How to define the storage and loss moduli for a rheologically nonlinear material?

A large amplitude oscillatory shear (LAOS) is considered in the strain-controlled regime, and the interrelation between the Fourier transform and the stress decomposition approaches is established. Several definitions of the generalized storage and loss moduli are examined in a unified conceptual scheme based on the Lissajous–Bowditch plots. An illustrative example of evaluating the generalized moduli from a LAOS flow is given.     

On secondary loops in LAOS via self-intersection of Lissajous–Bowditch curves

When the shear stress measured in large amplitude oscillatory shear (LAOS) deformation is represented as a 2-D Lissajous–Bowditch curve, the corresponding trajectory can appear to self-intersect and form secondary loops. This self-intersection is a general consequence of a strongly nonlinear material response to the imposed oscillatory forcing and can be observed for various material systems and constitutive models. We derive the mathematical criteria for the formation of secondary loops, quantify the location of the apparent intersection, and furthermore suggest a qualitative physical understanding for the associated nonlinear material behavior. We show that when secondary loops appear in the viscous projection of the stress response (the 2-D plot of stress vs. strain rate), they are best interpreted by understanding the corresponding elastic response (the 2-D projection of stress vs. strain). The analysis shows clearly that sufficiently strong elastic nonlinearity is required to observe secondary loops on the conjugate viscous projection. Such a strong elastic nonlinearity physically corresponds to a nonlinear viscoelastic shear stress overshoot in which existing stress is unloaded more quickly than new deformation is accumulated. This general understanding of secondary loops in LAOS flows can be applied to various molecular configurations and microstructures such as polymer solutions, polymer melts, soft glassy materials, and other structured fluids.     

Flow of concentrated solutions of starlike micelles under large-amplitude oscillatory shear

The non-linear viscoelasticity of concentrated solutions and glasses of soft starlike micelles has been studied by large-amplitude oscillatory shear (LAOS). The non-linear response has been analysed using current schemes of Fourier transform (FT) rheology, and its character has been determined by the phase of the third harmonic contribution to the stress. The limitations of FT rheology and related analysis methods are discussed, and an alternative method is presented that takes into account all the higher harmonics. This method reveals a strain-hardening character of intracycle non-linearities at large strain amplitudes for all volume fractions. We also show that, although the relation of LAOS with steady shear measurements works qualitatively, due to inherent limitations of LAOS, steady shear data cannot be reproduced quantitatively.     

Large-amplitude oscillatory shear flow from the corotational Maxwell model

Using the single relaxation time corotational Maxwell fluid, we derive explicit analytical expressions for the first, third, and fifth harmonics of the alternating shear stress response in large-amplitude oscillatory shear (LAOS). We also derive corresponding expressions for the zeroth, second, and fourth harmonics of both the first and second normal stress differences. These harmonics are found to depend upon just two dimensionless groups: the Deborah and Weissenberg numbers, each of which causes non-Newtonian behavior. The form of the solution for the corotational Maxwell model in LAOS matches the forms of the analytical solutions for two molecular models for dilute solutions and one for concentrated solutions or melts. We also derive an analytical solution for the corotational Maxwell model after startup of LAOS. For this we find that both small and large amplitude cases approach a periodic limit cycle (alternance) at the same rate for both the shear stress response and for the normal stress differences. For molten high density polyethylene that is lightly filled with carbon black, we find good quantitative agreement with measured LAOS behavior when our analytical solution is superposed for multiple relaxation times.     

Onset of nonlinearity and yield strain of a model soft solid

For soft solids with their low modulus, small stress already results in large strain, which may cause nonlinearity and yielding. These potentially competing effects were studied on a clay/polybutadiene (clay/sPB) composite, which is a soft physical gel. Structural changes were introduced by oscillatory shear using large amplitude (LAOS). LAOS beyond a critical limit reduced the internal connectivity. This softened the already soft solid even further, thereby moving it closer to its physical gel point. For clay/sPB, the shear-induced changes were irreversible so that they could get probed using small amplitude shear (SAOS) frequency sweeps. Sequences of SAOS-LAOS-SAOS (SLS) where repeated with increasing LAOS amplitude and increasing duration. The flow-induced structural changes in the soft solid were attributed to yielding, which began to occur at about the same stress/strain values as found for the onset of nonlinearity in traditional SAOS to LAOS (StL) stress amplitude sweeps. The onset of nonlinearity and the yielding seem to be a strain activated process since the characteristic strain amplitude is independent of frequency and temperature, but not so for the characteristic stress amplitude. The duration of LAOS in a SLS experiment beyond yielding is an important parameter since flow-induced structural changes require time to grow.     

Prediction of normal stresses under large amplitude oscillatory shear flow

The dynamic response of viscoelastic fluids under large amplitude oscillatory shear (LAOS) has been a subject of long history. In the LAOS flow, the analysis has been mostly focused on shear stress, possibly due to the lack of accurate measurement of normal stress. However, the normal stress may become larger than shear stress at high-strain amplitudes, and thus it is important that we have a good understanding of the normal stress behavior. Furthermore, with the advancement in the instrumentation, it has become possible to get more reliable data. The purpose of this paper is to develop a research platform to analyze and to understand the normal stress behavior of complex fluids under LAOS flow. In this study, we utilized the Giesekus model as a representative constitutive model, and investigated its diverse responses. We defined the dynamic properties corresponding to normal stress, in a similar way to define dynamic moduli from shear stress, and examine their behavior with various analyzing tools. Experimental data were also compared with model predictions. Despite the fact that it is not yet possible to compare all of the predictions because of instrumental limitation, the prediction has been found to fit well with the experimental data. This study is expected to provide a useful framework for further understanding the nonlinear behavior of complex fluids at large deformation.     

Large amplitude oscillatory shear of pseudoplastic and elastoviscoplastic materials

We explore the utility of strain-controlled large amplitude oscillatory shear (LAOS) deformation for identifying and characterizing apparent yield stress responses in elastoviscoplastic materials. Our approach emphasizes the visual representation of the LAOS stress response within the framework of Lissajous curves with strain, strain rate, and stress as the coordinate axes, in conjunction with quantitative analysis of the corresponding limit cycle behavior. This approach enables us to explore how the material properties characterizing the yielding response depend on both strain amplitude and frequency of deformation. Canonical constitutive models (including the purely viscous Carreau model and the elastic Bingham model) are used to illustrate the characteristic features of pseudoplastic and elastoplastic material responses under large amplitude oscillatory shear. A new parameter, the perfect plastic dissipation ratio, is introduced for uniquely identifying plastic behavior. Experimental results are presented for two complex fluids, a pseudoplastic shear-thinning xanthan gum solution and an elastoviscoplastic invert-emulsion drilling fluid. The LAOS test protocols and the associated material measures provide a rheological fingerprint of the yielding behavior of a complex fluid that can be compactly represented within the domain of a Pipkin diagram defined by the amplitude and timescale of deformation.     

Normal stress differences in large-amplitude oscillatory shear flow for the corotational ��ANSR�� model

Using the integral form of a nonlinear corotational model, we derive explicit analytical expressions for the zeroth, second, and fourth harmonics of the second normal stress difference in large-amplitude oscillatory shear (LAOS). This model yields an arbitrary normal stress ratio (ANSR) in any simple shearing deformation, including LAOS. This corotational ANSR model adds one parameter to the corotational Maxwell model, a time constant ?? ₀ controlling the ratio ??₂/??₁ for both the real and imaginary parts of each harmonic of the normal stress difference. The explicit analytical expressions for all harmonics of the alternating shear stress and first normal stress difference responses in LAOS match those obtained previously for the corotational Maxwell model. We evaluate the corotational ANSR model for the case of a single Maxwell relaxation time fluid.     

A new approach for calculating the true stress response from large amplitude oscillatory shear (LAOS) measurements using parallel plates

The parallel plates geometry is often deemed unsuitable for nonlinear viscoelasticity measurements because the strain field, and thus the nonlinear response, varies across the sample. Although cone–plate and Couette geometries are designed to circumvent this problem by ensuring a uniform strain field, it is not always easy to shape the material to the complex shapes that is required for these geometries. This has motivated the development of techniques to accurately determine the nonlinear stress response using the more convenient plate–plate geometry. Here, we introduce a new approach to obtain this true material response in large amplitude oscillatory shear (LAOS) experiments using the plate–plate geometry. By tracing the Fourier components of the torque response and their derivatives with respect to the maximum applied deformation, we accurately obtain the material’s true stress–strain response from parallel plate measurements. The approach does not require any assumptions about the material’s viscoelastic behavior. We test our approach experimentally on fibrin biopolymer gels, as well as numerically on a Giesekus model. We confirm in both cases that our approach captures the detailed shape of the true stress response in LAOS measurements. Moreover, we also show that our method is less sensitive to experimental noise present in the data than the previous standard method. Our approach for obtaining the true stress response from parallel plate measurements is directly applicable to measurements on a wide range of solid-like nonlinear materials, including biological networks, tissues, or hydrogels.     

Differences between stress and strain control in the non-linear behavior of complex fluids

Various techniques have been proposed to characterize the behavior in the non-linear regime. A new theoretical framework, as proposed recently by Ewoldt et al. (J Rheol 52(6):1427–1458, 2008), provides a quantitative analysis of Lissajous figures during large-amplitude oscillatory shear (LAOS). Intra- and intercycle non-linearities, strain stiffening and softening, and shear thinning and thickening are described and can be distinguished. The new LAOS framework from Ewoldt et al. has been extended to a sinusoidal stress input. Measurements on two different samples reveal significant different results for sinusoidal strain or sinusoidal stress input. For both sinusoidal inputs, the results have been verified by cyclic stress and strain loading tests. The sinusoidal input tests are analyzed as an oscillatory test by the rheometer software and firmware, whereas the cyclic loading tests are purely rotational tests. Since both types of testing give the same results, any instrumental artifacts can be excluded. This implies that complex fluids can behave differently whether periodic stress or strain input functions outside the linear visco-elastic range are applied. All tests in controlled strain and stress in rotational and oscillatory modes have been performed with the same rheometer based on an air bearing-supported electrically commutated synchronous motor.     

Large amplitude oscillatory shear behavior of PEO-PPO-PEO triblock copolymer solutions

Rheological properties of poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEO-PPO-PEO) triblock copolymer solution in both linear and nonlinear regions have been investigated. PEO-PPO-PEO triblock copolymer solution shows a dramatic change in mechanical properties as temperature changes. PEO-PPO-PEO triblock copolymer undergoes a transition from sol to gel with increase of temperature. During this transition the copolymer solution passes through three different stages, namely sol, soft gel, and hard gel. In our previous research (Hyun et al. in J Non-Newtonian Fluid Mech 55:51–65, 2002), large amplitude oscillatory shear (LAOS) behavior was found to be very sensitive to the generated microstructures. In this study, we investigated the relationship between the LAOS type and the microdomain structure. Newtonian behavior is observed in sol region, while there appear two kinds of LAOS types in the soft gel region. One is type I (G′, G′′ decreasing) and the other is a combination of type I and type IV (G′, G′′ increasing followed by decreasing). Type III (G′ decreasing, G′′ increasing followed by decreasing) is observed in the hard gel region. We compared the shape of stress curves, Lissajous pattern, and Fourier transform (FT) rheology of hard gel and soft gel under LAOS, and tried to relate the complex LAOS behavior with the microstructural change. From these investigations, it was found that the LAOS behavior and the stress pattern at large strain are closely related to the microdomain structure of PEO-PPO-PEO triblock copolymer, and provide a lot of useful information on the microstructures induced by large deformation.     

Fluid inertia in large amplitude oscillatory shear

Homogeneous shearing is required in sliding plate flow experiments with one plate fixed and the other oscillating. However, when fluid inertia becomes significant, the velocity gradient and the stress will not be uniform. MacDonald et al. (1969) and Schrag (1977) investigated this effect for a linear viscoelastic fluid. However, linear viscoelasticity does not describe the behavior of melts in large amplitude oscillatory shear (LAOS). Jeyaseelan et al. (1993) have shown that the Berkeley kinetic network model does accurately describe the LAOS behavior of polymer melts. In this work, the Berkeley model is solved for LAOS in sliding plate flow with fluid inertia, by numerical integration of spatially discretized forms of the governing equations. Nonlinear viscoelasticity is predicted to aggravate the effects of fluid inertia in LAOS and experiments confirm this. Specifically, fluid inertia amplifies the first harmonic and produces no even harmonics. Operating limits are presented graphically for minimizing inertial effects in LAOS experiments. Received: 2 January 1998 Accepted: 27 April 1998     

Large amplitude oscillatory shear behavior of graphene derivative/polydimethylsiloxane nanocomposites

Rheological properties of three different nanocomposites, consisting of graphene oxide (GO), reduced graphene oxide (rGO), and polyhedral oligomeric silsesquioxane grafted reduced graphene oxide (rGO-POSS) as nanofillers and polydimethylsiloxane (PDMS), were investigated by large amplitude oscillatory shear (LAOS). The viscoelastic nonlinearity of the three nanofluids groups was studied by Lissajous curves, local nonlinear viscoelastic moduli of an oscillatory shear cycle, and Fourier transform rheology as a function of filler concentration and increasing and decreasing strain magnitude. The nonlinear behavior of the nanofluids was compared to understand the variation of internal microstructures. Firstly, GO/PDMS composites behave with higher moduli and smaller linear viscoelastic range comparing to that of other two composites. Secondly, the elastic stress Lissajous curves of these composites changed from elliptic to rectangular with round the corner with increasing the filler level and strain amplitude. Thirdly, all these three nanofluids exhibited intra-cycle strain stiffening with increasing strains and shear thickening at intermediate strain and then shearing thinning with increasing strain further. Fourthly, higher harmonic intensity of rGO/PDMS increased with increasing strain and came to a plateau, while that of other two nanofluids reached a maximum and then decreased. It suggested that different surface functionalization of nanoparticles will present different rheological behavior due to formed different network and LAOS could be used as a potential helpful method to characterize rheological properties of nanocomposites, especially at higher shear strain.     

A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids

A constitutive model for elasto-viscoplastic thixotropic materials is proposed. It consists of two differential equations, one for the stress and the other for the structure parameter, a scalar quantity that indicates the structuring level of the microstructure. In contrast to previous models of this kind, the structure parameter varies from zero to a positive and typically large number. The lower limit corresponds to a fully unstructured material, whereas the upper limit corresponds to a fully structured material. When the upper limit is finite, the model represents a highly shear-thinning, thixotropic, and viscoelastic liquid that possesses an apparent yield stress. When it tends to infinity, the behavior of a true yield-stress material is achieved. Predictions for rheometric flows such as constant shear rate tests, creep tests, SAOS, and large-amplitude oscillatory shear (LAOS) are presented, and it is shown that, in all cases, the trends observed experimentally are faithfully reproduced by the model. Within the framework of the model, simple explanations are given for the avalanche effect and the shear banding phenomenon. The LAOS results obtained are of particular importance because they provide a piece of information that so far is absent in the literature, namely a quantitative link between the Lissajous–Bowditch curve shapes and rheological effects such as elasticity, thixotropy, and yielding.     

Nonlinear rheological behavior of diphenylmethylvinyl silicone gum: an example of homogeneous shear

The present work shows, based on an effective particle-tracking velocimetric (PTV) method, that a commercial diphenylmethylvinyl silicone gum displays nonlinear rheological responses to startup shear and large amplitude oscillatory shear (LAOS) in a homogeneous manner, in contrast to monodisperse melts. Using an effective cone-partitioned plate (CPP) setup along with PTV, rheological characterizations of shear thinning in continuous shear and strain softening in LAOS are carried out reliably without the inherent experimental complication associated with edge fracture. Conversely, based on the CPP, the rheological effects of edge fracture are also illustrated for both startup shear and LAOS.     

Comparison of stress-controlled and strain-controlled rheometers for large amplitude oscillatory shear

In linear viscoelastic region, it is well known that dynamic modulus and dynamic compliance can be converted to each other. However, it is questionable whether there exists an interconversion between large amplitude oscillatory shear (LAOS) data measured from different types of rheometers—stress-controlled and strain-controlled rheometers. Hence, we tried to prove the existence by use of polyethylene oxide (PEO) aqueous solutions with well-developed entanglements. From this experiment, we can conclude that a stress-controlled rheometer can simulate LAOS behavior measured from a strain-controlled rheometer under the conditions where inertia effect is not significant. Furthermore, it is investigated whether the LAOS data of the stress-controlled rheometer obey stress–frequency superposition as the strain–frequency superposition found by Cho et al. (J Rheol 54:27–63, 2010) from LAOS data measured by the strain-controlled rheometer. This scaling relation shows that the dimensionless stress amplitude is a function of zeta which is the product of the stress amplitude and linear viscoelastic function J′(ω). The plot shows that all of the data are superposed in a single curve without regard to frequency, molecular weight, and concentration of PEO aqueous solutions.     

Polymer motion as detected via dielectric spectra of 1,4-cis-polyisoprene under large amplitude oscillatory shear (LAOS)

In order to investigate the global polymer chain motion under large amplitude oscillatory shear (LAOS), the dielectric properties under LAOS are measured by a new rheo-dielectric combination. The design of the rheo-dielectric setup including a new fixture and modified oven is explained in detail. For 1,4-cis-polyisoprene, having type-A dipoles parallel to the backbone, the dielectric dipoles can detect the global polymer chain motion via the end-to-end vector. Thus rheological and dielectric (rheo-dielectric) properties reflect the dynamics of the polymer chain motion under LAOS. In this study, we investigate the rheo-dielectric properties under LAOS with 1,4-cis-polyisoprene as model component. As the strain amplitude was increased under LAOS, the relaxation strength from dielectric properties decreased for the whole spectra without changing the shape of the dielectric spectra. These results are analyzed on the basis of the molecular model of dynamic tube dilation (DTD) induced by the convective constraint release (CCR). It is found that the global chain motion under LAOS flow is affected by both rheological frequency and strain amplitude. It is also observed that segmental motion is affected via the oscillatory frequency under LAOS. This result differs from experiments under steady shear.     

Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure

The paper introduces a novel procedure to determine the non-linear parameter of the Giesekus model, in relation to the characterization of the non-linear oscillatory shear regime of viscoelastic polymer solutions based on polyacrylamide. Instead of using the shear-thinning viscosity as the representative non-linear effect, the third harmonic in the Fourier spectrum of the shear stress response signal is considered for computing the mobility factor. The fluid is subjected to large amplitude oscillatory shear (LAOS) and its response is recorded. Deviations of this signal from the sinusoidal form are specific to each material and gives both qualitative and quantitative measures of the non-linearity. By fitting the material response with the corresponding numerical solutions of the n-modes Giesekus constitutive relation, one can extract the values of the non-linear α_i-parameters that describe the fluid rheology. It is demonstrated that this procedure, which can be successfully applied to semi-concentrated polymer solutions, provides better results than the classical viscosity-fit method.     

Rheological behavior of fiber-filled polymers under large amplitude oscillatory shear flow

Small and large amplitude oscillatory shear measurements (SAOS and LAOS) were used to investigate the rheological behavior of short glass fibers suspended in polybutene and molten polypropylene. Raw torque and normal force signals obtained from a strain-controlled instrument (ARES rheometer) were digitized using an analog to digital converter (ADC) card to allow more precise data analysis. The fiber concentration did not affect the torque signal in the SAOS mode, except for its magnitude, whereas the normal force signal was too low to be measurable. With increasing strain amplitude, the magnitude of the torque became a function of time. Depending on the applied frequency and strain rate, the stress in the filled polybutene increased with time, whereas for reinforced polypropylene (viscoelastic matrix), the behavior was opposite, i.e. the stress decreased with time. These effects were more pronounced at high fiber content. In addition the primary normal stress differences were no longer negligible at large deformation amplitude and exhibited a non-sinusoidal periodic response. Fast Fourier transform (FFT) analysis was performed and the resulting spectra, along with Lissajous figures of the shear stress and the primary normal stress differences, are explained in terms of fiber orientation. The experimental results for the suspensions in polybutene are well predicted by the Folgar-Tucker-Lipscomb (FTL) model.