BE 105, Lecture 10 Geometric Properties II Part 1: Bone, continued
cranial post cranial, axial flexible rod that resists compression network of flexible linkages
How to make a fish fin head muscle ‘back bone’ active muscle inactive muscle laterally flexible, but resists compression
tunicate larva
Garstang Hypothesis
early tetrapods
How do bones articulate? joint types
Four bar system
e.g. 4 bar system Four bar system
4 bar system
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 LL L Force = force / cross sectional area = change in length / total length
force length Area LL 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
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 ( ) 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)
G = E 2(1+ ) L T TT LL 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
Mlle Vionnet ‘bias-cut’ dress gravity
fiber windings
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
Bone fractures
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!