Lecture # 10: Hydrostatic Skeletons
1 cell cellular sheetcellular bilayer
bilayered canister
ecto-derm
endo-derm
one way gut
mouthanus
cephalization
mesoderm
Body Plan Evolution
1 cell cellular sheetcellular bilayer
bilayered canister
ecto-derm
endo-derm
one way gut
mouthanus
cephalization
mesoderm
Body Plan Evolution
ectoderm
mesoderm
endoderm
gut
coelom
coelom
ectoderm
mesoderm
gut
pseudocoelom
endoderm
Consider a hollow spherical animal….
slice in half
d
rP
P= internal pressurer = radius
d= thickness
PP
what is stress in wall?
= force/area = (r2 p) / (2 r d) = (r p) / (2 d)
disk area ~ r2
rim area ~ 2 r d
Define tension, T, as force/length
then T = x d = r p / 2
T = ½ r p
LaPlace’s Law: Tension in wall of sphere is proportional to radius and pressure.
How does stress in a worm depend on geometry?
Consider a cylindrical animal….
Equivalent to spherical case,Thus longitudinal tension, TLis same as in sphere of equal radius:
TL = ½ r p
1) longitudinal slice 2) slice in half
3) cap withhemisphere
Consider a cylindrical animal….
1) transverse wedge
c = force/area = (2 r p) / (2 d ) = r p / d
Again, TC = c x d
TC = r p
r
c
slice area=2r
rim area=2 d
d
Circumferential or ‘hoop stress’is twice than longitudinal stress.
TC = 2 x TL
Implications of LaPlace’s Law:
Pierre-Simon Laplace 1749-1827
1) Small worm withstand greater pressure than large worms.
2) Large worms should have thicker walls.
3) Square cross sections should be rare.
P P
P P
tension is infinite
Consider a helical worm:
Volume = r2 L
Solve for volume in terms of (helical angle):
D = L cos r = D sin /(2 )
V = D3 sin2 cos 4
Solve for dV/d
Maximum volume at = 54.73o
L
L
Permissible Morpho-space
V = d3 sin2 cos 4
ellipticalprofile
circularsection
muscle action
Ontogenetic scaling of burrowing forces in the earthworm Lumbricus terrestris
• Kim Quillin • J Exp Biol 203, 2757-2770 (2000)