Louis J. Zapas

Creep and recovery behaviour of ultra-high molecular weight polyethylene in the region of small uniaxial deformations

Abstract The creep and recovery behaviour of an ultra-high molecular weight polyethylene (UHMWPE) has been studied in the region of small uniaxial deformations. At deformations as small as 5 × 10−4 the stress-strain behaviour is non-rectilinear and the recovery cannot be described by a theory of fading memory. A new one-dimensional constitutive relation is presented which describes quantitatively the multistep creep and recovery behaviour of this material in the case where the specimens are not mechanically preconditioned. The multistep in strain-stress relaxation behaviour of the UHMWPE has also been investigated for the case in which the second step in strain is approximately half the magnitude of the first step. Calculations of the strain necessary to give the observed stress in a two-step stress-relaxation experiment have been made assuming that the stress-relaxation experiment can be represented by a series of multistep creep experiments where in each step the stress is adjusted so as to maintain a constant deformation. The agreement between the experimental values and the calculated values are very good. The proposed equation, which describes plasto-viscoelastic behaviour, appears to be able to describe quantitatively the creep and recovery behaviour of a wide range of semicrystalline polymers.

An Analysis of the Corrections to the Normal Force Response for the Cone and Plate Geometry in Single‐Step Stress Relaxation Experiments

An analysis and experimental results are presented for the transient response in single‐step stress relaxation experiments in a cone and plate geometry. Results from experiments on a polyisobutylene solution show deviations from unity of the ratio of the first normal stress difference to the product of the shear strain times the shear stress. These are accounted for by including three important corrections in the analysis. First, it is shown that the finite time required to apply the step introduces errors in the normal stresses which are greater than those for the shear stress. Second, the machine compliance introduces errors in the normal force by causing an increased gap separation which subsequently relaxes as the normal force relaxes. Third, the constrained geometry of the cone and plate results in the compliance errors being “magnified” by some 1600 times, leading to the need for large corrections and apparent violations of the universal relation at long times. Experimental results for extension and compression in a parallel plate geometry are presented for different gap settings and used to demonstrate that the constrained cylinder problem in viscoelastic fluids is similar to that observed in elastic bodies.

Experiments on the small-strain behaviour of crosslinked natural rubber: 1. Torsion

The torsional behaviour of 1, 3 and 5 phr peroxide crosslinked natural rubber has been characterized over a range of strains from near the undistorted state (γ ≈ 0.017) to γ ≈ 1.0. Isochronal measurements of both torque and normal force were used to calculate values of the derivatives of the strain energy function W with respect to the first and second stretch invariants I1 and I2. In the course of our work we found that, contrary to many reports in the literature, ∂W∂I1 was affected significantly by the amount of crosslinking. Finally for the 1 phr peroxide crosslinked rubber it was found that, while ageing for 14 months at ambient conditions did not significantly affect the small-strain torsional modulus, G = 2(∂W∂I1 + ∂W∂I2), it did significantly affect the individual derivatives ∂W∂I1 and ∂W∂I2.

Experiments on the small-strain behaviour of crosslinked natural rubber: 2. Extension and compression

Abstract Experiments were carried out to characterize the small-strain tension and compression behaviour of dicumyl peroxide crosslinked natural rubber. Strains that were smaller by an order of magnitude than any reported previously on natural rubber were achieved. Our results support the contention that the compression and extension moduli of natural rubber are different. A new finding is reported: that is, the moduli in tension and compression do not become constant but rather they increase significantly as zero deformation is approached.

On the small strain behavior of peroxide crosslinked natural rubber

Abstract Torque and normal force measurements on a cylinder subjected to torsion at constant length were used to study the behavior of NR crosslinked with 5 phr dicumyl peroxide. The derivatives of the strain-energy density function ∂W/∂I1 and ∂W/∂I2 were calculated from the data using the scaling law of Penn and Kearsley. The new results extend the limit of small strains at which the strain-energy density function derivatives have been measured to γ<0.005 and further confirm our previous results that for peroxide-crosslinked NR, ∂W/∂I2 does not become negative at small strain, contrary to several reports in the literature. Reduced stress was determined for the rubber by using the approach of Kearsley and Zapas to calculate the derivative w′(λ) of the Valanis-Landel form of the strain energy function. The results were compared with the measured values for reduced stress in tension and compression at small strains. While the deviation between the predictions and the experimental behavior do not exceed 6%, ...

The time dependent strain potential function for a polymeric glass

Abstract Torsion and normal force measurements were made during single step stress relaxation experiments on a polymeric glass (PMMA). Isochronal data were analysed using an approach adapted from that developed by Penn and Kearsley 1 (for incompressible elastic materials) to determine the derivatives ∂W ∂I 1 , and ∂W ∂I 2 of the time dependent strain potential function. ∂W ∂I 1 and ∂W ∂I 2 are determined from existing solution to the torsion of an incompressible cylinder. A special solution to the torsion of a compressible cylinder is presented and it is shown that the values of ∂W ∂I 1 and ∂W ∂I 2 obtained using this solution to analyse the data do not differ greatly from those obtained using the incompressible solution. It is found from both solutions that ∂W ∂I 1 is negative and increases towards zero with increasing time and deformation while ∂W ∂I 2 is positive, greater in magnitude than ∂W ∂I 1 and decreases towards zero with increasing time and deformation. These results were unexpected and a full understanding of their meaning has yet to be reached.

Viscoelastic Behavior of Poly(Methyl Methacrylate): Prediction of Extensional Response from Torsional Data

Some years ago Rivlin1 showed that for certain deformations, one can treat isochronal data from single step stress relaxation experiments on viscoelastic materials in the same fashion as if the data were obtained for an elastic material. We have conducted single step stress relaxation experiments on cylinders of poly (methyl methacrylate) (PMMA) where we measured torque and normal force responses as functions of time and angle of twist. By assuming that torsion is an isocholic motion and that volume effects are separable, we obtained isochronal values for ∂W/∂Il and ∂W/∂I2. Our results showed that ∂W/∂I1 is negative while ∂W/∂I2 is positive and greater in magnitude than ∂W/∂I1. These findings led to the possibility of explaining the phenomenon observed by Sternstein and Ho3 that the single step stress relaxation response of PMMA is different in torsion than in simple extension. Specifically, it was found that the rate of decay of the stress with respect to time is significantly higher in torsion than in extension. This phenomenon was observed at small strains where the stress responses in both torsion and extension were linear in the appropriate strain measure. The difference could not be accounted for either in terms of a time dependent Poisson’s ratio or the material compressibility.

Approximate relations for the analysis of single-step stress-relaxation data in uniaxial extension from experiments involving a finite step time

Abstract In the determination of single-step stress-relaxation behaviour, a finite time is required to reach the desired strain. As a result, an uncertainty is introduced into the observed behaviour at the early times. In the region of linear behaviour, an approximation has previously been derived which can be applied to shear stress-relaxation experiments. In the present work, approximate relations are derived which can be applied to uniaxial extension experiments in the region of non-linear behaviour. The derivations are based on the assumption that, under the set of strain histories considered, one can use the Bernstein, Kearsley and Zapas theory as a one-dimensional description. To demonstrate the validity of the approximate relations, we have obtained data on a linear low-density polyethylene copolymer under conditions of a varied step time and strains well into the region of non-linear behaviour.

Mechanical behaviour of isotactic polypropylene subjected to various strain histories in uniaxial extension

Abstract The mechanical behaviour of a slowly quenched isotactic polypropylene has been studied for various strain histories in extension. Creep, constant rate of strain, and constant rate of loading experiments were carried out at deformations up to and beyond the point where necking occurs. A creep diagram, which includes the failure envelope, is presented. From the available data we have also obtained an extrapolated surface of the single step stress-relaxation behaviour. From this surface we can calculate, using the Bernstein and Zapas theory on the instability of viscoelastic bars, the deformation at which necking occurs for various strain histories.

A Phenomenological Theory of the Influence of Strain History on the Rate of Isothermal Stress Relaxation

Constitutive relations are formulated for a class of incompressible viscoelastic fluids for which internal structural changes occur at a rate that is influenced by the history of the strain. For the materials considered, the contribution to the stress at time t made by the strain at an earlier time τ is a function of that strain, the true elapsed time t−τ, and a quantity σ(t,τ) that can be interpreted as the elapsed time measured by a clock whose rate of advance, because it is tied to the rate of structural change, is affected by the history of the strain. The functional relating ∂σ(t,τ)/∂t to the history of the strain up to time t is assumed to have the same domain and a structure similar to that relating stress to strain history. The present theory reduces to the theory of BKZ fluids in the (extreme) special case in which σ(t,τ)≡t−τ, in other words, in which ∂σ(t,τ)/∂t≡1. It is shown that there is a sense in which constitutive relations recently found to account well for observed discrepancies between e...