1 mid-term exam

1

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 : myounglee@snu.ac.kr

TA: Chanmi Moon (문찬미)

30-521 (Office)

chanmi0705@snu.ac.kr (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 ??