III. Strain and Stress

1. III. Strain and Stress • Basics of continuum mechanics, Strain • Basics of continuum mechanics, Stress Reading Suppe, Chapter 3 Twiss&Moores, chapter 15 Additional References : Jean Salençon, Handbook of continuum mechanics: general concepts, thermoelasticity, Springer, 2001 Chandrasekharaiah D.S., Debnath L. (1994) Continuum Mechanics Publisher: Academic press, Inc.

2. Force • Force is the cause of deformation and/or motion of a body. • 2 kinds of force: • Contact forces - involve physical contact between objects. • Examples: the force involved in kicking a ball, pulling a wagon • Field forces - don't involve physical contact between objects. Examples: the gravitational force and the electromagnetic force

3. Stress Stress is force per unit area • Spreading out the weight reduces the stress with the same force. • Normal Stress is skier’s weight distributed over skis surface area. F=mg

4. Sign convention for σn • Geomechanics: normal stress is positive in compression (convention used here) • Continuum mechanics:normal stress is positive in extension

5. The stress (red vector) acting on a plane at M is the force exterted by one side over the other side divided by plane area…

6. The stress tensor The state of stress at a point can be characterizes from the stress tensor defined as …

7. Stress acting on a plane at point M… Let n be the unit vector defining an oriented surface with elementary area da at point M. (n points from side A to side B) Let dT be the force exerted on the plane by the medium on side B. It can be decomposed into a normal and shear component parallel to the surface. The stress vector is: n Side B Side A Normal stress Shear stress

8. Symmetry…

9. Principal stresses Because the matrix is symmetric, there is coordinate frame such that…. Engineering sign convention tension is positive,Geology sign convention compression is positive… Plane perpendicular toprincipal direction has no shear stress…

10. The deviatoric stress tensor… Stress tensor = mean stress + deviatoric stress tensor

11. The Mohr diagram 2-D stress on all possible internal planes… Sum of forces in 1- and 2-directions…

12. 2-D stress on all possible internal planes… Sum of forces in 1- and 2-directions…

13. Rearrange equations… Rearrange equations yet again… Get more useful relationship betweenprincipal stresses andstress on any plane….

14. The Mohr diagram

15. [1] What does a point on the circle mean? [2] What does the center of the circle tell you? [3] Where are the principle stresses? [4] What does the diameter or radius mean? [6] Where is the maximum shear stress?

16. Representation of the stress state in 3-D using the Mohr cirles. This circle represent the state of stress on planes parallel to σ3 The state of stress of a plane with any orientation plots in this domain This circle represent the state of stress on planes parallel to σ3 This circle represent the state of stress on planes parallel toσ1

17. Classification of stress state σ3 < σ1< 0 0 < σ3< σ1 • General tension • General compression • Uniaxial Compression • Uniaxial tension • Biaxial stress σ3=0 σ1=0 σ3 < 0 < σ1

18. Pure Shear(as a state of stress) • The exression ‘Pure shear’ is sometime used to characterize the a particular case of biaxial stress Do not confuse with ‘pure shear’ as a state of strain

19. Pole of the Mohr circle P A B σ3 σ1

20. Poles of the Mohr circle σ1 P A σ3 σ1

21. Poles of the Mohr circle σ3 σ1 P A σ3 σ1 A represent the state of stress on a facet with known orientation The geometric construction, based on the pole of the facet (P), allows to infer the state of stress on any orientation

22. Applications • Dip angle of a normal fault • Dip angle of a thrust fault • Stress ‘refraction’ across an interface.