Page 60 - Physical Principles of Sedimentary Basin Analysis

42 Linear elasticity and continuum mechanics

du 2

dx 2

dx 1 du 1
Figure 3.4. A rectangular box with initial sides dx 1 and dx 2 is displaced by a deformation, and
the difference in the displacement of the two parallel sides are du 1 and du 2 . The change in volume
is dV = dx 1 du 2 + du 1 dx 2 , when the second-order contribution du 1 du 2 to dV is ignored. (It is
already assumed that du i
dx i .)

u(x + dx)
du
x + dx
a
dx 2 dx 2

u(x)
x
dx 1 dx 1
Figure 3.5. Left: the two points x and x + dx are displaced by the vectors u(x) and u(x + dx),
respectively. Right: the two points are separated by dx + du after the displacement, where du =
u(x + dx)−u(x). The distance du + dx between the two points after the displacement is the initial
distance vector between the points dx rotated on an angle a.

and we get
dV ∂u i
≈ = ε ii (3.13)
V ∂x i
(where the summation convention is used). The relative volume change, also called the
dilatation, is simply the sum of the diagonal elements of ε ij .
We will now look at the antisymmetric part R ij in two dimensions to convince ourselves
that it really expresses rotation. The difference in the displacement du i between two points
initially separated by (dx 1 , dx 2 ) becomes

du 1 0 −a dx 1
= (3.14)
du 2 a 0 dx 2
where R 12 =−R 21 =−a. See Figure 3.5. The distance between the two points in the

deformed state is (dx , dx ) = (dx 1 + du 1 , dx 2 + du 2 ), which can be written
1 2

dx 1 −a dx 1
1 = . (3.15)
dx a 1 dx 2
2