1. Lecture # 7 Viscoelastic Materials
spring
Young’s modulus
(stiffness)
dashpot
viscosity
reminder:
solids resist strain: F = k1 x
fluids resist rate of change of length: F = k2 d(x)/dt
most biomaterials (including bone) are viscoelastic e s
time
solid e s
fluid e s
viscoelastic
step
responses
viscoelastic materials may be modeled with springs and dashpots.
e.g.
in series
= Maxwell Model
in parallel
= Voigt Model

2. Maxwell Model Voigt Model ‘isotonic’ response (constant e e stress) s
dashpot acts
as strut
acts as
spring
dashpot
relaxes s e
spring
expands
dashpot
contracts
‘isotonic’
response
(constant
stress) e s
dashpot
acts as strut
acts as
spring
relaxes
= stress
relaxation
curve e s
dashpot
relaxes
acts as strut
zero
stress
‘isometric’
response
(constant
strain)
= damper
or low pass filter

3. I) Harmonic Analysis of Materials
force
length
stuff
input:
e(t) = e0sin wt
output:
s(t) = s0sin wt + d s e
Case 1: input in phase with output:
input:
e(t) = e0sin wt
output:
s(t) = s0sin wt
stress and strain maximum
(and minimum) at same time.
material is acting as an elastic solid, described by single term:
E = s0/e0
E = Young’s modulus

4. material is acting like Newtonian fluid, described by single term:
Case 2: output phase advanced by 90o
stress is maximum
when de/dt is maximum s e
input:
e(t) = e0sin wt
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

5. Material is acting as a viscoelastic substance.
stress is maximum
at intermediate point s e
Case 3: -90o < output phase < 0o :
input:
e(t) = e0sin (wt)
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’’
in phase
component: s’
Input strain: e(t) = e0sin wt
Output stress: s(t) = s’0sin (wt) + s’’0cos(wt) Let s’=e0E’ and s’’=e0E’’
= e0 (E’ sin wt + E’’ cos wt)

6. E* = complex modulus = s0/e0
Case 3, continued
E* = complex modulus = s0/e0 E’
E’’ E* d
elastic,storage
in-phase axis
out-of-phase axis
viscous,loss
viscous
component
elastic
E’ = E* cos d
E’’ = E* sin d
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?

7. material starts to flow.
Creep
Harmonic Analysis is valid only for small stresses and strains.
What about large deformations and long time periods?
log time E
creep = slow decrease
in stiffness,
material starts to flow. s e
time
yield
creep
continuous
stress
‘necking’
creep
material makes slow ‘solid to fluid transition’

9. Phylum Cnidaria

10. nematocyst

12. Metridium

15. Prey (Stomphia)
Predator (Dermasterias)

16. Collagen

17. Part III: Collagen Most common protein in vertebrate body BY FAR!
20% of a mouse by weight.
33% glycine, 20% hydroxyproline

18. 1 2 3 1 Each tropo-collagen fiber held together by hydrogen bonds
involving central glycines:
glycine 1 2 3 1

19. fiber
within fiber
construction:

20. Julian Voss-Andreae's sculpture Unraveling Collagen (2005)