Consider a deformation of rubber balloon. Initially it is cylindrically shaped with the radius a and the
Question:
Consider a deformation of rubber balloon. Initially it is cylindrically shaped with the radius a and the length 2b (blue) with a very small initial thickness T (i.e., T << a < b). It is then blown up to shape as a sphere with the radius 2b (red). Its axisymmetrical (independent of angle) deformation in the polar/cylindrical coordinates is, Z, z
r=2b2−Z2R,θ=Θ,z 2Z= SideView
a B R,r
TopView
Ignore the deformation at top (dashed line) and bottom. The deformation gradient tensor in the polar coordinates is,
b a
A
R,r
Θ,θ a
∂r 1 ∂r ∂r
2b 2b
R is measured from the Z axis (not from the origin)
and points outward from the Z axis.
First find the deformed coordinates (r = ?, z = ?) of points A (R = a, Z = 0) and B (R = a, Z = b), (ignore Θ, θ). Then compute Fcyl and its Jacobian J. Express the area change ratio (ds/dS) along
∂R R∂Θ ∂Z
∂θ r∂θ ∂θ F=r r cyl ∂R R∂Θ ∂Z
∂z 1 ∂z ∂z
∂R R∂Θ ∂Z
Expert Answer:
Finding Deformed Coordinates r z Point A Given R a and Z 0 plug these values into the equation for r ... View the full answer
