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

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Presentation on theme: "Lecture 1. How to model: physical grounds Outline Physical grounds: parameters and basic laws Physical grounds: rheology."— Presentation transcript:

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

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

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

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

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

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

9 Strain tensor 1 2 3

10 1 2 3

11

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 Substantive time derivative Heat flux Shear heating

22 Physical grounds: constitutive laws

23 Rheology 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:  = E 

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 Brittle-ductileDuctile Modes of deformation

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

39 Full set of equations mass momentum energy

40 Final effective viscosity

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

42 Final effective viscosity if

43 Strength envelope Strength Depth (km) water crust Moho Mantle Strength Moho Mantle Crust


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