be 105, lecture 10 geometric properties ii part 1: bone, continued
TRANSCRIPT
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
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!