1. BE 105, Lecture 10 Geometric Properties II Part 1: Bone, continued

2. cranial post cranial, axial flexible rod that resists compression network of flexible linkages

3. How to make a fish fin head muscle ‘back bone’ active muscle inactive muscle laterally flexible, but resists compression

4. tunicate larva

5. Garstang Hypothesis

6. early tetrapods

7. How do bones articulate? joint types

8. Four bar system

9. e.g. 4 bar system Four bar system

10. 4 bar system

12. Part 2: Torsion and Shear E =  G =  E = Young’s modulus,  = stress,  = strain G = Shear modulus,  = shear stress,  = shear strain F A  shear stress,  = force/area shear strain,  = angular deflection For a given material, what is relationship between E and G? Area LL L Force  = force / cross sectional area  = change in length / total length

13. force length Area LL L stress (  ) = F / A 0 strain (  ) =  L / L 0 Force Engineering units But…what if strain is large? Area will decrease and we will underestimate stress. True units: stress (  ) = F / A (  ) strain (  ) = ln ( L / L 0 ) strain (  ) = dL = ln ( L / L 0 ) 1L1L ‘Engineering’ vs. ‘True’ stress and strain

14. x y z The ratio of ‘primary’ to ‘secondary’ strains is known as: Poisson’s ratio, :  =  2 /  1  measures how much a material thins when pulled. Simon Denis Poisson (1781-1840) Poisson’s ratio also tells us relationship between shear modulus, G, And Young’s modulus, E: G = E 2(1+ ) where is Poisson’s ratio for an isovolumetric material (e.g. water)

15. G = E 2(1+ ) L T TT LL Material  Incompressible materials (e.g. water) 0.5 Most metals 0.3 Cork 0 Natural rubber 0.5 Bone c. 0.4 Bias-cut cloth 1.0

16. Mlle Vionnet ‘bias-cut’ dress gravity

17. fiber windings

18. compression apply torsion shear tension compression tension cantilever beam EI = Flexural stiffness GJ = Torsional stiffness where J = polar second moment of area J =  r 2 dA = ½  r 4 (solid cylinder) r dA 0 R How to measure J?  = ML/(GJ) L F x  M = Fx

19. Bone fractures

20. compression apply torsion tension Bones fail easily in tension: G (compression) = 18,000 MPa G (Tension) = 200 MPa Bone is a a great brick, but a lousy cable!