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Tronc
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 Posted: Thu Jan 08, 2004 8:00 pm Post subject: Spherical membrane |
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Dear all
I>ve got a mechanics-type problem at the moment - don>t worry, it really
isn>t for homework.
Consider a liquid-filled sphere, surrounded by some sort of solid membrane
in space holding the liquid together, with no gravity. Assume the inner
(liquid) sphere has radius r_1, say, and the outer annulus has a radius r_2,
so that the total radius of the sphere is r_1 + r_2. As we allow r_1 to
increase, the mass increases proportionally to r_1^3 and the surface area
proportionally to r_1^2. But in order to keep the structural integrity (ie
so the bubble doesn>t just break apart), the membrane thickness r_2 needs to
increase proportionally to r_1^a, where a is some power. My problem is, I>m
not sure what a should be. I>m guessing 2 or 3 but I>m useless at mechanics.
Anyway, if anyone could help, it would be very much appreciated.
Tronc |
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David Wilkinson
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| Posted: Tue Jan 20, 2004 2:16 am Post subject: Re: Spherical membrane |
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The geometry is not clear. What is the outer annulus?
What is the loading on the membrane? There is no gravity so is it pressure
in the liquid? Does the pressure stay the same when r-1 changes?
"Tronc" <tronc_boy@hotmail.com> wrote in message
news:btjnqm$d17$1@news.ox.ac.uk...
[quote]Dear all
I>ve got a mechanics-type problem at the moment - don>t worry, it really
isn>t for homework.
Consider a liquid-filled sphere, surrounded by some sort of solid membrane
in space holding the liquid together, with no gravity. Assume the inner
(liquid) sphere has radius r_1, say, and the outer annulus has a radius
r_2,
so that the total radius of the sphere is r_1 + r_2. As we allow r_1 to
increase, the mass increases proportionally to r_1^3 and the surface area
proportionally to r_1^2. But in order to keep the structural integrity (ie
so the bubble doesn>t just break apart), the membrane thickness r_2 needs
to
increase proportionally to r_1^a, where a is some power. My problem is,
I>m
not sure what a should be. I>m guessing 2 or 3 but I>m useless at
mechanics.
Anyway, if anyone could help, it would be very much appreciated.
Tronc
[/quote] |
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Phil Smith
Guest
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| Posted: Wed Jan 21, 2004 5:08 pm Post subject: Re: Spherical membrane |
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On Jan 8 Tronc wrote in sci.mech.fluids:
[quote]Dear all
I>ve got a mechanics-type problem at the moment - don>t worry, it really
isn>t for homework.
Consider a liquid-filled sphere, surrounded by some sort of solid membrane
in space holding the liquid together, with no gravity. Assume the inner
(liquid) sphere has radius r_1, say, and the outer annulus has a radius r_2,
so that the total radius of the sphere is r_1 + r_2. As we allow r_1 to
increase, the mass increases proportionally to r_1^3 and the surface area
proportionally to r_1^2. But in order to keep the structural integrity (ie
so the bubble doesn>t just break apart), the membrane thickness r_2 needs to
increase proportionally to r_1^a, where a is some power. My problem is, I>m
not sure what a should be. I>m guessing 2 or 3 but I>m useless at mechanics.
[/quote]
As far as I can tell, you have a solid membrane of thickness r_2,
which you have filled with fluid so that in equilibrium the internal
radius of the membrane is r_1, and you are allowing the initial mass to
vary[1]. You wish to determine r_2 such that the elastic stresses in the
membrane caused by the uniform internal pressure loading are less than
the breaking stresses of the membrane.
The simplest answer is r_2 = r_1 * f(E, v) where E and v are respectively
the Young>s and Poisson>s moduli of the (classical Hookean) solid membrane
and f is a positive-valued function, which must be determined by solving a
solid mechanics problem for the membrane subject to uniform internal
pressure loading given r_1, and finding the *initial* r_2 such that the
maximum stress in the body is less than the breaking stress; this will
give you a lower bound on r_2.
[1]: You seem to be assuming implicitly that the fluid is incompressible,
which is a fair enough assumption provided that you state it.
--
P.A.C. Smith
replying by email: s/NOSPAM//
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