1 stress iii mohr
TRANSCRIPT
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Mohr Circle In 2D space (e.g., on the s
1
s2
, s1
s3
, or s2
s3plane), the normal stress (sn) and the shear
stress (ss), could be given by equations (1)and (2) in the next slides
Note: The equations are given here in thes1s2 plane, where s1is greater than s2. If we were dealing with the s2s3 plane,
then the two principal stresses would be s2and s
3
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Normal StressThe normal stress, sn:
sn= (s1+s2)/2 + (s1-s2)/2 cos2q 1) In parametric form the equation becomes:
sn = c + r cosWhere
c= (s1
+s2
)/2 is thecenter, which lies on thenormal stress axis (x axis)
r = (s1-s2)/2is the radius = 2q
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Sign Conventionssnis compressive when it is +, i.e., when sn>0snis tensile when it is -, i.e., when sn< 0sn= (s1+s2)/2+(s1-s2)/2 cos2q
NOTE:qis the angle froms1tothe normalto the plane!sn= s1 at q = 0
o (a maximum)
sn= s2 at q = 90o
(a minimum )
There is no shear stress on the three principal planes(perpendicular to the principal stresses)
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Resolved Normal and Shear Stress
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Shear Stress
The shear stressss= (s1-s2)/2 sin2q 2)
In parametric form the equation becomes:ss = r sin where = 2q
ss > 0 represents left-lateral shear
ss < 0 represents right-lateral shear
ss= 0 at q = 0oor 90o or 180o (a min)
ss= (s1-s2)/2 at q = + 45o
(maximum shear stress)
The maximum ssis 1/2 the differential stress
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Construction of the Mohr Circle in 2D
Plot the normal stress, sn, vs. shear stress, ss, on agraph paper using arbitrary scale (e.g., mm scale!)
Calculate:
Center c = (s1+s2)/2 Radius r = (s1-s2)/2
Note: Diameter is the dif ferential stress(s1-s2)
The circle intersects the sn(x-axis) at the twoprincipal stresses (s1ands2)
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Construction of the Mohr Circle
Multiply the physical angle qby 2
The angle 2qis from the cs1line to any point on thecircle
+2q(CCW) angles are read above the x-axis -2q(CW) angles below the x-axis, from the s
1axis
The sn andss of a point on the circle represent thenormal and shear stresses on the plane with the
given 2q angle NOTE: The axes of the Mohr circle have no
geographic significance!
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Mohr Circle for Stress
.
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Mohr Circle in 3D
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Maximum & Minimum Normal Stresses
The normal stress:sn= (s1+s2)/2 + (s1-s2)/2 cos2q
NOTE: q in physical space) is the angle from s1to the normalto the plane
When q = 0o thencos2q = 1 and sn=(s1+s2)/2 + (s1-s2)/2which reduces to a maximum value:
sn= (s1+s2 +s1-s2)/2 sn= 2s1/2 sn= s1When q = 90o thencos2q = -1 and sn= (s1+s2)/2 - (s1-s2)/2which reduces to a minimum
sn= (s1+s2-s1+s2)/2 sn= 2s2/2 sn= s2
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Special States of Stress - Uniaxial Stress
Uniaxial Stress(compression or tension) One principal stress (s1or s3) is non-zero, and
the other two are equal to zero
Uniaxial compression
Compressive stress in one direction: s1> s2=s3= 0| a 0 0|
| 0 0 0|| 0 0 0|
The Mohr circle is tangent to the ordinate at the
origin (i.e., s2=s3= 0) on the + (compressive) side
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Special States of Stress
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Uniaxial Tension
Tension in one direction:
0 = s1= s2 > s3|0 0 0|
|0 00|
|0 0-a|
The Mohr circle is tangent to the ordinate atthe origin on the -(i.e., tensile) side
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Special States of Stress - Axial Stress
Axial (confined) compression: s1> s
2= s
3> 0
|a 0 0||0 b 0||0 0 b|
Axial extension (extension): s1= s2 > s3 > 0|a 0 0||0 a 0|
|0 0 b|
The Mohr circle for both of these cases are to the
right of the origin (non-tangent)
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Special States of Stress - Biaxial Stress
Biaxial Stress:
Two of the principal stresses are non-zero and the other
is zero
Pure Shear:
s1= -s3and is non-zero (equal in magnitude but opposite insign)
s2= 0(i.e., it is a biaxial state) The normal stress on planes of maximum shear is zero
(pure shear!)
|a 0 0 ||0 0 0 ||0 0 -a|
The Mohr circle is symmetric w.r.t. the ordinate (center is at
the origin)
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Special States of Stress
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Special States of Stress - Triaxial Stress
Triaxial Stress:
s1, s2, ands3have non-zero values s1 > s2>s3and can be tensile or compressive
Is the most general state in nature|a 0 0 |
|0 b 0 |
|0 0 c|
The Mohr circle has three distinct circles
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Triaxial Stress
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Two-dimensional cases: General Stress
General Compression Both principal stresses are compressive
is common in earth)
General Tension
Both principal stresses are tensile
Possible at shallow depths in earth
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Isotropic Stress
The 3D, isotropic stresses are equal in magnitudein all directions (as radii of a sphere)
Magnitude = the mean of the principal stressessm= (s1+s2+s3)/3 = (s11+s22+s33)/3P = s1= s2= s3 when principal stresses are equal
i.e., it is an invariant (does not depend on aspecific coordinate system). No need to know the
principal stress; we can use any! Leads to dilation (+ev&-ev); but no shape change
ev=(v-vo)/vo= v/vo [no dimension]vand voare final and original volumes
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Stress in Liquids
Fluids (liquids/gases) are stressed equally in alldirections (e.g. magma); e.g.:
Hydrostatic, Lithostatic, Atmospheric pressure
All of these are pressure due to the column ofwater, rock, or air, respectively:
P = rgz zis thickness
ris density gis the acceleration due to gravity
H d t ti P H d t ti T i
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Hydrostatic Pressure- Hydrostatic Tension
Hydrostatic Pressure: s1= s2= s3= P|P 0 0|
|0 P 0||0 0 P|
All principal stresses are compressive and equal (P)
No shear stress exists on any plane All orthogonal coordinate systems are principal
coordinates
Mohr circle reduces to a point on the snaxis Hydrostatic Tension
The stress across all planes is tensile and equal
There are no shearing stresses
Is an unlikely case of stress in the earth
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Deviatoric Stress A total stress sTcan be divided into its components: isotropic(Pressure) or mean stress(sm)
Pressureis the mean of the principal stresses (may be
neglected in most problems). Only causes volume change.
deviatoric(sd) that deviates from the mean Deviators components are calculated by subtracting the
mean stress (pressure) from each of the normal stresses of
the general stress tensor (not the shear stresses!). Causesshape change and that it the part which we are most
interested in.
sT=sm+sd or sd=sT-sm
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Confining Pressure
In experimental rock deformation, pressureis called confining pressure, and is taken to
be equal to the s2 ands3 (uniaxial loading)
This is the pressure that is hydraulically
applied around the rock specimen
In the Earth, at any point z, the confining
pressure is isotropic (lithostatic) pressure:P = rgz
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Decomposition of Matrix
Decomposition of the total stress matrix into the
mean and deviatoric matrices
The deviator ic part of total stress leads to change inshape
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Example - Deviatoric & Mean stress
Given: s1 = 8 Mpa, s2 = 5 Mpa, and s3 = 2 MpaFind the mean and the diviatoric stresses
The mean stress (sm):sm = (8 + 5 + 2) / 3 = 5 MPaThe deviatoric stresses (sn):s1 = 8-5 = 3 Mpa (compressive)s2 = 5-5 = 0 Mpas3 = 2-5 = -3 Mpa (tensile)
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Differential Stress
The difference between the maximum and theminimum principal stresses (s1-s2)
Is always positive
Its value is:
twice the radius of the largest Mohr circle
It is twice the maximum shear stresses
Note: ss
= (s1
-s2
)/2 sin2qss= (s1-s2)/2 at q = + 45
o (a maximum)
The maximum ssis 1/2 the differential stress
Is an invariant of the stress tensor
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Effective Stress
Its components are calculated by subtracting the
internal pore fluid pressure (Pf) from each of thenormal stresses of the external stress tensor
This means that the pore fluid pressures opposesthe external stress, decreasing the effective
confining pressure
The pore fluid pressure shifts the Mohr circle
toward lower normal stresses. This changes the
applied stress into an effective stress
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Effective Stress
(applied stress - pore fluid pressure)= effective stress
|s11s12s13| | Pf 0 0 | |s11- Pf s12 s13 ||s21s22s23| - | 0 Pf 0 |=|s21 s22Pf s23 ||s
31
s32
s33
| | 0 0 Pf
| |s31
s32
s33
- Pf
|
Mechanical behavior of a brittle material depends
on the effective stress, not on the applied stress
P Fl id P
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Pore Fluid Pressure