Lecture # 7 Viscoelastic Materials
reminder:
solids resist strain: F = k1 xfluids resist rate of change of length: F = k2 d(x)/dt
springYoung’s modulus
(stiffness)
dashpotviscosity
most biomaterials (including bone) are viscoelastic
time
solid
fluid
viscoelastic
stepresponses
viscoelastic materials may be modeled with springs and dashpots.
e.g. in series
= Maxwell Model
in parallel
= Voigt Model
Maxwell Model Voigt Model
springexpands dashpot
expands
springcontracts
‘isotonic’response(constantstress)
dashpot acts as strut
acts asspring
dashpotrelaxes
dashpotacts as strut
acts asspring
dashpotrelaxes
= stress relaxation
curve
dashpotrelaxes
dashpotacts as strut
zerostress
‘isometric’response(constantstrain)
= damperor low pass filter
I) Harmonic Analysis of Materials
input:(t) = 0sin t
output:(t) = 0sin t +
force
lengthstuff
Case 1: input in phase with output:
input:(t) = 0sin t
output:(t) = 0sin t
stress and strain maximum (and minimum) at same time.
material is acting as an elastic solid, described by single term:
E = 0/0
E = Young’s modulus
Case 2: output phase advanced by 90o
input:(t) = 0sin t
output:(t) = 0sin t – 90o
stress is maximum when d/dt is maximum
material is acting like Newtonian fluid, described by single term:
= 0/(0)
using…
(t) = 0sin t d(t)/dt = 0cos t
= dynamic viscosity
Case 3: -90o < output phase < 0o :
input:(t) = 0sin (t)
output:(t) = 0sin (t – 0o < < 90o
stress is maximum at intermediate point
Material is acting as a viscoelastic substance.output waveform (t), can be described as the sum of two different waveforms:
in phase component = ’0 sin (t) out-of-phase component = ”0 sin (t – 90o)
= ”0 cos (t)
in phase component:
’out-of-phasecomponent:
’’
Input strain:(t) = 0sin tOutput stress:(t) = ’0sin (t) + ’’0cos(t)Let’0E’ and’’0E’’= 0 (E’ sin t + E’’ cos t)
E’ = E* cos = elastic, storage, in-phase, or real modulus
E’’ = E* sin = viscous, loss, out-of-phase, or imaginary modulus
tan = E’’/E’
Case 3, continued
E’
E’’E*
elastic,storagein-phase axis
visc
ous
,loss
out-
of-p
hase
axi
s
viscouscomponent
elasticcomponent
E’ = E* cos E’’ = E* sin
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?
E=complex modulus =
Creep
Harmonic Analysis is valid only for small stresses and strains.What about large deformations and long time periods?
time
yieldcreep
log time
E
creep = slow decreasein stiffness,
material starts to flow.
continuousstress
‘necking’creep creep
material makes slow ‘solid to fluid transition’
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:
1 2 3 1
glycine