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

3. Photomicrograph of ooid limestone. Grains are 0. 5-1 mm in size
Photomicrograph of ooid limestone.  Grains are mm in size.  Large grain in center shows well developed concentric calcite layers.

4. 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.

6. Deformed Ordovian trilobite
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.

7. 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.

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

9. 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
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

10. A. Describing the transformation of a body
Reference frame (coordinate system): R
Reference state (initial configuration): k0
State of the medium at time t: kt
Displacement (from t0 to t),
Velocity (at time t)
Position (at t), particle path (=trajectory)
Strain (changes in length of lines, angles between lines, volume)

11. A.1 Lagrangian parametrisation
Displacements
Trajectories
Streamlines

12. A.1 Lagrangian parametrisation
Volume Change

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

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

15. B. Homogeneous Tansformation
definition

16. Homogeneous Transformation
Changing reference frame

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

18. Homogeneous transformation
Convective transport of a volume

19. Homogeneous transformation
Convective transport of a surface

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

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

22. Tangent Homogeneous Deformation
Displacement field

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

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

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

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

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

28. 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.

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

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

31. Strain during homogeneous Deformation
Rigid Body Transformation

32. Strain during homogeneous Deformation
Rigid Body Transformation

33. Strain during homogeneous Deformation
Polar factorisation

34. Strain during homogeneous Deformation
Polar factorisation

35. Strain during homogeneous Deformation
Polar factorisation

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

37. D. Some properties of homogeneous Deformation

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

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

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

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

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

43. Homogeneous Transformation
x= R S X + c c
x1=S X
x2=R x1
x = x2+c

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

45. E. Infinitesimal transformation

46. E. Infinitesimal transformation
Infinitesimal strain tensor

47. Infinitesimal transformation

48. Relation between the infinitesimal strain tensor and displacement gradient

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

50. 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

51. Relation between the infinitesimal strain tensor and displacement gradient

52. 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.

53. Non-rotational transformationB b
x= S X + c

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

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

56. Uniaxial strain
NB: Uniaxial strain is a type a non-rotational deformation

57. Pure Shear
NB: Pure shear in is a type a non-rotational deformation (plane strain, l2=1)

58. Simple Shear
NB: Simple shear is rotational (Plane strain, l2=1)

60. Progressive simple Shear
Progressive simple shear is non coaxial

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