Geoscience Reference
In-Depth Information
simple shear into a rhomboid, both the sides and the diag-
onals of the square have experienced shear strain.
(a)
a
b
90°
3.14.5
Pure shear and simple shear
Pure shear and simple shear are examples of homogeneous
strain where a distortion is produced while maintaining
the original area (2D) or volume (3D) of the object. Both
types of strain give parallelograms from original cubes.
Pure shear
or homogeneous flattening is a distortion which
converts an original reference square object into a rectan-
gle when pressed from two opposite sides. The shortening
produced is compensated by a perpendicular lengthening
(Fig. 3.84a; see also Figs 3.81 and 3.83b). Any line in the
object orientated in the flattening direction or normal to it
does not suffer angular shear strain, whereas any pair of
perpendicular lines in the object inclined respect to these
directions suffer shear strain (like the diagonals or the rec-
tangle in Fig. 3.83b or the two normal to each other radii
in the circle in Fig. 3.84a).
Simple shear
is another kind of distortion that trans-
forms the initial shape of a square object into a rhomboid,
so that all the displacement vectors are parallel to each
other and also to two of the mutually parallel sides of the
rhomboid. All vectors will be pointing in one direction,
known as
shear direction
. All discrete surfaces which slide
with respect to each other in the shear direction are named
shear planes
, as will happen in a deck of cards lying on a
table when the upper card is pushed with the hand
(Fig. 3.84c). The two sides of the rhomboid normal to the
displacement vectors will suffer a rotation defining an
angular shear
Angular shear
γ
g
= tan
c
(3)
g
= tan
c
= tan -32° = -0.62
a
-32°
(b)
c
b
a
90°
c
(c)
30°
c
g
= tan 30° = 0.57
Fig. 3.83
Examples of measuring the angular shear in a square object
(a) deformed into a rectangle by pure shear (b) and a rhomboid (c)
by simple shear.
and will also suffer extension, whereas the
sides parallel to the shear planes will not rotate and will
remain unaltered in length as the cards do when we dis-
place them parallel to the table. Note the difference with
respect to the rectangle formed by pure shear whose sides
do not suffer shear strain. Note also that any circle repre-
sented inside the square is transformed into an ellipse in
both simple and pure shear. To measure strain, fossils or
other objects of regular shape and size can be used. If the
original proportions and lengths of different parts in the
body of a particular species are known (Fig. 3.85a), it is
possible to determine linear strain for the rocks in which
they are contained. Figure 3.85 shows an example of
homogeneous deformation in trilobites (fossile arthropods)
deformed by simple shear (Fig. 3.85b) and pure shear
(Fig. 3.85c). Note how two originally perpendicular lines
in the specimen, in this case the cephalon (head) and the
bilateral symmetry axis of the body, can be used to meas-
ure the shear angle and to calculate shear strain.
shows the object deformed by homogeneous flattening
(Fig. 3.83b), the sides of the square remain perpendicular
to each other, but notice that both diagonals of the square
(
a
and
b
in Fig. 3.83) initially at 90
have experienced
deformation by shear strain moving to the positions
a
and
b
in the deformed objects. To determine the angular
shear, the original perpendicular situation of both lines has
to be reconstructed and then the angle
can be measured.
In this case the line
a
has suffered a negative shear with
respect to
b
. The line
a
perpendicular to
b
has been plot-
ted and the angle between
a
and
a
defines the angular
shear. The shear strain is calculated by the tangent of the
angle
. The same procedure can be followed to calculate
the strain angle between both lines plotting a line normal
to
a
. Note that in this case the shear will be positive as the
angle between
b
and
a
. In the second
example (Fig. 3.83c) the square has been deformed by
is smaller than 90
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