^{1,a)}and Víctor Hugo Rolón-Garrido

^{1}

### Abstract

Characterizing elongational behavior of polymer melts in constant elongation rate or constant tensile stress experiments is hampered by the onset of “necking” instabilities according to the Considère criterion: For an elastic filament, homogeneous uniaxial extension is not guaranteed for strains larger than the strain at which a maximum occurs in the force versus extension curve. Although simulations and experiments seem to indicate that in viscoelastic fluids viscosity effects delay the onset of necking to higher strains, integral measurements of elongational viscosities in Meissner- or Münstedt-type elongational rheometers are often affected by inhomogeneous deformations. A simple means of avoiding the consequences of the Considère criterion consists in operating at constant tensile force, but little attention has been paid so far to constant force elongational rheometry since the pioneering work of Raible *et al.* [J. Non-Newtonian Fluid Mech. **11**, 239 (1982)]. Constant force elongation is also of great industrial relevance, since it is the correct analog of steady fiber spinning, as, e.g., exemplified by the so-called Rheotens test. We present experimental data for two long-chain branched polyethylene melts obtained in constant elongation rate and constant tensile force modes by use of a Sentmanat extensional rheometer in combination with an Anton Paar MCR301 rotational rheometer. The accessible experimental window and experimental limitations are discussed. Constant force elongation can be subdivided into three distinct deformation regimes: At small deformations (regime 1), constant force elongation is equivalent to creep at constant tensile stress. This is followed by regime 2, which is characterized by a steep increase in tensile stress at roughly constant strain rate, while regime 3 corresponds to elongation of a viscous power-law fluid. All three regimes can be modelled quantitatively by a single integral constitutive equation, and a constitutive analysis reveals substantial underestimation of the effective strain-hardening effect observed in constant strain-rate elongation in comparison to constant tensile force extension, which may be indicative of the effect of necking under constant elongation-rate conditions.

Financial support from the German Science Foundation (DFG) is gratefully acknowledged.

I. INTRODUCTION

II. EXPERIMENTS

III. THEORY

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

### Key Topics

- Viscosity
- 26.0
- Viscosity measurements
- 9.0
- Creep
- 7.0
- Experiment design
- 7.0
- Polymer melts
- 7.0

## Figures

Linear-viscoelastic data of LDPE 1840D (circles) and LDPE 3020D (triangles) at 170 °C. Storage modulus G′ (full symbols) and loss modulus G″ (open symbols). Continuous lines indicate fit by use of discrete relaxation spectrum of Table I.

Linear-viscoelastic data of LDPE 1840D (circles) and LDPE 3020D (triangles) at 170 °C. Storage modulus G′ (full symbols) and loss modulus G″ (open symbols). Continuous lines indicate fit by use of discrete relaxation spectrum of Table I.

Hencky strain *ɛ* as function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Hencky strain *ɛ* as function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Logarithm of Hencky strain *ɛ* as function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Logarithm of Hencky strain *ɛ* as function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

True stress *σ* as a function of time at constant tensile force deformation. Broken lines represent experimental data for LDPE 3020D. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right.

True stress *σ* as a function of time at constant tensile force deformation. Broken lines represent experimental data for LDPE 3020D. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right.

True stress as function of strain rate for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is increasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

True stress as function of strain rate for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is increasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity *η* as a function of strain rate for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is increasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity *η* as a function of strain rate for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is increasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Strain rate as a function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is increasing from bottom to top. (a) LDPE 3020D; (b) LDPE 1840D.

Strain rate as a function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is increasing from bottom to top. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity as a function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity as a function of time for constant force elongation with *F* _{0} = *σ* _{0} *A* _{0}. Broken lines represent experimental data. Continuous lines indicate predictions of Eq. (7). Engineering stress *σ* _{0} is decreasing from left to right. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity data (symbols) as a function of time for constant strain-rate elongation. Continuous and broken lines indicate predictions of Eq. (7) with parameters (Table II) for constant tensile force and constant strain-rate deformation modes, respectively. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity data (symbols) as a function of time for constant strain-rate elongation. Continuous and broken lines indicate predictions of Eq. (7) with parameters (Table II) for constant tensile force and constant strain-rate deformation modes, respectively. (a) LDPE 3020D; (b) LDPE 1840D.

Damping function calculated from elongational viscosity data (symbols) according to Eq. (6). Continuous and broken lines indicate predictions of Eq. (4) with parameters (Table II) for constant tensile force and constant strain-rate deformation modes, respectively. (a) LDPE 3020D; (b) LDPE 1840D.

Damping function calculated from elongational viscosity data (symbols) according to Eq. (6). Continuous and broken lines indicate predictions of Eq. (4) with parameters (Table II) for constant tensile force and constant strain-rate deformation modes, respectively. (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity data (symbols) as a function of time for constant strain-rate elongation. Continuous lines indicate predictions of Eq. (7) with parameters (Table II) for constant tensile force mode and engineering stresses *σ* _{0} from 5000 to 100 000 Pa (from right to left). (a) LDPE 3020D; (b) LDPE 1840D.

Elongational viscosity data (symbols) as a function of time for constant strain-rate elongation. Continuous lines indicate predictions of Eq. (7) with parameters (Table II) for constant tensile force mode and engineering stresses *σ* _{0} from 5000 to 100 000 Pa (from right to left). (a) LDPE 3020D; (b) LDPE 1840D.

## Tables

Molecular characterization and relaxation spectrum of LDPE 1840D and LDPE 3020D obtained by dynamic shear measurement at 170 °C.

Molecular characterization and relaxation spectrum of LDPE 1840D and LDPE 3020D obtained by dynamic shear measurement at 170 °C.

Nonlinear parameters fitted to constant tensile force and constant strain-rate data.

Nonlinear parameters fitted to constant tensile force and constant strain-rate data.

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