Geoscience Reference
In-Depth Information
Changes in length of lines
(+)
Extension
e
e =
l
1
-
l
0
Positive
shear strain
l
0
a
b
(1)
30 mm
l
0
Side a:
e =
(40-30)/30 = 0.33
c
c
−
90º
30 mm
l
0
b
33% lengthening
Side b:
e =
(22.5-30)/30 = -0.25
a
25% shortening
Negative
shear strain
Stretch
S
(
−
)
l
1
a
S
=
l
1
= 1 +
e
l
0
b
(2)
40 mm
Side a:
S
= 1 + 0.33 = 1.33
c
+ 90º
22.5 mm
l
1
b
c
Side b:
S
= 1 - 0.25 = 0.75
a
Fig. 3.81
Measuring the changes in the length of lines
a
and
b
in a
square transformed into a rectangle.
Fig. 3.82
Notation for negative and positive shear angles. The
dashed line represents the original nondeformed state of line
b
per-
pendicular to line
a
. Clockwise rotation is considered positive and
anticlockwise rotation negative.
are extension (
e
) and stretch (
S
).
Extension
is a nondimen-
sional parameter defined by subtracting the original non-
deformed length (
l
0
) from the final deformed length (
l
1
),
and normalizing it by dividing the result by the original
length (Fig. 3.81) so it becomes a proportion. All lines
which are longer than the original after deformation (as
line
a
in Fig. 3.81) have positive values of
e
, and all lines
suffering shortening (like line
b
in Fig. 3.81) have negative
values of
e
. Values of
e
range between
0/
l
0
0, although this situation is not likely to happen.
Maximum stretching will be produced when the deformed
length
l
1
is
. The square of the stretch is
also used to measure linear strain, when it is called
quad-
ratic elongation
or
quadratic extension
(
as
/
l
0
).
Angular changes between lines can be determined if the
object contains two mutually perpendicular lines before
deformation. When a line rotates and makes an angle dif-
ferent to 90
1 for maximum
shortening and
for maximum stretching, zero being
the value before deformation. Maximum shortening will
give a final deformed length
l
1
equal to zero (a very theo-
retical situation unlikely to happen, but even so it will be
the minimum possible value), and so (0
, the difference in angle from the original per-
pendicular position to the deformed position is called
angular shear
,
represents the
angular deformation which is called
shear strain
,
. The tangent of the angle
l
0
)/
l
0
l
0
/
(Fig. 3.82). Positive and negative angular shear has to be
defined to discriminate sense of rotation from the original
nondeformed state. Defining clockwise and anticlockwise
sense of rotation for reference has the same problem as for
rigid rotations described previously, the observation point
has to be defined. There is not a general agreement of
which sense is the positive or the negative and both
choices can be found in the literature. Another way of
defining the sign is to consider that when the resulting
deformed angle is bigger than 90
l
0
1. Maximum possible stretching (still very theoreti-
cal) will give a value for
l
1
of
and so the limit value for
the extension
e
is (
. Extension can also be
given in percentages multiplying
e
by 100.
Stretch, S
, is also
a nondimensional parameter used to measure shortening
or lengthening of lines contained in an object.
S
is the ratio
between the length of the line after deformation
l
1
and the
original length of the line
l
0
before deformation
(Fig. 3.81). The stretch can be also obtained by adding 1
to the extension
e
(since
e
l
0
)/
l
0
(
l
1
l
0
/
l
0
)
(
l
1
/
l
0
l
0
/
l
0
)
) the shear is
considered negative, and when the total deformed angle is
smaller than 90
(90
(
l
1
/
l
0
S
). The value of
S
for a nonstrained body is 1, as in this case
l
0
equals
l
1
.
Limiting values for
S
are 0 for maximum shortening to
1) and finally
e
1
l
1
/
l
0
) the shear will be defined as posi-
tive. Two examples of how to measure the shear strain are
shown in Figs 3.83b and c. The object before deformation
has a square shape (Fig. 3.83a). One of the examples
(90
for maximum stretching. Maximum shortening will hap-
pen when the final deformed length
l
1
became 0 as
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