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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    Characteristic behaviour ofviscoelastic materials

    Why are biological tissuesviscoelastic?

    Theory of viscoelasticity

    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.

    Viscoelastic response due to

    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

    Viscoelastic response due to

    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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    Characteristic behaviour ofviscoelastic materials

    Why are biological tissuesviscoelastic?

    Theory of viscoelasticity

    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

    Does the model creep?

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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

    Standard Linear Solid

    )(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

    Does the model creep?

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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

    Standard Linear Solid

    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