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# Viscoelastic Damping: Zener model

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Zener model of Viscoelastic Materials

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### Viscoelastic Damping: Zener model

1. 1. Viscoelastic Damping Zener Model Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
2. 2. Zener Model • The Zener model describes the material as a viscous damper in parallel with an elastic stiffness and both are in series with another stiffness. • The strain may be written as: Viscoelastic Damping Mohammad Tawfik ε=ε s +ε 1 WikiCourses# http://WikiCourses.WikiSpaces.com
3. 3. Stress-Strain Relation • The stress is is given by the relation: • From which we may write: • Or: σ =E s ε s =E p ε1 +C d ε 1 ˙ σ σ ε s = , ε1= Es E p +sC d σ σ ε= + E s E p + sC d ¿σ Viscoelastic Damping Mohammad Tawfik ( E p + sC d + E s E s ( E p + sC d ) ) WikiCourses# http://WikiCourses.WikiSpaces.com
4. 4. Stress-Strain Relation • Giving: E s ( E p +sC d ) ε=( E p +sC d + E s ) σ • Back to time domain: E s E p ε+ E s C d ε =( E p + E s ) σ +C d σ ˙ ˙ Eε +Eβ ε =σ +α σ ˙ ˙ Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
5. 5. Zener Model Characteristics • Creep: – For constant stress, we get: – Giving: Eε +Eβ ε =σ 0 ˙ σ 0 e−t / β ε= − E Es Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
6. 6. Zener Model Characteristics • Relaxation: – For constant strain, we get: – Which gives: Viscoelastic Damping Mohammad Tawfik Eε=σ +α σ ˙ σ =σ 0 + Eε 0 ( 1−e−t /α ) WikiCourses# http://WikiCourses.WikiSpaces.com
7. 7. Zener Model Characteristics • Storage and Loss Factors: σ =σ 0 e ε=ε 0 e – For harmonic stress: – Which drives the strain harmonically: – Giving: Eε o + jωE βε o =σ o + j ωασ o jωt jωt 2 1+ω αβ + jω ( β−α ) 1+ j ωβ σ o= E ε o= E εo 2 2 1+ j ωα 1+ω α Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
8. 8. Zener Model Characteristics ( ) 2 1+ ω αβ jω ( β−α ) σ o= E + εo 2 2 2 2 1+ω α 1+ω α ' σ o= E ( 1+ jη ) ε o Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
9. 9. Frequency Dependent Behavior 2 1.8 1.6 Modulus 1.4 1.2 1 0.8 0.6 E 0.4 u 0.2 0 0 1 2 3 4 Frequency Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com
10. 10. Other Models • Some, more accurate, models were developed to represent the behavior of viscoelastic material • The greatest concern was paid for the modeling in the time domain. • The most famous models are: – Golla-Hughes-McTavish – Augmented Temperature Field – Fractional Derivative Viscoelastic Damping Mohammad Tawfik WikiCourses# http://WikiCourses.WikiSpaces.com