Mohr Circle for stress

1. Mohr Circle for stress • In 2D space (e.g., on the s1s2 , s1s3, or s2s3 plane), 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 the s1s2 plane, where s1 is greater than s2. • If we were dealing with the s2s3 plane, then the two principal stresses would be s2 and s3

2. Normal Stress The normal stress, sn: sn= (s1+s2)/2 + (s1-s2)/2 cos2q (1) • In parametric form the equation becomes: sn = c + r cosω Where • c = (s1+s2)/2 is the center, which lies on the normal stress axis (x axis) • r = (s1-s2)/2 is the radius • ω= 2q

3. Sign Conventions sn is compressive when it is “+”, i.e., when sn>0 sn is tensile when it is “-”, i.e., when sn< 0 sn= (s1+s2)/2+(s1-s2)/2 cos2q NOTE: qis the angle froms1to the normal to the plane! sn = s1 at q = 0o(a maximum) sn = s2 at q = 90o(a minimum ) • There is no shear stress on the three principal planes (perpendicular to the principal stresses)

4. Resolved Normal and Shear Stress normal to plane plane

5. Shear Stress The shear stress ss = (s1-s2)/2 sin2q (2) • In parametric form the equation becomes: ss = r sinωwhere ω= 2q ss > 0 ‘+’ shear stress represents left-lateral shear ss < 0 ‘-’ shear stress represents right-lateral shear ss = 0 (a min)at q = 0o or 90o or 180o ss = (s1-s2)/2 (a max) at q = + 45o • The maximum ss is 1/2 the differential stress (diameter), i.e., it is the length of the radius!

6. Construction of the Mohr Circle in 2D • Plot the normal stress, sn, vs. shear stress, ss, on a graph paper using arbitrary scale (e.g., mm scale!) • Calculate: • Center c = (s1+s2)/2 • Radius r = (s1-s2)/2 • Note: Diameter is the differential stress(s1-s2) • The circle intersects the sn (x-axis) at the two principal stresses (s1 ands2)

7. Construction of the Mohr Circle • Multiply the physical angle q by 2 • The angle 2q is from the cs1 line to any point on the circle • +2q (CCW) angles are read above the x-axis • -2q (CW) angles below the x-axis, from the s1 axis • The sn andss of a point on the circle represent the normal and shear stresses on the plane with the given 2q angle • NOTE: The axes of the Mohr circle have no geographic significance!

8. Mohr Circle for Stress Max s .

9. Mohr Circle in 3D

10. Maximum & Minimum Normal Stresses The normal stress: sn= (s1+s2)/2 + (s1-s2)/2 cos2q NOTE: q (in physical space) is the angle from s1 to the normal to the plane When q = 0othencos2q = 1 and sn=(s1+s2)/2 + (s1-s2)/2 which reduces to a maximum value: sn= (s1+s2 +s1-s2)/2  sn= 2s1/2  sn= s1 When q = 90othencos2q = -1 and sn= (s1+s2)/2 - (s1-s2)/2 which reduces to a minimum sn= (s1+s2 -s1+s2)/2  sn= 2s2/2  sn= s2

11. Special States of Stress - Uniaxial Stress • Uniaxial Stress (compression or tension) • One principal stress (s1 or 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

12. Special States of Stress

13. Uniaxial Tension Tension in one direction: 0 = s1 = s2 > s3 |0 0 0| |0 0 0| |0 0-a| • The Mohr circle is tangent to the ordinate at the origin on the - (i.e., tensile) side

14. Special States of Stress - Axial Stress • Axial (confined) compression: s1 > s2 = s3 > 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)

15. Special States of Stress - Biaxial Stress • Biaxial Stress: • Two of the principal stresses are non-zero and the other is zero • Pure Shear: s1 = -s3 and is non-zero (equal in magnitude but opposite in sign) 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)

16. Special States of Stress

17. 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

18. Triaxial Stress

19. 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

20. Isotropic Stress • The 3D, isotropic stresses are equal in magnitude in all directions (as radii of a sphere) • Magnitude = the mean of the principal stresses sm= (s1+s2+s3)/3 = (s11+s22+s33 )/3 P = s1= s2= s3when principal stresses are equal • i.e., it is an invariant (does not depend on a specific 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] v´and vo are final and original volumes

21. Stress in Liquids • Fluids (liquids/gases) are stressed equally in all directions (e.g. magma); e.g.: • Hydrostatic, Lithostatic, Atmospheric pressure • All of these are pressure due to the column of water, rock, or air, respectively: P = rgz • z is thickness • ris density • g is the acceleration due to gravity

22. 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 sn axis • 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

23. Deviatoric Stress • A total stress sTcan be divided into its components: • isotropic (Pressure) or mean stress (sm) • Pressure is the mean of the principal stresses (may be neglected in most problems). Only causes volume change. • deviatoric (sd) that deviates from the mean • Deviator’s components are calculated by subtracting the mean stress (pressure) from each of the normal stresses of the general stress tensor (not the shear stresses!). Causes shape change and that it the part which we are most interested in. sT=sm+sd or sd=sT-sm

24. Confining Pressure • In experimental rock deformation, pressure is called confining pressure, and is taken to be equal to the 2 and3 (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

25. Decomposition of Matrix • Decomposition of the total stress matrix into the mean and deviatoric matrices • The deviatoric part of total stress leads to change in shape

26. Example - Deviatoric & Mean stress Given: s1 = 8 Mpa, s2 = 5 Mpa, and s3 = 2 Mpa Find the mean and the diviatoric stresses The mean stress (sm): sm = (8 + 5 + 2) / 3 = 5 MPa The deviatoric stresses (sn): s1= 8-5 = 3 Mpa (compressive) s2 = 5-5 = 0 Mpa s3 = 2-5 = -3 Mpa (tensile)

27. Differential Stress • The difference between the maximum and the minimum 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 sin2q ss = (s1-s2)/2 at q = + 45o(a maximum) • The maximum ss is 1/2 the differential stress • Is an invariant of the stress tensor

28. Effective Stress • Its components are calculated by subtracting the internal pore fluid pressure (Pf) from each of the normal stresses of the external stress tensor • This means that the pore fluid pressures opposes the 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

29. Effective Stress • (applied stress - pore fluid pressure)= effective stress |s11s12s13 | | Pf 0 0 | |s11- Pf s12s13 | |s21s22s23 | - | 0 Pf 0 |=|s21 s22 – Pf s23 | |s31 s32s33 | | 0 0 Pf | |s31 s32 s33- Pf | • Mechanical behavior of a brittle material depends on the effective stress, not on the applied stress

30. Pore Fluid Pressure