Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 90

of maximum and minimum extension or stretch corre-
spond to directions that do not experience (at that point)
shear strain (note the analogy with the stress ellipse in
which the principal stress axis are directions in which no

shear stress is produced). Shear strain can be determined in
the ellipse by two originally perpendicular lines, radii R of
the circle and the line tangent to a radius at the perimeter
(Fig. 3.87). In (a), before deformation, the tangent line to
the circle is perpendicular to the radius R . In (b), after
deformation, the lines are no longer normal to each other
and so an angular shear can be measured and the shear

strain calculated, as explained earlier.
In strain analysis two different kinds of ellipse can be
defined, (i) the instantaneous strain ellipse which defines
the homogeneous strain state of an object in a small incre-
ment of deformation and (ii) the finite strain ellipse which
represents the final deformation state or the sum of all the
phases and increments of instantaneous deformations that
the object has gone through. In 3D a regular ellipsoid will
develop with three principal axes of the strain ellipsoid,
namely S 1 , S 2 , and S 3 , being S 1

(a) Before

deformation

r = 1

Pure shear

S 3

(b)

S 3 .
Now that we have introduced the concept of the strain
ellipse we can return to the previous examples of homoge-
neous deformation and have a look at the behavior of the
strain axes. In the example of Fig. 3.84 the familiar square
is depicted again showing an inner circle (Fig. 3.88). Two
mutually perpendicular radius of the circle have been
marked as decoration. Note that a pure shear strain has
been produced in four different steps. The circle
has become an ellipse that, as the radius of the circle has
a value of 1, will represent the strain ellipsoid, with two
principal axes S 1 and S 3 . Note that when a pure shear is
produced the orientation of the principal strain axis
remains the same through all steps in deformation and so
it is called coaxial strain (Fig. 3.88). This means that the
directions of maximum and minimum extension are pre-
served with successive stages of flattening. A very different
situation happens when simple shear occurs (Fig. 3.89):
the axes of the strain ellipsoid rotate in the shear direction

S 2

S 1

(c)

Simple shear

S 3

S 1

Fig. 3.86 The stress ellipse in 2D strain analysis reflects the state of
strain of an object and represents the homogeneous deformation of
a circle of radius 1 transformed into an ellipsoid. As I 0 is 1,

S 1 I 1 which represents the stretch S of the long axis.

Similarly S 3

I 1 /1

I 1 giving the stretch S of the short axis.

(a) (b)

+ c

S 3

R

R '

R = 1

S 1

Fig. 3.87 Shear strain in the strain ellipse. In (a), before deformation, the tangent line to the circle is perpendicular to the radius R . In (b), after
deformation, both lines are not normal to each other, the angular shear can be obtained and the shear strain calculated by tracing a normal line

to the tangent to the circle at the point where R' intercepts the circle, and measuring the angle . The shear strain can be calculated as y tan .