Axisymmetric necking in a stretched circular membrane
Yibin Fu (Keele)
Necking is traditionally associated with plastic deformations and is commonly observed in the tension test of ductile materials. However, an increasing body of literature has demonstrated that elastic necking is also possible in soft materials due to multi-fields such as surface tension, residual stresses, and electric field. It is likely to become a more common phenomenon as more and more smart materials are synthesized (e.g. cellular materials and double-network hydrogels). We report our recent research effort on characterising axisymmetric necking of a circular membrane under an all-round tension (or equivalently a square membrane under equibiaxial tension). We derive the bifurcation condition, conduct a weakly nonlinear analysis to obtain an amplitude equation in the form of a fourth-order ODE with quadratic nonlinearity and variable coefficients, and develop a 1D reduced model to describe the fully nonlinear regime where the necking front propagates from the center towards the edge. We shall explain the methodology in the purely mechanical case, but our initial motivation has come from applications associated with dielectric elastomer actuators under the combined action of mechanical stretching and an electric field.
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