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’

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:

1 2 3 1

glycine

fiberwithin fiberconstruction:

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