Radial contraction of a stretched hyperelastic cylinder

[SerArthurRamShackle]

Hi, I'm trying to work out how much strain potential I can produce in a cylinder but I want to know how much is stored for any specific part of the cylinder. The problem then is this:

I have a cylinder of original dimensions r and h, where h is the original height and r is the radius. It is fixed on the surface f the Earth and its z axis is along the axis of the cylinder hanging down into a hole in the earth. Upon stretching along its axis only I have an h' = h + a for the stretched cylinder which will produce an r(z,a) deformation of the radius, that is, a non-constant one along its length but dependent upon the z coordinate of the cross-section and the displacement a of the free cylinder end.

I know this is something to do with the Poisson Ratio, but that suggests radial contraction is uniform which it simply could not be. I know that I can just find the total strain energy of the cylinder if I have r(z,a) by a solid of revolution method but I'm at a loss as to what type of function could describe the varying radius. Any help would be much appreciated, I'm trying to teach myself nonlinear elasticity and it's quite hard!

 

[Baluncore]

Details of the end connections are critical. Can we assume that the circular ends are adhered to rigid discs also of radius r ?
Can we assume that the volume of the elastomer remains constant ?

 

[SerArthurRamShackle]

The fixed end will have to have a radius of r but I don't place that restriction on the free stretched end, though it could be to simplify things.
I've tried volume invariance but for finding the equations of motion it makes no sense. Using the principal stretches of the deformation gradient tensor I can obtain a strain energy density but the motion of the band doesn't seem to really work. As a very crude approximation:

L = mv²/2 + Mv²/2 - πr'²h'W
2L = mv² + (πr²hρ)v² - 2πr'²h'W

where:
m is payload mass
M is cylinder mass
v is the known velocity I require of the mass
r is the undeformed radius
r' is deformed radius
h is undeformed length
h' is deformed length
W is a strain energy density function
ρ is the density of the cylinder material before deformation

Now if λ is the principal stretch along z and we stretch some amount x along z:

h' = h + x = λh so if we preserve volume:

πr²h = πr'²h'
πr²h = πr'²λh
r² = r'²λ
r'²/r² = 1/λ

hence 1/√λ is the radial principal stretch and so:

h' = λh = h +x = λx/(λ - 1) and r' = r/√λ = r√h/√(h + x)

Now I can shove my variable x into my Lagrangian and find the equation of motion using Euler-Lagrange equations:

2L = mv² + (πr²hρ)v² - 2πr'²h'W
2L = mv² + (πr²hρ)v² - 2πr'²h'W
2L = mv² + (πr²hρ)v² - 2π(r²h/(h + x))λx/(λ - 1)W
2L = mv² + (πr²hρ)v² - 2π(r²h/(h + x))λx/(λ - 1)W
2L = mv² + (πr²hρ)v² - 2πr²hλxW/((h + x)(λ - 1))

Now, just ignoring the factor of two in front of the Lagrangian for the moment I have to calculate d/dt d/dv (L) and d/dx (L), or I could take λ as a function of x and do it with respect to λ. The thing is, I need either λ or x only in the Lagrangian to do this but by substitution for one of them in terms of the other, I'll just end up back at πr²hW. Now I could have that, but strain energy density functions are very situation dependent and although they depend on λ, it won't actually get me anywhere. Intuitively I cannot see the equation of motion being independent of how the cylinder was strained.

 

[SerArthurRamShackle]

I can write, for the moment that:

2L = mv² + (πr²hρ)v² - 2πr'²h'W
2L = mv² + (πr²hρ)v² - 2πf(x)λxW/(λ - 1)

Essentially I have replaced an explicit functional dependence of r on x with an implicit one and have continued as normal so that I may perform the relevant differentiation when I do know it and can then plug it in.

but λ = (h+ x)/h so:

2L = mv² + (πr²hρ)v² - 2πf(x)((h+ x)/h)xW/((h+ x)/h) - 1)
2L = mv² + (πr²hρ)v² - 2πf(x)(h+ x)xW/(h + x - h)
2L = mv² + (πr²hρ)v² - 2πf(x)(h+ x)xW/x
2L = mv² + (πr²hρ)v² - 2πf(x)(h+ x)W

hence d/dx (2L) = d/dx(- 2πf(x)(h+ x)W) and so we just use product rule:

d/dx (2L) = d/dx(-2πf(x)hW - 2πf(x)xW)
d/dx (2L) = -2πf'(x)hW - 2πf'(x)xW - 2πf(x)W

d/dv (2L) = d/dv(mv² + (πr²hρ)v²)
d/dv (2L) = 2mv + 2πr²hρv

d/dt d/dv (2L) = d/dt(2mv + 2πr²hρv) = 2ma + 2πr²hρa

finally:d/dx (L) - d/dt d/dv (L) = 0
-πf'(x)hW - πf'(x)xW - πf(x)W - ma - πr²hρa = 0
ma + πr²hρa = -πf'(x)hW - πf'(x)xW - πf(x)W
a(m + πr²hρ) = -πf'(x)hW - πf'(x)xW - πf(x)W
a = (-πf'(x)hW - πf'(x)xW - πf(x)W)/(m + πr²hρ)

This also worries me because I'm not sure it should be negative.

 

[Nidum]

Not a complete answer but some useful information on here : http://www.qucosa.de/fileadmin/data/qucosa/documents/5995/data/Analysis_of_Hyperelastic_Materials_with_MECHANICA.pdf

You really need to study this subject in more depth before proceeding further .

 

[SerArthurRamShackle]

For the most part I know the material covered by the pdf in that link, although some of it less well than other parts because it's not particularly relevant to me. The stretch ratios I have derived above agree with the pdf but that is in the case of volume preservation and uniform radial contraction. There seems to be a model in the pdf of a stretched bar that contracts perpendicular to the direction of stretch though it offers no analytical explanation of what has happened.