Fractional acoustic wave equations from mechanical and. Sverre Holm

Transcription

1 Fractional acoustic wave equations from mechanical and thermal constitutive equations Sverre Holm

2 Outline 2 Perspectives Non fractional wave equations Low and high frequency regions Fractional viscoelastic models Kelvin Voigt Zener Fractional heat flow model 2

3 2 Perspectives 1. Attenuation Power Laws Wave Equations 2. Constitutive Equations Physics or mathematics? 3

4 1. Power Laws: P and S loss, ω y Similar power laws Longitudinal, pressure: Granite: y 1 Liver: y 1.3 Shear: YIG: y=2 (Yttrium indium garnet) Granite: y 1 db/cm Szabo and Wu, A model for longitudinal and shear wave propagation in viscoelastic media, JASA (2000). Data for shear and longitudinal wave loss which show power-law dependence over four decades of frequency. 4 4

5 1. Wave Equations with Power Law Solutions 2 2 u 1 u 2 c 2 0 t 2 + τ t ( 2 ) y/2 u = u u 2 y/2 2 (y+1)/2 c 2 0 t 2 +τ ( ) u+η( ) u = 0 t 2 u 1 2 u c 2 t 2 2α 0 y+1 u 0 cos(πy/2) t y+1 =0 2 u 1 2 u c 2 t 2 2α 0 y+1 u c cos(πy/2) t y+1 α 2 0 2y u 0 0 cos 2 (πy/2) t 2y =0 5

6 1. Wave Equations with Power Law Solutions 2 u 1 c u t 2 + τ α σ α t α 2 u =0 1 2 u α 2 τ α α+2 u 2 u c 2 0 t 2 + τ ασ t α 2 u ² c 2 = 0 0 tα+2 2 p 1 c p t 2 +τ α σ α t α 2 p+τ 2 γ th 2 γ t 2 γ 2 p = β NL 2 p 2 ρ 0 c 4 0 t 2 6

7 3. Constitutive Equations Non fractional σ(t) = E σ(t) σ(t) + τ ² t " # "²(t) + τ σ t Fractional (t) + ²(t)# σ(t) = E = E " # ²(t) ²(t) + τ σ t β σ(t)+τ² β σ(t) t β = E " α α ²(t) # ²(t) + τσ t α " α # ²(t)+τσ α ²(t) t α q = κ T q(t) = τ 1 γ th κi γ 1 T (t) 7

8 Kelvin Voigt constitutive equation Stress, σ vs strain ² E : elastic modulus τ σ : ratio of viscosity and elasticity retardation time Hooke + Newton = spring + damper 8 8

9 Viscous wave equation 1. Conservation of mass 2. Conservation of momentum 3. Kelvin Voigt constitutive equation => Dispersion equation, insert displacement u = exp{i(ωt kx)} k 2 ω 2 /c τ σiωk 2 =0 9

10 Attenuation, phase/group velocity 2 regions ω 2 ω ω0.5 ω Never a single power law Causal since constitutive equation was causal 11

11 Application dependent cut off Low frequency region: Pressure waves in water, τ σ 1 ns ωτ σ << 1 for frequencies less than about 1 GHz Application : Sonar (1. order model for water) High frequency region: Shear waves in human tissue, τ σ 0.1 sec ωτ σ >> 1 for frequencies larger than about 10 Hz Application: Elastography 12

12 Losses in a fluid Loss: Mechanical: η, ζ shear and bulk viscosity coefficients Thermal: κ thermal conductivity it c v, c p heat capacity per unit of mass at constant volume / pressure. Loss terms are additive, but only for low frequencies, Bhatia, Ultrasonic absorption, 1967 (ch. 4.5) 22

13 Fractional Kelvin Voigt Stress, σ vs strain ² Fractional Kelvin Voigt σ(t) = E ²(t) + τσ α α ²(t) t α M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Roy. Astr. S Y. Rossikhin and M. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 1997 α=1 is normal viscoelastic case (figure) 25

14 A physical fractional device Capacitor soakage, dielectric absorption i = Cd a u/dt a, a 1 Z = u/i = 1/C(jω) a Example in video: 220 uf/63 Volt 10 Volts for 60 sec Shorted for 6 sec 26

15 Wave equation from fractional K V Fractional loss operator Conservation laws => M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Roy. Astr. S Dispersion equation, u = exp{i(ωt kx)} imaginary part of k is loss: k 2 ω 2 /c (τ σiω) α k 2 =0. Holm & Sinkus, A unifying fractional wave equation for compressional and shear waves, JASA,

16 Attenuation, fractional Kelvin Voigt y = α+1 y = 1-α/2 Elastography shear waves, ω τ >> 1 0/0.5<y<1 Ultrasound compression waves ω τ << 1, 1<y<2 α = 0.1, 0.3, 0.7, and 1 α = 1: viscoelastic case 33 33

17 Attenuation, fractional Zener y = α+1, y = 1-α/2, y = 1-α y = α+1, y = 1-α/2, y = 1-α 1. Low frequency 2. Medium frequency = High frequency for Kelvin-Voigt model 3. High frequency α = β = 0.1, 0.3, 0.7, and

18 MR Elastography, brain Experimental and model wave speed and damping High dispersion Klatt et al, Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography: a study of brain and liver viscoelasticity, Phys. Med. Biol

19 Modifying Fourier s heat law Gurtin & Pipkin (Arch. Ration. Mech. Anal 1968): From q = κ T To q(t) = Z t K(t τ) T(τ)dτ. 47

20 Fractional Fourier law Power law memory kernel (1 < γ 2): K(t τ) = χ (t τ)γ 2 Γ(γ 1) Gives fractional Fourier law (fract. integral) q(t) = χi γ 1 T(t) Povstenko, Int. J. Solids Struct., 2007 γ 1 standard Fourier law (kernel impulse) Fractional diffusion: 1 1 γ T α γ T γ t = 2 T 48

21 Fract. Kelvin Voigt + Fourier law: Fract. Westervelt eq of 1. form, decoupled: u δ me α 2 δ th 2 γ 2 β 2 p 2 u c t2 c 2 u + 0 tα c 2 u = 0 t2 γ ρ 0 c 4 0 t2 = τ α 1 δ 4 me µζ + η ρ 0 3 Fract. Westervelt of 2. form, coupled γ=2 α: 2 u 1 c u t 2 + µ δme + δ th α t α 2 u = β ρ c c 2 0 c 2 0 Standard Westervelt equation for γ=α=1 2 p 2 t 2 49

22 Conclusion The constitutive equation, a material property, is what determines wave behavior Both the low and high frequency regions are important Which one which is relevant depends on application 55

23 Fractional viscoelastic wave equations Fractional mechanical models: Fract. Kelvin Voigt: Fract. Zener: Attenuation ω y Low frequencies: y = 1+α High/Intermediate frequencies: y = 1 α/2 High frequencies (for Zener): y= 1 α 56