Zener model of Viscoelastic Materials Can you add complexity? #WikiCourses http://wikicourses.wikispaces.com/Topic01+Viscoelastic+Materials

1. Viscoelastic Damping
Zener Model
Viscoelastic Damping
Mohammad Tawfik
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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
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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 )
)
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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
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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
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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 )
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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
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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
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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
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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
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