# Necking after extensional filament stretching of complex fluids and soft solids

## TL;DR

This paper develops analytical criteria for necking in complex fluids and soft solids during stress relaxation after extensional strain ramps, validated by simulations across various models, revealing delayed necking phenomena.

## Contribution

It introduces new analytical criteria for necking during stress relaxation in complex fluids, validated by simulations, and connects delayed necking to similar phenomena in shear banding.

## Key findings

- Two modes of necking identified: stress relaxation curvature and elastic force derivative.
- Criteria agree with simulations of Oldroyd B, Giesekus, and RoliePoly models.
- Delayed necking occurs after high strain ramps exceeding ~0.7, at certain ramp rates.

## 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. Here, 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 RoliePoly 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, 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 RoliePoly model with chain stretch, and with the scenario discussed by the Copenhagen group in Refs. [1, 2]. In particular, we see delayed necking after strain ramps with an accumulated strain exceeding ~ 0.7, 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 [3]. 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 [4] and of constant imposed tensile stress or constant imposed tensile force [5].