Necking after extensional filament stretching of complex fluids and soft solids

Abstract

We perform linear stability analysis and nonlinear slender filament simulations of extensional necking in complex fluids and soft solids, during the stress relaxation process following an interrupted strain ramp. We start by deriving analytical criteria for necking within a highly simplified and generalised scalar constitutive model. Within this, we find two different possible modes of necking: one associated with an upward curvature in the stress relaxation function on a log-linear plot, and another related to a carefully defined ‘elastic’ derivative of the tensile force with respect to an imagined sudden strain increment. We showed these two criteria to agree fully with simulations of the Oldroyd B and Giesekus models of polymeric solutions, and with the Rolie-Poly model of more concentrated polymeric solutions and melts, without polymer chain stretch. With chain stretch included, we find a slightly more complicated analytical criterion for necking during the stress relaxation, although with key ingredients that closely mirror counterpart ingredients of the simpler criteria obtained within the scalar model. We show this criterion to agree fully with slender filament simulations of the Rolie-Poly model with chain stretch, and with the scenario discussed by the Copenhagen group in Refs.[A Lyhne, H. K Rasmussen, and O Hassager. Simulation of elastic rupture in extension of entangled monodisperse polymer melts. Physical Review Letters, 102:1–4, 2009., H. K Rasmussen and K Yu. Spontaneous breakup of extended monodisperse polymer melts. Physical Review Letters, 107(12):1–4, 2011]. In particular, we see delayed necking after strain ramps with an accumulated strain exceeding View the MathML source for ramp rates exceeding the inverse chain stretch relaxation timescale. We discuss finally an analogy between this delayed necking following an interrupted extensional strain ramp and delayed shear banding following an interrupted shear strain ramp [R.L. Moorcroft and S.M. Fielding. Shear banding in time-dependent ows of polymers and wormlike micelles. Journal of Rheology, 58(1):103, 2013]. This work provides the counterpart, for interrupted extensional strain ramps, to earlier papers giving criteria for necking in the protocols of constant imposed Hencky strain rate [D.M. Hoyle and S.M. Fielding. Criteria for extensional necking instability in complex uids and soft solids. Part I: Imposed Hencky strain rate protocol. Journal of Rhe- ology, 60(6):1347–1375, 2016.] and of constant imposed tensile stress or constant imposed tensile force [D.M. Hoyle and S.M. Fielding. Criteria for extensional necking instability in complex uids and soft solids. Part II: Imposed tensile stress and force protocols. Journal of Rheology, 60(6):1377–1397, 2016.].