Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 89

Flattening direction

(a)

y

(c)

x

(b)

y

y

c

x x

Fig. 3.84 Pure shear (a) and simple shear (b) are two examples of homogeneous strain. Both consist of distortions (no area or volume changes
are produced; (c) Simple shear has been classically compared to the shearing of a new card deck whose cards slide with respect to each other
when pushed (or sheared) by hand in one direction.

(a) (b) (c)

ψ

Fig. 3.85 Homogeneous deformation in fossil trilobites: (a) nondeformed specimen; (b) deformed by simple shear. Note how two originally
perpendicular lines such as the cephalon base and the bilateral symmetry axis can be used to measure the shear angle and calculate shear strain
(c) deformed by pure shear. If the original size and proportions of three species is known, linear strain can be established.

3.14.6 The strain ellipse and ellipsoid

known as the principal axis of the strain ellipse and are
mutually perpendicular. The strain ellipse records not only
the directions of maximum and minimum stretch or exten-
sion but also the magnitudes and proportions of both
parameters in any direction. To understand the values of
the axis of the strain ellipse imagine the homogeneous
deformation of a circle having a radius of magnitude 1,
which will be the value of l 0 (Fig. 3.86a). Now, if we apply
the simple equation of the stretch S (Equation 2; Fig. 3.81)
whereas for a given direction, the stretch e is the difference
in length between the radius of the ellipse and the initial
undeformed circle of radius 1 it is easy to see that the major
axis of the ellipse will have the value of S 1 and the minor
axis the value of S 3 . An important property of the strain
axes is that they are mutually perpendicular lines which
were also perpendicular before strain. Thus the directions

We have seen earlier (Figs 3.80 and 3.84) that when homo-
geneous deformation occurs any circle is transformed into
a perfectly regular ellipse. This ellipse describes the change
in length for any direction in the object after strain; it is
called the strain ellipse. For instance, the major axis of the
ellipse, which is named S 1 (or e 1 ), is the direction of maxi-
mum lengthening and so the circle is mostly enlarged in
this direction. Any other lines having different positions on
the strained objects which are parallel to the major axis of
the ellipse suffer the maximum stretch or extension.
Similarly the minor axis of the ellipse, which is known as S 3
(or e 3 ) is the direction where the lines have been shortened
most, and so the values of the extension e and the stretch S
are minimum. The axis of the strain ellipse S 1

and S 3 are