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Viscoelasticity-KTH.pdf
Viscoelasticity
1
Mechanical properties of living tissues
F7
Soft tissues
F8
Muscles
F9
Viscoelasticity
F15
Bone
2
Lab
• Construct a Finite element
Model
• Dissection of deer spine
• Mechanical testing
• Analysis of the results and
implement properties into FEmodel
• Report
Group
number
Laboration
week 16
Fri
Fri
18/4
18/4
8 - 12
13 - 17
1-4
5-8
Dissec + mech test
Dissec + mech test
AN/DL
AN/DL
3
Viscoelasticity
Material from:
Kleiven S. An Introduction to Viscoelasticity, Lecture notes for
4E1150,
Division of Neuronic Engng., CTV, KTH, Sweden, 2003.
Fung, Y.C. Foundations of Solid Mechanics, Prentice-Hall Inc.,
Englewood Cliffs, New Jersey, USA, 1965.
Flügge, W. Viscoelasticity, Springer-Verlag, 1975.
4
Viscoelasticity
• Characteristic behaviour of
viscoelastic materials
• Why are biological tissues
viscoelastic?
• Theory of viscoelasticity
5
Linear Viscoelasticity
• 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 history of strain
• These properties can be modeled by the theory of
viscoelasticity
6
Stress relaxation
Load
Steel bolt through plastic…
The load will
decrease in time for
a constant applied
deformation
Time
7
Stress relaxation
Examples of Biological tissues in Stress relaxation
Stress relaxation function for Nucleus Pulposus
(Normalized) (Iatridis et al, 1997)
Stress relaxation curves for the Scalp
(Melvin et al., 1970)
8
Creep
Structure loaded by gravity…
Deformation
mg
Deformation will
continue in time
although the load is
constant
Time
9
Creep
Example of Biological tissues in Creep
Creep compliance of dura mater in tension
(Galford et al., 1970)
10
Hysteresis and strain-rate dependence
Increased strain-rate
leads to a stiffer response
Load
Load
Loading
Unloading
Deformation
Deformation
Area between curves
is proportional to
dissipated energy
11
Hysteresis and strain-rate dependence
Example of hysteresis & strain-rate
dependence in Biological tissues
Hysteresis loops for ankle
ligament
(Funk et al., 2000)
The stress-strain relationship for
bone
(McElhaney, 1966)
12
Viscoelasticity
• Characteristic behaviour of
viscoelastic materials
• Why are biological tissues
viscoelastic?
• Theory of viscoelasticity
13
Ligaments & Tendons
Rate dependent strength,
stiffness and energy
absorption.
Increased by a factor of 3 if the load
rate is increased from 8 to 2300 m/s.
Viscoelastic response due to
• Interactions between the
proteoglycans in the
groundsubstance and
collagen fibrils.
14
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
15
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 60º
– 78% water
Viscoelastic response due to
• Fluid flow during loading
• Shear forces between the matrix
and fibers during fiber
straightening.
16
Trabecular bone
higher density,
roughly prismatic cells
from the femural head
low-density, open
cell, rod-type
structure from the
femural head
intermediate density
parallel plate structure
with rods normal to the
plate
Viscoelastic response due to
• Fluid flow during loading
17
Importance of viscoelasticity
• Structure made of polymers (i.e. FRP)->
Creep, relaxation etc….
-> Collapse?
• Damping, dynamic modulus etc. of
Biological tissues
-> Protection!
18
Importance of viscoelasticity
Maximal shear strain for human head
including viscoelasticity
Maximal shear strain for human head
excluding viscoelasticity
19
Viscoelasticity
• Characteristic behaviour of
viscoelastic materials
• Why are biological tissues
viscoelastic?
• Theory of viscoelasticity
20
Basic elements
Spring (Hooke element)
E1
E1
Viscous Damper
(Newton element)

21
Simple Viscoelastic Models
Stress depends on strain and strain-rate:
Maxwell
Kelvin (aka Voigt)
Standard Linear Solid
• 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
22
Maxwell Model
Total strain = spring strain + dashpot strain:

E1
E1
E1

E1

E1
E1

E1

E1


E1
A linear first-order ordinary
differential equation (ODE)
23
Creep Solution
Does the model creep?

d
dt

E1
d
1 d
E1 dt
1
d
E1
dt
Integrating, for constant applied stress, σ0 :
(t )
d
(0)
1
E1
(t )
d
(0)
t
0
dt C
(t )
(0)
0
t
C
0
24
(t )
(0)
0t
Creep Solution
C
Only the Hooke element reacts initially:
(t )
0
0
t
0
E1
1
E1
(0)
C
0
J (t )
t
0
E1
Creep function
ε(t)
0
0
E1
σ0
0
t
25
Relaxation Solution
Does the model relax?
A constant strain 0 is instantaneously applied at time t=0, when =0
Constant strain  d /dt=0, when t>0
d
dt
1
d
E1
1 d
E1 dt
dt
0
0
d
E1
dt
Integrating:
(t )
( 0)
d
E1
t
dt C
ln( (t )) ln( (0))
E1
t C
0
26
Relaxation Solution
e
(
(ln( ( t )) ln( ( 0 )))
(t )
(0)
E1
t C)
e
e
E1
C
C1
t
(0)
C1e
E1
0
C1
1
(Only the Hooke element reacts initially):
σ(t)
0
E1
(t )
0
E1e
E1
t
0
E (t )
Relaxation function
ε0
0
t
27
Kelvin Model
Total stress = spring stress + dashpot stress:
E1
E1
E1

E1
E1

A linear first-order ordinary differential equation (ODE)
28
Creep Solution

E1
Does the model creep?
Constant stress, σ0 :
d H
dt
d
dt
E1
E1
0
d
0
H
E1
H
E1
H
(t )
N
(t )
E1
dt
H
(t )
t
C1e
H
E1
N
(t )
(t )
C2
E1
C1e
t
(t )
E1
C1e
E1
C1e
t
E1
C2
C2
E1
(t )
C1e
d (t )
dt
t
t
0
C2
E1
E1
C1e
0
E1
0
E1
29
t
Creep Solution
E1
(0)
0
0
C1
E1
(t )
0
E1
(1 e
t
)
0
J (t )
ε(t)
( )
0
E1
σ0
0
t
30
Relaxation Solution
E1
Does the model relax?
d
dt
An instantaneous change in the strain d =
0
and dt0 gives (0)∞
(0)
(t
σ(t)
0)
E1
0
ε0
0
t
31
Standard Linear Solid
E2
E2
E2
Kelvin
E1
E2
Kelvin
Kelvin

Kelvin
E 2
Kelvin
E 2
Kelvin

E2
Kelvin
(
E1
Kelvin
)/
32
Standard Linear Solid


E2
(
kelvin

E2


E2
E1
kelvin
)/
E2

E2
E1
(
E2
)
E2
E1
(1
)
E2
E1
33
Creep Solution
Does the model creep?


E2
E1
(1
)
E2
E1
A constant stress, σ0 is instantaneously applied at time t=0,
Constant stress dσ/dt=0, when t>0
d
dt
E1
0
E1
(1
)
E2
A linear first-order ordinary differential equation (ODE)
Solving, and using the fact that only
the Hooke element reacts initially
(0)
0
E2
34
Creep Solution
...
(t )
0
1
(1 e
E1
E1
t
)
1
E2
0
J (t )
ε(t)
( )
0
1
E1
1
E2
0
E2
σ0
0
t
35
Relaxation Solution
Does the model relax?


E2
E1
(1
)
E2
E1
A constant strain 0 is instantaneously applied at time t=0,
when =0, Constant strain d /dt=0, when t>0
0

E2

( E1
E1
(1
)
E2
E2 )
E1
0
E1 E2
0
A linear first-order
ordinary differential
equation (ODE)
Solving, and using the initial conditions…
36
Relaxation Solution
...
E
(t )
0
E1 E2
E1 E2
E2
E0
σ(t)
2
2
E E1 E2
E1 E2
E1
0
E1 E2
2
2
t
e
E1 E2
E1 E2
0
E (t )
Instantaneous modulus
E (t )
E
( E0
E )e
t
0
E
Asymptotic modulus
ε0
0
t
37
For the Lab
E0 Instantaneous modulus
σ(t)
E
G
Asymptotic modulus
E (t )
E
2(1 )
ε0
E
( E0
E )e
t
G )e
t
In LS-DYNA
0
t
G (t )
Use
E
G
(G0
to determine bulkmodulus
Assume:
K
≈ 0.45
38
Linear Viscoelastic Models,
Creep Functions
Maxwell
Kelvin
SLS
ε(t)
σ0
t
39
Linear Viscoelastic Models,
Relaxation Functions
Maxwell
Kelvin
SLS
σ(t)
ε0
t
40
Linear Viscoelasticity, Summary of Key Points
• In viscoelastic materials stress depends on strain and strain-rate
• They exhibit creep, relaxation and hysteresis
• Viscoelastic models can be derived by combining
springs with dampers, 3-parameter linear models (e.g. SLS) have
exponentially 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
41
What if the curve of the model
does not fit the curve of the
material we want to describe?
Generalized Models…
42
Generalized Kelvin Model
E1
E2
1
E3
3
2
Total strain =
1
En
n
(strains in each Kelvin element)
n
2
...
n
i
i 1
Creep function:
Ei
n
0
(t )
i 1
Ei
(1 e
i
t
)
43
Generalized Maxwell Model
E1
1
E2
E3
2
3
Total stress=
En
n
(stress in each Maxwell element)
n
1
...
2
n
i
i 1
Relaxation function:
Ei
n
(t )
0 Ei e
t
i
i 1
44
The Convolution Integral for Stress
2
1
t0
0
t0
2
2
1
1
0
0
E (t t2 )
E (t t1 )
E (t t0 )
45
The Convolution Integral
(t )
0
E (t t0 )
1
E (t t1 )
2
E (t t2 ) ...
Hence for continuously varying strain:
t
(t )
t
E (t
0
)d
E (t
0
d
) d
d
46
How are the viscoelastic
properies of soft biological
tissues determined?
47
Oscillations to determine viscoelastic properties
σ, ε
^
σ
ˆ sin( t )
^ε
φ
48
Harmonic Loading
y
x
z
x iy
A cos( t
) i sin( t
i( t
i
Ae
A
A sin( t
A cos( t
amplitude
)
)
Be
i t
B
B
phase angle
Ae e
i t
)
)
tan 1 (Im B / Re B )
angular frequency
2 f (rad / s)
49
Harmonic Stress and Strain History
i t
ˆ
ˆ cos( t )
e

i
ˆ
i
e
t
ˆ cos( t ) ...

i

i
Maxwell Model

i
i
(
E1
1
)
(i )
i
i (
E1

E1
1
)
1
E (i )
50
Complex modulus for the Maxwell Model
E (i )
1
i
i
E1
1
2
E (i )
(
E1
Re E (i )
) /(
2
1
2
1
2
E
Im E (i )
1
)
)
i
2
1
2
2
E1
i
1
E1
E (i )
2
i
(
E1
i
(
E1
)
E (i )
E ' iE' '
”Stiffness”
”Damping”
51
Dynamic modulus for the Maxwell Model
Dynamic modulus
E (i )
(Re E ) 2
amplitude
4
2
2
2
E1
E (i )
2
(
E1
2
2
E1
2
1
2
2
(Im E ) 2
2
1
2
2
2
E1
1
)
2
2
1
2
2
E1
E1
2
E1
2
2
1
2
2
E1
2
E1
E (i )
2
E1
2
2
2
52
Internal friction for the Maxwell Model
2
E (i )
(
E1
i
) /(
2
1
2
2
E1
)
Internal friction
D
Im E
Re E
tan
D
E1
D
53
Complex modulus for the Kelvin Model
Kelvin

E1

i
(i )
( E1 i
)
E (i )
E (i )
E1 i
Re E (i )
Im E (i )
”Stiffness”
”Damping”
54
Dynamic modulus and Internal Friction for the Kelvin Model
E (i )
E1 i
Dynamic modulus
E (i )
amplitude
(Re E )
2
(Im E )
2
E1
2
)2
(
Internal friction
D
tan
Im E
Re E
D
E1
55
Complex modulus for the SLS Model
Standard Linear Solid



i
i
i
E1 i
( E1 E2 ) i
( E1
(1
E2 )
E1
)
E2
i E2
E1
E1E2
E2
E (i )
E (i )

E2
E1 E 2 ( E1 E 2 ) (
( E1 E 2 ) 2 (
E (i )
E1 i
( E1 E2 ) i
...
2
2
) E2
)2
E2
i
( E1
E2
E2 ) 2 (
)2
56
Dynamic Modulus and Internal Friction in the SLS Model
Dynamic modulus
2
|E|
E1
(
( E1 E 2 ) 2
)2
E2
2
( )
D
Internal friction
D
tan
E1 ( E1
E2
E2 ) (
peak
)2
E1 ( E1
E2 ) /
57
Hysteresis-Frequency Behavior
58
But soft biological tissues
determined are not linear
elastic…
59
Quasilinear Viscoelasticity
• Soft tissues exhibit several viscoelastic properties:
– hysteresis
– stress relaxation
– creep
– strain-rate dependence
• Linear viscoelastic models also display many of these properties
• However, soft tissue elasticity is nonlinear
• Quasilinear viscoelasticity combines the time history
dependence of linear viscoelasticity with nonlinear elasticity
60
Quasilinear Viscoelasticity
• In soft tissue, the elastic response is nonlinear: σ = σ(e)(e)
• However, the creep and relaxation functions can be
normalized with reasonable accuracy, i.e. Jr(t) and Er(t), are
relatively independent of the initial strain or stress
• In quasilinear viscoelasticity, σ(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) σ(e)(e).
• Hence, the convolution integral for the stress is:
t
(t )
Er (t
0
t
(e)
)
t
d
(e)
Er (t
0
)
( )
t
d
61
Viscoelasticity, Summary of Key Points
• The stress-strain relation is not unique, it depends
on 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.
62
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
t
(t )
t
J (t
)d
J (t
0
0
d
)
d
d
• The stress response to an arbitrary strain history is
obtained from the E(t) by superposition
t
(t )
t
E (t
0
)d
E (t
0
d
) d
d
63
Summary (continued)
• 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 called
the internal friction and represents the damping due to
viscous elements
• Spring-damper models give rise to a finite number of peaks
in the Hysteresis-Frequency curve
• However, soft tissues have no discrete peaks. That is, soft
tissue behave as though they have an infinite number
of springs and dampers
• Quasilinear viscoelasticity combines the time history
dependence of linear viscoelasticity with nonlinear elasticity
64
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
65
Fly UP