Transcript BE105_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
How to make a fish
‘back bone’
head
fin
muscle
inactive muscle
active muscle
laterally flexible,
but resists compression
tunicate larva
Garstang Hypothesis
early tetrapods
How do bones articulate?
joint types
Four bar system
Four bar system
e.g. 4 bar system
4 bar system
Part 2: Torsion and Shear
Force
DL
Area
F
shear stress, t
= force/area
L
s = force / cross sectional area
e = change in length / total length
shear strain, g
= angular
deflection
g
A
E = s/e
E = Young’s modulus, s = stress, e = strain
G = t/g
G = Shear modulus, t = shear stress, e = shear strain
For a given material, what is relationship between E and G?
‘Engineering’ vs. ‘True’ stress and strain
Engineering units
force
Force
DL
length
Area
stress (s) = F / A 0
strain (e) = D L / L 0
L
But…what if strain is large?
Area will decrease and we will underestimate stress.
True units:
stress (s) = F / A (e)
strain (e) =
strain (e) = ln ( L / L 0)
1
dL = ln ( L / L 0)
L
x
x0 y0 z0  x1 y1 z1  volume
y
2
z
x0 y1 z1 y1

 2
x1 y0 z0 y0
for an isovolumetric
material (e.g. water)
x0
y0
ln( )  2 ln( )
x1
y1
e x  2e y
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)
Poisson’s ratio also tells us relationship between shear modulus, G,
And Young’s modulus, E:
G=
E
2(1+n)
where n is Poisson’s ratio
DT
DL
L
T
Material
Incompressible materials (e.g. water)
Most metals
Cork
Natural rubber
Bone
Bias-cut cloth
n
0.5
0.3
0
0.5
c. 0.4
1.0
G=
E
2(1+n)
Mlle Vionnet ‘bias-cut’ dress
gravity
fiber windings
cantilever
beam
compression
apply
torsion
tension
compression
tension
GJ =
Torsional stiffness
EI = Flexural stiffness
shear
where J = polar second moment of area
J =  r2 dA
R
r
dA
0
= ½ pr4
(solid cylinder)
q
L
How to measure J?
q = ML/(GJ)
x
M = Fx
F
Bone fractures
compression
apply
torsion
Bones fail easily in tension:
tension
G (compression) = 18,000 MPa
G (Tension) = 200 MPa
Bone is a a great brick,
but a lousy cable!