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.

## I. INTRODUCTION

Weakly nonlinear analysis of complex fluid rheology is excellent for relating rheology to structure and studying material-level physics of fluids. Weakly nonlinear rheological characterization, such as medium-amplitude oscillatory shear (MAOS) [1–9] and medium-amplitude parallel superposition (MAPS) [10], produces more information than linear viscoelastic measures while staying mathematically tractable to theoretical prediction. For example, recently, our group collaborated [11] to settle a 70-year debate and infer the nonlinear mechanisms of an aqueous viscoelastic liquid reversible polymer network by combining MAOS measurements with a novel asymptotically nonlinear viscoelastic model [12] and the Polymer Reference Interaction Site Model (PRISM) [13].

Here, we derive new analytical predictions for MAOS for a nonlinear viscoelastic fluid constitutive model, wherein the nonlinear parameter (which can be fit to experimental observations) can be interpreted via nonaffine deformation of the material structures that “slip” compared to the continuum deformation. Specifically, we consider the Johnson–Segalman (JS) integral model [14], which is equivalent, in the single-mode Maxwell relaxation limit, to the Gordon–Schowalter (GS) derivative applied to the Maxwell model [15,16]. Several MAOS solutions for various models are recovered as subsets of this JS/GS framework by tuning the nonlinear parameter [upper-convected, lower-convected, and corotational Maxwell (CM) [17]].

The results derived here provide a new model to be considered when fitting MAOS data, adding to the existing toolbox of known analytical results [2,18,19]. Not only does this support better checks on model credibility when fitting by enabling multiple possible models to fit rheological data [20], but also it provides a physical interpretation for inference of the fluid physics, in contrast to otherwise heuristic existing models including the CM model and generalized time-strain separable (TSS) integral model [21]. Recently, Song *et al*. expanded the library of MAOS solutions extensively by analytically deriving the MAOS material functions for several nonlinear viscoelastic constitutive models, including the JS/GS model in the special case of single-mode relaxation [19]. Here, we derive the analytical results of the general JS integral model that allows for any type of relaxation function and hence offers more flexibility in fitting data and expands the list of materials that can be studied using this model. For example, a power-law relaxation function $G(t)=St\u2212n$ that is observed for critical gels [22] can only be modeled with JS/GS nonaffine deformation using results presented here. In addition, this work offers a detailed analysis that illustrates the physical interpretation of this model and its nonaffine deformation.

The outline of our contributions is as follows. We theoretically derive the MAOS intrinsic material functions for the JS/GS model and show the results for the single-mode relaxation case. In the discussion, we first show that the model is TSS, thus providing a route to interpret, at least for a certain range of the nonlinear TSS parameter, the general TSS class of models that was not found before. We continue the discussion by introducing a visualization of the nonaffine deformation field, which acts on stress-generating elements to reinforce the physical interpretation of the JS/GS class of models. The discussion also includes a comment on interpreting the model physically and how it applies to different material classes. Finally, a case study is presented where previously published results, from heuristically fitting TSS models to MAOS data, can be reinterpreted based on the concept of nonaffine motion of the JS/GS framework.

## II. BACKGROUND

### A. Medium-amplitude oscillatory shear

Oscillatory deformation is preferred for weakly nonlinear rheometry, compared to step input forcing, since oscillations lock in at each frequency (Deborah number) to provide a high signal-to-noise ratio for the weakly nonlinear signals. Imposing a simple shear strain input $\gamma (t)=\gamma 0sin\omega t$, the shear stress response $\sigma 21$ can be written as

where $\sigma n\u2032and\sigma n\u2032\u2032$ are Fourier coefficients [23]. Even harmonics of *n* are excluded for time-periodic shear-symmetric responses, for which stress is an odd function of strain and only *n = *odd are required. The linear (small-amplitude oscillatory shear, SAOS) and weakly nonlinear MAOS regimes at any frequency $\omega $ are defined by a power-series expansion with respect to the amplitude ($\gamma 0$ or $\gamma \u02d90=\gamma 0\omega $) [1,24–26], for example, as in [1],

where the coefficients of the expansion depend on frequency but not on amplitude. Here, $G\u2032(\omega )$ and $G\u2032\u2032(\omega )$ are the two linear viscoelastic moduli, and four parameters are required to fully describe the asymptotic deviation from linearity: two for the first-harmonic response and two for the third-harmonic response. Here, we have used an expansion with respect to strain amplitude, $\gamma 03$, but this is a subjective choice and expansions can also be made with respect to strain-rate amplitude $\gamma \u02d903$, e.g., see [17], their Eq. (9). Noninteger power expansions have also been observed experimentally [27], though with limited theoretical prediction [28].

The measured stress harmonics are often plotted as a function of strain amplitude, as shown in Fig. 1 for the model we consider in this paper (see Secs. II B and III for model details). When normalized by strain amplitude, the first-harmonics approach asymptotic plateaus equal to the linear viscoelastic moduli *G′* and *G″* in the limit of amplitude → 0. At any nonzero strain amplitude, nonlinearities are always present, in the sense of a power-law expansion. These ever-present nonlinearities cause a deviation from the linear response first-harmonic plateau and additionally generate third harmonics in the shear stress signal, as in Eq. (2). The labeled SAOS limit in Fig. 1 is defined by the domain of strain amplitude where all normalized first harmonics of the fully nonlinear solution deviate by no more than 1% from the linear plateau. Similarly, the weakly nonlinear MAOS regime, which includes only the first deviation from linearity, is defined by comparing the deviation of the fully nonlinear solution to the truncated MAOS expansion defined in Eq. (2) (up to order $\gamma 03$). Since four independent signals are present in the MAOS solution, we define the end of the MAOS regime when any one of the four MAOS truncated stress signals deviates more than 1% from the fully nonlinear solution. The choice of 1% deviation is a subjective choice (0.1% or 10% may also be reasonable) and depends on the accuracy and resolution of the experimental measurement, theoretical model, or numerical simulation.

The four MAOS measures (expansion coefficients) can be represented in various ways. Two options are shown in Eq. (2), the latter of which involves the Chebyshev expansion coefficients $[e1](\omega ),[v1](\omega ),[e3](\omega ),and[v3](\omega )$ as defined in [1], where “*e*” is for elastic with SI units (Pa), “*v*” is for viscous with SI units (Pa s), and a notable negative sign appears in front of [*e*_{3}] due to the conversion of Chebyshev to Fourier coefficients [25]. The Chebyshev expansion coefficients describe oscillatory waveforms visualized as Lissajous–Bowditch curves (hysteresis loops) of stress-versus-strain and stress-versus-rate, but it is convenient to use the Fourier representation in Eq. (2) for signal processing. The Chebyshev coefficients offer physical interpretation for elastic and viscous nonlinearities of all four MAOS nonlinearities (see [1], their Figs. 6 and 7), e.g., in contrast to time-domain Fourier coefficients or magnitude nonlinearities alone [5]. The MAOS material functions of Eq. (2) can also be related to an asymptotic power expansion of the Sequence of Physical Processes (SPP) LAOS framework of Rogers and co-workers, as recently shown in [29], their Eqs. (F1) and (F2), which clearly show that the SPP metrics identify so-called “elastic” and “viscous” effects differently from the Chebyshev expansion of Eq. (2).^{1}

For example, a *purely elastic* weakly nonlinear response $\sigma 21(\gamma )$ will generate only elastic coefficients *G*′, [*e*_{1}], and [*e*_{3}] in the Chebyshev expansion, whereas it generates non-zero *G*′, *G _{t}*, and “viscous”

*η*in the SPP framework. Similarly, a purely viscous nonlinear response $\sigma 21(\gamma \u02d9)$ generates only viscous Chebyshev expansion coefficients, but generates

_{t}*η*′ and both

*η*and “elastic”

_{t}*G*in SPP.

_{t}### B. Constitutive model

We consider the JS nonaffine deformation model [14], or equivalently the generalized GS Maxwell model [15].

Gordon and Schowalter modified the molecular theory of elastic dumbbells (simplified polymer strands) to allow nonaffine deformation by adding slip between the velocity gradient felt by the dumbbells and the gradient imposed by the continuum deformation. The resulting continuum level stress-strain constitutive equation, derived by [30], is given by

where $\sigma __\u25a1$ is the Gordon–Schowalter convected derivative defined [16] as

where $D/Dt=\u2202/\u2202t+v_\u22c5\u2207_$ is the substantial derivative, $\omega __=12(\u2207_v_\u2212(\u2207_v_)T)$ is the vorticity tensor, and $\gamma \u02d9__=\u2207_v_+(\u2207_v_)T$ is the rate of deformation tensor in the continuum velocity field $v_(x_,t)$. This constitutive model is equivalent to the Maxwell model, but with replacing the time derivative with the GS convected derivative, and therefore it is referred to as the GS Maxwell model, which is a generalization of the upper-convected Maxwell (UCM) model.

A different but equivalent route used by Johnson and Segalman was to use the Lodge integral equation and replace the velocity gradient with a nonaffine velocity gradient, eventually resulting in an integral form model given by

where $S__$, defined in Sec. III, is a measure of the rate of strain associated with the nonaffine velocity and $GA(t)$ is a time-dependent kernel function that is equivalent to the stress relaxation modulus in the affine limit of the model. While Gordon and Schowalter were motivated by polymer solutions, Johnson and Segalman did not restrict their model to a specific microstructure as will be discussed more in Sec. IV. Moreover, the GS Maxwell model in Eq. (4) is a subset of the JS model in Eq. (5). In this work, we will use this integral form to arrive at the MAOS solutions for its simplicity and generality. There seems to be a naming confusion in the literature [16,19,31], where the single-mode GS Maxwell model is sometimes misnamed as the JS model. The mathematical response of the single-mode GS differential model is identical to the single-mode JS integral model, but the JS model in integral form is more general, allowing any relaxation kernel including those that are not easily represented as a sum of exponentials. We hope the background information presented above and in Sec. III clarifies the different origins of the two models.

In both cases, GS and JS, the nonaffine motion can be interpreted as introducing a nonaffine “effective” velocity gradient tensor

which is used for the calculation of stress [see Eqs. (5), (10), and (11)], where *a* is the nonlinear model parameter. In Eq. (6), the convention used to evaluate components of the velocty gradient tensor $\u2207_v_$ is $\u2207ivj=\u2202vj/\u2202xi$ (unlike the definition in the work of [14]). The deformation resulting from this gradient, illustrated in Fig. 1, “slips” compared to the affine velocity field but still involves stretching and rotation (or only rotation in the limit of *a* = 0, equivalent to the CM model). The affinity parameter (or slip parameter) *a* is the only nonlinear parameter of the model. In various limits, it recovers other known models as a subset, e.g., *a* = 1 recovers the Lodge integral model or UCM differential model, *a* = −1 is the lower-convected Maxwell model, and *a* = 0 is the CM model [16].

## III. RESULTS: THEORETICAL DERIVATION OF MAOS SIGNATURES

Before deriving the MAOS material functions of the JS/GS model, a brief outline of the procedure is given here. First, an oscillatory homogeneous simple shear velocity field is imposed as an input to the constitutive model to compute the output shear stress as a function of time. This equation of stress was derived in the work of Johnson and Segalman, and we repeat the steps below to avoid confusion related to the change in notation. In particular, here we use the rate of strain tensor $\gamma \u02d9__$ instead of the deformation tensor $D=12\gamma \u02d9__$, and we use $GA(t)$ as the affine relaxation modulus which is equivalent to $12$ *G ^{JS}*(

*t*) where

*G*(

^{JS}*t*) refers to what Johnson and Segalman [14] called “

*G*(

*t*)” in their Eq. 2.25. Our notation avoids unnecessary factors of $12$ when relating $GA(t)$ to the material function of stress relaxation modulus

*G*(

*t*) [see Eq. (23) in Sec. III A]. Next, the resultant stress solution is expanded in amplitude and in frequency. The order of the expansions does not change the results, but here we choose to expand first in amplitude (up to third order, i.e., $\gamma 03$) for mathematical convenience. To expand in frequency, the Euler–Fourier equations are applied to the amplitude-expanded stress solution to obtain the stress harmonics defined in Eq. (1) up to third order in $\gamma 0$. Using the result, the MAOS material functions are readily identified. The choice of MAOS material functions to be reported is not unique, but rather depends on the chosen representation of the stress expansion in both frequency and amplitude. Here, we adopt the MAOS material functions defined by Ewoldt and Bharadwaj; if desired these can be converted to other representations in the literature [1].

### A. General relaxation kernel result

Following the above procedure, we start by assuming a one-dimensional homogeneous simple shear flow and represent the components of the Cartesian velocity field as

where $\gamma \u02d9(t)$ is the rate of strain. The rate of strain tensor is calculated as $\gamma \u02d9__=\u2207_v_+(\u2207_v_)T$, furnishing

The choice of $\gamma (t)=\gamma 0sin\omega t$ follows the convention of an oscillatory characterization protocol [23], giving the strain rate input of the form

To compute the stress, we use the integral form of Johnson and Segalman from Eq. (5). The nonaffine rate of strain measure is defined as

where the measure of accumulated strain $E__(t\u2032,t)$ is obtained by solving the differential equation

with the initial condition $E__(t\u2032,t\u2032)=I__$. It is important to notice that in the linear limit of small strains, Eq. (5) reduces to the stress equation given by the Boltzmann superposition as

with the effective “Nonaffine” (NA) strain-rate being $\gamma \u02d9__NA(t\u2032)=a\gamma \u02d9__(t\u2032)$. Moreover, the stress equation reduces to the Lodge equation in the affine limit of $a=1$ [16].

For homogeneous unsteady simple shear flow [as defined in Eq. (7)], the set of differential equations in Eq. (11) can be solved to find the components of $E__(t\u2032,t)$ [see [14], but note the difference in notation for $\gamma \u02d9(t)$^{2}

We changed the symbol for shear rate from *k* to $\gamma \u02d9$.

where $\lambda =121\u2212a2$ and $s(t,t\u2032)=\u222bt\u2032t\gamma \u02d9(t\u2032\u2032)dt\u2032\u2032$. The components of the nonaffine rate of strain measure are then calculated from Eq. (10) as

Equation (14) holds for any unsteady homogeneous simple shear defined by $\gamma \u02d9(t)$. It is now possible to calculate the shear stress from the 21-component of $S__(t\u2032,t)$ from Eq. (14) as

that can be rewritten using Eq. (14) as

This same result was obtained in the work of Johnson and Segalman in their Eq. (3.8) [14], noting the difference in notation as previously mentioned, where $GA(t)$ in our equation replaces ½ *G*^{JS}(*t*) (*G*^{JS} refers to what Johnson and Segalman called “*G*”).

For an oscillatory deformation field defined in Eq. (9) the integral involving the shear rate in Eq. (16) is evaluated and the following expression is obtained for the shear stress:

Here, we assumed the oscillations started at $t=\u2212\u221e$ to obtain the steady time-periodic (alternance state) shear stress response. To evaluate the time integral in Eq. (17), it is convenient to carry out a variable transformation $t\u2212t\u2032=s$ that reduces it to

At this point, we reach the fully nonlinear time-dependent solution, and the necessary expansions are taken next. In the asymptotic limit of small strain amplitude, $\gamma 0\u21920$, it is possible to expand the cosine term involving $\gamma 0$ in Eq. (18) as a Taylor series about $\gamma 0=0,$ resulting in a simplified expression

Applying the Euler–Fourier equations results in the expanded stress harmonics, which are coefficients of the $cos\omega t,sin\omega t,cos3\omega t,sin3\omega t$ terms. This operation further simplifies Eq. (19) and allows the expression of shear stress as a third-order power-series expansion in strain amplitude $\gamma 0$, where each order is subsequently separated into orthogonal harmonics as

The SAOS and MAOS functions are identified by comparing Eq. (20) to the shear stress expansion given in Eqs. (1) and (2). The linear viscoelastic material functions, e.g., storage and loss moduli, appear as coefficients with the first power of $\gamma 0$

By comparing Eqs. (21) and (22) to the definition of the linear storage and loss moduli, we can deduce that the effective stress relaxation modulus in the nonaffine JS/GS model is given by

The relaxation modulus $G(s)$ is well-behaved and finite in the limit of $a\u21920$, but this requires the magnitude of the affine relaxation modulus $GA$ to tend to $\u221e$ in this limit that is equivalent to the CM model [14]. Here, we choose to follow the convention of [23] and use the intrinsic measures of MAOS as defined in Eq. (2), which for the JS model with generic kernel *G*(*s*) are

Equations (24)−(27) represent the first of three major theoretical contributions of our work here. As mentioned in Sec. I, recent results of [19] [Eq. (6) in that work] can be retrieved as a subset of our equations (24)−(27) by assuming a single-mode Maxwell effective relaxation modulus (also note a difference in notation, where their *ζ* is related to our *a* as $\zeta =1\u2212a$).

Another material function defined to study the MAOS regime was introduced by Hyun and Wilhelm [5] and it is related to the Chebyshev metrics above [1,24] by

The results of this work can be used to calculate this MAOS measure *Q*_{0}, but we prefer using Eqs. (24)−(27) as the MOAS material functions to study the independent contributions of the different nonlinearities present in the system, which have distinct interpretations [1].

It follows from Eqs. (24)−(27) that the MAOS material functions are linear combinations of the linear viscoelastic storage and loss moduli, evaluated at different frequencies. This is a general feature of TSS MAOS signatures [21], as will be discussed in Sec. IV A. Furthermore, the above expressions allow us to compute oscillatory shear material functions as a function of the effective time-dependent relaxation modulus $G(s)$ and the affinity parameter *a*. In addition, the term $1\u2212a2$ appears as a front factor and changes the magnitudes of the nonlinearities, but not their signs. Therefore, as expected, the JS/GS model is only able to predict shear thinning, not thickening.

A physically meaningful range for *a* is $0<a\u22641$ and considering the extremes of this limit is insightful. Although mathematically *a* can take any value, it is unreasonable to assume that the stress causing rate of strain tensor $\gamma \u02d9__NA(t\u2032)=a\gamma \u02d9__(t\u2032)$ is greater than the imposed rate or of opposite sign to it. Furthermore, the visualization of streamlines and material deformation, based on the results of Sec. IV B, for $a<0$ generates physically unreasonable deformation (see Appendix A). The first extreme of the physical range, $a=1$, which corresponds to the affine limiting case, is incapable of predicting shear stress nonlinearities, confirmed with all four nonlinearities vanishing at $1\u2212a2=0$. This result agrees with the fact that in the affine limit, the JS/GS model reduces to the UCM model, which is known to have no shear thinning. On the other hand, the limit of $a\u21920$ corresponds to the CM model of Goddard and Miller and has the most nonlinear behavior, with a deformation history consisting of pure rotation, as will be shown in Sec. III B. As Johnson and Segalman [14] note in their work, $aGA(s)$ should stay nonzero as that limit is taken, otherwise the stress tensor will go to zero as well.

### B. Single-mode response

From the general MAOS solution for any relaxation function, we now explore the single-mode exponential relaxation to illustrate the behavior of the model, with

where $G0$ is the elastic modulus magnitude and $\tau 0$ is the relaxation time. While $G(s)$ can take many different mathematical forms, the single-mode relaxation is a building block to understand more complex relaxation spectra, as $G(s)$ can typically be approximated by a sum of exponential modes (Prony series).

The MAOS material functions for the single-mode JS/GS response are found by substituting Eq. (29) into Eqs. (24)−(27),

where the Deborah number, $De=\omega \tau 0$, is a dimensionless measure of frequency.^{3}

Gordon and Everage derived analytical expressions for the first harmonics $\sigma 1\u2032and\sigma 1\u2032\u2032$ in the LAOS regime for the single mode relaxation limit [30]. These expressions can be expanded to obtain $[e1]and[v1]$ in Eq. (30).

^{,}

^{4}

We note that a more complicated route to Eq. (30) could have started from the fully nonlinear LAOS analytical results of the Oldroyd 8-constant model, which the single-mode JS/GS is a subset of, derived by [18]. In that work, the fully nonlinear (not truncated) oscillatory shear stress solution is given for the single-mode relaxation. The JS/GS limit of the Oldroyd 8-constant model could be taken, and this stress can then in theory be expanded around $\gamma 0=0$ to obtain the single-mode MAOS material functions in Eq. (30).

*a*, which can be compared to other MAOS signatures as surveyed in [2].

The signs and shapes of the MAOS material functions in Fig. 2(b) hold important physical interpretation [1]. First, it is clear that for any relaxation modulus considered, the model shows elastic softening and viscous thinning based on the negative sign of $[e1](\omega )$ and $[v1](\omega )$ across all time scales. Moreover, the sign change present in $[e3](\omega )$ means that as the Deborah number is increased above $De=1$, the elastic softening will be driven by large instantaneous strain rather than large rate-of-strain. Similarly, for $[v3](\omega )$, the viscous thinning is driven by large strains for $De>0.3$ and by a large strain rate for lower Deborah numbers. The frequency dependence (e.g., location of sign changes) is independent of both *G*_{0} and *a*, whereas the magnitudes of the nonlinear MAOS functions are all linearly proportional to *G*_{0} and further depend on the affinity parameter *a* as shown in Fig. 2. We re-emphasize here that although $a\u21920$ shows the highest magnitude of nonlinearity, the affine relaxation modulus of the material has to go to infinity in this limit which is a nonphysical assumption. Combining modes with different time scales can introduce the possibility of more than one sign change in $[e3](\omega )$ and $[v3](\omega )$, which can be observed in continuous spectra models as well. The reader is referred to the work of Martinetti and Ewoldt for the shapes introduced by those relaxation spectra [21].

While the results in terms of effective modulus *G*(*t*) are useful for fitting experimental data, it is insightful to consider the *affine* relaxation modulus *G _{A}*(t) [see Eq. (23)]. In Appendix B, we analyze this perspective and study how the nonlinear MAOS functions change with the affinity parameter

*a*while keeping the affine modulus

*G*(

_{A}*t*) fixed. Interestingly, although decreasing

*a*increases the nonaffinity and hence introduces more nonlinearity in the material element deformation (Sec. IV B), for fixed affine relaxation modulus, the slip first increases but then

*decreases*the magnitude of the resulting nonlinear functions. This perspective is important for microstructural interpretation, e.g., if slip is truly the cause of the nonlinearity, then for a fixed affine relaxation modulus, the maximum nonlinearity occurs with the affinity parameter $a=1/3$, and not $a=0$.

## IV. DISCUSSION

### A. Time-strain separability

In recent work, Martinetti and Ewoldt derived the general functional form of the MAOS material functions of a TSS model for viscoelastic fluids. There, they differentiate between a TSS MAOS and a general TSS category, where in the latter the stress relaxation modulus in step strain tests can be factored into two independent functions of time and strain. On the other hand, a model is TSS MAOS if the stress can be written as

where $m(t)=\u2212dG(t)dt$, $\gamma =\gamma (t,t\u2032)$ is the accumulated strain between time *t′* and *t*, where the damping function *h* can be expanded as

where *A* is the TSS MAOS nonlinear parameter and its magnitude is proportional to the magnitude of the nonlinear material functions. Given this TSS MAOS form, the MAOS functions will be a linear combination of the linear viscoelastic functions [Eqs. 27(a)−27(d) in [21] with $Geq=0$]

Comparing these Eqs. (33)−(36) to the results of solving the JS/GS model in Eq. (30) proves that the JS/GS model belongs to the TSS MAOS class. Furthermore, the nonlinear parameter *A* is related to the JS/GS nonlinear parameter as

Equation (37) represents the second of three major theoretical contributions of this work. This result shows that all the properties of the general TSS MAOS result discussed in [21] apply to the JS/GS model solution derived here. More importantly, Eq. (37) provides a physical interpretation (in terms of nonaffine deformation) for the TSS parameter *A*, within the range $\u221216<A\u22640$ for $0<a\u22641$.

Contextualizing the JS/GS model with other known TSS MAOS models provides a phenomenological understanding of its nonlinear behavior and demonstrates the added ability to interpret TSS MAOS data. Table I shows a collection of TSS MAOS constitutive models. All these models share the same mathematical structure but differ in the value of the nonlinear parameter *A* and in their physical interpretation. While molecular models, such as Doi–Edwards IAA, and semiempirical models, such as JS/GS, have a distinct physical interpretation, phenomenological models such as the general TSS MAOS model do not. Moreover, when MAOS experimental data are found to be the best fit using a TSS MAOS mathematical form with a specific *A* value, material property inference is only possible when this value corresponds to a specific value predicted from the molecular level theory.

Constitutive model . | A
. | Interpretation . |
---|---|---|

TSS | A ∈ ℝ | N/A |

JS/GS | $\u22120.16\xaf\u2264A<0$ | Nonaffine slip in effective velocity gradient |

Corotational Maxwell (CM) | $\u22120.16\xaf$ | Pure rotation of material elements having infinite relaxation mode strength |

Doi–Edwards IAA^{a} | −0.238 | Instantaneous retraction of deformed polymer chains with independent alignment approximation [32] |

L-MSF^{b} | −0.138 | Stretch of polymer chains by strain-dependent reptation tube “squeezing” [33] |

Q-MSF^{c} | −0.038 |

Constitutive model . | A
. | Interpretation . |
---|---|---|

TSS | A ∈ ℝ | N/A |

JS/GS | $\u22120.16\xaf\u2264A<0$ | Nonaffine slip in effective velocity gradient |

Corotational Maxwell (CM) | $\u22120.16\xaf$ | Pure rotation of material elements having infinite relaxation mode strength |

Doi–Edwards IAA^{a} | −0.238 | Instantaneous retraction of deformed polymer chains with independent alignment approximation [32] |

L-MSF^{b} | −0.138 | Stretch of polymer chains by strain-dependent reptation tube “squeezing” [33] |

Q-MSF^{c} | −0.038 |

^{a}

Doi–Edwards (DE) with independent alignment approximation (IAA).

^{b}

Linear molecular stress function (L-MSF).

^{c}

Quadratic molecular stress function (Q-MSF).

The number line of *A* shown in Fig. 3 illustrates which values of *A* are covered by an available theory with published MAOS solution. The result of this work covers a range of *A* values which did not have a physical interpretation before. In addition, comparing the magnitude of nonlinearity *A* of JS/GS with other TSS models shows that it can only be as nonlinear as the CM model. Furthermore, shear-stiffening materials have positive values of *A*, e.g., PVA-Borax transient networks can show a range from *A* = 0.08–0.2 (from reinterpretation of data in [11]^{5}),

In [11], experimental $[e3]$ and $[v3]$ material functions were provided for an extensive range of PVA-Borax compositions. We used here the plateaus of $[e3]$ at high frequency, reported in their Table 3, to calculate the apparent nonlinear TSS parameter as $A=29lim\omega \u2192\u221e\u2061[e3](\omega )/G0$ using Eq. (35) and assuming a high-frequency plateau *G*_{0}.

Finally, there are two clarifying notes regarding TSS MAOS models in the literature. First, although Song *et al*. [19] added the Larson model and White-Metzner with Carreau viscosity to their list of TSS models, these models are *not* TSS MAOS based on the definition of [21]. This confusion often occurs when a nonlinear parameter appears as a factorized front factor in their MAOS material functions. However, all TSS MAOS functions must have the same *frequency* dependence as Eqs. (33)−(36). Based on the analytic solutions shown in [19], the MAOS functions of the mentioned models do *not* satisfy those mathematical forms and therefore are not TSS MAOS. The second note is regarding a common fitting approach, where certain relaxation modes used to fit SAOS data are neglected when fitting MAOS nonlinear signals, as if the nonlinear parameter is not constant, but depends on the associated relaxation time scale. In this case, even if the model in its single-mode form is TSS MAOS, the considered model fit is not TSS MAOS, since Eqs. (33)–(36) will not hold anymore.

### B. Visualizing nonaffine deformation

Understanding the evolution of a material element in the nonaffine deformation model of JS/GS is critical to connect it to a clear molecular picture. Therefore, the goal of this section is to visualize this evolution by analyzing the model equations presented in Secs. II and III, and this forms the third major contribution of this work. This visualization is relevant to models outside the JS/GS family, such the Phan-Thien/Tanner (PPT) [34], where the same nonaffine deformation is assumed but with a different stress calculator. We note that previous attempts have been made to visualize the deformation history of the JS/GS model by Petrie [35], where the trajectory of a point is tracked back in time. Here, we compute the effective nonaffine velocity field $v_NA(x_,t)$ and show its Eulerian streamlines and Lagrangian pathlines, which allow us to track the deformation of material elements in the nonaffine model forward and backward in time.

The nonaffine slipping motion of the JS/GS model can be visualized by considering the effective velocity gradient $L__$ in Eq. (6). We write this as

where we interpret $v_NA(x_,t)$ as the nonaffine velocity field. Then, for an assumed continuum flow field $v_(x_,t)$, Eq. (6) is used to compute $L__$ and Eq. (38) is integrated to find $v_NA(x_,t)$. Here, we show two visualizations of $v_NA(x_,t)$: Eulerian streamlines and Lagrangian material element cubes.

Assuming the homogeneous simple shear flow defined in Eq. (7), it follows from Eq. (6) that the velocity gradient is given by

where $\gamma \u02d9(t)$ can have any time dependence. As expected for homogeneous simple shear flow, the velocity gradient is independent of position $x_$, and this is maintained in the JS/GS model. Let $v_NA(x_,t)$ be one possible velocity field giving rise to stress in the JS/GS model satisfying Eq. (39). It follows that the *x*-component of the velocity field *v*_{NA,x} is defined by three differential equations which are $\u2202vNA,x/\u2202y=\gamma \u02d9(t)(a+1)/2and\u2202vNA,y/\u2202x=\gamma \u02d9(t)(a\u22121)/2$. Similarly, the *y*-component is defined by $\u2202vNA,y/\u2202x=(a\u22121)/2(\gamma \u02d9(t))and\u2202vNA,y/\u2202y=\u2202vNA,y/\u2202z=0$ and the *z*-component by $\u2202vNA,z/\u2202x=\u2202vNA,z/\u2202y=\u2202vNA,z/\u2202z=0$. The general form of the velocity field satisfying these equations is

To observe the deformation of a material element without any translation, we choose a fixed zero velocity at the origin, $v_NA(0,t)=0$, to obtain

The Eulerian streamlines are defined by

The equation of the streamline passing through any point on the *y*-axis, $(0,yint)$, is given by

where $yint\u2208R$. The shapes of the streamlines are self-similar and independent of the strain or strain rate for a particular value $yint$ for this flow field. Nevertheless, the value of the stream function across each of the streamlines will depend on the instantaneous strain rate.

Figure 4(a) shows derived streamlines for the case of $a=0.5$ and the two limiting cases of the model. For all values of $0<a<1$, the streamlines will be ellipses. However, as *a* approaches the limit of affine motion, $a=1$, the streamlines approach straight lines, as would be expected for affine simple shear flow. On the other hand, as $a\u21920$, the streamlines become circles, which is the behavior of the corotational model. As mentioned in Sec. IV A, the negative *a* values are physically unrealistic, which is further proven by the associated nonaffine streamlines and deformation shown in Appendix A.

A second and perhaps more useful visualization is of Lagrangian material elements deformed by the field $v_NA(x_,t)$, as shown in Fig. 4(b). For this, consider cube-shaped elements centered at the origin, which omits any translation and isolates the deformation and rotation of the elements. The path of each material point $x_i(t)$ is defined by the differential equation

which can be integrated (numerically if needed) starting from any specific position $x_i(0)$. In our case, starting from multiple positions that define the boundaries of an initial material volume. The initial size of the material cube does not affect the relative shape of the deformation due to the self-similar streamlines and pathlines.

For homogeneous simple shear, $v_NA(x_,t)$ from Eq. (41) results in zero velocity in the *z*-direction, therefore a two-dimensional projection of the cube is sufficient. Starting from an undeformed unit square in the *x*–*y* plane, we integrate material points along the pathlines based on Eq. (44), taking snapshots at different points in time, which map to different values of imposed macroscopic shear strain $\gamma (t)=\u222b0t\gamma \u02d9(t\u2032)dt\u2032$. The time history of the shear rate does not affect the result, because the streamlines are independent of time [Eq. (43)] for this flow field, which makes the deformation dependent only on the accumulated strain.

Figure 4(b) shows how the element changes as the strain increases for three cases: affine (*a* = 1), corotational (*a* = 0), and *a* = 0.5. In the corotational limit, material stress elements are only rotated, keeping the unit square shape intact. On the other hand, in the affine limit, the square deforms along the horizontal streamlines defined by the affine velocity field. In the interesting range introduced by the JS/GS model, the material elements are stretched and rotated, depending on the affinity parameter *a*. Figure 4(b) shows that for small strains, the deviation of the nonaffine history from the affine limit is small and grows gradually as the strain amplitude is increased. Similarly, the deviation of the nonaffine stress prediction compared to the linear affine limit increases as the amplitude is increased as illustrated in Fig. 1 due to the deviation of the material element deformation.

The results presented above can be generalized for other types of flow fields. Here, we derive the results for uniaxial extension, for which the velocity field is defined as

where $\epsilon \u02d9$ is the extensional strain rate. The resulting nonaffine velocity gradient is

Assuming $v_NA(0_,t)=0_$, integrating Eq. (46) results in $v_NA=av_$. The streamline passing through $(xint,yint,z=0)$ for this velocity field is

Figure 5(a) shows the Eulerian streamlines for this flow, highlighting that the shape of the streamlines is not affected by the value of *a*. Nevertheless, the magnitude of the nonaffine velocity is affected by *a*, as seen in the Lagrangian material elements in Fig. 5(b) which extend more slowly for nonaffine flow.

### C. Model interpretation and applicability to different material classes

As mentioned throughout this work, the importance of the JS/GS model is that it offers a possibility for material-level inference from MAOS data. Nevertheless, the material elements, whose nonaffine deformation was demonstrated above in Fig. 4(b), do not correspond to a specific physical or molecular picture. Hence, the JS/GS model is a semiempirical model that consists of two components—the Lodgelike linear viscoelastic stress calculator determined by the affine relaxation modulus *G*_{A}(*t*) and the nonaffine material element deformation determined by the affinity parameter *a*. Moreover, it is the nonaffinity that generates the nonlinearity in this model, and it corresponds to a specific physical picture. On the other hand, the affine relaxation modulus can be related to molecular parameters through any known molecular model that satisfies the Lodgelike mathematical form. Examples include but are not restricted to the Green and Tobolsky model of transient polymer networks and the Rouse theory for dilute polymer solutions. Moreover, Winter and Mours [36] used the Lodge equation with power-law relaxation to model the linear viscoelastic response of critical gels. Consider that the JS/GS model can be used to fit MAOS data of a polymer solution with a specific $G(s)$ and *a*. In this case, the affine relaxation function $GA(s)=G(s)/a$ can be used to infer the quantitative properties of the polymer chains and solvent in the Rouse model. However, the important addition given by JS/GS is that *a* can be used to infer how the material elements are deforming (Sec. IV B).

The utility of the JS/GS model is not restricted to one microstructure or material. Therefore, if a new material satisfies Lodgelike behavior described above, it might be modeled using the JS/GS model to have a nonaffine deformation interpretation. However, without further modification, the JS/GS model cannot be used for materials that exhibit nonlinear behavior in a form other than nonaffine deformation such as finite extensibility, plasticity, or structure/network breaking.

## V. CASE STUDY

Here, we demonstrate the applicability of the JS/GS MAOS predictions to experimental MAOS data already in the public domain. Although several such studies are available, here we use published data for a linear polymer melt as described in the work of Singh [37]. The fitting results there are adopted as is but can be given a new interpretation based on the JS/GS model, which has the highest credibility score^{6} of all the models considered in the original work.

The analysis done in [37] shows that between the extensive list of TSS and non-TSS models considered, the most credible fit was found to be a TSS MAOS model having a fractional Maxwell spectrum and a nonlinear parameter *A* *= *−0.115. This value was interpreted by considering values of available constitutive models (Table I, Fig. 3), but no exact match was found, since the JS/GS results here and in [19] were not yet available.

Now, using the key result of Eq. (37), that value of *A* can be interpreted as nonaffine slip in the context of the JS/GS model. Figure 6 shows the relation between the TSS MAOS nonlinear parameter *A* and the JS/GS affinity parameter *a*, based on Eq. (37)*.* It further locates the result of *A* *= *−0.115, which corresponds to a JS/GS model having an affinity parameter $a=0.56$. Moreover, the referenced work also shows fitting results for different $G(s)$ forms, and all of them had an optimal BIC for a value of *A* that falls in the JS/GS range. In short, this is a clear illustration of the ability of the JS/GS model to fit MAOS data and assign to it a physical interpretation.

## VI. CONCLUSION

The MAOS regime viscoelastic response of the Johnson-Segalman/Gordon–Schowalter nonaffine deformation model has been derived for a general relaxation function. The model offers a physical interpretation of a subset of the TSS MAOS class which was not previously identified. The molecular picture of nonaffine motion was illustrated and visualized in terms of an effective nonaffine flow field giving rise to stress. The analysis showed that the internal slip introduced in the JS/GS model is the cause of nonlinearity and that for a fixed affine modulus $GA(t)$ the maximum magnitude of the nonlinear functions occurs at $a=1/3\u22480.577$. This result is related to previously derived MAOS solutions of the CM model, which is one limit of the JS/GS model. In particular, the molecular picture of the CM model from this perspective can be understood to be that of pure rotation of material elements that have an infinite relaxation modulus. Although the single-mode analytical results were derived previously, the thorough study of the generalized integral model and its nonaffine deformation presented here contribute to utilizing it in material inference.

One known limitation of the JS/GS model is that it fails for large strains as detailed by Petrie and Larson [16,35], and hence it is expected that the model will fail in the LAOS regime. Nevertheless, this limitation does not affect the utility of the model in explaining data in the MAOS regime, since MAOS is a power expansion defined in the limit of strain going to zero. In fact, this work makes it possible to reinterpret experimental data. The case study of Sec. V demonstrates how the JS/GS model may offer the most credible fit/explanation among many other considered models for MAOS data, suggesting the importance of nonaffine deformation in the initial growth of nonlinearities. Nevertheless, credibility of MAOS inference can be further strengthened as more theories and explanations are brought to the realm of MAOS measurements.

The JS/GS picture of nonaffine motion is applicable to many material classes [16] and holds promise in explaining experimental data. Experimental studies show the presence of nonaffine deformation in materials such as polymer hydrogels, synthetic polymers, and other types of soft matter [38]. Further experiments on these materials can show if the JS/GS model can be used to explain the observations, or whether other types of nonaffine deformation must be considered. For example, other models include nonaffine heterogeneous deformation of polymer strands in a network due to extensibility effects [39] and nonaffine deformation due to fluctuations in the end position of polymer strands around their affine positions [40]. Comparing our analytical solutions to weakly nonlinear MAOS experiments should assist in building our understanding of material behavior and to develop more credible nonaffine deformation models and make material-level inference.

## ACKNOWLEDGMENT

This research is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award Nos. DE-FG02-07ER46471 and DE-SC0020858, through the Materials Research Laboratory at the University of Illinois at Urbana-Champaign.

### APPENDIX A

The unphysical range of affinity parameter $a<0$ produces peculiar streamlines and material element deformation, as we show here. It is known that for $a=\u22121$ the GS convected derivative is equivalent to the lower-convected derivative and therefore studying this limit is instructive. To do this, we follow the same steps taken in Sec. IV B, but rewriting the streamline equation, Eq. (43), to show the equation of the streamline passing through a particular x-intercept $(xint,0)$, as

Figure 7 shows the streamlines and deformation for negative affinity parameter and compares it to the case of *a* = 1. For an imposed homogeneous simple shear deformation with nonzero x-velocity (the same as in Fig. 4), for *a* = −1 the material is deformed perpendicularly with a nonzero y-velocity. For *a* = −0.5, the material is rotated and deformed, but with the deformation also occurring perpendicularly to the imposed gradient. Both of these cases display unphysical behavior, adding to the reasons why this limit of the model should not be considered to fit experimental data.

### APPENDIX B

Peculiar results occur if we conceptualize a fixed *affine* modulus *G _{A}*(

*s*) while changing the slip parameter

*a*, as if the underlying material structure is the same and we only change the amount of slip. In this case, the MAOS nonlinear strength is a nonmonotonic function of

*a*, in contrast to the monotonic trend with

*a*as shown in Fig. 2 for a fixed

*effective*modulus

*G*(

*s*). Fixing the effective relaxation modulus $G(s)=aGA(s)$, as we analyzed in Sec. III B, is perhaps more relevant for experiments since it is measured directly. Nevertheless, fixing $G(s)$ hides the nonaffine conceptual picture of the microstructural nonlinearities, and we will see how fixing $GA(s)$ changes the analysis.

Starting from the general MAOS solution for any relaxation function (Sec. III A), we consider a single-mode exponential relaxation for the affine modulus

where $GA,0$ is the affine elastic modulus magnitude and $\tau 0$ is the relaxation time. The resulting MAOS material functions are

The results from Eq. (47) are plotted in Figs. 8(a) and 8(b). Two differences with Fig. 2 stand out. First, the linear moduli have a linear dependence on the affinity parameter, where both moduli are maximum for the affine limit (*a *= 1) and vanish in the corotational limit (*a *= 0). Second, the dependence of the magnitude of the nonlinear functions has a nonmonotonic dependence on the affinity parameter, unlike what was observed in Fig. 2(b). By examining Eq. (B2), the dependence of the nonlinearity strength on *a* can be captured by the factor $|a(a2\u22121)/6|$, compared to $|(a2\u22121)/6|$ for the fixed *G*(*s*) case from Eq. (30). This nonmonotonic dependence is illustrated in Fig. 8(c). Although the slip increases the nonlinearity of the deformation response, the stress response is more complicated. The slip introduced by nonaffinity decreases the effective linear modulus of the material and therefore weakens the overall stress response. Thus, the maximum magnitude of the nonlinear functions for this model will be observed for $a=1/3\u22480.577$ for the fixed affine relaxation function. Using a fixed effective modulus hides this finding.

## REFERENCES

Singh [37] used an effective Bayesian Information Criterion (BIC) $BIC=RSSmin+Nplog\u2061(Nd)$ to evaluate model credibility by penalizing excessive parameters, an approximation of the full Bayes factors [20], where $RSSmin$ is the root sum of squares of error between the model and the data at the $Nd$ data points for a model having $Np$ number of parameters.