Failure and nonfailure of fluid filaments in extension

doi:10.1016/S0377-0257(97)00115-8

Journal of Non-Newtonian Fluid Mechanics

Volume 76, Issues 1–3, April 1998, Pages 137-151

https://doi.org/10.1016/S0377-0257(97)00115-8 Get rights and content

Abstract

The phenomenon of ductile failure of Newtonian and viscoelastic fluid filaments without surface tension is studied by a 2D finite element method and by 1D non-linear analysis. The viscoelastic fluids are described by single integral constitutive equations. The main conclusions are: (1) Newtonian fluid filaments do not exhibit ductile failure without surface tension; (2) some viscoelastic fluids form stable filaments while other fluids exhibit ductile failure as a result of an elastic instability; (3) for large Deborah numbers, the Considère condition may be used to predict the Hencky strain of the elastic instability.

Introduction

The mechanisms of failure of fluid filaments in extension is a topic of both fundamental and industrial interest 1, 2. It is customary [3] to distinguish between ductile failure and fracture. We will not be concerned with the latter but merely with ductile failure, which is described by Ide and White [3] as a situation with ‘a 100% reduction in cross section within a neck caused by the high local stress levels’. We use this definition with the perhaps obvious addendum that the cross section must tend to zero in finite time for the effect to be observable.

In a real situation capillarity will be combined with the mechanism of ductile failure. However, in discussions of failure, reference is often made to theoretical analyses of ideal situations where these mechanisms are isolated. Here we wish to consider the mechanism of ductile failure in the absence of surface tension.

A number of approaches have been applied to the analysis of the stability of fluid filaments in extension. Chang and Lodge [4] considered two uniform cylindrical filaments of different initial cross-sectional areas connected by a ficticious deformable clamp capable of transmitting a force without interfering with each filament. In this study they find, that for a Newtonian fluid the diameter ratio rapidly tends to infinity. While this result may lead the reader to conclude that the Newtonian filament will fail in finite time, Chang and Lodge are careful not to draw this conclusion. Indeed, due to the use of the ficticious clamp and the resulting uniform diameter of the smaller cylinder, it is clear that an infinite Hencky strain is needed to reduce the cross sectional area of the smaller cylinder to zero. Hence, the ficticious clamp method can not be used to draw any conclusions about ductile failure according to our definition. The main message of Chang and Lodge was not related to the Newtonian fluid but to the ‘rubberlike liquid’ (a generalized Oldroyd-B fluid) for which they showed that the diameter ratio of the two cylinders tends to a constant value (different from 1) under specific conditions.

The postulate of a ficticious clamp was avoided by Ide and White 3, 5 who used the Matovich and Pearson thin filament equations [6] to study the transient development of an initial sinusoidal disturbance of the radius of a fluid filament in extension. Ide and White extrapolate the solution to the linearized disturbance equation to the time at which the disturbance becomes equal to the radius of the filament. This time is then denoted the time of ductile failure or time to break. This extrapolation led them to the conclusion that the Newtonian fluid model and the Maxwell model at Deborah numbers >1/2 will show ductile failure at some finite Hencky strain.

Pearson and Connelly [7] analyze failure in terms of the Considère criterion 8, 9 which asserts that ‘uniformity of stretching is guaranteed if the strain is less than that at which a maximum occurs in the force extension curve’. While the criterion does not apply to viscous filaments, we have found it to be useful for viscoelastic filaments in the limit of large Deborah numbers.

Another avenue to the study of failure and nonfailure was opened with the observations of filament extension [10] and the development of the transient filament extension apparatus 11, 12, 13, 14. For example Spiegelberg et al. [14] report measurements of the extension of Newtonian filaments up to a Hencky strain of 4. Numerical simulations of the filament extension apparatus were pioneered by Shipman et al. [15] who did not find any failure. Subsequently, Sizaire and Legat [16] performed simulations for Newtonian fluids up to a Hencky strain of 4.5 and Yao and McKinley [17] up to a Hencky strain of ∼5 without observations of failure.

Thus, experiments and simulations of transient filament extension seem to suggest that Newtonian fluid filaments do in fact not show ductile failure. It may be argued that perhaps the filament extension experiments and simulations have not been performed to sufficiently high Hencky strain. Indeed for high Deborah numbers the simulations of Sizaire and Legat and Yao and McKinley are limited in Hencky strain due to difficulties of mesh generation near the end plates. This difficulty was circumvented by Kolte et al. [18] who used a finite element method based on a Lagrangian kinematic specification [19]. In the next section we show typical simulations with this method for a Newtonian fluid, a strain-hardening constant viscosity fluid and a strain-hardening shear-thinning fluid none of which show signs of ductile failure up to a Hencky strain of 6. While these simulations provide further evidence to suggest that even Newtonian fluids do not show ductile failure, it is clear that finite element simulations can never be used to prove that a filament can not fail at some large Hencky strain. In Section 3we use a simple 1D model of a filament [20] to prove that the Newtonian fluid filament cannot fail in any finite time however large. In the final section we show simulations of one particular viscoelastic fluid model that does show ductile failure in finite time.

Section snippets

Rheology and initial simulations

The stress is modeled as explicitly split into a Newtonian solvent contribution τ_s and a polymer contribution τ_p. The polymer solution rheology is modeled as a memory weighted time integral over the finite strain tensor and a damping function: $=_{s} +_{p} =−η_{s} g ̇ r, z, t +∫_{−∞}^{t} M t−t′ φ I_{1}, I_{2} g_{0} r, z, t, t′ d t′.$ Here M(t−t′) is the linear viscoelastic memory function, φ(I₁, I₂) the damping function and γ_[0](r, z, t, t′) the finite strain tensor defined in terms of the Finger strain tensor, B(r, z, t, t′) as (Bird et al. [21]) $g_{0}$

Proof that the Newtonian filament does not fail in finite time

We consider a filament of a Newtonian fluid, which has uniform thickness in its reference configuration. It is assumed that stresses and deformations can be regarded as uniform across cross sections. While this assumption is not valid near the end plates, it is a very good assumption near the symmetry plane, where failure could occur. Let X be a Lagrangian coordinate, and let s(X, t) denote the stretch, i.e. the actual position of particles in space satisfies ∂x/∂X=s. We assume that only viscous

Simulations of a viscoelastic filament that does fail in finite time

Simulations were also made for the shear-thinning fluid with β-values different from zero. Model predictions of the transient Trouton ratio in ideal elongational flow as function of Hencky strain for selected β-values are shown in Fig. 3. According to this, β≠0 corresponds to having a steady elongational viscosity for t→∞. In Fig. 4 we show the mid-filament radius made dimensionless with the initial radius as function of Hencky strain for simulations of the filament with β=0.1 and all other

Discussion

We have used finite element simulations for three specific fluid models as a basis for a discussion of the phenomenon of ductile failure in the absence of surface tension. We prove that Newtonian fluid filaments will not exhibit ductile failure. Not surprisingly the same holds for our simulations of the extension of viscoelastic fluids at small Deborah numbers.

For viscoelastic fluids at large Deborah numbers we have simulated two types of behavior. Some fluid models (Boger fluids) predict a

Acknowledgements

MR was supported by ONR Grant N00014-92-J-1664 and NSF Grant DMS-9622735. MIK was supported by the Danish Polymer Centre. The finite element program was developed by Dr H. K. Rasmussen. The authors wish to thank Drs H.K. Rasmussen and P. Szabo for numerous useful discussions and advice.

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¹: Dedicated to the memory of Professor Gianni Astarita.

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Failure and nonfailure of fluid filaments in extension1

Abstract

Introduction

Section snippets

Rheology and initial simulations

Proof that the Newtonian filament does not fail in finite time

Simulations of a viscoelastic filament that does fail in finite time

Discussion

Acknowledgements

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

Some conditions for rupture of polymer liquids in extension

J. Rheol.

Elongational Flows

The spinning of polymer fluid filaments

J. Appl. Polym. Sci.

Rheol. Acta

Ind. Eng. Chem. Fundam.

Failure and nonfailure of fluid filaments in extension¹