BE 105, Lecture 10Geometric Properties II

Part 1: Bone, continued

cranial

post cranial, axial

flexible rod that resists compression

network of flexible

linkages

How to make a fish

finhead

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

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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?

AreaL

L

Force

= force / cross sectional area = change in length / total length

force

length

AreaL

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)

1L

‘Engineering’ vs. ‘True’ stress and strain

xy

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

yx

y

y

x

x

y

y

zy

zy

x

x

volumezyxzyx

2

)ln(2)ln(1

0

1

0

20

21

00

11

1

0

111000

for an isovolumetricmaterial (e.g. water)

G = E

2(1+)L

T

T LMaterial Incompressible materials (e.g. water) 0.5Most metals 0.3Cork 0Natural rubber 0.5Bone c. 0.4Bias-cut cloth 1.0

Mlle Vionnet ‘bias-cut’ dress

gravity

fiber windings

compression applytorsion

shear

tensioncompression

tension

cantileverbeam

EI = Flexural stiffness GJ = Torsional stiffness

where J = polar second moment of area

J = r2 dA

= ½ r4

(solid cylinder)

r dA0

R

How to measure J?

= ML/(GJ)

L

F

x

M = Fx

Bone fractures

compression applytorsion

tension

Bones fail easily in tension:

G (compression) = 18,000 MPaG (Tension) = 200 MPa

Bone is a a great brick, but a lousy cable!