FREE INFLATION OF AN ANISOTROPIC HYPERELASTIC CIRCULAR MEMBRANE

Abstract

Inflation of a thin elastic membrane is a well-studied classical problem in nonlinear continuum mechanics; however, the impact of material anisotropy on the inflation response of hyperelastic materials has yet to be thoroughly researched. Therefore, this research examines the axisymmetric axial deformation behaviour of a circular membrane composed of an incompressible anisotropic Mooney-Rivlin material being blown up against a uniform transverse pressure load, without any initial radial pre strain. The derivation of governing equations comes from the principle of stationary potential energy, with the strain energy density function being expressed using the two principal stretches from the plane of the membrane and an anisotropic invariant 𝐼4 = 𝜆1 2 for the thickness of the membrane. The invariant thus gives rise to one stretch in the thickness direction, since there can be no volume change under the assumption of incompressibility. Each of these principal stretches is identified by introducing the appropriate field variable so that the governing equations are reduced from two second-order ordinary differential equations into a system of three first-order ordinary differential equations. The converting or changing of the governing equations into a different format allows for easier numerical integration using standard initial value problem solvers. The two-point boundary value problem resulting from this research has been solved using the combination of the shooting method and the optimization procedure using fminsearch. The primary advancement in the numerical solution is the established verification of the scaling invariance property inherent to the equilibrium equations applies to the anisotropic equations, thus allowing for the elimination of the arc-length continuation methods in all cases where the pressure-stretch relationship has limit point instability. Presented here are numerical results for two different pairs of Mooney-Rivlin material parameters, 𝛼 = 𝐶2 𝐶1 =0.01,0.03,0.1 and four different levels of anisotropy, 𝜁 = 0,0.01,0.03,0.05, with no pre-stretch (𝛽 = 1). The results show that anisotropy causes a stiffening effect in the inflation response, through the pressure-deflection curves. The pressure-deflection curves for 𝛼 = 0.01 (close to Neo-Hookean behaviour) show that the limit point is always there regardless of connectivity level 𝜁. Critical pressures remain about the same for each value of 𝜁. When working with 𝛼 = 0.1 (which has a large component for 𝐶2), the response is monotonic and stable; but as the value of 𝜁 increases, maximum deflections will decrease by as much as 56 at a pressure of 10. The stress resultants show that inflation creates anisotropic behaviour in the meridional and circumferential components of the material properties (highest stresses at the clamped edges). The peak stress at the clamp reduces and circumferential stress increases at the interior of the structure when the anisotropy parameter is increased. The 3D surface reconstruction and membrane profiles have shown that hysteresis in the anisotropic membrane is less than that of the isotropic membrane at similar elevations (pressure). This study will benefit the designer of any inflatable structure requiring uniform strains and controlled directional stiffness.