Limit points in the free inflation of a magnetoelastic toroidal membrane

Abstract

One common phenomenon native to inflation of membranes is the elastic limit-point instability-a bifurcation point at which the membrane begins to deform enormously at the slightest increase of pressure. In the case of magnetoelastic materials, there is another possible phenomenon which we call magnetic limitpoint instability, a state referring to the non-existence of an equilibrium state -either stable or unstable. In this work, we are concerned with such instabilities in an incompressible isotropic magnetoelastic toroidal membrane with an initial circular cross-section. A non-uniform magnetic field is generated using a circular current carrying loop placed inside the membrane in addition to inflation by a uniform hydrostatic pressure. An energy formulation based on magnetization is used to model the magneto-mechanical coupling along with a Mooney-Rivlin constitutive model for the elastic strain energy density. Computations show that the magnetic field strongly influences the location of elastic limit points and in some cases can cause them to vanish. Multiple equilibrium states are obtained as solutions of the governing equations and a criterion based on second variation is employed to determine their stability. Existence and dependence of magnetic limit point on the magnetic field is demonstrated. While the quantitative results obtained here are specific to the toroidal geometry, the deformation behaviour can be generalised to any magnetoelastic membrane.

Instabilities in the free inflation of a nonlinear hyperelastic toroidal membrane

“…For example, behaviour of natural rubbers can usually be explained by the three-term Ogden model (Ogden, 1972) while that of certain soft biological tissues can be simulated by the neo-Hookean model (Horný et al, 2007). Our results for the Mooney-Rivlin model match those presented by Tamadapu and DasGupta (2014) for M = 0.3, γ = 0.2 and 0.5 cases, and those presented by Reddy and Saxena (2017) for M = 0.1, γ = 0.2 and 0.5; thus providing a validation of the formulation and the computations.…”

“…The torus is symmetric about the Y 1 − Y 2 plane, hence we constrain the solution space and study only the deformations of the toroidal membrane with respect to the upper half of the Y 1 − Y 2 plane. The calculations and notation below closely follow those in (Reddy and Saxena, 2017).…”

“…The governing equations (8) and (9) for fundamental solution are subjected to boundary conditions defined by equation (10). They are numerically solved for three elastic constitutive models (Ogden, Mooney-Rivlin, and neo-Hookean) by following a method used for similar problems by Tamadapu and DasGupta (2014) and Reddy and Saxena (2017). The two point boundary value problem is converted into an initial value problem with two unknown parameters ( (0) , η θ (0)) at a particular non-dimensional pressure P .…”

Coupled electro-elastic deformation and instabilities of a toroidal membrane

“…7,10,29,52,70]. Computation of accurate pressure-volume characteristics in this case require a path-following scheme due to the non-uniqueness of solution [59].…”

Modeling and analysis of a magnetoelastic annular membrane placed in an azimuthal magnetic field

“…An O ( h 3 ) formulation was subsequently developed for a nonlinear magnetoelastic plate by Santapuri and Steigmann [45]. Reddy and Saxena [46, 47] studied deformation of toroidal and cylindrical magnetoelastic membranes in the presence of external magnetic field. This approach was extended by Saxena et al [48] to study the limit point and wrinkling instabilities occurring in a circular MRE membrane.…”