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

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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´

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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´

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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

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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

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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

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Strain tensor 1 2 3

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1 2 3

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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 Substantive time derivative Heat flux Shear heating

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Physical grounds: constitutive laws

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Rheology 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: = E

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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 Brittle-ductileDuctile Modes of deformation

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

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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 Von Mises yield criterion Drucker-Prager yield criterion softening

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Final effective viscosity if

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

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