Stationary whirling of slender and homogeneous (continuous) elastic shafts rotating around their axis, with pin-pin boundary condition at the ends, is revisited by considering the complete deformations in the cross section of the shaft. The stability against a synchronous sinusoidal disturbance of any wave length is investigated and the analytic expression of the buckling amplitude is derived in the weakly non-linear regime by considering both geometric and material (hyper-elastic) non-linearities. The bifurcation is super-critical in the long wave length domain for any elastic constitutive law, and sub-critical in the short wave length limit for a limited range of non-linear material parameters.

Dicing soft solids with a sharp knife is quicker and smoother if the blade is sliding rapidly parallel to its edge in addition to the normal squeezing motion. We explain this common observation with a consistent theory suited for soft gels and departing from the standard theories of elastic fracture mechanics relied on for a century. The gel is assumed to fail locally when submitted to stresses exceeding a threshold σ 1. The changes in its structure generate a liquid layer coating the blade and transmitting the stress through viscous forces. The driving parameters are the ratio U=W of the normal to the tangential velocity of the blade, and the characteristic length ηW=σ 1 , with η the viscosity of the liquid layer. The existence of a maximal value of U=W for a steady regime explains the crucial role of the tangential velocity for slicing biological and other soft materials.

This paper is devoted to an unresolved model for the simulation of air invasion in immersed granular flows without interface reconstruction between the liquid and the gas. Experiments of air invading a granular bed immersed in ethanol were achieved in a Hele-Shaw cell to observe the gas invasion paths and to calibrate the numerical multiscale model. The grains movements are computed at a fine scale using the non-smooth contact dynamics method, a time-stepping method considering impenetrable grains. The fluid flow is modelled by equations averaged using the volume fraction of fluid and computed at a coarse scale with the finite element method. A phase indicator function is used to dissociate the gas and the liquid constituting the fluid and to compute the density and viscosity of the fluid at each position. It is moved using a convection equation at each time step. The fluid, solid and phase indicator function computations are validated on simple cases before being used to reproduce experiments of air invasion in immersed granular flows. The experiments are supported by simulations in two dimensions to refine the study and the understanding of the invasion dynamics at short times.