1. Lecture 1. How to model: physical grounds
Outline
Physical grounds: parameters and basic laws
Physical grounds: rheology

2. Physical grounds: displacement, strain and stress tensors

3. Strain ξi= ξi (x1,x2,x3,t) i=1,2,3 r´ r x3 t´ t x2 x1 ξ1,ξ2,ξ3

4. Strain Lagrangian ξi= ξi (x1,x2,x3,t) i=1,2,3 r´ r x3 t´ t x2 x1
ξ1,ξ2,ξ3 r
x1,x2,x3 x2 x1

5. Strain Lagrangian ξi= ξi (x1,x2,x3,t) i=1,2,3 xi= xi (ξ1,ξ2,ξ3,t)

6. Strain Lagrangian ξi= ξi (x1,x2,x3,t) i=1,2,3 xi= xi (ξ1,ξ2,ξ3,t)u
xi= xi (ξ1,ξ2,ξ3,t)
i=1,2,3 t r´
ξ1,ξ2,ξ3 r
x1,x2,x3 x2 x1
u= r´-r -displacement vector

7. Strain Lagrangian ξi= ξi (x1,x2,x3,t) i=1,2,3 xi= xi (ξ1,ξ2,ξ3,t)u
xi= xi (ξ1,ξ2,ξ3,t)
i=1,2,3 t r´
ξ1,ξ2,ξ3 r
x1,x2,x3
ui= ξi -xi= ui (x1,x2,x3,t)
i=1,2,3 x2 x1
u= r´-r -displacement vector

8. Strain Lagrangian ξi= ξi (x1,x2,x3,t) i=1,2,3 xi= xi (ξ1,ξ2,ξ3,t)u
xi= xi (ξ1,ξ2,ξ3,t)
i=1,2,3 t r´
ξ1,ξ2,ξ3 r
x1,x2,x3
ui= ξi -xi= ui (x1,x2,x3,t)
i=1,2,3 x2
ui= ξi -xi= ui (ξ1,ξ2,ξ3,t)
i=1,2,3 x1
u= r´-r -displacement vector
Eulerian

9. Strain tensor 1 3 2

10. Strain tensor 1 2 3

11. Strain tensor

12. Strain and strain rate tensors

13. Geometric meaning of strain tensor
Changing length
Changing angle
Changing volume

14. Stress tensor
is the i component of the force acting at the unit surface which is orthogonal to direction j

15. Tensor invariants, deviators
Stress deviator
P is pressure
Stress norm

16. Principal stress axes

17. Physical grounds: conservation laws

18. Mass conservation equation
Material flux
Substantive time derivative

19. Mass conservation equation

20. Momentum conservation equation
Substantive time derivative

21. Energy conservation equation
Shear heating
Heat flux
Substantive time derivative

22. Physical grounds: constitutive laws

23. Rheology – the relationship between stress and strain
Rheology for ideal bodies
Elastic
Viscous
Plastic

24. An isotropic, homogeneous elastic material follows Hooke's Law
Hooke's Law: s = Ee

25. For an ideal Newtonian fluid: differential stress = viscosity x strain rate viscosity: measure of resistance to flow

26. Ideal plastic behavior

27. Constitutive laws Elastic strain rate Viscous flow strain rate
Plastic flow strain rate

28. Superposition of constitutive laws;

29. Full set of equations
mass
momentum
energy

30. How Does Rock Flow?
Brittle – Low T, low P
Ductile – as P, T increase

31. Brittle
Modes of deformation
Brittle-ductile
Ductile

32. Deformation Mechanisms
Brittle
Cataclasis
Frictional grain-boundary sliding
Ductile
Diffusion creep
Power-law creep
Peierls creep

33. Cataclasis Fine-scale fracturing and movement along fractures.
Favoured by low-confining pressures

34. Frictional grain-boundary sliding
Involves the sliding of grains past each other - enhanced by films on the grain boundaries.

35. Diffusion Creep
Involves the transport of material from one site to another within a rock body
The material transfer may be intracrystalline, intercrystalline, or both.
Newtionian rheology:

36. Dislocation Creep
Involves processes internal to grains and crystals of rocks.
In any crystalline lattice there are always single atoms (i.e. point defects) or rows of atoms (i.e. line defects or dislocations) missing. These defects can migrate or diffuse within the crystalline lattice.
Motion of these defects that allows a crystalline lattice to strain or change shape without brittle fracture.
Power-law rheology :

37. Peierls Creep
For high differential stresses (about times Young’s modulus as a threshold), mixed dislocation glide and climb occurs. This mechanism has a finite yield stress (Peierls stress, ) at absolute zero temperature conditions.

38. Three creep processes Diffusion creep Dislocation creep Peierls creep
( Kameyama et al. 1999)

39. Full set of equations
mass
momentum
energy

40. Final effective viscosity

41. Plastic strain if if if Drucker-Prager yield criterion
Von Mises yield criterion
softening

42. Final effective viscosityif if

43. Strength envelope Strength 50 Depth (km) 100 water crust Moho Crust50
100
Strength
Depth (km)
water
crust
Moho
Mantle
Crust