J. Sweeney

A constitutive law for large deformations of polymers at high temperatures

Abstract The model of crosslinks and sliplinks of Ball et al. [(1981) Elasticity of entangled networks. Polymer 22, 1010–1018] was shown in a previous paper to return the appropriate behaviour of necking and strain hardening observed in many polymers. Stress-strain experiments on the stretching of polypropylene at 150 °C show that there is an initial stiff response preceding necking, which begins at a lower strain than that predicted by the model of Ball et al. We have introduced a modification such that, as deformation proceeds, the number of sliplinks decreases. In the modified model, necking can begin at strains comparable with those observed experimentally. The proposed model is of an elastic, but not hyperelastic, material. An alternative explanation of the observed early onset of necking in terms of rate-dependent plasticity is explored and found to be inadequate, suggesting that the observed effect is an inherent property of the network. Superficially similar proposals for the modelling of glassy polymers, which also involve a changing number of chain interactions, are compared, and found to be essentially different; the effects modelled by these theories can be equally well modelled by the unmodified Ball et al. approach. We demonstrate the applicability of the present proposal by incorporating it into a finite element scheme and modelling a uniaxially stressed necking specimen of polypropylene.

Application of an elastic model to the large deformation, high temperature stretching of polypropylene

A physically based constitutive law for the deformation of polymers is applied to the stretching of polypropylene to large deformations at elevated temperatures. In this deformation regime, which is applicable to many forming processes, necking of the material is a persistent feature. The theory is elastic in nature, but includes the necking phenomenon as an inherent property. It is incorporated into a commercial finite element code and used to model a number of different experimental modes of deformation, both uniaxial and biaxial. Comparison is made with the experiments and it is found that both strains and forces are represented realistically, even though the true nature of the material is viscoelastic. Some of the discrepancies in the model predictions are traceable to its elastic nature.

Rate dependent and network phenomena in the multiaxial drawing of poly(vinyl chloride)

Stress-strain data have been obtained for poly(vinyl chloride) from drawing experiments in three stretching modes: uniaxial, planar and equibiaxial extension. The experiments were conducted isothermally at 84 °C, which is in the glass transition region, and at 90 °C, above T g . The stress-strain behaviour is qualitatively different at the two temperatures. At the lower temperature, there is an initial stiff response followed by a decrease in slope suggestive of yielding, whereas this effect is absent at 90 °C. There is strain rate dependence at both temperatures, with stress simply related to the logarithm of the shear rate. To model the behaviour above the glass transition, strain rate dependence is included in a network model; the alternative of a time independent network acting in parallel with a yielding process is found to be inadequate. At the lower temperature, however, there is a need for such a process acting in parallel with the time dependent network in order to predict adequately the initial shapes of the stress-strain curves. The resulting constitutive equations provide very good representations at both temperatures.

Application of a large deformation model to unstable tensile stretching of polyethylene

Abstract A large deformation, rate dependent model is applied to high temperature stretching of polyethylene. The theory is physically motivated, consisting of a model of a network of chain molecules to represent regions dominated by amorphous polymer, with embedded rigid spheres to introduce strain concentration similar to that caused by hard crystalline regions. Dependence on time and rate is introduced via shear stress driven diminution of the sphere radii. Experimentally, the rate dependence of the stress is such that, under tensile deformation, there is no necking associated with the initial yield point. Necking occurs at higher strains; this is associated with a weakening of rate dependence with increasing strain, which is a natural feature of the theory. It provides a realistic model of large tensile deformations, which in general involve the evolution of necking instabilities. It is implemented in a finite element scheme by using the package ABAQUS.

A unified model of stress relaxation and creep applied to oriented polyethylene

The stress relaxation behaviour of high-modulus oriented polyethylene fibre has been studied with regard to the response to successive small strain increments imposed on an initial relatively large strain deformation. For isotropic polymers, the results of such experiments have previously been interpreted in terms of a single thermally activated process modified by strain hardening. It has been found that, although this approach can describe satisfactorily some of the stress relaxation experiments on the oriented polyethylene fibres, it is unsatisfactory once the strain increments have exceeded a certain size, and that it is at variance with stress recovery experiments. It is shown that both the present stress relaxation and stress recovery experiments can be interpreted in terms of a model comprising two thermally activated processes acting in parallel. Furthermore, the parameters obtained for the stress relaxation data are consistent with those required to fit creep data obtained in a comparable stress range. The essential feature of the mechanical behaviour which was previously attributed to strain hardening can now be seen to arise from the transfer of stress between the two thermally activated processes in the two-process model.

The measurement of transverse mechanical properties of polymer fibres

We report developments in the measurement of transverse elastic properties of a range of polymer fibres. The cylindrical fibre specimen is compressed across a diameter between a pair of parallel flat plates by a known load, while two quantities are measured; the width of the rectangular contact zone at one of the fibre-plate boundaries, and the degree of diametrical compression in the direction of the compressing force. Both measurements are used together with the relevant elastic solutions to give estimates of the transverse elastic modulus. For some fibres, plasticity is detected in the measurements of diametrical compression, and loading strategies are adopted to isolate this effect; the modulus values obtained from the two measurements are then in good agreement, provided the fibre is sufficiently large to permit accurate measurement of the contact zone. Finite-element modelling is used to show that the elastic-plastic behaviour is essentially understood and to estimate a value for yield stress. Results are presented for five fibres with diameters in the range 0.13–0.57 mm.

Application of a necking criterion to PET fibers in tension

A criterion to predict instability in rate-dependent materials is developed. It is implemented for PET fibers in tension at three temperatures: 60, 75, and 80°C. At 60°C, necking is always observed, whereas at 80°C, the deformation is uniform, and 75°C marks a transition region, where necking is observed only at higher speeds and the deformation is otherwise uniform. As a necessary tool in the implementation of this criterion, the stress-strain behavior of PET is modeled using a combination of an Eyring process, a Gaussian network, and a linear elastic element. The resulting instability model gives predictions that are generally consistent with the experimental observations at all temperatures.

The modeling of multiaxial necking in polypropylene using a sliplink–crosslink theory

We demonstrate that the model of [Polymer 22, 1010 (1981)] has the property of forming a model of a necking polymer. Numerical models of necked bodies are generated by incorporating the theory into a finite element scheme. Large multiaxial extensions are applied to polypropylene sheets at a high temperature to produce necked specimens. Values of the model parameters are chosen such that the observed necks—in uniaxial, planar, and equibiaxial extension—are predicted realistically. The drawing forces observed are also shown to be consistent with the predictions.

A comparison of three polymer network models in current use

Abstract The relationship between three theories of polymer network deformation is explored. The theories are: the eight-chain model of Arruda and Boyce; the full network model of Wu and van der Giessen; and the crosslink–sliplink model of Edwards and Vilgis. All have a history of use as the network component in theories of solid polymer deformation. Given results from either the eight-chain or full network models, least-squares fitting of the stresses is used to derive optimal parameters of the Edwards–Vilgis model. Both the eight-chain and the full network models can be closely approximated by an Edwards–Vilgis model, provided the finite chain extensibility limit is not approached too closely. The eight-chain model is found to be equivalent to an Edwards–Vilgis model with a small number of sliplinks, whereas the full network model corresponds to an Edwards–Vilgis model with no sliplinks. The physical interpretation of these findings is discussed.

The modelling of large deformations of pre-oriented polyethylene

Abstract High temperature reversion tests have revealed a state of pre-existing molecular orientation in extruded polyethylene sheet. This state is related to differences in stress-deformation behaviour when specimens of the sheet are stretched along different angles with respect to the extrusion direction. An established large deformation, rate-dependent constitutive equation has been developed to model this material, by incorporating the pre-orientation by the addition of a strained Gaussian network. The level of pre-orientation is deduced from the dimensional changes on shrinkage. The constitutive equation is incorporated into the finite element package abaqus , and the shapes and drawing forces of tensile specimens extended at various angles to the extrusion direction are modelled.