Geodynamics - Elasticity

Lecture Notes 12 Elasticity So far: Stress → angle of repose vs accretionary wedge Strain → reaction to stress → but how? Constitutive relations τ ij = τ ij (ε kl ) ; ε ij = ε ij (τ kl ) For example, Elasticity Isotropic Anisotropic Viscous flow Isotropic Anisotropic Power law creep Viscoelasticity Trade offs: simplicity ↔ realism constant variable isotropic anisotropic elastic, viscous viscoelastic history history dependent independent Tensors Most physical quantities that are important in continuum mechanics like temperature, force, and stress can be represented by a tensor. Temperature can be specified by stating a single numerical value called a scalar and is called a zeroth-order tensor. A force, however, must be specified by stating both a magnitude and direction. It is an example of a first-order tensor. Specifying a stress is even more complicated and requires stating a magnitude and two directions—the direction of a force vector and the direction of the normal vector to the plane on which the force acts. Stresses are represented by secondorder tensors. Tensors are quantities independent of coordinate system. Coordinate System x3' x3 P Direction Cosines x2' x2 x1 x1' Figure 12.1 x1 x2 x3 x1' �11 �12 �13 x2' �21 �22 �23 x3' �31 �32 �33 α ij = cos φij where φij is the angle of primed to original. xi ' = α ij x j xi = α ji x j ' α ij = ∂xi ' ∂x j = ∂x j ∂xi ' Tensors: a. 0th order (scalar) – quantity dependent only on position b. 1st order (31 components) Ai ' = α ij Aj c. 2nd order (32 = 9 components) Aij ' = α isα jk Ask d. 3rd order (33 = 27 components) Aijk ' = α isα jtα kp Astp e. 4th order (34 = 81 components) Aijkl ' = α isα jtα kpα lq Astpq Conventional moduli: 1. Hydrostatic comp. Figure 12.2 τ ij = − pδij τ ii = − pδii = 3λ ekk + 2µeii = −3p = (3λ + 2µ )eii − p VP =− ≡K eii ∆V where K = λ + 2 / 3µ is bulk modulus. 2. Uniaxial stress τ 11 = T other τ ij = 0 2µe1 = T − λ 2µ + 3λ T T ≡ E (sometimes Y ) e1 where E = µ (2µ + 3λ ) is Young’s modulus µ+λ Hook’s law: T = Ee e22 e33 = ≡ −ν e11 e11 2µe22 = − This is called Poisson’s ratio. λ 2µ + 3λ τ 11 ⇒ ν = λ 2(µ + λ ) θ = e11 + e22 + e33 = e11 (1 − 2ν ) fluid: µ → 0 ⇒ ν → 1 2 most material: ν = 0.2 − 0.3 1 ν= ⇒ λ =µ 4 It is Poisson solid. steel: ν ; 0.3 − 0.33 seismically measured λ + 2µ vp = ρ vs = µ ρ compare v p , vs → ν → discriminate rock types 3. Simple shear τ 12 = τ 21 = τ τ 12 = 2µe12 = 2Ge12 where G is shear modulus. Note: Among λ , µ, K , ν , E, G only two are independent.