Lecture # 7 Viscoelastic Materials
spring
Young’s modulus
(stiffness)
reminder:
solids resist strain:
fluids resist rate of change of length:
F = k1 x
F = k2 d(x)/dt
dashpot
viscosity
most biomaterials (including bone) are viscoelastic
step
responses
fluid
solid
s
e
time
s
viscoelastic
s
e
e
viscoelastic materials may be modeled with springs and dashpots.
e.g.
in series
in parallel
= Maxwell Model
= Voigt Model
Maxwell Model
spring
expands
‘isotonic’
response
(constant
stress)
Voigt Model
dashpot
relaxes
spring
contracts
dashpot acts
as strut
dashpot
expands
e
e
s
s
dashpot
acts as strut
dashpot
acts as strut
dashpot
relaxes
dashpot
relaxes
zero
stress
‘isometric’
response
s
(constant
strain)
s
e
e
= stress
relaxation
curve
acts as
spring
acts as
spring
= damper
or low pass filter
I) Harmonic Analysis of Materials
force
input:
e(t) = e0sin wt
stuff
length
output:
s(t) = s0sin wt + d
Case 1: input in phase with output:
stress and strain maximum
(and minimum) at same time.
input:
e(t) = e0sin wt
s
output:
s(t) = s0sin wt
e
material is acting as an elastic solid, described by single term:
E = s0/e0
E = Young’s modulus
Case 2: output phase advanced by 90o
input:
e(t) = e0sin wt
stress is maximum
when de/dt is maximum
s
e
output:
s(t) = s0sin wt – 90o
material is acting like Newtonian fluid, described by single term:
m = s0/(we0)
using…
e(t) = e0sin wt
de(t)/dt = we0cos wt
m = dynamic viscosity
Case 3: -90o < output phase < 0o :
stress is maximum
at intermediate point
input:
e(t) = e0sin (wt)
s
e
output:
s(t) = s0sin (wt – d)
{0o < d < 90o }
Material is acting as a viscoelastic substance.
output waveform s(t), can be described as the sum of two different waveforms:
in phase component = s’0 sin (wt)
out-of-phase component = s”0 sin (wt – 90o)
= s”0 cos (wt)
out-of-phase
component:
s’
s’’
in phase
component:
Input strain: e(t) = e0sin wt
Output stress: s(t) = s’0sin (wt) + s’’0cos(wt)
= e0 (E’ sin wt + E’’ cos wt)
Let s’=e0E’ and s’’=e0E’’
Case 3, continued
E* = complex modulus = s0/e0
E’ = E* cos d
E’’ = E* sin d
viscous,loss
out-of-phase axis
elastic
component
E’
E*
E’’
viscous
component
d
elastic,storage
in-phase axis
E’ = E* cos d = elastic, storage, in-phase, or real modulus
E’’ = E* sin d = viscous, loss, out-of-phase, or imaginary modulus
tan d = E’’/E’
Questions for reflection:
1) What similarities do springs and dashpots have with resistors and capacitors?
2) What would it mean to have a negative viscous modulus?
3) Could you repeat this analysis at different frequencies?
Creep
Harmonic Analysis is valid only for small stresses and strains.
What about large deformations and long time periods?
creep
yield
E
creep = slow decrease
in stiffness,
material starts to flow.
e
s
time
log time
‘necking’
creep
creep
continuous
stress
material makes slow ‘solid to fluid transition’
Phylum Cnidaria
nematocyst
Metridium
Prey (Stomphia)
Predator (Dermasterias)
Collagen
Part III: Collagen
Most common protein in vertebrate body BY FAR!
20% of a mouse by weight.
33% glycine, 20% hydroxyproline
Each tropo-collagen fiber held together by hydrogen bonds
involving central glycines:
glycine
1
2
3
1
fiber
within fiber
construction:
Julian Voss-Andreae's sculpture Unraveling Collagen (2005)