Topic G — A Pneumatic Soft Actuator

nearly incompressible hyperelasticity · inflation instability (limit point) · a bending actuator · tier: advanced

Motivation

Soft robots and stretchable devices move by inflating elastomer chambers. Two things make them hard to model: the rubber is nearly incompressible (standard elements lock), and inflation is unstable — a balloon reaches a maximum pressure and then bulges suddenly, which a load-controlled Newton solver cannot follow. You will build an actuator model that handles both, verified on the inflation of a cylinder and a sphere, and then compute the bending of a fibre-reinforced or two-layer actuator.

Related fields: soft materials and soft robotics: stretchable electronics, wearable sensors, adaptive soft actuators. Course weeks: 1 (hyperelasticity, locking, load steps, SNES), 7 (load control vs displacement control, continuation), 8 (thin structures and locking).

Model

Incompressible or nearly incompressible neo-Hookean (or Mooney–Rivlin) solid, \Psi=\frac\mu2(\bar I_1-3)+\frac\kappa2(J-1)^2\quad\text{or a mixed }(\mathbf u,p)\text{ formulation with }J=1\text{ enforced weakly}, inner pressure P as a follower load on the chamber wall (its work depends on the deformed area: -P\int J\mathbf F^{-T}\mathbf N\cdot\mathbf v\,dA).

  1. Thick-walled cylinder (plane strain or axisymmetric, Week 2) and sphere (axisymmetric): pressure–stretch curves, the limit point.
  2. Bending actuator: a rectangular elastomer block with a row of chambers (2D plane strain first, then 3D), a stiffer strain-limiting layer (paper/fabric: a stiff thin layer, or a bonded inextensible film) — tip angle against pressure.

Design outputs: the limit pressure of the chamber; the tip angle and blocked force of the actuator against pressure.

Suggested parameters

\mu = 0.1–1 MPa (Ecoflex/PDMS), \kappa/\mu = 10³ (or exact incompressibility), wall thickness/radius 0.05–0.5, pressures up to the limit point, strain-limiting layer 100–1000 times stiffer.

Required tasks

  1. Mixed (\mathbf u,p) or a locking-free alternative, with a test showing that P1 displacements lock at \kappa/\mu=10^3.
  2. Inflation of the thin and thick cylinder/sphere up to and past the limit point (volume control or arclength).
  3. Bending actuator in 2D: tip angle vs pressure; mesh independence.
  4. (3D) one chamber of the actuator, comparison with 2D.

Verification specific to this topic

  • Thin spherical membrane (incompressible neo-Hookean, wall H\ll R): P=\frac{2\mu H}{R}\big(\lambda^{-1}-\lambda^{-7}\big), maximum at \lambda=7^{1/6}\approx1.38. Thin cylinder (plane strain, axial stretch fixed): derive the analogous formula and check that it has no maximum — the instability needs the biaxial membrane of the sphere or a free axial stretch.
  • Thick-walled tube/sphere: the exact pressure–stretch relation is a one-dimensional integral over the wall (Ogden, Non-linear Elastic Deformations): evaluate it with NumPy quadrature and compare the whole curve.
  • Incompressibility: report \max|J-1| and its convergence; the mixed solution must not show pressure checkerboards (inf-sup stable pair, e.g. P2–P1).
  • MMS for the hyperelastic problem (a smooth isochoric displacement field, body force from UFL).

Stretch goals

  1. Snap-through of a hemispherical cap (a soft “popper”) with arclength continuation.
  2. Liquid-metal conductor channel: resistance change of a stretched channel (strain → geometry → conductance).
  3. Fibre-reinforced (anisotropic) chambers: twisting/extension instead of bending.

Milestones

M1 locking test and mixed formulation · M2 cylinder/sphere against the exact curves · M3 limit point with volume control or arclength · M4 2D bending actuator · M5 3D chamber.

Pitfalls

  • Load control stalls at the limit point (SNES diverges just before P_{\max}) — that is physics, not a solver bug: switch to volume control (prescribe the enclosed volume, solve for P as an unknown) or pseudo-arclength.
  • The follower pressure makes the Jacobian unsymmetric: use LU (MUMPS) or GMRES, not CG.
  • Inverted elements in the first load step (Week 1, reason −4): smaller steps near zero load where the linear predictor is large.

References

  • R. W. Ogden, Non-linear Elastic Deformations, Dover 1997 — inflation of tubes and spheres.
  • G. A. Holzapfel, Nonlinear Solid Mechanics, Wiley 2000 — mixed formulations for incompressibility.
  • F. Ilievski, A. D. Mazzeo, R. F. Shepherd, X. Chen, G. M. Whitesides, “Soft robotics for chemists”, Angew. Chem. Int. Ed. 50 (2011) 1890–1895.