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# Viscoelastic Damping: Classical Models

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What is viscoelastic damping?
What are the classical models?
What is creep?
What is Relaxation?
What is complex modulus?

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http://wikicourses.wikispaces.com/Topic01+Viscoelastic+Materials

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### Viscoelastic Damping: Classical Models

1. 1. Viscoelastic Damping Classical Models Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
2. 2. Objectives • Recognize the nature of viscoelastic material • Understand the damping models of viscoelastic material • Dynamics of structures with viscoelastic material Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
3. 3. What is Viscoelastic Material? • Materials that Exhibit, both, viscous and elastic characteristics. • The material may be modeled in many different ways. Classical models include: – Mawxell Model – Kalvin-Voight Model Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
4. 4. Maxwell Model • The Maxwell model describes the material as a viscous damper in series with an elastic stiffness. • When stress is applied, it is uniform through the element. • The strain may be written as: Viscoelastic Damping Mohammad Tawfik ε =ε s + ε d WikiCourses# http://WikiCourses.WikiSpaces.com
5. 5. Stress-Strain Relation • The stress is equal in both elements and is given by the relation: • From which we may write: • Or: Viscoelastic Damping Mohammad Tawfik σ =E s ε s =C d ε d ˙ σ σ ε s = , ε d =∫ dt Es Cd σ σ σ σ ˙ ε = +∫ dt ∧ ε = + ˙ Es Cd Es Cd WikiCourses# http://WikiCourses.WikiSpaces.com
6. 6. Three Main Characteristics • Creep Strain changing with time for the same stress • Relaxation Stress changing with time for constant strain • Storage and Loss Moduli Effective modulus of elasticity in response to frequency excitation Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
7. 7. Maxwell Model Characteristics • Creep: – For constant stress, we get: σ σ ˙ ε= + ˙ E s Cd ⏟ zero – Which gives: σ ε= t Cd • Which indicates that the strain will grow to an unbound value as time increases! Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
8. 8. Maxwell Model Characteristics • Relaxation: – For constant strain, we get: σ σ ˙ + 0= Es Cd – Which gives: −tE s /C d σ =σ 0 e • Which means that the stress will decrease as time grows for the same strain Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
9. 9. Maxwell Model Characteristics • Storage and Loss Factors: σ =σ 0 e ε=ε 0 e – For harmonic stress: – Which drives the strain harmonically: – Giving: ( ) jω 1 j ωε o = + σo E s Cd Viscoelastic Damping Mohammad Tawfik σ o= jωt jωt E s C d jω E s + jωC d εo WikiCourses# http://WikiCourses.WikiSpaces.com
10. 10. Maxwell Model Characteristics C σ o= ( σ o= 2 d2 Es ω + E E C E s d 2 2+ ω s E s ω2 2 2 +ω 2 C d 2C d s C 2 εo d2 E +j jω E s 2Cd s 2 +ω 2 ω C d 2 ) εo ' σ o= E ( 1+ jη ) ε o Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
11. 11. Storage and Loss Moduli • The stress strain relation of the ' viscoelastic material appears to σ o= E ( 1+ jη ) ε o contain a complex modulus of elasticity! • The real part is called the storage modulus • The imaginary part is called the loss modulus • And their ratio is called the loss factor Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
12. 12. Frequency Dependent Behavior 1 0.9 0.8 0.7 M o d u lu s 0.6 E 0.5 u 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 Frequency Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
13. 13. Notes on the Maxwell Model • Under static loading, the stiffness, storage modulus is zero, and the loss factor is infinity! • For very high frequencies, the loss factor becomes zero! Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
14. 14. Kalvin-Voigt Model • The Kalvin-Voigt model describes the material as a viscous damper in parallel with an elastic stiffness. • When stress is applied, it is distributed through the elements. • The stress strain relation may be written as: Viscoelastic Damping Mohammad Tawfik σ =σ s + σ d σ =E s ε s +C d ε d ˙ WikiCourses# http://WikiCourses.WikiSpaces.com
15. 15. Kalvin-Voigt Model Characteristics • Creep: – For constant stress, we get: −E s t /C d σ ) ε= (1−e Es • Which indicates that the strain will grow to a constant value as time increases! Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
16. 16. Kalvin-Voigt Model Characteristics • Relaxation: – For constant strain, we get: σ =E s ε 0 • Which means that the stress will stay constant as time grows for the same strain! Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
17. 17. Creep Relaxation Summary Maxwell Kalvin-Voigt Creep Bad Good Relaxation Good Bad Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
18. 18. Kalvin-Voigt Model Characteristics • Storage and Loss Factors: σ =σ 0 e ε=ε 0 e – For harmonic stress: – Which drives the strain harmonically: – Giving: jωt jωt σ =( E s + jωC d ) ε o Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
19. 19. Frequency Dependent Behavior 14 12 Modulus 10 8 6 E 4 u 2 0 0 2 4 6 8 10 Frequency Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
20. 20. Notes on the Kalvin-Voigt Model • Under all loading, storage modulus is equal to the stiffness of the spring, and the loss factor is zero. • For very high frequencies, the loss factor becomes grows unbound! Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
21. 21. Assignment • Study the creep, relaxation, and frequency response characteristics of the Zener model shown in the following sketch Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com