  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.
