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We study the necking of a filament of complex fluid or soft solid subject touniaxial tensile stretching, under conditions of constant imposed stress andforce, by means of linear stability analysis and nonlinear simulations. Wedemonstrate necking to be a flow instability that is an unavoidable consequenceof the constitutive behaviour of essentially any viscoelastic material. Wederive criteria for the onset of necking that can be reported in terms ofcharacteristic signatures in the shapes of the experimentally measured materialfunctions, and that should therefore apply universally to all materials. Toconfirm their generality, we show them to hold numerically in six constitutivemodels. Under conditions of constant stress, we find two distinct dynamicalregimes as a function of time. In the first regime the strain rate quicklyattains a value prescribed by the fluid's underlying homogeneous constitutivecurve, at the given stress. In this first regime, no appreciable neckingarises. A second regime then ensues in which the homogeneous flow destabilisesto form a neck. This necking instability can occur via two distinct modes. Thefirst mode is relatively gentle and arises in any regime where the slope of theconstitutive curve is positive. It has a rate of necking per accumulated strainunit set by the inverse of the slope of the constitutive curve. The second modesets in when a carefully defined `elastic derivative' of the tensile forcefirst slopes down as a function of the time. We discuss the way in which thesemodes of instability manifest themselves in entangled polymeric fluids,demonstrating four distinct regimes of necking behaviour as a function ofstress. Under conditions of constant imposed force, typically the flow sweepsup the underlying constitutive curve of the fluid in question, with instabilityto necking in any regime where that curve is positively sloping.

Criteria for extensional necking instability in complex fluids and soft solids. Part II imposed tensile stress and force protocols