Criteria for extensional necking instability in complex fluids and soft   solids. Part II imposed tensile stress and force protocols
  We study the necking of a filament of complex fluid or soft solid subject to uniaxial tensile stretching, under conditions of constant imposed stress and force, by means of linear stability analysis and nonlinear simulations. We demonstrate necking to be a flow instability that is an unavoidable consequence of the constitutive behaviour of essentially any viscoelastic material. We derive criteria for the onset of necking that can be reported in terms of characteristic signatures in the shapes of the experimentally measured material functions, and that should therefore apply universally to all materials. To confirm their generality, we show them to hold numerically in six constitutive models. Under conditions of constant stress, we find two distinct dynamical regimes as a function of time. In the first regime the strain rate quickly attains a value prescribed by the fluid's underlying homogeneous constitutive curve, at the given stress. In this first regime, no appreciable necking arises. A second regime then ensues in which the homogeneous flow destabilises to form a neck. This necking instability can occur via two distinct modes. The first mode is relatively gentle and arises in any regime where the slope of the constitutive curve is positive. It has a rate of necking per accumulated strain unit set by the inverse of the slope of the constitutive curve. The second mode sets in when a carefully defined `elastic derivative' of the tensile force first slopes down as a function of the time. We discuss the way in which these modes of instability manifest themselves in entangled polymeric fluids, demonstrating four distinct regimes of necking behaviour as a function of stress. Under conditions of constant imposed force, typically the flow sweeps up the underlying constitutive curve of the fluid in question, with instability to necking in any regime where that curve is positively sloping. 
David M. Hoyle and Suzanne M. Fielding
cond-mat.soft