Viscoelasticity and Wave Propagation Part I Gaurav Dutta King Abdullah University of Science and Technology September 6, 2016
Outline Introduction to viscoelasticity Stress-strain relationship and complex modulus Strain energy density Q Summary
Outline Introduction to viscoelasticity Stress-strain relationship and complex modulus Strain energy density Q Summary
Viscoelasticity 𝜎(𝑡)= 𝑀 𝑒 𝜖(𝑡) Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation 𝜎(𝑡)= 𝑀 𝑒 𝜖(𝑡) Stress Strain Purely elastic material Stress Strain Viscoelastic material Loading Loading Energy Loss Unloading Unloading Hysteresis plot
Stress-strain Relationship Constant Strain Stress Relaxation 𝜎(𝑡)=𝜓(𝑡)∗ 𝜕𝜖(𝑡) 𝜕𝑡 Stress relaxation function Observed decrease in stress in response to the same amount of strain
Stress-strain Relationship Constant Stress Creep 𝜖(𝑡)=𝜉(𝑡)∗ 𝜕𝜎(𝑡) 𝜕𝑡 Creep function Observed increase in strain in response to the same amount of stress
Viscoelastic Materials
Viscoelastic Materials
Outline Introduction to viscoelasticity Stress-strain relationship and complex modulus Strain energy density Q Summary
Complex Modulus for viscoelastic media Stress-strain relationship 𝜎= Stress, 𝜖= Strain 𝜓(𝑡)= relaxation function 𝜎(𝑡)=𝜓(𝑡)∗ 𝜕𝜖(𝑡) 𝜕𝑡 For a lossless media 𝐻 𝑡 = Heaviside function 𝜓 𝑡 = 𝑀 𝑒 𝐻(𝑡) Hooke’s law 𝜎=𝜓 𝑡 ∗ 𝜕𝜖 𝜕𝑡 = 𝜕𝜓 𝑡 𝜕𝑡 ∗𝜖= 𝑀 𝑒 𝛿 𝑡 ∗𝜖= 𝑀 𝑒 𝜖 𝑀 𝑒 = Elastic Modulus
Complex Modulus for viscoelastic media Stress-strain relationship 𝜎= Stress, 𝜖= Strain 𝜓(𝑡)= relaxation function 𝜎 𝑡 =𝜓 𝑡 ∗ 𝜕𝜖 𝑡 𝜕𝑡 = 𝜕𝜓 𝜕𝑡 ∗𝜖 𝑡 = 𝜕 𝑡 𝜓 𝑡 ∗𝜖(𝑡) After Fourier transform 𝐹= Fourier transform operator 𝐹 𝜎 𝜔 =𝑀 𝜔 𝐹[𝜖(𝜔)] 𝑀 𝜔 =𝐹 𝜕 𝑡 𝜓 𝑡 = −∞ +∞ 𝜕 𝑡 𝜓 𝑡 exp −𝑖𝜔𝑡 𝑑𝑡 𝑀 𝜔 = Complex modulus
Complex Modulus for viscoelastic media For a Heaviside function 𝑓(𝑡) 𝑀 𝜔 = −∞ +∞ 𝜕 𝑡 𝜓 𝑡 exp −𝑖𝜔𝑡 𝑑𝑡 𝑓 𝑡 = 𝑓 𝑡 𝐻(𝑡) If 𝜓(𝑡) is of Heaviside type 𝜓(𝑡)= causal function 𝜓 𝑡 = 𝜓 (𝑡) for 𝑡>0 𝜓= 𝜓 𝐻(𝑡) 𝜕 𝑡 𝜓=𝛿 𝑡 𝜓 + 𝜕 𝑡 𝜓 𝐻(𝑡) 𝑀 𝜔 = 𝑀 1 𝜔 +𝑖 𝑀 2 (𝜔) 𝑀 𝜔 =𝜓 ∞ +𝑖𝜔 0 ∞ 𝜓 𝑡 −𝜓(∞) exp −𝑖𝜔𝑡 𝑑𝑡
Complex Modulus for viscoelastic media 𝑀 𝜔 =𝜓 ∞ +𝑖𝜔 0 ∞ 𝜓 𝑡 −𝜓(∞) exp −𝑖𝜔𝑡 𝑑𝑡 𝑀 𝜔 = 𝑀 1 𝜔 +𝑖 𝑀 2 (𝜔) Storage Modulus 𝑀 1 𝜔 =𝜔 0 ∞ 𝜓 𝑡 sin 𝜔𝑡 𝑑𝑡 Loss Modulus M 2 𝜔 =𝜔 0 ∞ 𝜓 𝑡 −𝜓(∞) cos 𝜔𝑡 𝑑𝑡
Outline Introduction to viscoelasticity Stress-strain relationship and complex modulus Strain energy density Q Summary
Strain Energy Density Strain energy density: Energy stored by a system per unit volume while undergoing deformation 𝐸= 1 2 𝑉𝜎𝜖= 1 2 𝑉 𝑀 𝑒 𝜖 2 Strain energy: 𝑉= Volume 𝑀 𝑒 = Elastic modulus 𝑈= 𝐸 𝑉 = 1 2 𝜎𝜖= 1 2 𝑀 𝑒 𝜖 2 Strain energy density: 𝑈 𝑡 = 1 2 𝜎 𝑖𝑗 𝜖 𝑖𝑗 = 1 2 𝐶 𝑖𝑗𝑘𝑙 𝜖 𝑖𝑗 𝜖 𝑘𝑙
Q For elastic media: 𝑈 𝑡 = 1 2 𝜎 𝑖𝑗 𝜖 𝑖𝑗 = 1 2 𝐶 𝑖𝑗𝑘𝑙 𝜖 𝑖𝑗 𝜖 𝑘𝑙 𝑈 𝑡 = 1 2 𝜎 𝑖𝑗 𝜖 𝑖𝑗 = 1 2 𝐶 𝑖𝑗𝑘𝑙 𝜖 𝑖𝑗 𝜖 𝑘𝑙 Loss Modulus Q ⇒ quantifies dissipation ⇒ 𝑠𝑡𝑟𝑎𝑖𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑜𝑟𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑙𝑜𝑠𝑡 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑐𝑦𝑐𝑙𝑒 = 𝐸 Δ𝐸 𝑀 𝜔 = 𝑀 1 𝜔 +𝑖 𝑀 2 (𝜔) Storage Modulus 𝑈= 𝐸 𝑉 = 1 2 𝜎𝜖= 1 2 𝑀 𝑒 𝜖 2 Strain energy density: Energy stored in the volume: 𝐸= 1 2 𝑀 1 𝜖 2 𝑄= 𝐸 Δ𝐸 = 1 2 𝑀 1 𝜖 2 1 2 𝑀 2 𝜖 2 = 𝑅𝑒𝑎𝑙 (𝑀(𝜔)) 𝐼𝑚(𝑀(𝜔)) Energy lost:Δ𝐸= 1 2 𝑀 2 𝜖 2
Strain Energy Density For elastic media: 𝑈 𝑡 = 1 2 𝜎 𝑖𝑗 𝜖 𝑖𝑗 = 1 2 𝐶 𝑖𝑗𝑘𝑙 𝜖 𝑖𝑗 𝜖 𝑘𝑙 For viscoelastic media: 𝜎(𝑡)=𝜓(𝑡)∗ 𝜕𝜖(𝑡) 𝜕𝑡 𝐶 𝑖𝑗𝑘𝑙 ⇒𝜓( 𝜏 1 + 𝜏 2 ) 𝜖 𝑖𝑗 ⇒ 𝜕 𝜏 1 𝜖(𝑡− 𝜏 1 ) 𝜖 𝑘𝑙 ⇒ 𝜕 𝜏 2 𝜖(𝑡− 𝜏 2 ) 𝑈 𝑡 = 1 2 0 ∞ 0 ∞ 𝜓 𝜏 1 + 𝜏 2 𝜕𝜖 𝑡− 𝜏 1 𝜕𝜖 𝑡− 𝜏 2 𝑑 𝜏 1 𝑑 𝜏 2
Strain Energy Density Strain energy density: 𝑈 𝑡 = 1 2 0 ∞ 0 ∞ 𝜓 𝜏 1 + 𝜏 2 𝜕 𝜏 1 𝜖 𝑡− 𝜏 1 𝜕 𝜏 2 𝜖 𝑡− 𝜏 2 𝑑 𝜏 1 𝑑 𝜏 2 1) Take the time-average over a period 2𝜋/𝜔 2) Use the relation: <𝑅𝑒 𝒂 𝑇 ⋅𝑅𝑒 𝒃 > = 1 2 𝑅𝑒( 𝒂 𝑇 ⋅ 𝒃 ∗ ) Time-averaged strain energy density: <𝑈 𝑡 > = 1 4 𝜖 2 𝑀 1 𝑀 𝜔 = 𝑀 1 𝜔 +𝑖 𝑀 2 (𝜔) Time-averaged rate of dissipated strain energy density: < 𝑉 𝑡 > = 1 2 𝜔 𝜖 2 𝑀 2
Outline Introduction to viscoelasticity Stress-strain relationship and complex modulus Strain energy density Q Summary
Definition of Q 𝑄 = Quality factor 𝑄 −1 = Dissipation factor Time-averaged strain energy density: <𝑈 𝑡 > = 1 4 𝜖 2 𝑀 1 Time-averaged rate of dissipated strain energy density: < 𝑉 𝑡 > = 1 2 𝜔 𝜖 2 𝑀 2 Define time-averaged dissipated strain energy density: <𝑉 𝑡 > = 𝜔 −1 < 𝑉(𝑡) > Q ⇒ quantifies dissipation ⇒ 2∗ 𝑡𝑖𝑚𝑒−𝑎𝑣𝑔 𝑠𝑡𝑟𝑎𝑖𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑡𝑖𝑚𝑒−𝑎𝑣𝑔 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑄= 2×𝑈(𝑡) 𝑉(𝑡) = 2× 1 4 𝜖 2 𝑀 1 1 2 𝜖 2 𝑀 2 = 𝑀 1 𝑀 2 𝑀 𝜔 = 𝑀 1 𝜔 +𝑖 𝑀 2 (𝜔) 𝑄 = Quality factor 𝑄= 𝑅𝑒(𝑀) 𝐼𝑚(𝑀) 𝑀 1 𝜔 , 𝑀 2 (𝜔) ≥0 𝑄 −1 = Dissipation factor
Outline Introduction to viscoelasticity Stress-strain relationship and complex modulus Strain energy density Q Summary
Properties of the relaxation function 1) 𝜓 𝑡 = 𝜓 𝑡 𝐻(𝑡) 𝜓 𝑡 = causal function 2) Strain energy density: 𝑈 𝑡 = 1 2 𝜎 𝑖𝑗 𝜖 𝑖𝑗 = 1 2 𝐶 𝑖𝑗𝑘𝑙 𝜖 𝑖𝑗 𝜖 𝑘𝑙 Elastic 𝑈 𝑡 = 1 2 𝜓 𝑖𝑗𝑘𝑙 (𝑡) 𝜖 𝑖𝑗 𝜖 𝑘𝑙 Viscoelastic 𝑈 𝑡 ≥0 𝜓 𝑖𝑗𝑘𝑙 𝑡 ≥0 𝜓= 𝜓 𝐻(𝑡) 𝜓 𝑡 = positive real function
Properties of the relaxation function 3) Fading memory hypothesis: The value of the stress depends more strongly upon the recent history than upon the remote history of the strain. 𝜓 𝑡 = decreasing function of time 𝜕𝜓 𝜕𝑡 t= t 1 ≤ 𝜕𝜓 𝜕𝑡 t= 𝑡 2 , 𝑡 1 > 𝑡 2 >0
Mechanical models and viscoelasticity Part II Gaurav Dutta King Abdullah University of Science and Technology September 7, 2016
Mechanical models and wave propagation For elastic medium: Spring 𝜎= 𝑀 𝑒 𝜖 For viscoelastic medium: 𝜎= 𝜕 𝑡 𝜓∗𝜖 Dashpot (dissipates energy) Spring (stores energy) +
Mechanical models and wave propagation Maxwell Model Kelvin-Voigt Model Zener or SLS Model
Maxwell Model 𝜎= 𝑀 𝑈 𝜖 1 𝜕 𝑡 = 𝜕 𝜕𝑡 𝜎=𝜂 𝜕 𝑡 𝜖 2 𝜖= 𝜖 1 + 𝜖 2 𝜎=𝑀𝜖 𝜎= 𝑀 𝑈 𝜖 1 Spring 𝜕 𝑡 = 𝜕 𝜕𝑡 𝜎=𝜂 𝜕 𝑡 𝜖 2 Dashpot 𝜖= 𝜖 1 + 𝜖 2 𝑀 𝑈 = elastic const. of spring 𝜂= viscosity 𝜕 𝑡 𝜎 𝑀 𝑈 + 𝜎 𝜂 = 𝜕 𝑡 𝜖 𝜎=𝑀𝜖 𝜏= 𝜂 𝑀 𝑈 𝑀 𝜔 = 𝜔𝜂 𝜔𝜏−𝑖 𝐹 𝜎 𝜔 =𝑀 𝜔 𝐹[𝜖(𝜔)] Relaxation time Complex Modulus
Relaxation function for Maxwell model 𝜏= 𝜂 𝑀 𝑈 𝑀 𝜔 = 𝜔𝜂 𝜔𝜏−𝑖 𝜖(𝑡)=𝐻(𝑡) Measure the stress after applying a constant unit strain 𝜎 𝑡 = 𝜕 𝑡 𝜓 𝑡 ∗𝜖 𝑡 =𝜓 𝑡 ∗ 𝜕 𝑡 𝜖 𝑡 =𝜓 𝑡 ∗𝛿 𝑡 =𝜓(𝑡) 𝜕 𝑡 𝜎 𝑀 𝑈 + 𝜎 𝜂 = 𝜕 𝑡 𝜖 𝜕𝜓 𝜕𝑡 + 𝜓 𝜏 = 𝑀 𝑈 𝛿 𝑡 ⇒𝜓 𝑡 = 𝑀 𝑈 exp −𝑡/𝜏 𝐻(𝑡) Relaxation function
Creep function and Q for Maxwell model Measure the strain after applying a constant unit stress 𝜎(𝑡)=𝐻(𝑡) 𝜖 𝑡 =𝜉 𝑡 ∗ 𝜕 𝑡 𝜎 𝑡 =𝜉 𝑡 ∗𝛿 𝑡 =𝜉(𝑡) 𝜕 𝑡 𝜎 𝑀 𝑈 + 𝜎 𝜂 = 𝜕 𝑡 𝜖⇒𝜉 𝑡 = 1 𝑀 𝑈 1+ 𝑡 𝜏 𝐻(𝑡) Creep function Q: 𝑄= 𝑅𝑒(𝑀(𝜔)) 𝐼𝑚(𝑀(𝜔)) = 𝑀 1 𝑀 2 =𝜔𝜏 𝑀 𝜔 = 𝜔𝜂 𝜔𝜏−𝑖 = 𝑀 1 +𝑖 𝑀 2 𝜏= 𝜂 𝑀 𝑈
Relaxation/Creep function for Maxwell model 𝜉 𝑡 = 1 𝑀 𝑈 1+ 𝑡 𝜏 𝐻(𝑡) 𝜓 𝑡 = 𝑀 𝑈 exp −𝑡/𝜏 𝐻(𝑡) 𝜏= 𝜂 𝑀 𝑈 1) Creep function resembles the creep function of a viscous fluid. 2) There is no asymptotical residual stress as seen in real solids !!!
Kelvin-Voigt Model 𝜎= 𝜎 1 + 𝜎 2 = 𝑀 𝑅 𝜖+𝜂 𝜕 𝑡 𝜖 Relaxation function: 𝜓 𝑡 = 𝑀 𝑅 𝐻 𝑡 +𝜂𝛿(𝑡) Creep function: Complex Modulus: 𝑀 𝜔 = 𝑀 𝑅 +𝑖𝜔𝜂 𝜖 𝜉 𝑡 = 1 𝑀 𝑅 1− exp − 𝑡 𝜏 𝐻(𝑡) Relaxation time: 𝜏= 𝜂 𝑀 𝑅 𝑄 𝜔 = 𝑅𝑒𝑎𝑙 𝑀 𝜔 𝐼𝑚𝑎𝑔(𝑀(𝜔)) = 1 𝜔𝜏
Relaxation/Creep function for KV model 𝜉 𝑡 = 1 𝑀 𝑅 1− exp − 𝑡 𝜏 𝐻(𝑡) 𝜓 𝑡 = 𝑀 𝑅 𝐻 𝑡 +𝜂𝛿(𝑡) 𝜏= 𝜂 𝑀 𝑅 Relaxation function has no time dependence. Creep function lacks the instantaneous response of real solids !!!
Experimental Creep function for solids
Standard Linear Solid Model Zener or SLS Model 𝜎= 𝑘 1 𝜖 1 𝜎 1 =𝜂 𝜕 𝑡 𝜖 2 𝜎 2 = 𝑘 2 𝜖 2 𝜎= 𝜎 1 + 𝜎 2 𝜖= 𝜖 1 + 𝜖 2 Stress-strain relationship: 𝜎+ 𝜏 𝜎 𝜕𝜎 𝜕𝑡 = 𝑀 𝑅 (𝜖+ 𝜏 𝜖 𝜕𝜖 𝜕𝑡 ) 𝑀 𝑅 = 𝑘 1 𝑘 2 𝑘 1 + 𝑘 2 𝜏 𝜎 = 𝜂 𝑘 1 + 𝑘 2 𝜏 𝜖 = 𝜂 𝑘 2 Strain relaxation time Relaxed Modulus Stress relaxation time
Standard Linear Solid Model Zener or SLS Model Stress-strain relationship: 𝜎+ 𝜏 𝜎 𝜕𝜎 𝜕𝑡 = 𝑀 𝑅 (𝜖+ 𝜏 𝜖 𝜕𝜖 𝜕𝑡 ) After taking Fourier transform on both sides 𝑀 𝜔 = 𝑀 𝑅 1+𝑖𝜔 𝜏 𝜖 1+𝑖𝜔 𝜏 𝜎 𝜏 𝜎 = 𝜂 𝑘 1 + 𝑘 2 𝑀 𝑅 = 𝑘 1 𝑘 2 𝑘 1 + 𝑘 2 𝜏 𝜖 = 𝜂 𝑘 2 𝜖(𝑡)=𝐻(𝑡) 𝜓 𝑡 = 𝑀 𝑅 1− 1− 𝜏 𝜖 𝜏 𝜎 exp − 𝑡 𝜏 𝜎 𝐻(𝑡) Relaxation function: 𝜉 𝑡 = 1 𝑀 𝑅 1− 1− 𝜏 𝜎 𝜏 𝜖 exp − 𝑡 𝜏 𝜖 𝐻(𝑡) Creep function: 𝜎(𝑡)=𝐻(𝑡)
Relaxation/Creep function for SLS model 𝜉 𝑡 = 1 𝑀 𝑅 1− 1− 𝜏 𝜎 𝜏 𝜖 exp − 𝑡 𝜏 𝜖 𝐻(𝑡) 𝜓 𝑡 = 𝑀 𝑅 1− 1− 𝜏 𝜖 𝜏 𝜎 exp − 𝑡 𝜏 𝜎 𝐻(𝑡) 𝜏 𝜎 = 𝜂 𝑘 1 + 𝑘 2 𝜏 𝜖 = 𝜂 𝑘 2
Experimental Creep function for solids
Q for SLS model 𝑀 𝜔 = 𝑀 𝑅 1+𝑖𝜔 𝜏 𝜖 1+𝑖𝜔 𝜏 𝜎 = 𝑀 1 +𝑖 𝑀 2 𝑣 𝜔 = 𝑀 𝜔 𝜌 𝑄(𝜔)= 𝑅𝑒(𝑀(𝜔)) 𝐼𝑚(𝑀(𝜔)) = 𝑀 1 𝑀 2 = 1+ 𝜔 2 𝜏 𝜖 𝜏 𝜎 𝜔( 𝜏 𝜖 − 𝜏 𝜎 ) 𝑑𝑄 𝜔 𝑑𝜔 =0 𝜔 0 = 1 𝜏 0 = 𝜏 𝜖 𝜏 𝜎
Q for SLS model 𝑄(𝜔)= 1+ 𝜔 2 𝜏 𝜖 𝜏 𝜎 𝜔( 𝜏 𝜖 − 𝜏 𝜎 ) 𝑄(𝜔)= 1+ 𝜔 2 𝜏 𝜖 𝜏 𝜎 𝜔( 𝜏 𝜖 − 𝜏 𝜎 ) 𝜔 0 = 1 𝜏 0 = 𝜏 𝜖 𝜏 𝜎 ⇒𝑄 0 = 2 𝜏 0 𝜏 𝜖 − 𝜏 𝜎 𝜏 𝜖 = 𝜏 0 𝑄 0 𝑄 0 2 +1 +1 𝜏 𝜎 = 𝜏 0 𝑄 0 𝑄 0 2 +1 −1
Generalized SLS model Stress-strain relationship: Total stress: 𝜎 𝑙 + 𝜏 𝜎𝑙 𝜕 𝜎 𝑙 𝜕𝑡 = 𝑀 𝑅𝑙 (𝜖+ 𝜏 𝜖𝑙 𝜕𝜖 𝜕𝑡 ) 𝑙 =1,…,L 𝑀 𝑙 𝜔 = 𝑀 𝑅𝑙 1+𝑖𝜔 𝜏 𝜖𝑙 1+𝑖𝜔 𝜏 𝜎𝑙 𝑀 𝑅𝑙 = 𝑘 1𝑙 𝑘 2𝑙 𝑘 1𝑙 + 𝑘 2𝑙 𝜏 𝜎𝑙 = 𝜂 𝑙 𝑘 1𝑙 + 𝑘 2𝑙 Total stress: 𝜎= 𝑙=1 𝐿 𝜎 𝑙 𝜏 𝜖𝑙 = 𝜂 𝑙 𝑘 2𝑙
Generalized SLS model Total stress: 𝑀 𝑅 = 𝐿𝑀 𝑅𝑙 Relaxation function: 𝜎= 𝑙=1 𝐿 𝜎 𝑙 = 𝑙=1 𝐿 𝑀 𝑙 𝜖 = 𝑙=1 𝐿 𝑀 𝑅𝑙 1+𝑖𝜔 𝜏 𝜖𝑙 1+𝑖𝜔 𝜏 𝜎𝑙 𝜖 Relaxation function: 𝜓 𝑡 = 𝑀 𝑅 1− 1 𝐿 𝑙=1 𝐿 1− 𝜏 𝜖𝑙 𝜏 𝜎𝑙 exp − 𝑡 𝜏 𝜎𝑙 𝐻(𝑡) 𝑀 𝑙 𝜔 = 𝑀 𝑅𝑙 1+𝑖𝜔 𝜏 𝜖𝑙 1+𝑖𝜔 𝜏 𝜎𝑙 𝜏 𝜖𝑙 = 𝜂 𝑙 𝑘 2𝑙 𝜏 𝜎𝑙 = 𝜂 𝑙 𝑘 1𝑙 + 𝑘 2𝑙 𝑀 𝑅 = 𝐿𝑀 𝑅𝑙
Viscoacoustic wave-equation Zener or SLS Model 𝜓 𝑡 =𝐾 1− 1− 𝜏 𝜖 𝜏 𝜎 exp − 𝑡 𝜏 𝜎 𝐻(𝑡) 𝜎= 𝜕 𝑡 𝜓∗𝜖 − 𝑝 =𝐾 1− 1− 𝜏 𝜖 𝜏 𝜎 𝑣 𝑥 + 𝐾 𝜏 𝜎 1− 𝜏 𝜖 𝜏 𝜎 exp − 𝑡 𝜏 𝜎 𝐻 𝑡 ∗ 𝑣 𝑥 𝜎=−𝑝 𝜕𝑣 𝜕𝑥 =𝑣 𝑥 = 𝜖 𝑟 (memory variable)
Viscoacoustic wave-equation Zener or SLS Model 𝑟= 𝐾 𝜏 𝜎 1− 𝜏 𝜖 𝜏 𝜎 exp − 𝑡 𝜏 𝜎 𝐻 𝑡 ∗ 𝑣 𝑥 𝜕𝑟 𝜕𝑡 =− 𝑟 𝜏 𝜎 + 𝐾 𝜏 𝜎 1− 𝜏 𝜖 𝜏 𝜎 𝑣 𝑥 Viscoacoustic wave-equations − 𝜕𝑝 𝜕𝑡 =𝐾 1− 1− 𝜏 𝜖 𝜏 𝜎 𝑣 𝑥 +𝑟 Newton’s law: − 1 𝜌 𝜕𝑝 𝜕𝑥 = 𝜕 𝑣 𝑥 𝜕𝑡
Viscoacoustic wave-equation 𝜕 𝑟 𝑙 𝜕𝑡 =− 𝑟 𝑙 𝜏 𝜎𝑙 + 𝐾 𝜏 𝜎𝑙 1− 𝜏 𝜖𝑙 𝜏 𝜎𝑙 𝑣 𝑥 Generalized SLS Model 𝑙 =1,…,L − 𝜕𝑝 𝜕𝑡 =𝐾 1− 1 𝐿 𝑙=1 𝐿 1− 𝜏 𝜖𝑙 𝜏 𝜎𝑙 𝑣 𝑥 + 1 𝐿 𝑙=1 𝐿 𝑟 𝑙 − 1 𝜌 𝜕𝑝 𝜕𝑥 = 𝜕 𝑣 𝑥 𝜕𝑡