1 mid-term exam
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Mid-term Exam
4월26일(월)
15:30~17:00
33동 225,226,327,328,330,331
2021-04-05
2021-04-05
재료의기계적거동(Mechanical Behavior of Materials)
VISCOELASTICITY (점탄성)
Myoung-Gyu Lee (이명규)
Department of Materials Science & Engineering
Seoul National University
Seoul 151-744, Korea
email : [email protected]
TA: Chanmi Moon (문찬미)
30-521 (Office)
[email protected] (E-mail)
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Viscoelasticity
• Elastic materials deform with stress and quickly return to their
original state if the stress is removed due to the bond stretching along
crystallographic planes in an ordered solid
• Viscous materials, like honey, resist shear flow and strain with time
when a stress is applied due to the diffusion of atoms or molecules
inside an amorphous material.
• Viscoelasticity is the property of materials that exhibit both viscosity
and elasticity during deformation and time-dependent strain.
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Viscoelasticity
• Creep: Increase in strain with time as the stress or load is kept
constant. Typical creep behavior shows that strain increases with
time at a decreasing rate followed by a constant rate and finally
increasing rate.
• Recovery: When the applied load is reduced (or instantly decreased),
the strain decreases with time, partially or completely. i.e., anelastic,
inelastic, elastic aftereffect
• Relaxation: Stress decreases with time when a strain is kept constant
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Viscoelasticity
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Creep
Relaxation
Recovery
Constant stress rate
Constant strain rate
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• If the stress is held constant, the strain increases with time (creep)
• If the strain is held constant, the stress decreases with time (stress relaxation)
• If a cyclic loading is applied, hysteresis occurs, leading to a dissipation of
mechanical energy ׯ𝝈𝒅𝜺
t
t
σ
ε
Constant stress
Creep
Creep
t
t
σ
ε
Constant strain
Stress relaxation
Stress Relaxation Hysteresisσ
ε
Energy Loss
Phenomenon of Viscoelastic Materials
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Constitutive models for linear viscoelasticity
Since its viscous component,
the stress-strain relation of viscoelastic materials is
time-dependent!
( )t ( )t
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Constitutive models for linear viscoelasticity
Viscoelasticity can be divided to elastic components and viscous
components. We can model viscoelastic materials as linear
combinations of springs and dashpots.
The springs represent the
elastic components.
The dashpots represent the viscous
components (perfect viscous fluid).
E
where η is the viscosity of the
material and dε/dt is the strain rate.
d
dt
where E is the elastic
modulus of the material.
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* No immediate extension takes place at
zero time when a sudden load is applied
(like a rigid body)
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Maxwell Model
s d
s E d
s d
s dE
A purely viscous damper
and purely elastic spring
connected in series.
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Maxwell Models d
E
( ) 1 1( )M
tJ t t
E
( )( ) e
E t
M
tG t E
t
G(t)
Creep Compliance J(t)
t
J(t)
1
E
1
Elastic component
Viscous component
In creep, actual strain rate
decreases with time!
Relaxation Modulus G(t)
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Creep
(𝜎 = 𝑐𝑜𝑛𝑠𝑡. ሶ𝜎 = 0 )Stress Relaxation
(𝜀 = 𝑐𝑜𝑛𝑠𝑡. ሶ𝜀 = 0 )
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Maxwell Model
d Edt
0
( )( ) e
E t
M
tG t E
Creep Compliance J(t) Relaxation Modulus G(t)
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0ln ln
E E
t C t
0 0exp( ) exp( )
E E
t E t
0
d dt
0 0 0( )
t t C tE
( )( ) e
E t
M
tG t E
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Maxwell Model
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RelaxationCreep & recovery
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Voigt-Kelvin (V-K) Model
s d
s E d
s d
s d E A purely viscous damper
and purely elastic spring
connected in parallel.
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* When a constant stress is applied, the dashpot prevent instantaneous
extension of spring and each component supports a portion of applied
stress
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V-K Model
( ) 1( ) (1 e )
E t
V K
tJ t
E
( )
( )V K
tG t E
Creep Compliance Function J(t)Relaxation Modulus G(t)
s d E
t
G(t)
E
t
J(t)
1
E
Actual stress is not constant
in viscoelastic materials.
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Creep
(𝜎 = 𝑐𝑜𝑛𝑠𝑡. ሶ𝜎 = 0 )Stress Relaxation
(𝜀 = 𝑐𝑜𝑛𝑠𝑡. ሶ𝜀 = 0 )
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V-K Model
0 0 E const
Creep Compliance J(t) Relaxation Modulus G(t)
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0
E
d dt dt
0( ) 1 exp( )
Et t
E
( ) 1( ) (1 e )
E t
V K
tJ t
E
( )( )V K
tG t E
No relaxation is predicted
by the Kelvin model
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V-K Model
Recovery
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0
d dt
0( ) 1 exp( )
Et t
E
At time t=t1, stress is suddenly
reduced to zero!!
10 t t
1
E1
t t t
1t t
1
1( ) exp( )
t t 0 1 1
( ) exp( ) 1 exp( )
tt t
E1t tor
Let,
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V-K Model
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Creep & recovery
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Standard Linear Solid(Zener) Model
A Maxwell model and a purely
elastic spring connected in parallel
(three-parameter standard model)
1 2 1 2 2( )E E E E E
1 2
2 3
1 2 3
1 1,
2 2, 3 3,
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SLS Model
1 2( )1 22
1 1 2
1( ) 1 e
E E
E Et
SLS
EJ t
E E E
2
1 2( ) eE
t
SLSG t E E
Creep Compliance Function J(t)
Relaxation Modulus G(t)
1 2 1 2 2( )E E E E E
It matches well to real linear viscoelastic behaviors!
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Homework #1 – Derive the following two equations
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Comparison of Several Models
Model Creep compliance function J(t) Relaxation modulus G(t)
Maxwell
Voigt-Kelvin
Standard Linear
Solid (Zener)
eE t
E
11 e
E t
E
( )E t
11
Et
E
1 2( )1 22
1 1 2
11 e
E E
E EtE
E E E
2
1 2 eE
tE E
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Comparison of Several Models
Y.C. Fung, “Biomechanics : mechanical properties of living tissues”, Second edition, New York : Springer-Verlag, 1993.2021-04-05
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Homework #2
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2E
1E
Homework #1
1) Derive relaxation modulus and creep compliance
2) Discuss the recovery response of the unit
Due on April ??