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Anelasticity and Attenuation Part 1: Generalities Attenuation a Range Of Anelasticity and Attenuation Part 1: Generalities SIO 227C Bernard Minster April 2004 1 2 Attenuation A range of rheologies . Energy is lost to heat . Elasticity . Not explained by theory of elasticity . Inelasticity . Many mechanisms, most of which operate at . Plasticity the microscopic level . Viscosity . “slight” departure from elastic behavior . Viscoelasticity . Rheology based on simple considerations of . Visco-plasticity near-equilibrium thermodynamics. Elasto-plasticity . As much as possible, stick to linear theory . Anelasticity 3 4 1 Elasticity Viscosity . Linear elasticity (Green) . Linear viscosity Stress Stress . Hooke’s law σ = Kε σ = κε˙ . Nonlinear elasticity . Nonlinear viscosity Strain . Example. Strain rate σ = F(ε) σ ∝ε˙1 / N Stress Stress . Fully reversible deformation: . For instance, for water ice,N=3 Loading and unloading paths . Perhaps for the mantle as well are the same . Large stress leads to low effective . Equilibrium reached instantly Strain viscosity Strain rate . Deformation nonrecoverable 5 6 Plasticity Inelasticity . Instantaneous plasticity Stress . Nonlinear, . Rigid-plastic nonrecoverable . Time-dependent Strain . (example: material Stress Stress failure) . Elasto-plasticity Strain Strain rate 7 8 2 Anelasticity postulates Some Comments 1. For every stress there is a unique . Anelasticity is just like elasticity, plus the equilibrium value of strain and vice time dependence postulate versa . There can be an elastic component of the deformation in addition to the time-dependent 2. The equilibrium response is achieved component only after the passage of sufficient . Recovery is also time dependent time . Linearity is taken in the mathematical sense: 3. The stress-strain relationship is linear σ(αε1 + βε2) = ασ(ε1) + βσ(ε2) 9 10 Comparison of Rheologies Thermodynamic Substance Unique equilibrium Instan- Linear . A thermodynamic substance is one which can (complete recoverability) taneous assume a continuous succession of unique Ideal Elasticity Yes Yes Yes equilibrium states in response to a series of infinitesimal changes in an external variable. Nonlinear Yes Yes No . Such a substance satisfies the first postulate of elasticity anelasticity Instantaneous No Yes No . Plastic and visco-elastic solids do not qualify as plasticity thermodynamic solids Anelasticity Yes No Yes . The time dependence is captured in the word relaxation, (often anelastic relaxation), a Linear visco- No No Yes thermodynamic phenomenon. elasticity 11 12 3 Anelastic responses Anelastic responses OAE= OCE= finite-time instantaneous application of application of stress σE stress σE OBE= ODE= finite-time instantaneous application of application of strain εE strain εE 13 14 Anelastic responses Anelastic responses . Apply stress and . Apply strain and monitor strain as monitor stress a function of as a function of time time . A: Instantaneous . B: Instantaneous load load . C: progressive . D: progressive load load 15 16 4 Creep: definitions Elastic Aftereffect . Creep experiment: . Recovery experiment ⎧0 , for t < 0 ⎧0, for t < −t1 σ = ⎨ ⎪ ⎩σ 0 for t ≥ 0 σ = ⎨σ 0 , for − t 1≤ t < 0 ⎪0 for t 0 . Creep function: Unrelaxed compliance, ⎩ ≥ relaxed Compliance, Compliance relaxation . Aftereffect function or creep J(t) = ε(t)/σ 0 recovery function. Depends on t1 J(0) = JU ; J(∞) = JR ; δJ = JR − JU N (t) (t)/ for t 0 t1 = ε σ 0 ≥ 17 18 Creep and Recovery Stress Relaxation . Stress relaxation experiment ⎧0 , for t < 0 ε = ⎨ ⎩ε 0 for t ≥ 0 . Stress relaxation function: Unrelaxed modulus, relaxed modulus, modulus defect M(t) = σ(t)/ε0 M(0) = MU ; M(∞) = MR ; δM = MU − MR 19 20 5 Normalized Creep and Relationships Relaxation Functions . Normalized creep function M R = 1 / JR ; MU = 1 / JU J(t) = JU + δ Jψ (t) = JU [1+ Δψ (t)] JR > JU so that M R < MU ψ (0) = 0 ; ψ (∞) = 1 . Normalized stress relaxation function δ M = δ J / JU JR M (t) = M R + δ Mϕ(t) = M R[1+ Δϕ(t)] . Relaxation Strength (dimensionless) ϕ(0) = 1 ; ϕ(∞) = 0 . So that Δ = δ J / J = δ M / M R JR = JU (1+ Δ) ; MU = M R (1+ Δ) 21 22 Linearity? Periodic Square-Wave hysteresis . (large) 1mm amplitude wave for 10km wavelength gives strain O(10-7) . Laboratory data shows rocks to be linear for strains < O(10-6) . Losses measured by area of hysteresis loops. Square strain deformation Square stress load 23 24 6 Sinusoidal loads/deformations Hysteresis implications Linearity implies elliptical hysteresis loops . The modulus (or compliance) depends on frequency. Stiff material at high frequencies . Compliant material at low frequencies . Wave velocity depends therefore on frequency: Physical Dispersion Low frequency High frequency 25 26 Balance Equations . Away from flow singularities (eg discontinuities) a quantity of the flow F is Part 2: Thermodynamics conserved if: ∂F + ∇x i(FV − J) = K ∂t Where V is the particle velocity at spatial position x, FV is a “convection” (advection) flux,J is a “conduction” flux, and K is an intrinsic source strength 27 28 7 ∂F Conservation Laws + ∇x i(FV − J) = K Internal Energy balance ∂t Law F J K . By combination of conservation laws: Mass ρ 0 0 ∂ρu + (ρuV + q ) = T D + ρh Momentum ρV T ρb ∂t i i ,i ij ij Energy ρE TiV - q ρ(biV + h) . Where Dij = (Vi. j + Vj.i ) 2 ρ = density b = body force density is the rate of deformation tensor V = particle velocity q = heat flux vector . Equations may be generalized to include T = Cauchy stress tensor h = heat source density 1 other, nonmechanical fluxes (e.g. electrical, E = u + ViV u = internal energy density etc.) q must then be generalized as well. 2 29 30 Linearized Near-Equilibrium L.N.E.T. Postulate 1 Thermodynamics (LNET) . Postulates: . Even though a system may not be in a 1. Local thermodynamic equilibrium state of global equilibrium, infinitesimal elements may be in local equilibrium, 2. Linearity of phenomenological and the functional dependence of state equations functions on state parameters is the 3. Onsager-Casimir symmetry relations same as in the case of equilibrium in (invariance of equations of motions classical thermodynamics. under time reversal) . Leads to Gibbs relations 31 32 8 Caloric Equation of State Gibbs relations . Energy density determined by thermodynamic . Gibbs relation reads state, ie entropy density and state variables m n u = u(s,ν ,...,ν ,ξ ,...,ξ ) 1 m 1 n du = Tds + ∑τ idνi − ∑ Ajdξ j . Macroscopically observable variables and i=1 j =1 conjugate thermodynamic tensions: . T is temperature, s is entropy. For νi , i = 1,...,m conjugate to τ i , i = 1,...,m locally adiabatic processes, then . Internal (unobservable) variables, and du = τidν − Αidξ associated affinities ξ j , j = 1,...,n associated with Aj , j = 1,...,n 33 34 Thermal Equations of State Maxwell’s Relations . Expand tensions and affinities near . u is a perfect differential. The order of equilibrium differentiation is immaterial, so the matrix P is symmetric: 0 0 ⎡ τ ⎤ ⎡ τ ⎤ ⎡P11 P12 ⎤ ⎡ν − ν ⎤ = + T T T ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ P = P ; P = P ; P = P ⎣−Α⎦ ⎣−Α ⎦ ⎣P21 P22 ⎦ ⎣ξ − ξ ⎦ 11 11 22 22 12 12 . Equilibrium values (0 superscript) will be . Stable equilibrium: u must be a set to zero (WLOG) minimum for given τ, A. So P must be positive-definite. 35 36 9 Nota bene.. Entropy balance (1) . Introduce dynamic tensions D as the . τ are thermodynamic tensions for which u is τ observable quantities,then a potential. If ν are strains, then τ are stresses only if the external stress work is recoverable ∂ρu + (ρuV + q ) = T D + ρh (eg adiabatic or isothermal processes with ∂t i i ,i ij ij resersible heat transfer) is of the general form: . In general, the dynamic stress tensor is not du m ρ = −∇iq + K where K = τ Dν identifiable with the thermodynamic tensions. dt u u ∑ i i The difference is called the viscous stress i=1 tensor. Hence from the Gibbs relation, the entropy balance reads: 37 38 Entropy balance (2) Entropy balance (3) ds q 1 ρ = −∇ + K i s D = ∑Χi Hi dt T 2 i . Where the entropy source strength is By comparison with the earlier expressions, Xi are 1 1 ∇T iq called generalized thermodynamic forces and Hi K = τ D − τ iν + Aiξ − s T ( ) T T 2 generalized fluxes The 2nd Law requires that D be a nonnegative . The Dissipation function is function that vanishes only for reversible processes. This is expressed through the Clausius-Duhem 1 1 D = TKs of the general form D = ∑Χi Hi inequality 2 2 i 39 40 10 2nd Law of Thermodynamics L.N.E.T. Postulate 2 • Strong form of Clausius-Duhem inequality: we . Near equilibrium the generalized fluxes require that, separately: are related to the generalized forces ⎧2D = (τ D − τ)iν + Αiξ ≥ 0 ⎪ local through linear phenomenological ⎨ ∇T iq ⎪ 2Dconduction = − ≥ 0 equations: ⎩ T 1 T Hi = Lij X j so that D = X LX . The second inequality constrains the heat ∑ 2 j conductivity tensor. The first one leads to anelastic relaxation 2nd Law requires L to be nonnegative 41 42 L.N.E.T. Postulate 3 Anelastic Evolution Equations (controversial) . In the absence of pseudo-forces such . Assume all rates dν/dt and dξ/dt as Coriolis or Lorentz forces (which may participate in entropy production (i.e. change sign under time reversal), the eliminate algebraically the reversible equations of motion of individual ones), then L is positive definite particles are invariant under time symmetric and possesses a Cayley reversal, i.e. L is symmetric (Onsager- inverse D=L-1 , itself symmetric. Casimir reciprocal relations): . Go back to Ks and identify terms… LT = L 43 44 11 Anelastic Evolution Equations Anelastic Evolution Equations D 1 1 ∇T iq ⎡D11 D12 ⎤ ⎡ν ⎤ ⎡P11 P12 ⎤ ⎡ν⎤ ⎡τ ⎤ K = τ D − τ iν + Aiξ − + = s ( ) 2 ⎢ T ⎥ ⎢ ⎥ ⎢ T ⎥ ⎢ ⎥ ⎢ ⎥ T T T ⎣D12 D22 ⎦ ξ ⎣P12 P22 ⎦ ⎣ξ⎦ ⎣ 0 ⎦ ⎣ ⎦ D adopt a common notation χ for all state variables ⎡ τ ⎤ ⎡D11 D12 ⎤ ⎡ν ⎤ ⎡τ ⎤ = − + (macro and internal), and define the quadratic form ⎢ ⎥ ⎢ T ⎥ ⎢ ⎥ ⎢ ⎥ −Α D12 D22 ξ 0 1 Τ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ P = χ Pχ 2 Combine with Thermal Equations of State, viz we find the familiar Lagrangean equations of evolution 0 0 ⎡ τ ⎤ ⎡ τ ⎤ ⎡P11 P12 ⎤ ⎡ν − ν ⎤ = + Where χi are generalized displacements ⎢−Α⎥ ⎢ 0 ⎥ ⎢P P ⎥ ⎢ 0 ⎥ ∂D ∂P ⎣ ⎦ ⎣−Α ⎦ ⎣ 21 22 ⎦ ⎣ξ − ξ ⎦ + = Qi and i are generalized forces, and a ∂χ ∂χ Q P i i potential function 45 46 Solving the equations (1) Solving the equations (2) .
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