Stress relaxation,creep, and internal stresses in high density polyethylene filled with calcium carbonate

The effect of filling high density polyethylene (HDPE) with calcium carbonate (up to 50% by weight) on the stress relaxation and the creep in uniaxial extension at room temperature was investigated. The addition of CaCO₃ was found to have a strong influence on the flow behaviour of HDPE. In particular, it was observed that the internal stress level, calculated from relaxation data, increased markedly with the filler content. The reduction in creep rate of the filled samples suggested that the CaCO₃-particles induce a change in the structure of the HDPE-interphase close to the filler surface. This was supported by dynamic mechanical measurements performed at low temperatures on swollen HDPE-CaCO₃ samples.     

Stress relaxation due to chain fractures in transient polymer networks

Aspects of a network model for concentrated dispersions are applied to polymer networks. It is shown how network deformation caused by network fracture affects the macroscopic stress.     

Stress relaxation,creep and set recovery of elastomers

The stress relaxation, creep and recovery behaviour of a cross-linked unfilled natural rubber has been investigated at moderate stresses in tension. The aim being to extend the idea, initially developed by Alan Gent in his seminal 1962 paper on the relaxation behaviour of rubber, in order to understand and examine the time dependent mechanisms that are present in elastomers under strain. A method based upon the Boltzmann superposition principle was used to compare the creep compliance with a measurement of its recovery after release from a range of constant loads held for different times. The creep behaviour was seen to exhibit the usual linear dependence on the logarithm of time. The recovery data was also seen to reduce onto a single recovery curve for any given applied tensile stress for a range of loading times using the Boltzmann superposition principle. The differences between the relative rates of the creep and the recovery behaviour can in part be attributed to the non-linearity in the stress–strain behaviour exhibited in tension of the elastomer.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

Dynamic study of gelatin gels by creep measurements

We have investigated the dynamical properties of gelatin gels using creep measurements. A commercial apparatus (Carrimed CSL500) was modified in order to increase the deformation of the gel and to take advantage of the inertia of the system. When a step stress is applied, the very first response of these materials is an oscillating strain owing to a coupling of the high elasticity of the gelatin gels and the inertia of the apparatus. From these damped oscillations, we have extracted the elastic and loss moduli as a function of frequency, which allows us to widen the frequency range (toward high frequencies) of measurement. After subtraction of the oscillations, we have obtained the compliance funtion from which, using Ferry's formalism, we can calculate the relaxation time distribution function over a very large time range (10^–3–10⁴ s). We show that the dynamics of gelatin gels is governed by two very different characteristic times. We interpret the faster relaxation time as relaxation at the scale of the gel network mesh-size, while the slower time we assign to relaxations involving the lifetime of the gelatin gel cross-links. It is now possible to use creep measurements as an alternative to the forced oscillatory function determination, as the same data can be obtained but, more quickly, and over a large frequency range. This gives us more indication of the gel's structure (gel network behaviour, kinetics of ageing) than all the laborious methods previously necessary.     

Transient behaviour of the mechanical loss factor of high density polyethylene during creep and relaxation

The mechanical loss factor of high density polyethylene has been measured during creep and stress relaxation experiments. The application of a constant stress (creep) or strain (relaxation) resulted in an instantaneous increase in tan above the value obtained in absence of the creep or relaxation process. During the flow the tan-value decreased and approached an apparent equilibrium value slightly above the value obtained without the static load or deformation. This behaviour was observed both at 1 and 10 Hz and at 23 and 60 °C. It is suggested that this time dependence of the mechanical loss factor is associated with the basic mechanisms responsible for the creep or relaxation process itself.     

Stress relaxation of large amplitudes and long timescales in soft thermoplastic and filled elastomers

We propose a model for describing mesoscale relaxation mechanisms in soft thermoplastic elastomers and also in the high-temperature regime of filled rubbers. The model consists of hard spheres embedded in an elastic matrix. It is solved by dissipative particle dynamics. We study the response of the model to deformations of various amplitudes. We show that it displays slow relaxation processes of large amplitudes that are related to irreversible reorganizations at a mesoscopic scale. We characterize these reorganizations as buckling of instabilities that change the local environment of the hard inclusions. Paper presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Evaluation by means of stress relaxation (after a step strain) experiments of the viscoelastic behavior of polymer melts in uniaxial extension

 As is widely acknowledged, morphology in most materials is far more sensitive to extensional than to shear deformations but, unfortunately, due to the experimental difficulties involved, there are no non-destructive, morphology probing techniques in such flows, i.e., the equivalent of stress relaxation and oscillatory experiments in shear flows. This paper tries to overcome some of those drawbacks by proposing an experimental technique that allows stress relaxation experiments after a step strain in uniaxial extension to be performed. The benefits of this technique are twofold: (a) while the deformation is small enough for the response to be in the linear viscoelastic regime it constitutes a probe of the microstructure of the material and (b) it allows the departure to the non-linear regime to be studied, useful, for example, for the definition of the damping function in uniaxial extensional flow or for the study of the response of materials to fast transient flows with a strong extensional component, such as contraction flows. In this work the proposed technique, which requires a correction to the apparent (theoretical) strain rate in order to allow the calculation of the true Hencky strains attained during the strain step, is tested and validated for two polyisobutylene melts. Received: 9 April 2001 Accepted: 26 July 2001     

Uniaxial deformation and relaxation of polymer blends: relationship between flow and morphology development

Uniaxial elongational flow followed by stress relaxation of a dilute mixture of polystyrene/polymethylmethacrylate) PS/PMMA with PS (5 wt%) as a dispersed phase was investigated. The behavior of the blend was found to be dominated by the PMMA matrix during elongation and by the interface during the relaxation at long time. Such a behavior was related to drop deformation and shape recovery during the relaxation process as was confirmed by morphological analyses on samples quenched within the rheometer just after elongation and at various times during the relaxation process. The morphology and the rheological material functions variation were compared to the Yu model (Yu W, Bousmina M, Grmela M, Palierne JF, Zhou C (2002) Quantitative relationship between rheology and morphology in emulsions. J Rheol 46(6):1381–1399).     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

Shear relaxation characteristics of rock joints under stepwise loadings

The stress relaxation characteristic of rock mass is an important aspect of rheology and has important practical significance for rock engineering. In order to investigate the relaxation characteristic of rock joints with different slope ratios and normal stresses, a series of shear stress relaxation tests were conducted on artifical rock joints poured by cement mortar. Test results show that the relaxation curves can be divided into three stages, i.e. instantaneous relaxation stage, attenuation relaxation stage, and stable relaxation stage. Furthermore, the nonlinear Maxwell relaxation equation was obtained by using the relation between the viscosity coefficient and time, and the theoretical curves based on the empirical equation agreed well with the test results. Moreover, the change law of the initial viscosity coefficient was investigated. Accordingly, a stress relaxation method, termed as relaxation stress peak method, was proposed to determine the long-term strength of rock joints.     

Mathematical and numerical modelling of large creep deformations for annular rotating disks

A simulation model is presented for the creep process of the rotating disks under the radial pressure in the presence of body forces. The finite strain theory is applied. The material is described by the Norton-Bailey law generalized for true stresses and logarithmic strains. A mathematical model is formulated in the form of a set of four partial differential equations with respect to the radial coordinate and time. Necessary initial and boundary conditions are also given. To make the model complete, a numerical procedure is proposed. The given example shows the effectiveness of this procedure. The results show that the classical finite element method cannot be used here because both the geometry and the loading (body forces) change with the time in the creep process, and the finite elements need to be redefined at each time step.     

Shear and normal stress relaxation in polymer blends: stress predictions from interface velocity and Laplace pressure terms

We derived for the first time the relationships among shear stress and normal stress differences for ellipsoidal interfaces under large step shear strains considering interface velocity term and Laplace pressure term in the expression of the stress tensor for mixtures of two Newtonian fluids. In the derivation, orientation angle of the interface is assumed to be given by the affine deformation assumption and is independent of time based on experimental results for blends with 0.048 ≤ K ≤ 0.54 where K is the ratio of droplet viscosity to matrix viscosity. For ellipsoidal droplets, the shear stress is only proportional to the first normal stress difference. On the other hand, for spheroidal droplets, proportionality among the shear stress, the first and the second normal stress differences was derived, and the ratio of the second normal stress difference to the first normal stress difference was given as a function of step strain. The shear stress and the first normal stress difference obtained experimentally satisfy the derived relationship, indicating applicability of the stress expression for polymer blends.     

Viscoelasticity of brain corpus callosum in biaxial tension

Computational models of the brain rely on accurate constitutive relationships to model the viscoelastic behavior of brain tissue. Current viscoelastic models have been derived from experiments conducted in a single direction at a time and therefore lack information on the effects of multiaxial loading. It is also unclear if the time-dependent behavior of brain tissue is dependent on either strain magnitude or the direction of loading when subjected to tensile stresses. Therefore, biaxial stress relaxation and cyclic experiments were conducted on corpus callosum tissue isolated from fresh ovine brains. Results demonstrated the relaxation behavior to be independent of strain magnitude, and a quasi-linear viscoelastic (QLV) model was able to accurately fit the experimental data. Also, an isotropic reduced relaxation tensor was sufficient to model the stress-relaxation in both the axonal and transverse directions. The QLV model was fitted to the averaged stress relaxation tests at five strain magnitudes while using the measured strain history from the experiments. The resulting model was able to accurately predict the stresses from cyclic tests at two strain magnitudes. In addition to deriving a constitutive model from the averaged experimental data, each specimen was fitted separately and the resulting distributions of the model parameters were reported and used in a probabilistic analysis to determine the probability distribution of model predictions and the sensitivity of the model to the variance of the parameters. These results can be used to improve the viscoelastic constitutive models used in computational studies of the brain.     

Comparison of the rheology of polymer melts in shear,and biaxial and uniaxial extensions

The experimental properties of different polymer melts, polystyrene, high density polyethylene and low density polyethylene are compared for the first time in three different deformations: step shear, step biaxial extension and steady uniaxial extension. Properties of three other melts are also studied in step biaxial and shear experiments. For our comparative purposes some data of Laun and Winter from the literature are used, as well as new data reported here. In all the step strain experiments, the stresses can be factored into a time dependent relaxation modulus and a strain dependent damping function. The data are interpreted using a differential constitutive equation of Larson which satisfies this time-strain separability and has a single parameter that describes the strain softening character of the material. Results show that differences in the properties of the melts are most pronounced in uniaxial extension and least in biaxial extension. All melts follow the Doi-Edwards prediction relatively closely in biaxial extension. In uniaxial extension, the branched material shows a strong strain hardening effect although its shear and biaxial properties are similar to the other melts. The constitutive model gives a reasonably good fit to the data in all three deformations for unbranched materials for the same value of the adjustable parameter; the model, however, fails for the branched low density polyethylene.     

Effect of sample dimensions,lubrication and deformation rate on uniaxial compression of gelatin gels

The response of 10% gelatin gels to uniaxial compression is determined in part by frictional effects at the gel-platen interface. By using teflon-coated plates, lubricated with paraffin or silicone oil, these frictional effects are effectively eliminated. The stress-strain response can then be described by the two-constant Mooney-Rivlin relation, the sum of the two parameters (C ₁ +C ₂ ) being about 25% lower in lubricated compression than the value obtained in simple shear and torsion. Cross-head speed (for total testing times of 0.2–3 min) had no effect on material response, but long-term stress relaxation does occur over periods of about 30 min and longer. Sample radius did not affect the response in lubricated compression but had a major effect under unlubricated conditions. No systematic change in response was seen with sample diameter to height (aspect) ratios between 9.6 and 3.1 in lubricated compression, but data scatter for a given sample diameter was worst at the lowest heights (highest aspect ratio). Agreement of all true stress versus strain data was within about ± 7% regardless of sample height or deformation rate.     

Stress concentration of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension

Summary This paper deals with the stress concentration problem of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknowns are densities of body forces distributed in the r- and z-directions in semi-infinite bodies having the same elastic constants as the ones of the matrix and inclusion. In order to satisfy the boundary conditions along the ellipsoidal boundary, four fundamental density functions proposed in [24, 25] are used. The body-force densities are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries even when the inclusion is very close to the free boundary. The effect of the free surface on the stress concentration factor is discussed with varying the distance from the surface, the shape ratio and the elastic modulus ratio. The present results are compared with the ones of an ellipsoidal cavity in a semi-infinite body.accepted for publication 11 November 2003     

Flow birefringence study of sharkskin and stress relaxation in polybutadiene melts

In this paper we report rheo-optical and rheological observations made through a transparent slit die attached to a capillary rheometer. We find that the flow birefringence signal oscillates periodically near the die exit when sharkskin-like extrudate distortion is present. In contrast, steady behavior is observed in the die inland region. Specifically, the flow birefringence varies at the die exit with a period identical to that measured directly from the sharkskin extrudate. We also show that the exit flow instability leading to sharkskin can be observed directly through cross-polarizers in terms of the temporal change of the retardation order. We demonstrate that the same kind of interfacial flow instability can occur at a boundary discontinuity within the die land where the upper portion of a clean die wall meets the lower portion of a polysiloxane-coated die wall. Finally, stress relaxation upon the cessation of the slit die flow of two polybutadiene melts is studied through time-dependent flow birefringence measurements. The stress relaxation is then correlated with sharkskin time scales to describe the role of relaxation in sharkskin ridge formation. Received: 8 February 1999 Accepted: 28 July 1999     

Structure and rheology of heat-set gels of globular proteins

The structure and the rheology of systems resulting from heating at 80°C isoelectric solutions of bovine serum albumin (BSA) in the concentration range 10–200mg/ml were studied.  Small-angle neutron scattering measurements view the systems as being formed of large aggregates of micrometric size with a close packed arrangement of denatured protein molecules. No indication of a fractal structure stands out.  The viscoelastic behaviour is linear up to about 5% strain, except in the BSA concentration range 30–90mg/ml where the linearity limit is below 1% strain. The viscoelastic response was analysed in the linear domain, or as close as possible to it, by combining the results of dynamic and creep recovery measurements. The dependence on concentration of the steady state viscosity, of the steady state compliance, and of the average retardation time shows a marked change around a concentration C ₀∼50mg/ml, corresponding probably to a percolation threshold. Received: 10 March 1998 Accepted: 1 May 1998     

Poroelastic toughening in polymer gels: A theoretical and numerical study

We explore the Mode I fracture toughness of a polymer gel containing a semi-infinite, growing crack. First, an expression is derived for the energy release rate within the linearized, small-strain setting. This expression reveals a crack tip velocity-independent toughening that stems from the poroelastic nature of polymer gels. Then, we establish a poroelastic cohesive zone model that allows us to describe the micromechanics of fracture in gels by identifying the role of solvent pressure in promoting poroelastic toughening. We evaluate the enhancement in the effective fracture toughness through asymptotic analysis. We confirm our theoretical findings by means of numerical simulations concerning the case of a steadily propagating crack. In broad terms, our results explain the role of poroelasticity and of the processes occurring in the fracturing region in promoting toughening of polymer gels.