Criteria for extensional necking instability in complex fluids and soft solids. Part I: Imposed Hencky strain rate protocol

Abstract

We study theoretically the necking dynamics of a filament of complex fluid or soft solid in uniaxial tensile stretching at constant imposed Hencky strain rate ϵ̇ ϵ̇, by means of linear stability analysis and nonlinear (slender filament) simulations. We demonstrate necking to be an intrinsic flow instability that arises as an inevitable consequence of the constitutive behavior of essentially any material (with a possible rare exception, which we outline), however carefully controlled the experimental conditions. We derive criteria for the onset of necking that are reportable simply in terms of characteristic signatures in the shapes of the experimentally measured rheological response functions, and should therefore apply universally to all materials. As evidence of their generality, we show them to hold numerically in six popular constitutive models: The Oldroyd B, Giesekus, FENE-CR, Rolie-Poly, and Pom-pom models of polymeric fluids, and a fluidity model of soft glassy materials. Two distinct modes of necking instability are predicted. The first is relatively gentle, and sets in when the tensile stress signal first curves downward as a function of the time t (or accumulated strain ϵ=ϵ̇ tϵ=ϵ̇t) since the inception of the flow. The second is more violent, and sets in when a carefully defined “elastic derivative” of the tensile force first slopes down as a function of time t (or strain ϵ). In the limit of fast flow ϵ̇ τ→∞ϵ̇τ→∞, where τ is the material's characteristic stress relaxation time, this second mode reduces to the Considère criterion for necking in solids. However, we show that the Considère criterion fails to correctly predict the onset of necking in any viscoelastic regime of finite imposed ϵ̇ τϵ̇τ, despite being widely discussed in the complex fluids literature. Finally, we elucidate in detail the way in which these modes of instability manifest themselves in entangled polymeric fluids (linear polymers, wormlike micelles and branched polymers). In particular, we demonstrate four distinct regimes of necking behavior as a function of imposed strain rate, consistent with master curves in the experimental literature.