Physical Processes in Earth and Environmental Sciences Phần 4 pdf

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 is smaller than 90 . In the second

example (Fig. 3.83c) the square has been deformed by

simple shear into a rhomboid, both the sides and the diag￾onals of the square have experienced shear strain.

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  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.

88 Chapter 3

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.

90°

90° c

c

c

Angular shear γ

g = tan c

g = tan 30° = 0.57

g = tan c = tan –32° = –0.62

30°

–32°

(a)

(b)

(c)

a

a

b

a b

(3)

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3.14.6 The strain ellipse and ellipsoid

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 S1 (or e1), 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 S3

(or e3) 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 S1 and S3 are

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 S1 and the minor

axis the value of S3. An important property of the strain

axes is that they are mutually perpendicular lines which

were also perpendicular before strain. Thus the directions

Forces and dynamics 89

y

y

y

x

x

x

c

Flattening direction (a)

(c)

(b)

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.

ψ

(a) (b) (c)

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.

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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 S1, S2, and S3, being S1S2 S3 .

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 S1 and S3. 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

90 Chapter 3

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 I0 is 1,

S1 I1/1 I1 which represents the stretch S of the long axis.

Similarly S3 I1 giving the stretch S of the short axis.

S3

S3

r = 1

S1

S1

Before

deformation

Pure shear

Simple shear

(a)

(b)

(c)

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 .

R = 1

S3

S1

+c

R

R‘

(a) (b)

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