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I am trying the understand the failure modes of braided tubes containing high pressure gas. Here is an example of such a braid with a variable radius:

A braided tube

The tubes have an internal liner for sealing purposes, but nearly all stiffness (pressure resistance) is supplied by the high strength fibers. Therefore, to first order we can imagine that all stiffness is supplied by the fibers, which are allowed to reshape themselves at will to define the shape the tube. Specifically, if $t \in [0,L]$ is the material arc length of a "standard fiber", the geometry is described by

  • A radius $r(t)$
  • An axial distance along the tube $z(t)$
  • An angle $\phi(t)$ around the tube

so that the fiber position is $(r \cos \phi, r \sin \phi, z)$. If we pressurize the tube to a pressure $p$, freeze $r,z,a$ at both ends, give the fibers a collective cross sectional stiffness $k$ with a quadratic stress/strain curve, and assume the fibers do not interact, we arrive at the energy $$U = \int_{t_0}^{t_1} \left[\frac{1}{2} k \epsilon^2 - p \pi r^2 z_t \right] dt$$ where the strain $\epsilon$ is given by $$(1+\epsilon)^2 = r_t^2 + z_t^2 + r^2 \phi_t^2$$ If we start with a perfectly cylindrical tube, set the pressure to a small but positive value, and minimize $U$, the fibers deform to maximize volume while keeping arc length roughly constant. A numerical result is shown below:

Energy minimizing geometry for small positive pressure

For most of the tube, the fibers are straight (not rotating around the tube). Since the angles at the ends are clamped, angle change must occur somewhere; this is accomplished with jump discontinuities in $\phi(t)$ at points where $r(t) = 0$. However, this behavior is not what is observed in practice: initially straight tubes seem to hold their shape quite well even for high pressures (until the liner material fails).

Thus, my conclusion is that assuming no fiber interaction does not work. This is not a particularly surprising conclusion, since the fibers have a nonzero cross sectional shape, but I am not sure exactly what interaction phenomenon dominates or how best to represent it. The braid lock that prevents ropes from arbitrarily elongating is similar, but may or may not be slightly different in character.

Questions: Does anyone have suggestions for energy terms or constraints to capture fiber interaction, without leaving easy 1D nonlinear elliptic land? Are there existing models of braid interactions I could pull from? My nonlinear elliptic solver only requires a formula for the energy, so I can quickly check any suggestions for terms.

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