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We derive new analytical solutions for the nonaffine Johnson–Segalman/Gordon–Schowalter (JS/GS) constitutive equation with a general relaxation kernel in medium-amplitude oscillatory shear (MAOS) deformation. The results show time-strain separable (TSS) nonlinearity, therefore providing new physically meaningful interpretation to the heuristic TSS nonlinear parameter in MAOS [Martinetti and Ewoldt, Phys. Fluids 31, 021213 (2019)]. The upper-convected, lower-convected, and corotational Maxwell models are all subsets of the results presented here. The model assumes that the microscale elements causing stress in the material slip compared to the continuum deformation. We introduce a visualization of the nonaffine deformation field that acts on stress-generating elements to reinforce the physical interpretation of the JS/GS class of models. Finally, a case study is presented where previously published results, from fitting TSS models to MAOS data, can be reinterpreted based on the concept of nonaffine motion of the JS/GS framework. [ABSTRACT FROM AUTHOR]

We construct a microstructure-based constitutive model that successfully predicts experimental rheology signatures that no other model has previously described. The experimental observations are the low-dimensional descriptions of asymptotically nonlinear oscillatory shear [Ewoldt and Bharadwaj, Rheol. Acta 52, 201-209 (2013)], also known as medium-amplitude oscillatory shear, characterized by four frequency-dependent material measures: [e1]ω, [e3]ω, [υ1]ω and [υ3]ω. These slightly nonlinear rheological measurements are the systematic step beyond linear viscoelastic characterization. The material is a transiently crosslinked polymeric hydrogel of aqueous polyvinyl alcohol cross-linked by sodium tetraborate (borax) [Bharadwaj and Ewoldt, J. Rheol. 59, 557-592 (2015)], which shows nonlinear elastic stiffening inferred from [e1]ω>0. Here, we hypothesize that the appropriate physical model is a transient network of strain-stiffening elastic elements. We rationalize that all nonlinearities are driven by the instantaneous stretch magnitude Q between junctions, either through strain-stiffening of network elements or through deformation-assisted network structuring. These two physical aspects are embedded into a single nonlinear parameter that successfully captures both elastic energy storage ([e1]ω and [e3]ω]) and viscous energy dissipation ([υ1]ω and [υ3]ω), including frequency-dependent sign changes. Analytical results are derived for all four asymptotic nonlinearities. The quantitative agreement provides fit parameters that are related to molecular features and network architecture. While the work here is focused on a specific polymeric system, it represents the broad potential contribution of asymptotic, leading-order nonlinearities to enable structurerheology insight, constitutive model development, and model selection for soft materials in general. [ABSTRACT FROM AUTHOR]

Rheologists have expectations for signatures of linear viscoelastic properties, such as shapes of G'(ω) and G?(ω). Medium amplitude (or asymptotically-nonlinear) oscillatory shear (MAOS) provides additional nonlinear rheological information with low dimensional, well-defined material functions [Ewoldt and Bharadwaj, Rheol. Acta 52, 201-209 (2013)]. Here, we develop expectations of signatures (or fingerprints) for the four asymptotically-nonlinear material functions associated with MAOS, [e1](ω), [e3](ω), [v1](ω), [v3](ω). Although the linear fingerprints may be identical for different models, the asymptotically-nonlinear fingerprints may be different in magnitude, frequency-scaling, curve shapes, and sign changes. To perform the analysis, we collect/translate a library of available analytical strain-controlled MAOS fingerprints for seven different constitutive models. Using this library, we identify general trends and highlight key differences of asymptotic-nonlinear viscoelasticity. Asymptotic nonlinearities for all models considered here obey the terminal regime inter-relations and frequency scaling predicted by Bharadwaj and Ewoldt [J. Rheol. 58, 891-910 (2014)]. Unlike the positive linear viscoelastic measures, at least one of the four asymptotic nonlinearities changes signs with Deborah number (De). Following sign interpretations of Ewoldt and Bharadwaj [Rheol. Acta 52, 201-209 (2013)], we show that nonlinearities tend to be driven by strain-rates at small De, and by strains at large De, a trend observed for nearly all the constitutive models studied here, the exception being the model for dilute rigid dumbbell suspensions of Bird et al. [J. Chem. Phys. 140, 074904 (2014)]. Some constitutive models exhibit multiple sign changes at intermediate De and there may be no universal behavior of asymptotically-nonlinear fingerprints in this regime. Therefore, frequency-dependent signatures can be material-specific. This will allow inverse problems to infer structure, select models, and fit model parameters using asymptotically-nonlinear signatures. To illustrate this aspect, we demonstrate a fingerprint matching exercise with experimental measurements on a transiently cross-linked hydrogel system. We find that currently available model fingerprints can match the qualitative magnitudes and frequency dependence, but not the signs of the experimental transient network response. [ABSTRACT FROM AUTHOR]

We use a fourth-order fluid expansion to make general predictions for asymptotically nonlinear material functions in oscillatory shear, a characterization protocol sometimes known as mediumamplitude oscillatory shear. The calculation applies to any viscoelastic fluid in the terminal regime defined by the limit of Deborah number much less than one. Two viscous nonlinearities appear at third order, with shear stress scaling as ω³, and are interrelated by a constant multiplicative factor. Two elastic nonlinearities appear at fourth order, with shear stress scaling as ω4, and are also interrelated by a constant multiplicative factor. These nonlinear measures are decoupled from the linear material functions G' and G" because they depend on different expansion coefficients. Experimental measurements are presented for all four asymptotic shear material functions using a linear and well-entangled homopolymer of polyisoprene. The experimental observations are consistent with both the predicted frequency scaling and the predicted interrelations in the terminal regime. Signs and magnitudes cannot be universally predicted, leaving these as free parameters that depend on the specifics of the material microstructure or constitutive model. These general results explain previous observations involving different materials and constitutive models, and provide an important reference for future experiments, analytical results, and numerical computations of these rheological fingerprints. [ABSTRACT FROM AUTHOR]