1 stress iii mohr

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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