Oscillatory Shear Strain Research Articles

Linear Rheology of Entangled Six-Arm and Eight-Arm Polybutadienes

Relaxation dynamics of various model entangled six-arm (A3-A-A3) and eight-arm (A3-A-A2-A-A3) polybutadiene melts are investigated using low-amplitude oscillatory shear and time-dependent step strain measurements. The frequency (time) and temperature range covered in these experiments are sufficiently broad to characterize the entire liquid-state relaxation spectrum of the materials. Several new findings about multiarm polymer dynamics are reported. First, the mean segmental relaxation time of multiarm polymers is a function of crossbar (A) molecular weight and polymer architecture. Second, for polymers with fixed arm (A) molecular weight, but variable crossbar molecular weight, terminal relaxation time (λ) and limiting shear viscosity (η0) scale quite strongly with crossbar molecular weight Mb (λ ∼ Mb∼6.8-7, η0 ∼ Mb∼8). When the crossbar tube length is renormalized by dilution of relaxed arms and the relaxation time and viscosity are rescaled to remove the inherent Mb dependence of segmental scale properties, these scaling exponents become closer to values expected for crossbar reptation in a dilated tube. Finally, relaxation dynamics of eight-arm A3-A-A2-A-A3 polybutadienes are found to be quite different from those of six-arm A3-A-A3 polymers with comparable arm molecular weight. Specifically, the slowest relaxation mode in well-entangled eight-arm polymers appears to be dominated by Rouse-like fluctuation effects, which blur the transition from high-frequency arm to terminal backbone relaxation.

The effect of tuber pre-heating temperature and storage time on the mechanical properties of potatoes

Changes in mechanical properties of potatoes are known to affect the quality of the products processed from the tubers. This study set out to ascertain the extent to which the changes in mechanical properties were attributable to turgor changes in the course of a storage season versus physiological changes that are expected to occur due to tissue ageing. Specimens taken from two potato cultivars ( Russet Burbank and Shepody) of two different solids contents were subjected to small strain oscillatory shear tests and wedge-penetration fracture tests. Prior to mechanical testing, specimens prepared from control (8°C) and pre-heated (33, 47 or 60°C) potato tubers were osmotically adjusted in a mannitol solution of 3, 5 or 7%. The tests were repeated every month from November to June. Turgor pressure and specific gravity were also measured during storage: turgor pressure slightly decreased, while specific gravity increased slightly throughout the storage period. Control and pre-heating temperatures of 33 and 47°C had little or no influence on shear modulus or fracture toughness; at a given mannitol concentration, pre-heating at 60°C caused both parameters to decrease. As mannitol concentration increased, the mechanical properties investigated decreased for the control, as well as for 33 and 47°C pre-heating treatments. Mannitol concentration had no effect on shear modulus and toughness for tubers that had been pre-heated at 60°C. Storage time had only a slight effect on the parameters of interest. Solids content and cultivar had no significant effect on either shear modulus or fracture toughness. To conclude, mechanical properties of tubers did change in storage, but the changes were much less pronounced than the changes caused by adjustment of turgor pressure. Thus, cell moisture content and its effect on turgor pressure affects tissue mechanical properties more dramatically than changes associated with ageing.

The crossover between linear and non-linear mechanical behaviour in polymer solutions as detected by Fourier-transform rheology

The application of oscillatory shear strain leads, in the non-linear regime, to the appearance of higher harmonic contributions in the shear stress response. These contributions can be analyzed as spectra in Fourier space, with respect to different frequencies, amplitudes and phase angles. In this article, we present an application of this new characterization method to a solution of the linear homopolymer polyisobutylene. The degree of non-linear response during oscillatory shear is quantified using the normalized intensity of the third harmonic contribution. We were able to show experimentally on polyisobutylene that there is an immediate onset of the non-linear response even for very small shear strain amplitudes.

High sensitivity Fourier-transform rheology

The application of oscillatory shear strain leads, in the non-linear regime, to the appearance of higher harmonic contributions in the shear stress response. These contributions can be analyzed as spectra in Fourier space with respect to their frequencies, amplitudes and phase angles. In this article, we present several theoretical and practical aspects of measuring Fourier rheology spectra with high sensitivity. Using the hardware of a conventional rheometer, Fourier rheology spectra with a signal-to-noise ratio of about 18,000:1 for a single acquisition were obtained. This allowed the observation of harmonics up to the 21st harmonic. Signal averaging can further increase the sensitivity.

Does plastic deformation proceed near thermodynamic equilibrium? The case made for shear-strained lamellar diblock copolymers

Observations on kink bands in lamellar diblock copolymers (SEP 40–70), caused by unidirectional or oscillatory shear strain, are interpreted in terms of the low-energy structure (LES) hypothesis, to wit: “In a material subject to mechanical stresses, that structure will be approached which has the lowest free energy among all structures which are in equilibrium with the tractions and are accessible to the system.” This is the generalization of the low-energy dislocation structure (LEDS) hypothesis applicable to dislocation structures in crystalline materials. In agreement with the LES hypothesis, moderate fatigue cycling of initially disordered material establishes an order such that the plane of the lamellae is parallel to the plane of shear stress application, being the orientation of lowest shear modulus and, hence, for fixed fatigue amplitude, of lowest strain energy. At fatigue strain amplitudes above about 40% the material develops kink bands on account of the compressive stress along the body diagonal of the samples. The geometry of these kink bands shows that the plane parallel to the lamellae serves as preferred slip plane with the lowest resistance against sliding among all possible directions. Also the kink band morphology conforms with the LES hypothesis. Specifically, on average the ratio of kink band length (L) to the square of kink band width (W), i.e., L/W2, is nearly constant as expected from the minimization of kink band boundary energy and the elastic strain energy on account of the strain discontinuity at the ends of the bands. Subsequent experiments on a different copolymer in a range of temperatures additionally verify the LES hypothesis through establishing that, throughout, large-amplitude cycling causes the lamella orientation of lowest shear modulus.

The effect of osmotic adjustment on the mechanical properties of potato parenchyma

Changes in the mechanical properties of potato tubers can be primarily attributed to physiological changes which affect structural components (cell wall, middle lamella) and to turgor pressure changes within the cells (which are affected by water loss and membrane integrity). In order to separate the effect of physiological changes from those associated with water loss, the cell turgor pressure of potato parenchyma was adjusted by hyper- and hypotonic solutions of mannitol. Tubers of the cv. Russet Burbank were stored for long (10 months) and short (1 month) periods. Slices and cylinders of parenchyma were obtained from the centres of the tubers along three axes; slices were also taken from the outer part of the tuber along two of the axes. Slices were subjected to small strain oscillatory shear at 0.02, 0.2 and 2 Hz and the storage and loss moduli (shear stiffnesses) obtained. Cylinders were compressed to 10.5% strain at 2 and 20 cm min −1 and initial and end moduli (longitudinal stiffnesses), and energy absorption terms obtained. Variability in measured mechanical properties was less when cells were fully turgid. Anisotropicity in shear stiffness was evident, although anisotropicity in longitudinal stiffness was dependent on turgor pressure. Shear stiffness was greater in outer parts of the tuber when slices were adjusted to common turgor pressures. Longitudinal stiffness was affected much more by osmotic adjustment than was shear stiffness. Although quantitatively different, hypotonic solutions caused the load-deformation curves of long term storage tubers to resemble those of short-term storage tubers. Similarly, hypertonic treatment brought about a load-deformation response in new tubers that made them appear as though they had been stored for ten months. The results indicate the importance of water content when analysing the mechanical properties of potato parenchyma.

Finite Element Formulation of a Sandwich Beam with Embedded Electro-Rheological Fluids

In this study, the amplitude-dependent dynamic characteristics of the ER sandwich beam was investigated. The nonlinear constitutive relation of the ER fluid in quasi-static shear was modeled by an exponential function. With the assumed quasi-static model, the hysteresis loop of ER fluid subjected to oscillatory shear strain was constructed. Thereby, the linearized complex moduli of the fluid at different amplitude of strain were derived by an energy approach. With this derived complex moduli, a linear finite element formulation of the sandwich beam structure based on Hamilton's principle was presented first. Then, an iterative process was presented to take the material nonlinearity of the ER fluid into account. The numerical examples of the sandwich beams in different boundary conditions demonstrated that the present modeling predicted qualitatively the changes of resonant frequencies and damping factors with respect to the amplitude of excitation as observed in the literature.

Structural response of nematic liquid crystals to weak transient shear flows

Using conoscopy, we have investigated the dynamic director response of nematics to oscillatory shear and step shear strain. In oscillation, the ratio of the amplitude of the director rotation to the strain amplitude and the phase difference between the director rotation and the applied strain are measured as functions of frequency for both 5CB and 8CB. Comparison with the Leslie–Ericksen theory allows extraction of several important material parameter ratios. In particular, data in the high frequency limit yield λ, the material parameter which indicates flow‐aligning (λ≳1) or tumbling (λ<1) behavior. For 5CB at 32.5 °C, λ≊1.03, while 8CB at 34 °C gives λ≊0.42. An analysis is presented for director field relaxation following distortion induced by a step strain. Step strain experiments on 5CB and 8CB provide an independent measurement of viscoelastic properties. Aside from confirming the flow classification of these model nematics, these new experimental methods establish a framework for the study of director dynamics in polymeric nematic liquid crystals.

Experimental Characterization of an Electrorheological Material Subjected to Oscillatory Shear Strains

The coupled electrical and mechanical dynamic properties of electrorheological (ER) materials comprised of alumino-silicate in fluorinated liquids are experimentally studied to gain insight into their effectiveness for application in the area of vibration control. A prototype test device is built to subject the test materials to oscillatory shear strains over a frequency range of 1 to 50 Hz. Energy dissipation in the material is determined as a function of volume fraction, electric field, strain amplitude, and frequency. Both the pre-yield and post-yield dynamic characteristics of the material under electric field are evaluated. Results of dynamic testing showed linear visco-elastic solid behavior for the pre-yield state with the shear modulus independent of the electric field and frequency. Post-yield energy dissipation of the materials tested is found to parallel that of Coulomb damping in which the energy dissipated is independent of frequency and increases linearly with increasing strain. The method of equivalent linearization is applied to determine equivalent viscous damping and stiffness expressions in terms of the device geometry, electric field, frequency of oscillation, and amplitude of oscillation for the nonlinear post-yield behavior.

Modelling the linear viscoelasticity of unfilled and carbon black loaded elastomers

Cyclic small strain deformation of unfilled and carbon black loaded vulcanised elastomers was investigated over a range of strain amplitudes, frequencies, and temperatures in order to determine and model the response of these materials. The elastomer used was a butadiene-acrylonitrile base polymer, KRYNAC 806. The carbon black filler was SRF N774 at a loading of 50 phr (parts per hundred by weight). Experiments were conducted in oscillatory shear using a Weissenberg rheogoniometer. Complex modulus data were obtained for a range of oscillatory shear strain amplitudes not exceeding 0.03 rads, for frequencies in the range 5 Hz–60 Hz and at temperatures between −20°C and 20°C. Time-temperature superposition was obtained for data above −10°C, with the same shifts applicable to both unfilled and carbon black loaded materials. Shifts were represented using the WLF equation. It was found that reduced data over a range of ten decades of log frequency were well represented by a Huet model. Varying the Huet model parameters thus, in principle, affords a means of modelling linear deformations of elastomers containing different loadings of carbon black filler.

Differential Dynamic Shear Moduli of Carbon Black-Filled Styrene-Butadiene Rubber Subjected to Large Shear Strain Histories

Abstract Combined measurements of shear-stress relaxation and differential dynamic storage and loss shear moduli G′ and G″ following a single-step shear strain of 0.4, as well as measurements of dynamic moduli in on-off strain and stress histories, have been made on styrene-butadiene rubber (type 1502) filled with carbon black (N299) at loadings of 40, 50, 60, and 70 phr, with 10 phr Sundex 790 oil. Both cured and uncured compounds were studied at temperatures of 25.0° and −0.5°C respectively. The maximum oscillatory shear strain was 0.005, and the frequency was from 0.4 to 1.8 Hz. The storage shear modulus G′(ω, 0) measured without imposition of static strain was approximately proportional to the fourth power of the volume fraction of black. With imposition of single-step strain, the differential storage modulus G′(ω, γ; t) fell 25% to 35% but slowly recovered somewhat while the strain was maintained for 4 to 5 h. During this period, the static stress relaxed continuously. At the highest content of black, the drop in log G′ was the least, and the final recovery was closest to the initial value of G′(ω, 0). In on-off experiments on uncured compounds, when the strain was “on” for 250 s and then “off” (either stress or strain returned to zero), G′ decreased when the strain was imposed as before and decreased further when it was removed. In the “off” state, G′ recovered partially but did not attain the initial value of G′(ω, 0) even after 7 d. In on-off experiments on cured compounds, removal of stress caused G′ to either increase or decrease depending on the content of black; in any case, in the “off” state, G′ recovered completely to its initial value. Other strain histories involved on-off sequences with different “on” periods and multiple on-off sequences with different “on” periods and multiple on-off sequences. The results are interpreted in terms of a network of black particle aggregates whose contacts can slowly rearrange even in the absence of stress as shown by stress relaxation at very small strains in earlier studies. In large strains, it is postulated that some contacts are broken but can partially reform, especially in the stress-free state; the rate of reformation is similar to that of small-strain stress relaxation. Only in cured compounds is the network fully recovered, presumably because in these the particles are imbedded in a crosslinked matrix and have crosslinked bridges that facilitate reestablishment of interparticle contacts, whereas in uncured compounds the matrix has no crosslinks and the bound rubber on adjacent particles may be merely entangled.

Stress Relaxation and Differential Dynamic Modulus of Carbon Black-Filled Styrene-Butadiene Rubber in Large Shearing Deformations

Abstract Combined measurements of shear stress relaxation and differential dynamic storage and loss shear moduli G′ and G″ have been made on styrene-butadiene rubber (type 1502) containing 50 phr N299 carbon black and 10 phr Sundex 790 oil, both cured and uncured, and compared with similar measurements on the cured and uncured gum rubber. The range of temperature was −22.5° to 63°C, and of static shear strain 0.01 to 0.40; the maximum oscillatory shear strain was about 0.005 and the frequency was usually 0.64 Hz. Dynamic measurements of G′ and G″ with no superposed static strain and stress relaxation measurements at small strains on the filled samples could be reduced by frequency-temperature superposition with αT shift factors that were somewhat larger than those that applied to the raw or crosslinked gum. At about 0.6 Hz, the filler increased the storage modulus by about a factor of 5 for the cured sample and about 10 for the uncured; the moduli for the cured and uncured filled samples were almost identical. The order of rates of relaxation was unfilled uncured &amp;gt; filled uncured &amp;gt; filled cured &amp;gt; unfilled cured. At small static strains, the stress relaxation of the filled samples (both uncured and cured) was substantial, but the differential dynamic moduli G′ and G″ remained nearly equal to their values with zero static strain throughout the relaxation process. With increasing static strain γ, the strain-dependent relaxation modulus G(γ;t) decreased by as much as a factor of 3 for the filled, cured sample and 4 for the filled, uncured sample. The modulus G(γ;t) could not be factored into a strain-dependent and a time-dependent function. During stress relaxation of the filled samples (both cured and uncured) at large strains, the differential storage modulus G′(ω,γ;t) experienced a drop followed by slow recovery toward its initial value at 25°C; the loss modulus G″(ω,γ;t) was unchanged. For the filled, cured sample, complete recovery to the initial value G′(ω,0) (where 0 refers to zero static strain) was accomplished by removing the shearing stress and heating for one day at 63° followed by return of the temperature to 25°C. The results are interpreted in terms of a structure which combines a crosslinked (or, in the case of uncured, an entangled) polymer network and a network of carbon black particles. At small strains, stress relaxation is thought to be accomplished primarily by rearrangements of carbon particles and/or polymer molecules bridging them, without structural damage in the sense that the rearranged structure has the same properties as the original. These rearrangements are impeded by crosslinking in the cured vis-a-vis the uncured filled rubber. The kinetics of rearrangement may be governed by configurational changes of the polymer molecules since the temperature shift factors do not differ greatly from those for the gum rubber. At large strains, the particle network can be damaged but can regain its structure by a healing process which is accelerated at higher temperatures. The conclusions apply to the particular compound studied here and not necessarily to other rubber-black compounds, which according to the literature show great diversity of properties.

Influence of superimposed steady shear flow on the dynamic properties of polyethylene melts

AbstractExperiments in which an oscillatory shear flow is superimposed on a steady state shear flow were performed on polyethylene melts by the use of a cone and plate type rheogoniometer. The phase difference between oscillatory shear stress and shear strain increases in all cases and for all frequencies with the increase of the superimposed shear rate. Between ω0, the frequency at which the phase difference is π/2 and the steady shear rate \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document}, as found by Booij for polymer solution, the relation ω0 = 1/2 \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma$\end{document}. holds also for polyethylene melts. The significance of this relation is discussed briefly from the viewpoint that the entanglement density decreases with the increase of the imposed shear rate.