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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
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