Download presentation

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 viscosity**

if if

43
**Strength envelope Strength 50 Depth (km) 100 water crust Moho Crust**

50 100 Strength Depth (km) water crust Moho Mantle Crust

Similar presentations

OK

AOE 5104 Class 7 Online presentations for next class: Homework 3

AOE 5104 Class 7 Online presentations for next class: Homework 3

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on construction of 3 phase induction motor Download ppt on gender discrimination in india Ppt on smart tv technology Animated ppt on magnetism worksheets Ppt online downloader without java Ppt on charge coupled device manufacturer Ppt on obesity prevention center Ppt on barack obama leadership Download ppt on direct and inverse proportion Teamwork for kids ppt on batteries