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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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**Physical grounds: displacement, strain and stress tensors**

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

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**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

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

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**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

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**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

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**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

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Strain tensor 1 3 2

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Strain tensor 1 2 3

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Strain tensor

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**Strain and strain rate tensors**

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**Geometric meaning of strain tensor**

Changing length Changing angle Changing volume

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

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**Tensor invariants, deviators**

Stress deviator P is pressure Stress norm

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Principal stress axes

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**Physical grounds: conservation laws**

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**Mass conservation equation**

Material flux Substantive time derivative

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**Mass conservation equation**

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**Momentum conservation equation**

Substantive time derivative

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**Energy conservation equation**

Shear heating Heat flux Substantive time derivative

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**Physical grounds: constitutive laws**

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**Rheology – the relationship between stress and strain**

Rheology for ideal bodies Elastic Viscous Plastic

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**An isotropic, homogeneous elastic material follows Hooke's Law**

Hooke's Law: s = Ee

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

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**Ideal plastic behavior**

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**Constitutive laws Elastic strain rate Viscous flow strain rate**

Plastic flow strain rate

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**Superposition of constitutive laws**

;

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Full set of equations mass momentum energy

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

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Brittle Modes of deformation Brittle-ductile Ductile

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**Deformation Mechanisms**

Brittle Cataclasis Frictional grain-boundary sliding Ductile Diffusion creep Power-law creep Peierls creep

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**Cataclasis Fine-scale fracturing and movement along fractures.**

Favoured by low-confining pressures

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**Frictional grain-boundary sliding**

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

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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:

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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 :

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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.

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**Three creep processes Diffusion creep Dislocation creep Peierls creep**

( Kameyama et al. 1999)

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Full set of equations mass momentum energy

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**Final effective viscosity**

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**Plastic strain if if if Drucker-Prager yield criterion**

Von Mises yield criterion softening

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**Final effective viscosity**

if if

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

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

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