1. Viscoelasticity and Wave Propagation
Part I
Gaurav Dutta
King Abdullah University of Science and Technology
September 6, 2016

2. Outline Introduction to viscoelasticity
Stress-strain relationship and complex modulus
Strain energy density Q
Summary

3. Outline Introduction to viscoelasticity
Stress-strain relationship and complex modulus
Strain energy density Q
Summary

4. 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

5. 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

6. 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

7. Viscoelastic Materials

8. Viscoelastic Materials

9. Outline Introduction to viscoelasticity
Stress-strain relationship and complex modulus
Strain energy density Q
Summary

10. 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

11. 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

12. 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

13. 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

14. Outline Introduction to viscoelasticity
Stress-strain relationship and complex modulus
Strain energy density Q
Summary

15. 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

16. 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 = 𝑀 1 𝜖 𝑀 2 𝜖 2 = 𝑅𝑒𝑎𝑙 (𝑀(𝜔)) 𝐼𝑚(𝑀(𝜔))
Energy lost:Δ𝐸= 1 2 𝑀 2 𝜖 2

17. 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 = ∞ 0 ∞ 𝜓 𝜏 1 + 𝜏 2 𝜕𝜖 𝑡− 𝜏 1 𝜕𝜖 𝑡− 𝜏 2 𝑑 𝜏 1 𝑑 𝜏 2

18. Strain Energy Density Strain energy density:
𝑈 𝑡 = ∞ 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 > = 𝜖 2 𝑀 1
𝑀 𝜔 = 𝑀 1 𝜔 +𝑖 𝑀 2 (𝜔)
Time-averaged rate of dissipated strain energy density:
< 𝑉 𝑡 > = 𝜔 𝜖 2 𝑀 2

19. Outline Introduction to viscoelasticity
Stress-strain relationship and complex modulus
Strain energy density Q
Summary

20. Definition of Q 𝑄 = Quality factor 𝑄 −1 = Dissipation factor
Time-averaged strain energy density:
<𝑈 𝑡 > = 𝜖 2 𝑀 1
Time-averaged rate of dissipated strain energy density:
< 𝑉 𝑡 > = 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× 𝜖 2 𝑀 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

21. Outline Introduction to viscoelasticity
Stress-strain relationship and complex modulus
Strain energy density Q
Summary

22. 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

23. 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

24. Mechanical models and viscoelasticity
Part II
Gaurav Dutta
King Abdullah University of Science and Technology
September 7, 2016

25. 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) +

26. Mechanical models and wave propagation
Maxwell Model
Kelvin-Voigt Model
Zener or SLS Model

27. 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

28. 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

29. 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

30. 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 !!!

31. 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

32. 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 !!!

33. Experimental Creep function for solids

34. 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

35. 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)

36. 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

37. Experimental Creep function for solids

38. 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

39. 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 = 𝜏 𝜖 𝜏 𝜎
⇒𝑄 0 = 2 𝜏 0 𝜏 𝜖 − 𝜏 𝜎
𝜏 𝜖 = 𝜏 0 𝑄 𝑄
𝜏 𝜎 = 𝜏 0 𝑄 𝑄 −1

40. 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

41. 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

42. 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)

43. 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

44. 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