Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet or all equations you will probably ever need Definitions 1. Coordinate system. x,y,z or x 1,x 2,x define points in D space. 2. Field variable. T x,y,z - temperature field - temperature varying in space. Indexed variables. v i,i = 1,2, implies v 1,v 2,v, i.e. three functions. 4. Repeated indices rule also called Einstein summation convention,i = 1,2,,implies i=1 = v 1 1 2 5. Tensor = indexed variable + the rule of transformation to another coordinate system. 6. Useful tensor - Kronecker delta, in D: δ i j = 1 0 0 0 1 0 0 0 1 7. Traction = a force per unit area acting on a plane a vector. 8. Traction sign convention. Compression is negative in mechanics, but positive in geology. 9. Mean stress/strain: σ = σ ii / = trσ/, ε = ε ii / = θ also called dilatation 10. Deviatoric stress/strain: σ = σ i j σδ i j, ε = ε i j εδ i j Stress tensor a matrix, two indexed variables, tensor of rank two: σ i j = σ i j = σ11 σ 12 2D σ 21 σ 22 σ 11 σ 12 σ 1 σ 21 σ 22 σ 2 D σ 1 σ 2 σ 2. Meaning of the elements: each row are components of the traction vectors acting on the coordinate planes. diagonal elements are normal stresses. off diagonal elements are shear stresses.. Special properties: Symmetric for homogeneous material.
4. What for? It is a magic tool: if you multiply the stress tensor treated as a simple matrix by a unit vector, n j, which is normal to a certain plane, you will get the traction vector on this plane Cauchy s formula: 5. How do you get it? Usually by solving the equilibrium equations T ni = σ i j n j = 6. NB: The number of equilibrium equations is less than the number of unknown stress tensor components. Strain and strain rate tensors a matrix, two indexed variables, tensor of rank two. 2. Meaning of the elements: diagonal elements are elongation relative changes of length in coordinate axes directions. off diagonal elements are shears deviations from 90 of the angles between lines coinciding with the coordinate axes directions before deformation.. Special properties: symmetric j=1 σ i j n j 4. What for? It is a measure of the deformation. It will be used in the rheological relationships. 5. How do you get it? via velocity/displacements: ε i j = 1 2 = vi + v j j v 1 1 1 v2 2 + v 1 1 1 v 2 + v 1 1 2 1 v1 2 2 1 v 2 2 12 v 2 12 v1 1 v2 2 2 v 1 6. NB: The number of velocity components is smaller than the number of strain rate components. Rheology, Stress-strain relationships A functional relationship between second rank tensors: incompressible viscous: σ i j = Pδ i j + 2η ε i j elastic: σ i j = λε kk δ i j + 2µε i j Maxwell visco-elastic for deviators: ε i j = σ i j 2G + σ i j 2η 2
2. Meaning of the elements: λ,η,µ - are parameters material properties. What for? In the equilibrium equations: first substitute stress via strainrate, than strainrate via displacement velocities, which results in a closed system of equations, meaning that the number of equations is equal to the number of unknowns velocities or displacement. 4. How to find? measure rheology in the lab. 5. NB. There are three major classes of rheologies: Reversible elastic rheology at small stresses and strains. Rate dependent creep - irreversible. Examples are Newtonian viscous or power-law rheology, which is usually thermally activated, pressure independent. Intermediate stress levels. Rate independent instantaneous ultimate yielding at large stresses. Frequently pressure sensitive, temperature independent. Also called: plastic or frictional brittle behavior. General continuum mechanics recipe: How to derive a closed system of equations. 1. Conservation laws Conservation of mass: ρ t + ρv i = 0 Conservation of momentum equilibrium force balance vi ρ where g i is the gravitational acceleration vector. Conservation of energy: j = σ i j j + ρg i E t + v E j + q i = ρq j where E is energy, q i the energy flux vector, and Q an energy source heat production. 2. Thermodynamic relationships Equations of state 1: the caloric equation where c p is heat capacity, and T is temperature. E = c p ρt Equations of state 2: relationships for the isotropic parts of the stress/strain tensors ρ = f T,P where P is pressure note: ρ = ρ 0 ε kk ;P = σ = σ kk /.
. Rheological relationships for deviators ε i j = R σ i j, σ i j 4. Energy flux vector vs. temperature gradient q i = k T where k is the thermal conductivity. Summary: The general system of equations for a continuum media in the gravity field. vi ρ E ρ t + ρv i = 0 1 i E x j x j = σ i j j + ρg i 2 + q i = ρq E = c p ρt 4 ρ = f T,P 5 ε i j = R σ i j, σ i j 6 q i = k T 7 where ρ is density, v i velocity, g i gravitational acceleration vector, E energy, q i heat flux vector, Q an energy source heat production, e.g. by radioactive elements, c is heat capacity, T temperature, P pressure and k thermal conductivity. Known functions, tensors and coefficients: g i,c p, f..,ρ 0,R...,k Unknown functions: ρ,v i, σ, σ i j,q i,e,t. The number of unknowns is thus equal to the number of equations. Example: The Stokes system of equations for slowly moving incompressible linear viscous Newtonian continuum materials. ρ 0 c p T T = x j σ i j = 0 8 + ρ 0 g i = 0 9 j k T + ρ 0 Q 10 ε i j = σ i j 2η 11 σ i j = Pδ i j + σ i j 12 Known functions, tensors and coefficients: g i,c p,q,ρ 0,η,k Unknown functions: v i,p, σ i j,σ i j,t. The number of unknowns is thus equal to the number of equations. 4
2D version, spelled out Choice of coordinate system and new notation for 2D: σxx σ g i = 0, g,x i = x,z,v i = v x,v z,σ i j = xz σ zx σ zz Note that σ zx = σ xz. The 2D Stokes system of equations the basis for basically every mantle convection/lithospheric deformation code: v x + v z = 0 1 σ xx + σ xz = 0 14 σ xz + σ zz ρg = 0 15 T ρ 0 c p t + v T x + v z σ xx = P + 2η v x σ zz = P + 2η v z vx σ xz = η + v z 2 T = k 2 + 2 T 2 T Known: g,q,c p,ρ 0,η,k. Unknown: v x,v z,p,σ xx,σ xz,σ zz,t. Number of equations? 16 17 18 + ρ 0 Q 19 5