Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 82

Fig. 3.72. Initial data are 1 40 MPa, 2 20 MPa, in the equation [( 1 3 )/2] sin 2 , the value of sin

and 3 are marked in

the x -axis at 40 and 20 MPa respectively; then both points
are joined by a circle with extreme values 20 and 40 MPa,
centered in the x -axis at the mean stress value
(

60 . First the positions of 1 and 2 1. In the numerical example (Fig. 3.72) the maxi-

mum value for corresponding to planes at 2 90 will

be 10 MPa.

1 3 )/2 30 MPa. The value n and of the surface

3.13.8 States of stress

orientated 60 1 can be obt-

ained by plotting the line measuring a double angle
(2

with respect to the normal to

), in this case in the upper half of the circle, as
the considered angle is positive (anticlockwise). Once the
line is plotted, the point p (the intersection of the line with
the circle), has coordinates (

120 Different states of stress are possible depending on the val-

ues of the nine tensor components (three normal tractions
and six shear tractions). Some of those will be discussed
briefly; they are represented by distinctive Mohr circles
(Fig. 3.74). Definitions of the states of stress are relevant
for rheological studies and fracture mechanics (Sections
3.15 and 4.14).
Hydrostatic stress or hydrostatic pressure (Section 3.5) is
the state of stress characteristic of fluids in which all the
tractions have the same value in all directions of space and
so it is not possible to define directions for the principal
stress axis as

) which can be obtained
by reading the respective values at the coordinate axes.
The values of

n ,

can also be calculated by using the
fundamental stress equations explained earlier. Resulting
values are

n and

n 25 MPa and 8.7 MPa. The differential

stress can be easily calculated as 20 MPa.

Considering the fundamental stress equations and look-
ing at the Mohr circles (Fig. 3.73) it is obvious that the
maximum possible value for the shear stress is given by
the value of the Mohr circle radius which has the value of
the deviatoric stress

1 3

pressure (Fig. 3.74a). All
tractions are compressive and normal stresses. Hydrostatic
stress is characterized by the absence of shear stress and
consequently, in the Mohr circle this particular kind of
stress is represented by a point in the x -axis corresponding
to the value of the pressure. The lithostatic stress , applied to

1 2 3

1 3 /2. Two planes of maximum
shear stress occur, each at 45 from 1 ( is 45 too) since

(a)

s 1

s 1

u = 45º u = − 45º 40 MPa

60º

u

s 3 s 3

s n

s 3

s 3

t

20 MPa

s 1

t

(b)

t

s 1 20

p

Max absolute
value for
shear stress t

10

2 u = +90°

t = 8.7

120º

s n

s n

s 1 u

0

s 3

2

0

-10 10 20 30 40 50

2 u = − 90°

s n = 25

Max absolute
value for
shear stress t

-10

Mean normal stress = 30 MPa
Differential stress = 20 MPa
Maximum value for shear stress = 10 MPa

-20

Fig. 3.72 The planes of maximum hear stress are those located at
an angle 45 (and also are 45 apart from the main principal

1 ). Fig. 3.73 A numerical example for the use of the Mohr circle.

stress