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BE 105, Lecture 10 Geometric Properties II

BE 105, Lecture 10 Geometric Properties II. Part 1: Bone, continued. post cranial, axial. cranial. flexible rod that resists compression. network of flexible linkages. inactive muscle. laterally flexible, but resists compression. active muscle. How to make a fish. ‘back bone’.

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BE 105, Lecture 10 Geometric Properties II

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

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

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

  4. tunicatelarva

  5. Garstang Hypothesis

  6. early tetrapods

  7. How do bones articulate? joint types

  8. Four bar system

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

  10. 4 bar system

  11. F Force shear stress, t = force/area shear strain, g = angular deflection Area g DL A L s = force / cross sectional area e = change in length / total length E = s/e G = t/g E = Young’s modulus, s = stress, e = strain G = Shear modulus, t = shear stress, e = shear strain Part 2: Torsion and Shear For a given material, what is relationship between E and G?

  12. Engineering units Force force Area stress (s) = F / A 0 strain (e) = D L / L 0 DL L length But…what if strain is large? Area will decrease and we will underestimate stress. True units: stress (s) = F / A (e) strain (e) = ln ( L / L 0) 1 L dL = ln ( L / L 0) strain (e) = ‘Engineering’ vs. ‘True’ stress and strain

  13. for an isovolumetric material (e.g. water) where n is Poisson’s ratio E G = 2(1+n) Poisson’s ratio also tells us relationship between shear modulus, G, And Young’s modulus, E: y x z • The ratio of ‘primary’ to ‘secondary’ strains is known as: • Poisson’s ratio, n: • n = e2/e1 • n measures how much a material thins when pulled. Simon Denis Poisson (1781-1840)

  14. DT E DL G = 2(1+n) L T Material n 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

  15. Mlle Vionnet ‘bias-cut’ dress gravity

  16. fiber windings

  17. compression apply torsion tension shear r dA where J = polar second moment of area J = ò r2 dA = ½ pr4 (solid cylinder) R 0 q L How to measure J? q = ML/(GJ) x F M = Fx cantilever beam tension compression GJ = Torsional stiffness EI = Flexural stiffness

  18. Bone fractures

  19. 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!

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