Lecture # 7 Viscoelastic Materials

1. spring Young’s modulus (stiffness) dashpot viscosity fluid viscoelastic solid s s s e e e time in series in parallel = Maxwell Model = Voigt Model Lecture # 7 Viscoelastic Materials 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 step responses viscoelastic materials may be modeled with springs and dashpots. e.g.

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

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

4. s e Case 2: output phase advanced by 90o stress is maximum when de/dt is maximum 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. s e s’’ out-of-phase component: s’ in phase component: stress is maximum at intermediate point 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) 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. elastic component E’ E’ = E* cos d E’’ = E* sin d E* viscous,loss out-of-phase axis viscous component E’’ d elastic,storage in-phase axis Case 3, continued E*=complex modulus = s0/e0 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. creep yield creep = slow decrease in stiffness, material starts to flow. e E s time log time ‘necking’ creep creep continuous stress Creep Harmonic Analysis is valid only for small stresses and strains. What about large deformations and long time periods? 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. 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)