1. III. Strain and Stress Strain Stress Rheology Reading Suppe, Chapter 3 Twiss&Moores, chapter 15

3. Stretched belemnite boudins with quartz and calcite infill. The space between the broken pieces of the belemnite are filled with precipitated material. The more translucent material in the middle of the gaps is quartz, the material closer to the pieces is calcite. Photo from the root zone of the Morcles nappe in the Rhone valley, Switzerland by Martin Casey http://www.see.leeds.ac.uk/structure/strain/gallery/belpart.html

5. Photomicrograph of ooid limestone. Grains are 0.5-1 mm in size. Large grain in center shows well developed concentric calcite layers.

6. Thin section (transmitted light) of deformed oolitic limestone, Swiss Alps. The approximate principal extension (red) and principal shortening directions (yellow) are indicated. The ooids are not perfectly elliptical because the original ooids were not perfectly spherical and because compositional differences led to heterogeneous strain. The elongated ooids define a crude foliation subperpendicular to the principal shortening direction. Photo credit: John Ramsay. http://www.rci.rutgers.edu/~schlisch/structureslides/oolite.html

7. http://www.alexstrekeisen.it/english/meta/deformedoolite.php

8. Deformed Ordovian trilobite. Deformed samples such as this provide valuable strain markers. Since we know the shape of an undeformed trilobite of this species, we can compare that to this deformed specimen and quantify the amount and style of strain in the rock.

10. III. Strain and Stress Strain –Basics of Continuum Mechanics –Geological examples 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.

11. Deformation of a deformable body can be discontinuous (localized on faults) or continuous. Strain: change of size and shape of a body

12. Basics of continuum mechanics, Strain A. Displacements, trajectories, streamlines, emission lines 1- Lagrangian parametrisation 2- Eulerian parametrisation B Homogeneous and tangent Homogeneous transformation 1- Definition of an homogeneous transformation 2- Convective transport equation during homogeneous transformation 3- Tangent homogeneous transformation C Strain during homogeneous transformation 1- The Green strain tensor and the Green deformation tensor Infinitesimal vs finite deformation 2- Polar factorisation D Properties of homogeneous transformations 1- Deformation of line 2- Deformation of spheres 3- The strain ellipse 4- The Mohr circle E Infinitesimal deformation 1- Definition 2- The infinitesimal strain tensor 3- Polar factorisation 4- The Mohr circle F Progressive, finite, and infinitesimal deformation : 1- Rotational/non rotational deformation 2- Coaxial Deformation G Case examples 1- Uniaxial strain 2- Pure shear, 3- Simple shear 4- Uniform dilatational strain

13. Reference frame (coordinate system): R Reference state (initial configuration):  0 State of the medium at time t:  t –Position (at t), particle path (=trajectory) –Displacement (from t 0 to t), –Velocity (at time t) –Strain (changes in length of lines, angles between lines, volume) A. Describing the transformation of a body

14. A.1 Lagrangian parametrisation Displacements Trajectories Streamlines

15. Volume Change A.1 Lagrangian parametrisation

16. A.2 Eulerian parametrisation Trajectories: Streamlines: (at time t)

17. A.3 Stationary Velocity Field Velocity is independent of time NB: If the motion is stationary in the chosen reference frame then trajectories=streamlines

18. B. Homogeneous Tansformation definition

19. Homogeneous Transformation Changing reference frame

20. Homogeneous Transformation Convective transport of a vector Implication: Straight lines remain straight during deformation

21. Homogeneous transformation Convective transport of a volume

22. Homogeneous transformation Convective transport of a surface

23. Tangent Homogeneous Deformation Any transformation can be approximated locally by its tangent homogeneous transformation

24. Tangent Homogeneous Deformation Any transformation can be approximated locally by its tangent homogeneous transformation

25. Tangent Homogeneous Deformation Displacement field

26. D. Strain during homogeneous Deformation The Cauchy strain tensor (or expansion tensor)

27. Strain during homogeneous Deformation Stretch (or elongation) in the direction of a vector

28. Strain during homogeneous Deformation Stretch (or elongation) in the direction of a vector Extension (or extension ratio), relative length change

29. Strain during homogeneous Deformation Change of angle between 2 initially orthogonal vectors Shear angle

30. Strain during homogeneous Deformation Signification of the strain tensor components

31. Strain during homogeneous Deformation An orthometric reference frame can be found in which the strain tensor is diagonal. This define the 3 principal axes of the strain tensor.

32. Strain during homogeneous Deformation The Green-Lagrange strain tensor (strain tensor)

33. Strain during homogeneous Deformation The Green-Lagrange strain tensor

34. Rigid Body Transformation Strain during homogeneous Deformation

35. Rigid Body Transformation Strain during homogeneous Deformation

36. Polar factorisation Strain during homogeneous Deformation

37. Polar factorisation Strain during homogeneous Deformation

38. Polar factorisation Strain during homogeneous Deformation

39. Pure deformation: The principal strain axes remain parallel to themselves during deformation Strain during homogeneous Deformation

40. D. Some properties of homogeneous Deformation

41. Some properties of homogeneous Deformation The strain tensor is uniquely characterized by the strain ellipsoid (a sphere with unit radius in the initial configuration)

42. Some properties of homogeneous Deformation The strain tensor is uniquely characterized by the strain ellipsoid (a sphere with unit radius in the initial configuration)

43. Some properties of homogeneous Deformation The Mohr Circle for finite strain

44. Some properties of homogeneous Deformation The Mohr Circle for finite strain

45. The strain tensor associated to an homogeneous transformation does not define uniquely the transformation (the translation and the rotation terms remain undetermined) Some properties of homogeneous Deformation

46. x= R S X + c x1=S Xx1=S Xx2=R x1x2=R x1 X = x 2 +c Homogeneous Transformation c

47. Classification of strain The Flinn diagram characterizes the ellipticity of strain (for constant volume deformation: with

48. E. Infinitesimal transformation

49. Infinitesimal strain tensor

50. Infinitesimal transformation

51. Relation between the infinitesimal strain tensor and displacement gradient

52. NB: The representation of principal extensions on this diagram is correct only for infinitesimal strain only The strain ellipse

53. Rk: For an infinitesimal deformation the principal extensions are small (typically less than 1%). The strain ellipse are close to a circle. For visualisation the strain ellipse is represented with some exaggeration

54. Relation between the infinitesimal strain tensor and displacement gradient

55. F. Finite, infinitesimal and progressive deformation –Finite deformation is said to be non-rotational if the principle strain axis in the initial and final configurations are parallel. This characterizes only how the final state relates to the initial state –Finite deformation of a body is the result of a deformation path (progressive deformation). –There is an infinity of possible deformation paths to reach a particular finite strain. –Generally, infinitesimal strain (or equivalently the strain rate tensor) is used to describe incremental deformation of a body that has experienced some finite strain –A progressive deformation is said to be coaxial if the principal axis of the infinitesimal strain tensor remain parallel to the principal axis of the finite strain tensor. This characterizes the deformation path.

56. If A and B are parallel to a and b respectively the deformation is said to be non-rotational (This means R= 1) Rotational vs non-rotational deformation

57. x= S X + c Non-rotational transformation a b A B (a//A and b//B)

58. Non-rotational non-coaxial progressive transformation Stage 1: Stage 2: a b A B

59. NB: Uniaxial strain is a type a non-rotational deformation (pure strain) Uniaxial strain

60. Pure Shear NB: Pure shear in is a type a non-rotational deformation (plane strain,  2 =1)

61. Simple Shear NB: Simple shear is rotational (Plane strain,  2 =1)

62. (NB: These are two example of stationnary transformations)

63. Progressive simple Shear Progressive simple shear is non coaxial

64. Progressive pure shear Progressive pure shear is a type of coaxial strain