Viscoelasticity KTH

7/22/2019 Viscoelasticity KTH

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Viscoelasticity


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Mechanical properties of living tissues

F7 Soft tissues

F8 Muscles

F9 Viscoelasticity

F15 Bone


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Lab Construct a Finite elementModel

Dissection of deer spine

Mechanical testing

Analysis of the results andimplement properties into FE-model

Report

LaborationGroup

number

week 16 Fri 18/4 8 - 12 1-4 Dissec + mech test AN/DL

Fri 18/4 13 - 17 5-8 Dissec + mech test AN/DL


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Viscoelasticity

Material from:

Kleiven S.An Introduction to Viscoelasticity, Lecture notes for4E1150,

Division of Neuronic Engng., CTV, KTH, Sweden, 2003.

Fung, Y.C. Foundations of Solid Mechanics, Prentice-Hall Inc.,Englewood Cliffs, New Jersey, USA, 1965.

Flgge, W. Viscoelasticity, Springer-Verlag, 1975.


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Characteristic behaviour ofviscoelastic materials

Why are biological tissuesviscoelastic?

Theory of viscoelasticity

Viscoelasticity


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Soft tissues exhibit several Viscoelastic properties:

stress relaxation at constant strain

creep at constant stress

hysteresis during loading and unloading

strain-rate dependence

i.e., in general, stress in soft tissues depends on strain and

the historyof strain

These properties can be modeled by the theory of

viscoelasticity

Linear Viscoelasticity


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Stress relaxation

Steel bolt through plastic

Time

Load

The load willdecrease in time fora constant applieddeformation


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Stress relaxation function for Nucleus Pulposus(Normalized) (Iatridis et al, 1997)

Stress relaxation curves for the Scalp(Melvin et al., 1970)

Stress relaxation

Examples of Biological tissues in Stress relaxation


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Creep

Structure loaded by gravityDeformation

Time

mg

Deformation willcontinue in timealthough the load isconstant


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Creep compliance of dura mater in tension

(Galford et al., 1970)

Creep

Example of Biological tissues in Creep


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Deformation

Load

Hysteresis and strain-rate dependence

Deformation

Load

Loading

Unloading

Area between curvesis proportional todissipated energy

Increased strain-rateleads to a stiffer response


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The stress-strain relationship for

bone(McElhaney, 1966)

Hysteresis loops for ankle

ligament(Funk et al., 2000)

Hysteresis and strain-rate dependence

Example of hysteresis & strain-ratedependence in Biological tissues


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Viscoelasticity


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Ligaments & Tendons

Viscoelastic response due to

Interactions between theproteoglycans in thegroundsubstance and

collagen fibrils.

Rate dependent strength,stiffness and energyabsorption.

Increased by a factor of 3 if the load

rate is increased from 8 to 2300 m/s.


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Cartilage low friction layer in the joints

Damping between two bones, for example in the knee. 70-80% water, collagen fibers and ground substance.


Fluid flow during loading


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The intervertebral disc

Nucleus pulposus (NP)

Hydrophilic gel

90% water (decreases with age to 70%).

Annulus fibrosus (AP)

Composite of collagen fibers in a

ground substance. Approximately 90 unidirectional

laminae

Fiber direction 60

78% water Viscoelastic response due to

Fluid flow during loading Shear forces between the matrix

and fibers during fiberstraightening.


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Trabecular bone

low-density, opencell, rod-typestructure from thefemural head

higher density,

roughly prismatic cellsfrom the femural head

intermediate densityparallel plate structurewith rods normal to theplate


Fluid flow during loading


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Importance of viscoelasticity

Structure made of polymers (i.e. FRP)->

Creep, relaxation etc.

-> Collapse? Damping, dynamic modulus etc. ofBiological tissues

-> Protection!


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Maximal shear strain for human headincluding viscoelasticity

Maximal shear strain for human headexcluding viscoelasticity

Importance of viscoelasticity


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Viscoelasticity


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Basic elements

Spring (Hooke element) Viscous Damper(Newton element)

11EE


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Simple Viscoelastic ModelsStress depends on strain and strain-rate:

Elastic stress depends on strain (spring)

Viscous stress depends on strain-rate (damper)

- Maxwell: Strains add in series, stresses are equal

- Kelvin: Stresses add in parallel, strains are equal

- SLS: Combination of Maxwell and Kelvin

Maxwell Kelvin (aka Voigt) Standard Linear Solid


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Maxwell Model

Total strain = spring strain + dashpot strain:

1E

1E

1E

1E

11 1 EE E

A linear first-order ordinarydifferential equation (ODE)

1

1 EE


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Creep Solution

1E

dt

d

Edt

d

1

1

dtdE

d1

1

ttt

CdtdE

d0

0

)(

)0(1

)(

)0(

1

Integrating, for constant applied stress, 0 :

Ct

t 0)0()(

Does the model creep?


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Only the Hooke element reacts initially:

1

0)0(E

C

)(1

)( 01

00

1

0 tJt

E

t

Et

0

(t)

0 t

1

0

E

0

Creep SolutionC

tt 0)0()(

Creep function


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A constant strain 0 is instantaneously applied at time t=0, when =0Constant strain d /dt=0, when t>0

01

1 dt

d

Edt

d

01

1

dtdE

Integrating:)(

)0( 0

1

t t

CdtEd

dtEd 1

CtE

t 1))0(ln())(ln(

Relaxation SolutionDoes the model relax?


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(Only the Hooke element reacts initially):

)()))0(ln())((ln(

1 CtE

t

eet

E

eCt

1

1)0(

)(

0

(t)

0 t

10E

1CeC

1)0( 101 CE

Relaxation Solution

)()( 010

1

tEeEt

tE

Relaxation function


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Total stress = spring stress + dashpot stress:

A linear first-order ordinary differential equation (ODE)

1E

1E 1E

11EE

Kelvin Model


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Constant stress, 0 : 01E

dtd

1E

)()()( ttt NH

tE

H eCt1

1)(01

HH E

dt

ddt

Ed

H

H 1

2)( CtN 21

1

)( CeCttE tE

eCE

dt

td 1

11)(

0

2

1

1

1

1

1

11

C

E

eC

E

eC

E tE

tE

1

0

2 EC

Creep Solution

1

01

1

)(E

eCtt

E



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)()1()( 01

01

tJeE

ttE

0

(t)

0 t

1

0)(E

1

010)0(

EC

Creep Solution


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dt

dE

1

An instantaneous change in the strain d = 0 and dt0 gives (0)

0

(t)

0 t

01)0( Et

)0(

Relaxation SolutionDoes the model relax?


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KelvinE2

KelvinKelvinKelvin E

1

KelvinE

2

/)( 1 KelvinKelvin E

22 E

E

22 E

E

Standard Linear Solid

KelvinE2


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/)( 12kelvinEE

2Ekelvin

22 E

E

1

2

1

2

)1( EEE

E


)(2

1

2E

E

E


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Solving, and using the fact that onlythe Hooke element reacts initially

2

0)0(E

A constant stress, 0 is instantaneously applied at time t=0,Constant stress d/dt=0, when t>0

)1(2

101

E

EE

dt

d

A linear first-order ordinary differential equation (ODE)

Creep Solution

1

2

1

2

)1( EEE

E



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)(

1

)1(

1

)(... 021

0

1

tJEeEt

tE

0

(t)

0 t

2

0

E

21

0

11)(

EE

Creep Solution


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Solving, and using the initial conditions

A linear first-orderordinary differential

equation (ODE)

A constant strain 0 is instantaneously applied at time t=0,

when =0, Constant strain d /dt=0, when t>0

01

2

1

2

)1(0 E

E

E

E

02121 )( EEEE

Relaxation Solution

1

2

1

2

)1( EEE

E

Does the model relax?


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)()(... 021

21

21

2

20

21

tEEE

EEeEE

Et tEE

0

(t)

0 t

0

21

21

EE

EE

teEEEtE )()( 0

0E

E

021

21

2

2

EE

EEE

Relaxation Solution

Instantaneous modulus

Asymptotic modulus


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teEEEtE )()( 0

0

(t)

0t

0E

E

)1(2

EG

teGGGtG )()( 0

Assume: 0.45

In LS-DYNA

For the Lab

Use to determine bulkmodulusE K

Instantaneous modulus

Asymptotic modulus


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Linear Viscoelastic Models,

Creep Functions

Maxwell Kelvin SLS

0

(t)

t


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0

(t)

t

Linear Viscoelastic Models,

Relaxation Functions

Maxwell Kelvin SLS


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Linear Viscoelasticity, Summary of Key Points

In viscoelasticmaterials stress depends on strain and strain-rate

They exhibit creep, relaxation and hysteresis

Viscoelastic models can be derived by combiningsprings with dampers, 3-parameter linear models (e.g. SLS) haveexponentially decaying creep and relaxation functions;time constants are the ratio of elasticity to damping

The instantaneous modulus, E0, is the stress-strain ratio at t=0

The asymptotic modulus, E , is the stress-strain ratio as t


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What if the curve of the modeldoes not fit the curve of thematerial we want to describe?

Generalized Models


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Generalized Kelvin Model

Total strain = (strains in each Kelvin element)

1 2 3 n

E1 E2 E3 En

n

i

in

1

21 ...

)1()( 0

1

tE

i

n

i

i

i

eE

t

Creep function:


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1 2 3 n

E1 E2 E3 En

Total stress= (stress in each Maxwell element)n

i

in

1

21 ...

Relaxation function:t

E

i

n

i

i

i

eEt 01

)(

Generalized Maxwell Model


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The Convolution Integral for Stress

t0

t0

0

)( 000 ttE

)( 111 ttE

1

)( 222 ttE

2


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...)()()()( 221100 ttEttEttEt

Hence for continuously varying strain:

dd

dtEdtEt

tt

00

)()()(

The Convolution Integral


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How are the viscoelasticproperies of soft biological

tissues determined?


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Oscillations to determine viscoelastic properties

,

^

^

)sin( t


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Harmonic Loading

)cos( tAx

)sin( tAy

iyxz )sin()cos( titA

)( tiAe tii eAe tiBe

B

BamplitudeA

)Re/(Imtan 1 BBphase angle

angular frequency f2 )/( srad


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Harmonic Stress and Strain History

)cos( t tie

tiei

i

i...)cos( t

Maxwell Model

1E

)1(1E

ii 11

)1

()(E

iii

)(iE


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Complex modulus for the Maxwell Model

1

1

)(

1E

iiiE)

1(

1E

i

)1

(

1

E

i

22

1

21

2

1)(

E

i

EiE

)

1

/()()( 221

2

1

2

E

i

EiE

)(Re iE )(Im iE

''')( iEEiE

Stiffness Damping


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22 )(Im)(Re)( EEamplitudeiE

Dynamic modulus

)1

()(

22

1

2

2

2

2

1

4

E

E

iE

22

1

2

22

1

2

22

1

22

11

1

EE

E

22

1

2

2

1

E

22

1E22

1E222

1

1)(E

EiE

Dynamic modulus for the Maxwell Model


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Internal friction for the Maxwell Model

Internal friction

E

ED

Re

Imtan

1EDD

)1

/()()(22

1

2

1

2

E

i

E

iE


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Complex modulus for the Kelvin Model

Kelvin

i

1E

)()( 1 iEi

)(iE

)(Re iE )(Im iE

iEiE 1)(

Stiffness Damping


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Dynamic modulus and Internal Friction for the Kelvin Model

iEiE 1)(

22

1

22 )()(Im)(Re)( EEEamplitudeiE

Dynamic modulus

Internal friction

1

Re

Imtan

EE

ED

D


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Complex modulus for the SLS Model

1

2

1

2

)1( EEE

E


i

i21221 )( EEEiEEi

)(iE

...

)(

)( 221

1 E

iEE

iEiE

22

21

2

2

22

21

2

2

2121

)()()()(

)()()(

EE

Ei

EE

EEEEEiE

2

21

1

)(E

iEE

iE


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Dynamic modulus

Internal friction

222

21

22

1 EEE

EE

)()(

)(||

2

211

2

)()(tan

EEE

ED

Dynamic Modulus and Internal Friction in the SLS Model

D

/)( 211 EEEpeak


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Hysteresis-Frequency Behavior


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But soft biological tissuesdetermined are not linear

elastic


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Quasilinear Viscoelasticity

Soft tissues exhibit several viscoelasticproperties:

hysteresis

stress relaxation

creep

strain-rate dependence

Linear viscoelastic models also display many of these properties

However, soft tissue elasticity is nonlinear

Quasilinear viscoelasticitycombines the time history

dependence of linear viscoelasticitywith nonlinearelasticity


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In soft tissue, the elasticresponse is nonlinear: = (e)(e)

However, the creep and relaxation functions can be

normalizedwith reasonable accuracy, i.e. Jr(t) and Er(t), are

relatively independent of the initial strain or stress

In quasilinear viscoelasticity, (e)(e) can be nonlinear, but

linear superposition still holds i.e., we separate the time and load dependence of the

relaxation or creep response, e.g. E(e,t) = Er(t) (e)(e).

Hence, the convolution integral for the stress is:

dt

tEdt

tEtt e

r

t e

r

0

)(

0

)( )()()()(

Quasilinear Viscoelasticity


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Viscoelasticity, Summary of Key Points

The stress-strain relation is not unique, it dependson the load history.

The elastic modulus depends on the load history.

Simple spring-damper models give rise to one or

more first-order linear ODEs which can be

conveniently formulated and solved.

Creep, relaxation and hysteresis are all properties

of linear viscoelastic models.

S ( ti d)


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Summary (continued)

Creep solution can be normalized by the initial strain to

give the reduced creep function Jr(t). Jr (0)=1.

Relaxation solution can be normalized by the initial stress

to give the reduced relaxation function Er (t). Er (0)=1.

The strain response to an arbitrary stress history is

obtained from J(t) by superposition

The stress response to an arbitrary strain history is

obtained from the E(t) by superposition

dddtJdtJt

tt

00

)()()(

dd

dtEdtEt

tt

00

)()()(

S ( ti d)


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The complex function E(iw) depends on the frequency w/2p,

and isccalled the complex modulus of elasticity The magnitude of E is the dynamic modulus of elasticity

The tangent of the phase angle Im(E)/Re(E) = tanf is calledthe internal friction and represents the damping due toviscous elements

Spring-damper models give rise to a finite number of peaksin the Hysteresis-Frequency curve

However, soft tissues have no discrete peaks. That is, softtissue behave as though they have an infinite numberof springs and dampers

Quasilinear viscoelasticitycombines the time history

dependence of linear viscoelasticitywith nonlinear elasticity

Summary (continued)


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Change in schedule

TUESDAY 15/4 at 10-12 in E31

Injury mechanisms: Head

Svein Kliven

WEDNESDAY 16/4 at 13-15 in D34

Energy absorption

Peter Halldin

Viscoelasticity KTH

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