference in
displacement of the two diagonal opposite corners x and dx, see Figure 3.4. The difference
in volume of the box caused by the deformation is then

dV = du 1 dx 2 dx 3 + dx 1 du 2 dx 3 + dx 1 dx 2 du 3

du 1 du 2 du 3
= + + dx 1 dx 2 dx 3 (3.12)
dx 1 dx 2 dx 3