Solid Mechanics Part I Kelly
109
Volumetric Strain
The volumetric strain is the unit change in volume due to a deformation. It is an
important measure of deformation and is discussed in what follows.
4.3.1 Two-Dimensional Volumetric Strain
Analogous to Eqn 3.5.1, the strain invariants are
1
2
2
xx yy
xxyy xy
I
I
Strain Invariants (4.3.1)
Using the strain transformation formulae, Eqns. 4.2.2, it will be verified that these
quantities remain unchanged under any rotation of axes.
The first of these has a very significant physical interpretation. Consider the deformation
of the material element shown in Fig. 4.3.1a. The unit change in volume, called the
volumetric strain, is
()()
(1 )(1 ) 1
xx yy
xx yy xx yy
Vaabbab
Vab
(4.3.2)
If the strains are small, the term xxyy
will be very much smaller than the other two
terms, and the volumetric strain in that case is given by
xxyy
V
V
Volumetric Strain (4.3.3)
Figure 4.3.1: deformation of a material element; (a) normal deformation, (b) with
shearing
Since by Eqn. 4.3.1 the volume change is an invariant, the normal strains in any
coordinate system may be used in its evaluation. This makes sense: the volume change
x
y
a
b
a
b
x
y
a
b
a
b
(a) (b)
d
c
