We derive and study equations for the weakly nonlinear medium-amplitude oscillatory shear (MAOS) response of materials exhibiting time-strain separability. Results apply to constitutive models with arbitrary linear memory function m(s) and for both viscoelastic liquids and viscoelastic solids. The derived equations serve as a reference to identify which models are time-strain separable (TSS) and which may appear separable but are not, in the weakly nonlinear limit. More importantly, we study how the linear viscoelastic (LVE) relaxation spectrum, H(τ), affects the frequency dependence of the TSS MAOS material functions. Continuous relaxation spectra are considered that are associated with analytical functions (log-normal and asymmetric Lorentzian distributions), fractional mechanical models (Maxwell and Zener), and molecular theories (Rouse and Doi-Edwards). TSS MAOS signatures reveal much more than just the perturbation parameter A in the shear damping function small-strain expansion, h(γ)=1+Aγ2+Oγ4. Specifically, the distribution of terminal relaxation times is significantly more apparent in the TSS MAOS functions than their LVE counterparts. We theoretically show that this occurs because TSS MAOS material functions are sensitive to higher-order moments of the relaxation spectrum, which leads to the definition of MAOS liquids. We also show the first examples of MAOS signatures that differ from the liquid-like terminal MAOS behavior predicted by the fourth-order fluid expansion. This occurs when higher moments of the relaxation spectrum are not finite. The famous corotational Maxwell model is a subset of our results here, for which A = −1/6, and any LVE relaxation spectrum could be used.

Bird,1 Giacomin, and co-workers2 ignited renewed interest in analytical solutions to constitutive models in medium-amplitude oscillatory shear (MAOS) in 2011 with their derived results for the corotational Maxwell (CM)2–5 model. Earlier derivations for various models began in the 1950s6,7 (for reviews of continued theoretical work, see Table II in the work of Saengow and Giacomin,8 Table III in the work of Saengow et al.,9 and Table I in the work of Bharadwaj and Ewoldt5), but the 2011 paper2 came at a fertile time when experimental measurements were improving10,11 and interpretations of the four MAOS shear stress signatures were being proposed.11 Despite this progress, open questions remain about how the linear viscoelastic (LVE) relaxation spectrum influences MAOS frequency-dependent signatures and what is meant by time-strain separable (TSS) MAOS. Here we help answer these questions by deriving equations from a general framework of TSS models in MAOS, of which the famous corotational Maxwell2–5 model is a subset.

Experimental evidence for time-strain separability dates back to the early studies on elastomers conducted by Tobolsky and Andrews12,13 and Guth et al.14 with experiments of extensional stress relaxation and by Smith15,16 with tensile tests at constant rate of (nominal) strain. Its applicability to polymer melts and solutions was later suggested by White and Tokita,17 Zapas and Phillips,18 and Einaga et al.19 Experimental and theoretical definitions of time-strain separability are discussed in Sec. II A.

The objective of this work is to investigate the consequences of time-strain separability in MAOS. To this end, the first step is to determine how TSS MAOS material functions depend on the LVE relaxation spectrum when time-strain separability applies. Specifically, we consider here the four frequency-dependent MAOS material functions proposed by Ewoldt and Bharadwaj11 in 2013: [e1](ω), [e3](ω), [v1](ω), and [v3](ω) (defined and discussed in Sec. II C); results also apply to subsets of these measures such as amplitude ratios from Fourier-transform analysis. We will show that (i) frequency dependence and sign changes of TSS MAOS functions are uniquely determined by LVE behavior, (ii) their magnitude and sign are set by a single nonlinear parameter, and (iii) there are three equivalent ways in which these TSS MAOS equations can be written (Sec. III A). One form of these equations represents the asymptotically nonlinear counterpart in oscillatory shear of the TSS nonlinear response in step-shear deformations.

The single nonlinear parameter for weakly nonlinear time-strain separability, which we call A, is defined here to represent the dimensionless magnitude of the leading-order nonlinearity encoded in the small-strain power series expansion of the shear damping function,20,21 i.e., h(γ)=1+Aγ2+Oγ4where γ is a short-hand notation for the shear strain accumulated between times t′ and t, γt,tγtγt; Sec. II A 3. We will derive TSS MAOS equations applicable to any LVE model, and our approach will encompass both LVE liquidsGeqlimtG(t)=0,M1G0sG(s)ds<+22 and LVE solidsGeq>022,23 (here, G(s) is the linear shear relaxation modulus3,4,23–27 and Geq is its value at equilibrium23,24). We will describe how our derived equations are more general than prior work,5,28–30 which can be considered as a subset of our results here (Sec. III B). The derivation will be based on the memory-integral expansion3 for third-order Coleman-Noll31–37 “simple materials” (the Coleman-Noll class is briefly reviewed in the  Appendix).

The equations presented herein provide a simple means (i) to assess whether a given set of MAOS analytical solutions obeys time-strain separability (Sec. IV A) and (ii) to predict the frequency-dependent behavior of the four TSS MAOS signatures from knowledge of the linear response alone (Sec. IV B). The value of our equations will be demonstrated by considering the TSS MAOS response associated with a variety of LVE relaxation spectra H(τ), originating from (i) simple analytical functions [log-normal (LN)23,38 and asymmetric Lorentzian (AL)23,39 distributions, Sec. IV B 1], (ii) fractional mechanical models [Maxwell (FM)40–46 and Zener (FZ),43,44,46 Sec. IV B 2], and (iii) molecular theories [Rouse (R)3,24,47,48 and Doi-Edwards (DE),49–56 Sec. IV B 3]. This will reveal terminal (De ≪ 1) MAOS signatures that differ, and starkly so, from the liquid-like terminal MAOS behavior predicted by Bharadwaj and Ewoldt57 (Secs. IV B 1 and IV B 2). The conditions that need to be satisfied for liquid-like terminal MAOS behavior to be exhibited will lead to the definition of MAOS liquids: Geq = 0 and M3G0s3G(s)ds<+ (Sec. III C). Our results will show that weakly nonlinear oscillatory perturbations from the material equilibrium state are substantially more sensitive than the LVE response to the distribution of terminal relaxation times (Secs. IV B 1 and IV B 2). Underlying this behavior is the inherent dependence of the TSS MAOS material functions on higher-order moments of the relaxation spectrum H(τ) than the ones controlling the response in the LVE regime (Sec. III C).

The paper is organized as follows. First, we will discuss three definitions of time-strain separability (Sec. II A) and review the definitions of linear (Sec. II B) and weakly nonlinear (Sec. II C) material functions in oscillatory shear. When presenting the definition of time-strain separability for third-order Coleman-Noll “simple materials”31–37 (Sec. II A 3), we will show the relationship [Eq. (9)] that needs to be obeyed by the nonlinear memory functions for third-order time-strain separability to apply. This relationship is analogous to the one [Eq. (A9)] that leads to complete time-strain separability (Sec. II A 2). Since we want our derivation of TSS MAOS functions to encompass both LVE liquidsGeq=0,M1G<+22 and LVE solidsGeq>0,22,23 the Coleman-Noll31–37 multiple integral expansion4 (or memory-integral expansion3) [Eqs. (4) and (7)] will include the “delayed elastic stress.”36 In the small-strain (i.e., LVE) limit, this will correspond to analytical expressions for Boltzmann’s superposition integral24,58 [Eq. (12a)], the LVE shear stress σSAOS(t) [Eqs. (14a) and (14b)], and the storage modulus G′(ω) [Eqs. (15) and (17)] embracing the response of LVE solids.

We will then proceed to (i) derive the analytical expressions for TSS MAOS material functions (Sec. III A), (ii) compare our derivation and equations with previous studies (Sec. III B), (iii) use our derived equations to show that TSS MAOS functions depend on higher-order moments of the relaxation spectrum (Sec. III C), and (iv) discuss the validity of the TSS MAOS equations outside of the Coleman-Noll class (Sec. III D).

Finally, we will show (i) how to determine if a MAOS model within the Coleman-Noll class is time-strain separable (Sec. IV A) and (ii) how the LVE relaxation spectrum affects the frequency dependence of the TSS MAOS material functions (Sec. IV B).

To understand the framework of our derivation and put our work into context, it is necessary to consider three definitions of time-strain separability, based on (i) the measured or predicted stress relaxation moduli3,4,23–27 (Sec. II A 1; Subsection 4 of the  Appendix), (ii) the complete Coleman-Noll31–37 memory-integral expansion3 out to any order (Sec. II A 2; Subsections 1, 3, and 4 of the  Appendix), and (iii) the third-order Coleman-Noll31–37 integral constitutive equation (the so-called finite third-order viscoelasticity31–37) (Sec. II A 3; Subsections 2 and 3 of the  Appendix; Fig. 11). The derivation of our TSS MAOS equations, presented in Sec. III A, will be based on the third definition of time-strain separability.

1. Stress relaxation moduli

If the shear strain excitation γ(t) is a step of strain of height γ0, i.e., if γ(t) = γ0s(t), where s(t) is the Heaviside unit step function,23 the nonlinear viscoelastic (NLVE) stress response σ(t) is said to exhibit time-strain separability when the nonlinear shear relaxation modulus,3,4,24–27G(t, γ0) ≡ σ(t)/γ0, is factorized into a function of time and a function of strain as3,4,24,26,27

(1)

where G(t)limγ00G(t,γ0) is the linear shear relaxation modulus,3,4,23–27 which characterizes the LVE response, and the strain-dependent factor h(γ0) is the so-called shear damping function.20,21 When time-strain separability applies, G(t, γ0) thus exhibits the same time dependence, G(t), which is elicited in the LVE regime (in other words, the time behavior of G(t, γ0) is set by the LVE response; cf. Sec. III A). Equations analogous to Eq. (1) exist for TSS NLVE behavior in flows other than simple shear (e.g., uniaxial and biaxial extension).4,24,26,27 A more general definition of damping function, which does not require time-strain separability, is discussed in the review by Rolón-Garrido and Wagner.21 

Equation (1) has been extensively verified17–21,59–66 and is known to apply over limited ranges of time and strain.19,26,27,61,63–66 Furthermore, Eq. (1) is predicted by the original Doi-Edwards49–56 reptation theory for polymer melts, formulated with (DE-IAA)49–52,56 and without (DE)53–56 the independent alignment approximation (IAA). For a zero value of the “link-tension coefficient” ε, the Curtiss-Bird3,48,67–70 theory for polymer melts is, within a numerical prefactor,4,71,72 equivalent to the DE-IAA theory and therefore also prescribes Eq. (1). In terms of classes of constitutive equations, Eq. (1) can be derived within the mathematical framework for isotropic, incompressible, viscoelastic “simple materials” of the integral type formulated by Coleman and Noll31–37(Sec. II A 2; Subsections 1 and 4 of the  Appendix). Still, Eq. (1) remains the most general and simplest definition of time-strain separability in simple shear because it embraces the TSS NLVE response also of continuum mechanical models and molecular theories outside of the Coleman-Noll class, as exemplified by the Doi-Edwards theory. In either version, the Doi-Edwards nonlinear shear relaxation modulus51,52,54–56 obeys Eq. (1). However, while the DE-IAA theory yields a constitutive equation that belongs to the Coleman-Noll class,4 the same is not true for the DE theory,4 whose constitutive equation is not even obtainable in closed form4,56,71–73 (cf. Secs. II A 2 and III B; Subsection 1 of the  Appendix). What Eq. (1) implies in terms of a TSS response in MAOS will be shown in Sec. III A.

The shear damping function, h(γ0), was first defined by Wagner20 [under the assumption ϕ2I1,I2=0 in Eq. (4)]. In the framework of tube models, h(γ0) can be interpreted in terms of strain-induced disentanglement due to chain retraction4,21,27,74–77 and therefore as a survival probability of temporary network strands.76 As such, h(γ0) should decrease monotonically with strain from its small-strain limit h(γ0 → 0) = 1 [cf. Eqs. (A10) and (A13)], approaching zero at large deformations. Various empirical analytical expressions for h(γ0) have been proposed (see, e.g., the work of Petrie,60 Larson,4 Graessley,27 and Rolón-Garrido and Wagner21). A popular functional form is59,62,78,79

(2)

which closely approximates80 the behavior of h(γ0) predicted by the Doi-Edwards reptation theory (for both DE-IAA and DE constitutive equations, as they yield similar predictions for h(γ0)).26,27,56,72,81 Furthermore, since its Taylor expansion about γ0 = 0 is

(3)

Eq. (2) approaches its small-strain limiting value of one with a zero slope and prescribes that the leading-order deviation of h(γ0) from unity is quadratic in the strain.26,28,82 As will be shown in Sec. II A 3, this functional behavior is required by the Coleman-Noll class of “simple materials”. The exponential form for h(γ0) originally proposed by Wagner,20 h(γ0)=eaγ0, leads to h(γ0)=1aγ0+Oγ02 and therefore prescribes a linear deviation of h(γ0) from unity at small strains.26 

2. Complete Coleman-Noll TSS response

For Coleman-Noll “simple materials”,31–37 the NLVE stress response exhibits complete time-strain separability when two conditions are met. First, the effects on the stress at present time t of strains at all different historic times t′ must be independent of each other3,4,83 (Subsection 3 of the  Appendix). Second, the resulting nonlinear memory functions3,4,83ψ1tt,I1,I2 and ψ2tt,I1,I2 must be factorizable as a product of time- and strain-dependent terms3,4,17–19 [Subsection 4 of the  Appendix, Eq. (A9)]. In this case, the Coleman-Noll31–37 multiple integral expansion4 (or memory-integral expansion3) out to any order yields the factorized or time-strain separable Rivlin-Sawyers (TSS-RS)3,4,84 equation,

(4)

In Eq. (4), the left-hand side is the difference between the extra stress tensor, σ(t), and the “delayed elastic stress,”36,σel(t) ≡ limtσ(t) (with σel(t) = 0 for “simple fluids”;31,37σel(t) ≠ 0 for “simple solids”31,36,37). In the right-hand side of Eq. (4), mttGtt/t is the linear memory function,3,4,24,26,27stt′ is the elapsed time, C1t,t and Ct,t are, respectively, the Finger and Cauchy measures3,4,27,83 of the strain accumulated between times t′ and t, ϕiI1,I2 are strain-dependent functions [Eqs. (A9) and (A10)], and I1=IC1 and I2=IIC1 are the first and second invariants3,4,27,83 of C−1. If ψ1 and ψ2 are given by a potential function, Vtt,I1,I2, and this can be expressed as Vtt,I1,I2=mttWI1,I2, Eq. (4) yields the factorized or time-strain separable Kaye-Bernstein-Kearsley-Zapas (TSS-K-BKZ)3,4,85,86 equation. The DE-IAA constitutive equation is of the TSS-K-BKZ type,3,4,26 with σel(t) = 0 and a deformation-dependent potential function3,4,26,56,87WI1,I2 such that ϕi = W/Ii. It is straightforward to show that Eq. (1) is obtained from Eq. (4) when the shear strain history is a step of strain of height γ0 (Subsection 4 of the  Appendix).

In simpler form, Eq. (4) can be written as4 

(5)

where Et,t, a symmetric tensor, is the generic elastic measure of the strain accumulated between times t′ and t. The fact that Rolón-Garrido and Wagner21 and Wagner et al.30 wrote the DE constitutive equation in the form of Eq. (5) (clearly, with σel(t) = 0) seems to have been an oversight, which led to inaccurate DE MAOS equations reported in the work of Wagner et al.30 [their Eqs. (19)–(22)], later used by Bharadwaj and Ewoldt.5 As mentioned in Sec. II A 1, without invoking the IAA, the constitutive equation of the Doi-Edwards theory4 is not obtainable in closed form and therefore cannot be written in the form of Eq. (5)4,56,71–73 (cf. Sec. III B and Subsection 1 of the  Appendix).

As discussed by Larson,4 it also follows from Eqs. (1) and (5) that4 

(6)

where E is the shear component of the generic tensorial measure of deformation Et,t in Eq. (5). The relationship between the damping function and the generic strain measure, Eq. (6), will be used in Eq. (11).

3. Third-order Coleman-Noll TSS response

Continuing our discussion for simple shear deformations, we now restrict our attention to third-order “simple materials”31–37 [Subsection 2 of the  Appendix, Eq. (A2)], which are fully characterized by four independent memory functions [Eqs. (A3)–(A5)]. When interactions of past kinematic events are neglected up to the third order (cf. Subsection 3 of the  Appendix), the shear stress σ(t) is given by

(7)

where the single integrals follow from Eqs. (A3), (A6), and (A7), the Oγ5 term stems from the higher-order integrals σ(i≥4)(t) in Eq. (A1), and3,4,27

(8)

is the shear strain accumulated between times t′ and t. Therefore, if the nonlinear memory functions m2(s, s), m3(s, s, s), and m4(s, s, s) obey [cf. Eq. (A9)]

(9)

for arbitrary constants c1 and c2, the number of independent material functions is reduced from four to one and third-order time-strain separability is obtained. Equation (7) then yields [cf. Eq. (A13)]

(10)

with σel(t) ≡ limtσ(t) = GeqE(t, −), where Geq ≡ limtG(t) is the equilibrium shear modulus,23,24 and with the generic measure Et,t of the shear strain accumulated between times t′ and t given by

(11)

where γ=γt,t and A is an arbitrary constant setting magnitude and sign of the leading-order nonlinearity. Thus, the sign of A indicates whether a softening (A < 0) or stiffening (A > 0) response is elicited for weakly nonlinear perturbations from equilibrium (e.g., in stress relaxation experiments; a similar interpretation also applies to MAOS experiments, as will be shown in Sec. III A). Clearly, Eq. (11) implies that (i) Et,t is an odd function of the shear strain (as required by shear symmetry; cf. Sec. II C) and (ii) the functional form of h(γ) for third-order “simple materials” is identical to the asymptotic small-strain behavior, Eq. (3), of Eq. (2) (with A = −a). Note that the functional form of Eq. (11) is not being assumed a priori but follows from Eq. (A1). A Coleman-Noll TSS constitutive equation analogous to Eq. (10) but to the fourth or higher order can be obtained by considering higher-order terms σ(i≥4)(t) in Eq. (A1) and by neglecting coupling effects from past deformations up to the chosen order (cf. Subsections 3 and 4 of the  Appendix).

For large perturbations from equilibrium, corresponding to the general NLVE region in Fig. 11, neglecting terms Oγ5 and higher in Eq. (11) is not possible. Indeed, Giacomin et al.82 showed that a truncated series expansion leads to physically unacceptable behavior when used to model the NLVE stress response in large-amplitude oscillatory shear (LAOS).

In the small-strain γ0 → 0 limit, i.e., when terms Oγ3 and higher in Eq. (11) can be neglected, Eq. (10) reduces to Boltzmann’s superposition integral24,58

(12a)
(12b)

as it must. The equivalency of Eqs. (12a) and (12b) is proven via integration by parts, considering the definition of the linear memory function, mttGtt/t and that γt,t/t=γ̇t from Eq. (8). Notice that the inclusion of the delayed elastic stress σel(t) in Eq. (10) is consistent with Boltzmann’s superposition integral24,58 embracing the response of LVE solids, Eq. (12a) [see, e.g., Ferry’s comment to Eq. (7) on p. 7 of his book24].

In this work, we are concerned with material functions resulting from a homogeneous oscillatory shear strain excitation γ(t) of amplitude γ0 and radian frequency ω, defined by a sinusoid of the form88 

(13)

which imposes an out-of-phase strain rate γ̇(t) of amplitude γ̇0γ0ω. Here, we define the LVE material functions characterizing the small-amplitude oscillatory shear (SAOS) response.3,23,24 The material functions pertaining to the MAOS response10,11,30,89 will be defined in Sec. II C.

At steady-state (i.e., when the stress signal has achieved time periodicity), and with γ(t) = 0 for t < 0, Pipkin90 showed that the LVE stress response σSAOS(t) from Eqs. (12a) and (12b) is

(14a)
(14b)
(14c)

with the familiar LVE material functions, the storage modulus G′(ω) and the loss modulus G″(ω), given by23,24,90

(15a)
(15b)
(16a)
(16b)

Note that for γ(t), γ̇(t), and σSAOS(t), the functional dependence on ω and γ0 is implied. Equations (15) and (16) are the general expressions of G′(ω) and G″(ω) in terms of the linear memory function m(s) and the linear relaxation modulus G(s). In terms of a continuous (viscosity-weighted)23,91 LVE relaxation spectrum H(τ), G′(ω) and G″(ω) are calculated as23 

(17)
(18)

Outside of the Coleman-Noll class, an even more general expression for G″(ω) can be obtained by adding a second, purely dissipative term to the right-hand side of Eqs. (16a), (16b), and (18) [e.g., ωηsol, where ηsol is a solvent viscosity;3,24,27,48,91,92 see Sec. IV B 3, Eq. (43b), and Fig. 9(a)].

At steady-state, and under the assumption of shear symmetry e.g., σ(γ0)=σ(γ0),93 the NLVE shear stress response σ(t) up to the third harmonic can be expressed as11 

(19)

where the functional dependence of the shear stress on ω and γ0 is again implied, and the NLVE material functions are based on the Chebyshev coefficients94,95 of a stress decomposition96 into in-phase-with-strain (“elastic”) and in-phase-with-strain-rate (“viscous”) components. In Eq. (19), the linear amplitude scaling (∼γ0) defines the SAOS response,3,23,24σSAOS(t), represented here by the storage modulus, G′(ω), and the loss viscosity, η′(ω), where η′(ω) ≡ G″(ω)/ω. The cubic amplitude scaling   γ03 represents the leading-order deviation from linearity and defines the MAOS response,10,11,30,89σMAOS(t), fully characterized by four asymptotically nonlinear material functions:11 the first-harmonic measures [e1](ω) and [v1](ω), and the third-harmonic measures [e3](ω) and [v3](ω). All six material functions have a functional dependence on frequency only. Because of their amplitude-intrinsic characteristics, the four MAOS material functions are denoted by brackets, [·], by analogy with the nomenclature adopted for the intrinsic viscosity,97,98 [η]. Unlike the LVE material functions, the asymptotically nonlinear material functions [e1](ω),[e3](ω),[v1](ω),[v3](ω) can be negative and can change sign with frequency.

Geometrically (Fig. 6 of Ewoldt and Bharadwaj11), the MAOS material functions are related to rotation [e1](ω),[v1](ω) and distortion [e3](ω),[v3](ω) of Lissajous-Bowditch curves95 decomposed into in-phase-with-strain and in-phase-with-strain-rate components.96 The physical interpretation of the first-harmonic MAOS functions is unambiguous:5,11 [e1](ω) and [v1](ω) describe asymptotic changes to the average (i.e., inter-cycle) stored and dissipated energy (respectively). If the asymptotically nonlinear stress response can be decomposed into elastic and viscous components,96 the third-harmonic MAOS functions represent intra-cycle changes of elasticity [e3](ω) and viscosity [v3](ω).5,11

As a result, the signs of the MAOS functions can identify the type of nonlinear response (Fig. 7 of Ewoldt and Bharadwaj11): elastic softening[ei](ω)<0 or stiffening[ei](ω)>0 and viscous thinning[vi](ω)<0 or thickening[vi](ω)>0.11,95 Ewoldt and Bharadwaj11 also proposed to interpret the combined signs of first- and third-harmonic MAOS functions as indicative of the driving cause for the linear-to-nonlinear transition (Fig. 7 of Ewoldt and Bharadwaj11), i.e., whether nonlinearity is driven primarily by large (instantaneous) strains [e3](ω)[e1](ω)>0,[v3](ω)[v1](ω)<0 or strain-rates [e3](ω)[e1](ω)<0,[v3](ω)[v1](ω)>0. This sign interpretation is consistent with the frequency dependence of the critical strain expected for materials and models belonging to the Coleman-Noll class of “simple fluids”:31,37 strain-rate-controlled at low De (γ̇0γ0ω=const.) and strain-controlled at high De (γ0 = const.) (as shown in Figs. 1, 3, 5, and 11).

FIG. 1.

Examples of TSS MAOS signatures associated with a single-mode Maxwell3,23,24,99 LVE behavior, as prescribed by the corotational Maxwell (CM)2–5 model, and the molecular stress function (MSF)30,100–103 model, both linear (L-MSF) and quadratic (Q-MSF). The (a) LVE response determines [via Eqs. (28a)(28d)] the frequency dependence and sign changes of the (b) TSS MAOS functions and the (c) critical strains for the linear-to-nonlinear elastic and viscous transitions. Magnitude and sign of the TSS MAOS functions, and magnitude of the critical strains, are set by a single nonlinear parameter, A, defined here to represent the leading-order deviation from unity of the shear damping function, i.e., h(γ)=1+Aγ2+Oγ4. The same is true for more complex types of LVE behavior, e.g., those considered in Sec. IV B. A gray, thinner line is used for G(ω) [(a), top left subplot].

FIG. 1.

Examples of TSS MAOS signatures associated with a single-mode Maxwell3,23,24,99 LVE behavior, as prescribed by the corotational Maxwell (CM)2–5 model, and the molecular stress function (MSF)30,100–103 model, both linear (L-MSF) and quadratic (Q-MSF). The (a) LVE response determines [via Eqs. (28a)(28d)] the frequency dependence and sign changes of the (b) TSS MAOS functions and the (c) critical strains for the linear-to-nonlinear elastic and viscous transitions. Magnitude and sign of the TSS MAOS functions, and magnitude of the critical strains, are set by a single nonlinear parameter, A, defined here to represent the leading-order deviation from unity of the shear damping function, i.e., h(γ)=1+Aγ2+Oγ4. The same is true for more complex types of LVE behavior, e.g., those considered in Sec. IV B. A gray, thinner line is used for G(ω) [(a), top left subplot].

Close modal

We are now in the position of deriving the sought-after analytical expressions for the TSS MAOS material functions. In what follows, we show that there are three equivalent ways in which they can be written: (i) integrals involving the linear memory function m(s), (ii) integrals involving the linear shear relaxation modulus G(s), and (iii) linear combinations of SAOS material functions at different integer-multiple frequencies. Notably, the following derivation is made while leaving the functional form for the linear memory function m(s) ≡ −∂G(s)/∂s unspecified, and no assumptions are made regarding the long-time (i.e., terminal) behavior of G(s). For ease of comparison to MAOS analytical solutions available in the literature, we also provide the functional relationship of TSS MAOS functions associated with a single-mode Maxwell3,23,24,99 LVE behavior (shown in Figs. 1 and 3).

The first step is to consider the third-order, TSS stress response σ(t) in oscillatory shear predicted by Eqs. (10) and (11) at steady-state and with90γ(t) = 0 for t < 0 (cf. Secs. II B and II C),

(20)

where σSAOS(t) is still given by Eq. (14a), and the TSS MAOS stress σMAOS(t), identified by the cubic scaling   γ03 in Eq. (19), is

(21)

Since γt,t=γ0sinωtγ0sinωt [Eq. (8)], Eq. (21) gives

(22)

By expanding Eq. (22) through the trigonometric identities94 

(23a)
(23b)
(23c)
(23d)
(23e)

and comparing the result with the generic expression of the MAOS stress response from Eq. (19),

(24)

we obtain the analytical expressions for the TSS MAOS functions in terms of the linear memory function m(s),

(25a)
(25b)
(25c)
(25d)

Equations (25a)–(25d) are the main theoretical results of this study. These equations allow the four TSS MAOS functions to be calculated from any linear memory function m(s) and are valid for both LVE liquidsGeqlimtG(t)=0,M1G0sG(s)ds<+22 and LVE solids(Geq>0).22,23 They rigorously apply to Coleman-Noll “simple materials” with no coupling effects from past deformations up to third order, i.e., m2s,s=m3s,s,s=m4s,s,s=0 unless s′ = s″ = s‴ = s = tt′ (cf. Subsection 3 of the  Appendix) and with nonlinear memory functions m2(s, s), m3(s, s, s), and m4(s, s, s) that obey Eq. (9). Obviously, Eqs. (25a)–(25d) can be converted to any other representation of MAOS shear stresses (for interrelations with other MAOS measures, see, for instance, the work of Ewoldt and Bharadwaj11).

Frequency dependence and sign changes of the TSS MAOS functions given by Eqs. (25a)–(25d) are uniquely determined by the LVE response: they are set by the time-dependent behavior of m(s) modulated by the corresponding trigonometric kernel functions [cf. Eqs. (1), (5), and (10)]. The proportionality constant A is the dimensionless magnitude of the leading-order nonlinearity encoded in h(γ)=1+Aγ2+Oγ4 [Eq. (11)]. Thus, the sign of A determines whether a softening/thinning (A < 0) or stiffening/thickening (A > 0) response is elicited for weakly nonlinear oscillatory perturbations from the material equilibrium state (cf. Secs. II A 3 and II C). It also follows that the value of A can be determined from asymptotically nonlinear deviations from LVE behavior measured in either stress relaxation or MAOS experiments. Importantly, a complete TSS response, i.e., over the entire experimental or theoretical time/strain domain considered (Sec. II A 2), is not required. A TSS nonlinear response in step-shear deformations over a limited experimental range is common,19,26,27,61,63–66 and the molecular stress function (MSF)100–102 model (Fig. 1) provides an example of a constitutive equation exhibiting time-strain separability only up to third order.30,103

Via integration by parts, Eqs. (25a)–(25d) can be written in terms of the linear shear relaxation modulus G(s),

(26a)
(26b)
(26c)
(26d)

Finally, through the definitions of G′(ω) and G″(ω) given by Eqs. (15) and (16), Eqs. (25) and (26) can be recast in a more compact form,

(27a)
(27b)
(27c)
(27d)

Equations (27a)–(27d) represent the asymptotically nonlinear counterpart in oscillatory shear of the TSS nonlinear response in step-shear deformations predicted by Eq. (1) (Sec. II A 1). The TSS relationship between nonlinear and linear shear relaxation moduli evaluated at time t is a simple proportionality governed by the shear damping function h(γ0). Equations (27a)–(27d) prescribe that the four TSS MAOS functions at frequency ω are linear combinations of the SAOS functions G′(ω) and η′(ω) evaluated at frequencies , with k = {1, 2} for [e1](ω) and [v1](ω) and k = {1, 2, 3} for [e3](ω) and [v3](ω). The fact that the elastic TSS MAOS functions [Eqs. (27a) and (27b)] only depend on G′(), and the viscous TSS MAOS functions [Eqs. (27c) and (27d)] only depend on η′(), lends support to the idea that they are, respectively, measures of stored and dissipated energy.29 

When the LVE response is that of a single-mode Maxwell3,23,24,99 model, with linear elastic modulus G0 ≡ limt→0G(t) and relaxation time τ0 = η0/G0 (where η0 is the steady-state zero-shear viscosity),3,23,24 the analytical expressions for the TSS MAOS functions are [cf. Eqs. (35a)–(35d)]

(28a)
(28b)
(28c)
(28d)

as obtained from Eqs. (25a)–(25d), (26a)–(26d), or (27a)–(27d). Equations (28a)–(28d) represent the simplest TSS MAOS signatures of any Coleman-Noll “simple fluid.”

Examples of TSS MAOS models associated with a single-mode Maxwell LVE behavior are shown in Fig. 1; here, and in Figs. 3, 5–10, the normalization adopts model-specific characteristic modulus, G, viscosity, η, and relaxation time, τchar, with dimensionless angular frequency (Deborah number26) defined as De ≡ ωτchar. In Fig. 1, G = G0 and η = η0 = G0τ0, while the value of the nonlinear parameter A is set by the TSS MAOS models being considered: A = −1/6 for the corotational Maxwell (CM)2–5 model, A = −29/210 for the linear molecular stress function (L-MSF)30,100–103 model, and A = −4/105 for the quadratic molecular stress function (Q-MSF)30,100–103 model. The case A = +1 is shown in Fig. 3(b) (black solid lines). Figure 1(c) re-interprets the MAOS nonlinear strengths in terms of a critical strain amplitude,11 showing a constant critical strain-rate amplitude at low De (strain-rate-driven nonlinearities) and a constant critical strain amplitude at high De (strain-driven nonlinearities). These linear-to-nonlinear elastic and viscous transitions are characteristic of Coleman-Noll “simple fluids”31,37 (cf. Fig. 11) and consistent with the combined sign interpretation of Ewoldt and Bharadwaj:11 [e3](ω)·[e1](ω) < 0 and [v3](ω)·[v1](ω) > 0 at low De; [e3](ω)·[e1](ω) > 0 and [v3](ω)·[v1](ω) < 0 at high De.

For LVE solidsGeq>0,22,23 the low-frequency behavior predicted for the elastic TSS MAOS functions, according to Eqs. (25a), (25b), (26a), (26b), (27a), and (27b), deserves a comment. When the storage modulus G′(ω) approaches a low-frequency plateau, limω→0G′(ω) = Geq, low-frequency plateaus are also predicted for the TSS [e1](ω) and [e3](ω) (cf. Fig. 8),

(29a)
(29b)

This is analogous to the corresponding long-time behavior of the TSS nonlinear stress relaxation modulus, limtG(t,γ0)=Geqh(γ0)=Geq1+Aγ02+Oγ04, as predicted by Eqs. (1) and (11). That the equilibrium shear modulus Geq explicitly enters Eqs. (25a), (25b), (26a), (26b), and (27a) is dictated by the inclusion of the delayed elastic stress σel(t) in the third-order Coleman-Noll TSS constitutive equation, Eq. (10) (Sec. II A 3).

In the case of LVE liquidsGeq=0,M1G<+,22 previous researchers5,28–30 have presented equations that can be compared with Eqs. (27a)–(27d) (Pearson and Rochefort;28 Cho et al.;29 and Wagner et al.30) and Eqs. (26a)–(26d) (Bharadwaj and Ewoldt5). The comparison, proceeding chronologically, is as follows.

In 1982, Pearson and Rochefort28 calculated the asymptotically nonlinear measures for the DE-IAA theory as coefficients of a power series expansion of the shear stress in both strain amplitude and frequency [their Eq. (13)]. Although they did not express their results as linear combinations of SAOS material functions, the analytical expressions [their Eqs. (B5)–(B10)] suggest, once the misprints are taken into account, that they can be recast in the form of Eqs. (27a)–(27d). Indeed, considering the tensorial strain measure of the DE-IAA theory,51,52,54,56 the DE-IAA MAOS functions are given by Eqs. (27a)–(27d) with Geq = 0 and A = −5/21 [Sec. IV B 3, Fig. 10(b)].

In 2010, Cho et al.29 considered the liquid-like TSS response in LAOS predicted by Eq. (10), with linear memory function mtt represented by a sum of discrete relaxation modes (i.e., a generalized Maxwell model3,4,99) and a power series expansion for the damping function h(γ) [equal to Eq. (11) to third order]. They adopted a power series expansion of the shear stress different from the one used by Pearson and Rochefort,28 but their Eqs. (B16a)–(B16b) can be translated into Eqs. (27a)–(27d).

A year later, under the same assumptions made by Cho et al.29 but focusing on the MAOS regime [i.e., Eq. (11)], Wagner et al.30 presented equations analogous to Eqs. (27a)–(27d) but written in terms of the Pearson-Rochefort power expansion stress coefficients [i.e., their Eqs. (19)–(22)]. In this case, the numerical prefactors (A = −29/210 and A = −4/105, in our nomenclature; Fig. 1) were set by the shear strain measure, respectively, of the linear and quadratic MSF30,100–102 models. Unfortunately, Wagner et al.30 also claimed that their Eqs. (19)–(22) described the MAOS response of the Doi-Edwards theory without the IAA, with A = −191/420 in our terminology. This claim is incorrect. Without invoking the IAA, the constitutive equation of the Doi-Edwards theory does not belong to the Coleman-Noll class and cannot be written in the form of Eq. (10) (which Wagner et al.30 used as the premise of their DE equations; cf. Secs. II A 1 and II A 2 and Subsection 1 of the  Appendix). As a result, incorrect is the conclusion reached by Wagner et al.30 (shown in Fig. 3 of Wagner et al.30 and Fig. 4 of Bharadwaj and Ewoldt5) that the DE-IAA and DE MAOS responses possess the same frequency dependence but exhibit a relative vertical shift set by different values of A. That this is not the case was discussed in 1982 by Helfand and Pearson104 (see their Fig. 3), who derived, via a non-trivial mathematical calculation, the correct Pearson-Rochefort power expansion stress coefficients for the DE theory [Sec. IV B 3, Fig. 10(b)].

Rather than obtaining equations equivalent to Eqs. (27a)–(27d) via a molecular theory (as in the work of Pearson and Rochefort28), or via Eqs. (10) and (11) with σel(t) = 0 and a Prony-Dirichlet series23 for G(s) (as in the work of Cho et al.29 and Wagner et al.30), Bharadwaj and Ewoldt5 derived Eqs. (26a)–(26d) for the special case of the corotational Maxwell (CM)2–5 model [their Eqs. (21)–(24)], using the tensorial shear measure calculated by Giacomin et al.2 (which corresponds to A = −1/6; Fig. 1). As opposed to earlier approaches, they kept the functional form of G(s) unspecified.

In summary, Eqs. (25a)–(25d) have not been presented earlier. As for Eqs. (26a)–(26d) and (27a)–(27d), they have not been previously derived for any model within the Coleman-Noll31–37 class, with unspecified linear memory function m(s) ≡ −∂G(s)/∂s and no assumptions on the long-time behavior of G(s), thereby embracing both liquid-likeGeq=0,M1G<+22 and solid-likeGeq>022,23 LVE behavior. These general results are particularly important when the sensitivity to long time scales is key, as we discuss next.

Although the frequency dependence of the TSS MAOS functions is set by the LVE response, weakly nonlinear oscillatory perturbations from the material equilibrium state are substantially more sensitive to the distribution of terminal relaxation times than their LVE counterparts. Underlying this behavior is the inherent dependence of the TSS MAOS functions on higher-order moments of the relaxation spectrum, H(τ), than the ones controlling the response in the LVE regime. This can be illustrated by considering the terminal (ω → 0) behavior of SAOS and TSS MAOS functions for LVE liquidsGeq=0,M1G<+22 (Fig. 2), as predicted by Eqs. (15b) and (16b) (SAOS) and Eqs. (26a)–(26d) (TSS MAOS).

FIG. 2.

TSS MAOS sensitivity to long time scales, illustrated by comparing terminal (De1) TSS MAOS behavior for LVE liquidsGeq=0,M1G<+22 with finite (black solid lines) and non-finite (red dashed-dotted lines) higher moments of G(s). (a) Long-time end of G(s) or H(τ) [cf. Eq. (33)], with M3G<+ (black solid line) and M2G+ (red dashed-dotted line). (b) Familiar scaling expected for LVE liquidsGeq=0,M1G<+22 (both black solid lines and red dashed-dotted lines). (c) Liquid-likeGeq=0,M3G<+ (black solid lines) and non-liquid-like (red dashed-dotted lines) terminal TSS MAOS behavior. For the former, see Figs. 1, 3, 5, 9, and 10; for the latter, see Figs. 6–8 and Table II.

FIG. 2.

TSS MAOS sensitivity to long time scales, illustrated by comparing terminal (De1) TSS MAOS behavior for LVE liquidsGeq=0,M1G<+22 with finite (black solid lines) and non-finite (red dashed-dotted lines) higher moments of G(s). (a) Long-time end of G(s) or H(τ) [cf. Eq. (33)], with M3G<+ (black solid line) and M2G+ (red dashed-dotted line). (b) Familiar scaling expected for LVE liquidsGeq=0,M1G<+22 (both black solid lines and red dashed-dotted lines). (c) Liquid-likeGeq=0,M3G<+ (black solid lines) and non-liquid-like (red dashed-dotted lines) terminal TSS MAOS behavior. For the former, see Figs. 1, 3, 5, 9, and 10; for the latter, see Figs. 6–8 and Table II.

Close modal

By expanding the trigonometric functions in Eqs. (15b) and (16b) in powers of ω, the asymptotic low-De behavior of G′(ω) and G″(ω) is

(30a)
(30b)

where

(31)

is the non-normalized i-th moment of G(s) about the origin.22 Equations (30a) and (30b) represent the familiar scaling of a liquid-like terminal SAOS response [Fig. 2(b); both black solid lines and red dashed-dotted lines], expected when Geq = 0 and M1G<+.

A similar power series expansion in Eqs. (26a)–(26d) gives

(32a)
(32b)
(32c)
(32d)

Equations (32a)–(32d) provide the scaling and relative magnitudes of a liquid-like terminal TSS MAOS response [Fig. 2(c); black solid lines], which clearly requires Geq = 0 and M3G<+. These requirements Geq=0,M3G<+ may be regarded as the definition of MAOS liquids.

Both scaling [ei](ω0)ω4,[vi](ω0)ω2 and relative magnitudes limω0[e3](ω)/[e1](ω)=1,limω0[v3](ω)/[v1](ω)=+1/3 in Eqs. (32a)–(32d) are consistent with the liquid-like terminal MAOS predictions obtained in 2014 by Bharadwaj and Ewoldt57 through a fourth-order fluid expansion (exhibited by the models in Figs. 1, 3, 5, 9, and 10). As opposed to the fourth-order fluid expansion derivation, assuming time-strain separability allows us to identify the front factors of a liquid-like terminal MAOS response with moments of G(s). In turn, this shows that, when Geq = 0, the asymptotic low-De behavior of TSS MAOS functions depends on higher moments of this distribution M3G,M2G than the SAOS functions M1G,M0G. If M3G is not finite, Eqs. (32a) and (32b) no longer apply; in this case, the terminal behavior of the elastic TSS MAOS functions [Fig. 2(c); red dashed-dotted lines] is not universal: scaling and relative magnitudes are dictated by how fast G(s) approaches zero in the t limit. If M2G also diverges, Eqs. (32c) and (32d) do not apply and the terminal behavior of the viscous TSS MAOS functions [Fig. 2(c); red dashed-dotted lines] ceases to be universal as well. Non-liquid-like terminal TSS MAOS behavior is shown in Figs. 6–8.

FIG. 3.

Comparison between TSS and non-TSS MAOS behavior associated with the same LVE model. (a) SAOS behavior of the single-mode Maxwell3,23,24,99 model and the single-mode SSTN109 model. A gray, thinner line is used for G(ω). (b) TSS MAOS behavior associated with the single-mode Maxwell model (black solid lines), according to Eqs. (28a)(28d) with A=+1, and non-TSS MAOS behavior of the SSTN model (red dashed-dotted lines), according to Eqs. (35a)(35d) with χ0=+(54/5)A (values of A and χ0 chosen to overlap the high-De plateaus of [e1](ω)). In either case, liquid-like terminal MAOS behavior57 is exhibited, in terms of scaling [ei](ω0)ω4,[vi](ω0)ω2 and relative magnitudes limω0[e3](ω)/[e1](ω)=1,limω0[v3](ω)/[v1](ω)=+1/3. (c) Linear-to-nonlinear elastic and viscous transitions: strain-rate-controlled at low De (γ̇0γ0ω=const.) and strain-controlled at high De (γ0=const.). These are characteristic of Coleman-Noll “simple fluids”31,37 (cf. Fig. 11) and consistent with the combined signs interpretation of Ewoldt and Bharadwaj:11[e3](ω)[e1](ω)<0 and [v3](ω)[v1](ω)>0 at low De; [e3](ω)[e1](ω)>0 and [v3](ω)[v1](ω)<0 at high De.

FIG. 3.

Comparison between TSS and non-TSS MAOS behavior associated with the same LVE model. (a) SAOS behavior of the single-mode Maxwell3,23,24,99 model and the single-mode SSTN109 model. A gray, thinner line is used for G(ω). (b) TSS MAOS behavior associated with the single-mode Maxwell model (black solid lines), according to Eqs. (28a)(28d) with A=+1, and non-TSS MAOS behavior of the SSTN model (red dashed-dotted lines), according to Eqs. (35a)(35d) with χ0=+(54/5)A (values of A and χ0 chosen to overlap the high-De plateaus of [e1](ω)). In either case, liquid-like terminal MAOS behavior57 is exhibited, in terms of scaling [ei](ω0)ω4,[vi](ω0)ω2 and relative magnitudes limω0[e3](ω)/[e1](ω)=1,limω0[v3](ω)/[v1](ω)=+1/3. (c) Linear-to-nonlinear elastic and viscous transitions: strain-rate-controlled at low De (γ̇0γ0ω=const.) and strain-controlled at high De (γ0=const.). These are characteristic of Coleman-Noll “simple fluids”31,37 (cf. Fig. 11) and consistent with the combined signs interpretation of Ewoldt and Bharadwaj:11[e3](ω)[e1](ω)<0 and [v3](ω)[v1](ω)>0 at low De; [e3](ω)[e1](ω)>0 and [v3](ω)[v1](ω)<0 at high De.

Close modal

Since23 

(33)

where Γ(·) is the complete gamma function,105–107 Eqs. (30a), (30b), (32a)–(32d) can be equivalently written in terms of moments of H(τ) about the origin, MiH0τiH(τ)dτ, because Mi0G=Γ(i+1)Mi0H when Geq = 0. Thus, the alternative definitions of LVE liquids Geq=0,M1H<+ and MAOS liquids Geq=0,M3H<+ are equivalent to those based on moments of G(s). This implies that the TSS MAOS sensitivity to long time scales is inherently related to non-normalized higher-order moments of the relaxation spectrum, H(τ), when Geq = 0.

We prefer to use moments of G(s), rather than H(τ), in our definitions of LVE and MAOS liquids because the former, but not the latter, always diverge when the terminal viscoelastic behavior is non-liquid-like. Consider, for instance, the case of LVE solidsGeq>0.22,23 In terms of classical mechanical models, the simplest solid-like terminal SAOS behavior is exhibited by the standard linear solid,23 exemplified by the single-mode Zener108 (or 3-parameter Maxwell,23 in Tschoegl’s terminology) model [Fig. 8(a); black solid lines]. In this case, all moments Mi0H are finite (because H(τ) is a delta function23), whereas Mi0G.

We commented earlier on the generality of Eq. (1). Although it can be derived from the memory-integral expansion3 (Sec. II A 2 and Subsection 4 of the  Appendix), its validity transcends the mathematical framework of the Coleman-Noll31–37 class ( Appendix), as indicated by both experimental observations and theoretical considerations. Experimentally, Eq. (1) has been verified for a large variety of material systems,17–21,59–66 although its validity is often restricted to specific ranges of time and strain.19,26,27,61,63–66 Theoretically, Eq. (1) is predicted by molecular theories, such as the DE theory,53–56 which yield constitutive equations that do not belong to the Coleman-Noll class (Sec. II A 1 and Subsection 1 of the  Appendix).

By analogy, it seems plausible to expect that the validity of Eqs. (25)–(27) may extend beyond the type of LVE and NLVE behavior exhibited by Coleman-Noll “simple materials”.31–37 Accepting this premise, it is then physically sensible to use Eqs. (25)–(27) to predict TSS MAOS signatures independently of the LVE requirements imposed by the Coleman-Noll class (Subsection 1 of the  Appendix). Even though Eqs. (25)–(27) have been derived from the multiple integral expansion for “simple materials”, they do not inherently impose the restrictions on m(s), G(s), G′(ω), and η′(ω) necessary to satisfy all the mathematical requirements formulated by Coleman and Noll. For instance, expressions for m(s) or G(s) associated with power-law distributions of relaxation times may violate the Coleman-Noll definition of fading memory (Subsection 1 of the  Appendix), and η′(ω) may in general include a purely dissipative term (e.g., a solvent contribution; Sec. II B) that inevitably violates the Coleman-Noll assumption of high-De purely elastic behavior (Subsection 1 of the  Appendix). If used in Eqs. (25)–(27), general expressions for m(s), G(s), G′(ω), or η′(ω) may therefore lead to results that deviate from the NLVE behavior expected for Coleman-Noll “simple materials” (Figs. 6–9). This permits to explore a larger diversity of TSS MAOS responses, which may describe experimental data and/or be obtained in the future on the basis of new continuum or statistical mechanical theories. The usefulness of this approach, for both practical and theoretical standpoints, will be shown in Sec. IV B. This will be accomplished by considering a variety of LVE models, beyond those (i) satisfying the LVE Coleman-Noll assumptions or (ii) associated with liquid-like and solid-like viscoelastic behavior.

As a first example of using our results, we show how to determine if a MAOS model within the Coleman-Noll class is time-strain separable. This is non-obvious when the nonlinear model parameters appear as front factors in the MAOS analytical solutions, which is a necessary but not sufficient condition for time-strain separability. In this case, TSS MAOS behavior is exhibited only if the frequency dependence of the MAOS functions is that predicted by Eqs. (25)–(27). Examples of TSS and non-TSS MAOS models are provided in Table I.

TABLE I.

Examples of TSS vs. non-TSS MAOS models.

TSSNon-TSSNon-TSS
(non-obvious from MAOS equations)(obvious from MAOS equations)
Corotational Maxwell (CM)2–5  Strain-stiffening transient network (SSTN)109  Giesekus5,110–114 
Molecular stress function (MSF)5,30,100–103 Simple emulsion5,115 Curtiss-Bird (ε ≠ 0)3,5,48,67–69 
Curtiss-Bird (ε = 0)3,5,48,67–70   
Doi-Edwards with IAA28,49–52,56   
Doi-Edwards without IAA53–56,104a   
TSSNon-TSSNon-TSS
(non-obvious from MAOS equations)(obvious from MAOS equations)
Corotational Maxwell (CM)2–5  Strain-stiffening transient network (SSTN)109  Giesekus5,110–114 
Molecular stress function (MSF)5,30,100–103 Simple emulsion5,115 Curtiss-Bird (ε ≠ 0)3,5,48,67–69 
Curtiss-Bird (ε = 0)3,5,48,67–70   
Doi-Edwards with IAA28,49–52,56   
Doi-Edwards without IAA53–56,104a   
a

The DE constitutive equation does not belong to the Coleman-Noll class (cf. Secs. II A 1, II A 2, and III B; Subsection 1 of the  Appendix); therefore, its TSS MAOS signatures [shown in Fig. 10(b)] are different from those given by Eqs. (25)–(27).

Without loss of generality, the task at hand can be explained as follows. Let us consider a set of N Maxwell mechanical elements99 in parallel, i.e., a discrete distribution (relaxation spectrum)23,24,116 of N relaxation times τi’s each associated with modulus Gi and viscosity ηi = Giτi. The stress contributions from each relaxation mode to the four MAOS material functions can then be written as91 

(34a)
(34b)
(34c)
(34d)

where f[e1](), f[e3](), f[v1](), and f[v3]() are dimensionless, model-specific functions of frequency ω, relaxation times τi’s, and q nonlinear model parameters represented by the q-dimensional vector ξ(τi)=ξ1(τi),,ξq(τi). The linear superposition of the modal stress contributions, Eqs. (34a)–(34d), gives [e1](ω), [e3](ω), [v1](ω), and [v3](ω) (for the case involving a continuous relaxation spectrum, see the work of Martinetti et al.91). Since the functions f[e1](), f[e3](), f[v1](), and f[v3]() in Eqs. (34a)–(34d) depend on both frequency and nonlinear parameters, the MAOS response in this case is clearly not time-strain separable. The Giesekus110–113 model provides an example of a time-strain inseparable constitutive equation, and its MAOS response is inevitably in the form of Eqs. (34a)–(34d) [see Eqs. (34)–(37) of Bharadwaj and Ewoldt5].

However, even if the nonlinear parameters ξ(τi) appeared as front factors in Eqs. (34a)–(34d), this would not necessarily imply time-strain separability as defined by either Eq. (1) or Eqs. (10) and (11). This behavior, which may be referred to as an apparent TSS MAOS response, is exemplified by the strain-stiffening transient network (SSTN) theory recently developed by Bharadwaj et al.109 The resulting SSTN constitutive equation yields an LVE response109 equivalent to that of a single-mode Maxwell3,23,24,99 model [Fig. 3(a)]. A single dimensionless nonlinear parameter χ0 determines the strength of nonlinearity and appears as a front factor in the SSTN model predictions for the four MAOS material functions,109 

(35a)
(35b)
(35c)
(35d)

The comparison between Eqs. (34a)–(34d) and Eqs. (35a)–(35d) reveals that fg(·) = χ0Kg(·) for each MAOS material function g(ω), i.e., the dependence on frequency, set by Kg(·), is decoupled from the effect of the nonlinear model parameter χ0. Nonetheless, the SSTN model, and therefore its MAOS response [Eqs. (35a)–(35d)], are not time-strain separable in the sense of Eq. (1) or Eqs. (10) and (11). Indeed, considering the SSTN equation for the stress tensor [Eq. (19) of Bharadwaj et al.109], subject to the shear strain history γ(t) = γ0s(t), we obtain the analytical expression for the SSTN nonlinear relaxation modulus,

(36)

where G(t)=G0et/τ0, as required by the single-mode Maxwell LVE behavior, and the solution has been expanded through a procedure similar to the one presented in Appendix 1 of Bharadwaj et al.109 Clearly, the time-dependent behaviors of G(t, γ0) and G(t) are not the same (cf. Sec. II A 1). In the t → 0 elastic limit, Eq. (36) reduces to Eqs. (61) and (62) of Bharadwaj et al.109 

Equations (35a)–(35d) are plotted in Fig. 3(b) (red dashed-dotted lines). Since the SSTN constitutive equation does belong to the Coleman-Noll class, that the SSTN MAOS functions given by Eqs. (35a)–(35d) are not time-strain separable can be deduced simply by comparing their De-dependence with that of Eqs. (28a)–(28d). This comparison makes it clear that, despite χ0 being a front factor in Eqs. (35a)–(35d), the nonlinearity of the SSTN model still affects the frequency dependence of the kernel functions Kg(·) to produce a response that deviates from the TSS MAOS response [i.e., Eqs. (28a)–(28d)]. On the time domain, the extent of such deviation is identified by the time-dependent behavior of the ratio G(t, γ0)/G(t), given by the term in square brackets in Eq. (36).

The equations derived in Sec. III A, Eqs. (25)–(27), allow one to study how LVE behavior affects the frequency dependence of TSS MAOS material functions. To this end, we considered several LVE models: (i) simple analytical expressions for H(τ) (associated with log-normal23,38 and asymmetric Lorentzian23,39 distributions, Sec. IV B 1), (ii) fractional mechanical models (Maxwell40–46 and Zener,43,44,46 Sec. IV B 2), and molecular theories (Rouse3,24,47,48 and Doi-Edwards,49–56 Sec. IV B 3). These choices were dictated by our desire to explore a large diversity of TSS MAOS responses (Sec. III D), regardless of the LVE requirements imposed by the Coleman-Noll class (Subsection 1 of the  Appendix). The relaxation spectra shown in Fig. 4 govern the time-varying part of these LVE models. The corresponding LVE material functions may also contain time-independent viscoelastic constants, associated with purely viscous (e.g., ηsol) or purely elastic (e.g., Geq) behavior [cf. Sec. II B and Eqs. (17) and (18)].

FIG. 4.

Continuous relaxation spectra of the LVE models considered in this work, originating from (i) simple analytical functions, i.e., log-normal (LN)23,38 and asymmetric Lorentzian (AL)23,39 distributions; (ii) mechanical models, i.e., fractional Maxwell (FM)40–46 and fractional Zener (FZ);43,44,46 and (iii) molecular theories, i.e., Rouse (R)3,24,47,48 and Doi-Edwards (DE).49–56 The discrete spectra predicted by the molecular theories are approximated here by continuous distribution functions of the power-law type: HR(τ)τ1/2 for the Rouse24,117 theory and HDE(τ)τ+1/2 for the Doi-Edwards51,118 theory.

FIG. 4.

Continuous relaxation spectra of the LVE models considered in this work, originating from (i) simple analytical functions, i.e., log-normal (LN)23,38 and asymmetric Lorentzian (AL)23,39 distributions; (ii) mechanical models, i.e., fractional Maxwell (FM)40–46 and fractional Zener (FZ);43,44,46 and (iii) molecular theories, i.e., Rouse (R)3,24,47,48 and Doi-Edwards (DE).49–56 The discrete spectra predicted by the molecular theories are approximated here by continuous distribution functions of the power-law type: HR(τ)τ1/2 for the Rouse24,117 theory and HDE(τ)τ+1/2 for the Doi-Edwards51,118 theory.

Close modal

Since the frequency-dependent behavior of the TSS MAOS functions can be predicted without any restrictions on the linear relaxation modulus G(s) (Secs. III A and III D), the variety of TSS MAOS signatures accessible through Eqs. (25)–(27) provides a means to describe experimental MAOS data. This use of Eqs. (25)–(27) was outside the scope of this work and remains for future investigations. Here, we focus instead on the theoretical implications of Eqs. (25)–(27). Specifically, the following examples will show the potential benefits of studying weakly nonlinear oscillatory perturbations from equilibrium for systems dominated by long-range spatial or temporal correlations and for model selection. In addition, MAOS signatures will be revealed that differ from the liquid-like terminal MAOS behavior predicted by Bharadwaj and Ewoldt,57 even in the presence of liquid-like terminal SAOS behavior (cf. Fig. 2). As discussed in Sec. III C, this occurs when higher moments of G(s) (or H(τ)) are not finite.

The easiest way to calculate TSS MAOS signatures from our results in Sec. III A is to use Eqs. (27a)–(27d), which require knowledge of G′(ω) and η′(ω) for each given LVE model under consideration. When analytical expressions for G′(ω) and η′(ω) are not directly available, G′(ω) and η′(ω) can be determined from the relaxation spectrum H(τ), through Eqs. (17) and (18). This approach was adopted for LVE behavior associated with log-normal23,38 and asymmetric Lorentzian23,39 distributions. Explicit expressions for G′(ω) and η′(ω) in terms of model parameters were used for the remaining LVE models in Fig. 4. Specifically, we employed the following equations for LVE spectra and material functions.

1. Simple distributions of relaxation times: Log-normal and asymmetric Lorentzian

The continuous (viscosity-weighted)23,91 relaxation spectrum HLN(τ) derived from the log-normal distribution is23,38

(37)

where QLN(τ) is the (modulus-weighted)23,91 log-normal distribution of relaxation times, τmax is the log-median relaxation time scale, Hmax is the peak of the spectrum HLN(τ), and σ is the standard deviation of the log-normal distribution. First used by Wiechert38 in 1893 to model the relaxation behavior in glasses, the distribution of Eq. (37) can be rationalized, for instance, for the terminal spectrum of transient networks.91,119 For this class of materials, the average lifetime of the inter-chain non-covalent associations can be thought of as the mean time for an association bond to break, which is an activated process characterized by a free energy barrier of height ε = ΔG/kBT ≫ 1 (where kBT is the thermal energy).120–125 Therefore, according to Kramers theory,126,127τ0eε. Assuming a Gaussian distribution of energy barriers yields a log-normal distribution of escape times.128 We used Eq. (37) to study the sensitivity of TSS MAOS functions to the spread of a simple single-peak distribution of relaxation times.

Figure 5 shows SAOS and TSS MAOS signatures for log-normal-based LVE relaxation spectra HLN(τ) [inset of Fig. 5(a)] of increasing breadth σ, therefore increasing “polydispersity index (PDI) of terminal relaxation times”27,92 [based on Eq. (37), PDI=eσ2].91 Note that σ = 0.96 (red dashed-dotted lines) corresponds to PDI ≈ 2.5, i.e., the “universal” terminal spectrum exhibited by melts and solutions of nearly monodisperse and well-entangled linear polymer chains.27,56,72,92,129 As the spectrum breadth increases, the liquid-like terminal regime behavior is pushed to lower frequencies (De). This shift is less dramatic for SAOS [Fig. 5(a)] and more dramatic for TSS MAOS [Figs. 5(b) and 5(c)] due to the higher sensitivity to long time scales of TSS MAOS functions [Eq. (32)] compared to their LVE counterparts [Eq. (30)]. Increasing σ separates the low- and high-De limits, but the terminal SAOS and TSS MAOS slopes are unchanged since the relevant moments in Eqs. (30) and (32) are finite for any σ.

FIG. 5.

Sensitivity of TSS MAOS functions to the increasing spread (σ) of log-normal-based23,38 LVE relaxation spectra HLN(τ) [Eq. (37)]. (a) SAOS, Eqs. (17) and (18); (b) TSS MAOS, Eqs. (27a)(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. A wider polydispersity of relaxation times shifts significantly the location of the sign change of [v3](ω) to lower frequencies.

FIG. 5.

Sensitivity of TSS MAOS functions to the increasing spread (σ) of log-normal-based23,38 LVE relaxation spectra HLN(τ) [Eq. (37)]. (a) SAOS, Eqs. (17) and (18); (b) TSS MAOS, Eqs. (27a)(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. A wider polydispersity of relaxation times shifts significantly the location of the sign change of [v3](ω) to lower frequencies.

Close modal

Notably, a wider polydispersity of modes shifts significantly the location of the zero crossing of [v3](ω) to lower frequencies [Fig. 5(b)]. Consistent with recent experimental observations,91 this suggests that the sign change location for the third-harmonic viscous nonlinearity may be sensitive to large-scale structural features (such as molar mass distribution) that control the polydispersity of terminal relaxation times.

A more flexible mathematical function to model the relaxation spectrum is the asymmetric Lorentzian distribution,23,39

(38)

where τmax and Hmax are again the location and value of the peak, r and r′ control the steepness on either side of the peak [the steepness at short (long) times increases as r (r′) increases], and ρ affects the overall spread of the distribution independently of r and r′. Greater flexibility comes at a cost: Eq. (38) cannot, in general, be integrated analytically.23 Nonetheless, the asymmetric Lorentzian distribution allowed us to study the sensitivity of TSS MAOS functions to the steepness (ρr′) of the long-time end of the spectrum.

Figure 6 shows SAOS and TSS MAOS signatures for asymmetric Lorentzian distributions HAL(τ) [inset of Fig. 6(a)] of decreasing steepness (ρr′) at long times. As the steepness decreases from ρr′ = 4.25 (black solid lines) to ρr′ = 2.01 (red dashed-dotted lines), G′(ω) and G″(ω) change slightly in magnitude while maintaining the familiar liquid-like terminal slopes. The terminal behavior of the TSS MAOS functions, on the other hand, changes dramatically: no longer universal, it depends on the steepness ρr′ of the long-time end of the spectrum. The non-liquid-like terminal TSS MAOS slopes are accompanied by a significant change in magnitude compared to the change observed for the SAOS functions over the same frequency range. Moreover, the sign change location for the third-harmonic viscous nonlinearity, [v3](ω), shifts by about two (!) orders of magnitude to lower frequencies. This occurs over a frequency range that would be experimentally accessible even without the application of time-temperature superposition. When the steepness of the long-time end of HAL(τ) is decreased even further (ρr′ = 1.25, blue dashed lines), G′(ω) ceases to exhibit the liquid-like terminal slope of two and the sign change of [v3](ω) disappears ([v3](ω) < 0 over the entire frequency range).

FIG. 6.

Sensitivity of TSS MAOS functions to the decreasing steepness (ρr) of the long-time end of LVE relaxation spectra HAL(τ) represented by the asymmetric Lorentzian23,39 distribution [Eq. (38)]. (a) SAOS, Eqs. (17) and (18); (b) TSS MAOS, Eqs. (27a)(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. For ρr=2.01 (red dashed-dotted lines) and ρr=1.25 (blue dashed lines), non-liquid-like terminal TSS MAOS behavior is exhibited because M2G and M3G in Eq. (32) diverge (Sec. III C), and [v3](ω) is the material function most sensitive to the decreasing steepness of the long-time end of the spectrum. For ρr=1.25, M1G also diverges and the terminal slope of G(ω) is no longer two [Eq. (30a)]. Terminal SAOS and TSS MAOS scaling conditions are more generally outlined in Table II.

FIG. 6.

Sensitivity of TSS MAOS functions to the decreasing steepness (ρr) of the long-time end of LVE relaxation spectra HAL(τ) represented by the asymmetric Lorentzian23,39 distribution [Eq. (38)]. (a) SAOS, Eqs. (17) and (18); (b) TSS MAOS, Eqs. (27a)(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. For ρr=2.01 (red dashed-dotted lines) and ρr=1.25 (blue dashed lines), non-liquid-like terminal TSS MAOS behavior is exhibited because M2G and M3G in Eq. (32) diverge (Sec. III C), and [v3](ω) is the material function most sensitive to the decreasing steepness of the long-time end of the spectrum. For ρr=1.25, M1G also diverges and the terminal slope of G(ω) is no longer two [Eq. (30a)]. Terminal SAOS and TSS MAOS scaling conditions are more generally outlined in Table II.

Close modal
TABLE II.

Terminal (ω → 0) scaling of SAOS and TSS MAOS functions for a relaxation spectrum HAL(τ) [Eq. (38)] represented by the asymmetric Lorentzian23,39 distribution (cf. Fig. 6).

0 < ρr′ < 1aρr′ = 11 < ρr′ < 2bρr′ = 22 < ρr′ < 3c3 ≤ ρr′ < 44 ≤ ρrd
G′(ω∼ ωρr ωρr ωρr ω2 ω2 ω2 ω2 
G″(ω∼ ωρr ω ω ω ω ω ω 
[e1](ω∼ ωρr ωρr ωρr ωρr ωρr ωρr ω4 
[e3](ω∼ ωρr ω4e ωρr ωρr ωρr ωρr ω4 
[v1](ω∼ ωρr′−1 ωρr′−1 ωρr′−1 ωρr′−1 ωρr′−1 ω2 ω2 
[v3](ω∼ ωρr′−1 ωρr′−1 ωρr′−1 ω2e ωρr′−1 ω2 ω2 
0 < ρr′ < 1aρr′ = 11 < ρr′ < 2bρr′ = 22 < ρr′ < 3c3 ≤ ρr′ < 44 ≤ ρrd
G′(ω∼ ωρr ωρr ωρr ω2 ω2 ω2 ω2 
G″(ω∼ ωρr ω ω ω ω ω ω 
[e1](ω∼ ωρr ωρr ωρr ωρr ωρr ωρr ω4 
[e3](ω∼ ωρr ω4e ωρr ωρr ωρr ωρr ω4 
[v1](ω∼ ωρr′−1 ωρr′−1 ωρr′−1 ωρr′−1 ωρr′−1 ω2 ω2 
[v3](ω∼ ωρr′−1 ωρr′−1 ωρr′−1 ω2e ωρr′−1 ω2 ω2 
a

Same power-law behavior exhibited by the fractional Maxwell40–46 model (Fig. 7) and the fractional Zener43,44,46 model (Fig. 8).

b

Shown by the blue dashed curves in Fig. 6.

c

Shown by the red dashed-dotted curves in Fig. 6.

d

Shown by the black solid curves in Fig. 6.

e

Special cases.

The terminal SAOS and TSS MAOS slopes can again be understood by considering the conditions for which the moments MiG in Eqs. (30) and (32) diverge (Sec. III C). A comprehensive analysis of the terminal SAOS and TSS MAOS scaling conditions is provided in Table II. For the cases considered in Fig. 6, [v3](ω) is the material function most sensitive to the decreasing steepness of the long-time end of the spectrum.

2. Fractional mechanical models: Maxwell and Zener

Complex systems characterized by long-range spatial or temporal correlations are often modeled by fractional viscoelastic models. We consider here the fractional Maxwell (FM)40–46 and fractional Zener (FZ)43,44,46 models and their TSS MAOS responses. The FM40–46 model captures double power-law relaxation patterns and allowed us to study the sensitivity of TSS MAOS functions to slight deviations from liquid-like terminal MAOS scaling (Fig. 7). The FZ43,44,46 model describes the so-called Z- or S-shaped transitions23 between two pseudo-plateaus. It includes, as a special case, the single-mode Zener23,108 model, which exemplifies solid-like LVE behavior with the fewest model parameters (Sec. III C). Therefore, the FZ model allowed us to (i) simulate the first and simplest solid-like TSS MAOS signatures and (ii) investigate the sensitivity of TSS MAOS functions to slight deviations from a solid-like terminal MAOS response (Fig. 8). We remind the reader that this was possible because our derivation of TSS MAOS functions in Sec. III A encompasses both LVE liquidsGeq=0,M1G<+22 and LVE solidsGeq>0.22,23

FIG. 7.

Effect of fractional Maxwell (FM)40–46 LVE relaxation spectra on TSS MAOS signatures: another example of non-liquid-like terminal TSS MAOS behavior. (a) SAOS, Eqs. (40a) and (40b); (b) TSS MAOS, Eqs. (27a)–(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. FM model with power-law exponents 0β<α1, for increasing departure from classical mechanical behavior: {β=0,α=1} (single-mode Maxwell3,23,24,99 model; black solid lines; the same as in Fig. 3), {β=0,α=0.999} (red dashed-dotted lines), and {β=0.250,α=0.750} (blue dashed lines). The slight deviation from liquid-like terminal SAOS and TSS MAOS scaling (red dashed-dotted lines) is detectable in MAOS at frequencies larger (10 times or so) than in SAOS, especially through [v1](ω) and [v3](ω); note that the latter exhibits an additional sign change at a frequency, De102, which would be easily accessible experimentally.

FIG. 7.

Effect of fractional Maxwell (FM)40–46 LVE relaxation spectra on TSS MAOS signatures: another example of non-liquid-like terminal TSS MAOS behavior. (a) SAOS, Eqs. (40a) and (40b); (b) TSS MAOS, Eqs. (27a)–(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. FM model with power-law exponents 0β<α1, for increasing departure from classical mechanical behavior: {β=0,α=1} (single-mode Maxwell3,23,24,99 model; black solid lines; the same as in Fig. 3), {β=0,α=0.999} (red dashed-dotted lines), and {β=0.250,α=0.750} (blue dashed lines). The slight deviation from liquid-like terminal SAOS and TSS MAOS scaling (red dashed-dotted lines) is detectable in MAOS at frequencies larger (10 times or so) than in SAOS, especially through [v1](ω) and [v3](ω); note that the latter exhibits an additional sign change at a frequency, De102, which would be easily accessible experimentally.

Close modal
FIG. 8.

Effect of fractional Zener (FZ)43,44,46 LVE relaxation spectra on TSS MAOS signatures: first example of solid-like terminal TSS MAOS behavior and deviations from it. (a) SAOS, Eqs. (42a) and (42b); (b) TSS MAOS, Eqs. (27a)–(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. FZ model with power-law exponents 0γ=β<α1 and ratio between characteristic moduli Gγ/GFM=103, for increasing departure from classical mechanical behavior: {β=γ=0,α=1} (single-mode Zener23,108 model; black solid lines), {β=γ=0.001,α=1} (red dashed-dotted lines), and {β=γ=0.125,α=0.750} (blue dashed lines). The solid-like terminal SAOS behavior G(ω0)=Geq (black solid lines) is manifested in (i) non-vanishing elastic MAOS nonlinearities at low De [ei](ω0)=[ei]eq, per Eqs. (29a) and (29b), and (ii) the appearance of an additional sign change for [e3](ω) at De101 (cf. Fig. 7). The slight deviation from solid-like terminal SAOS and TSS MAOS scaling (red dashed-dotted lines) is detectable in MAOS (all functions except [e3](ω)) at frequencies much larger 104timesorso than in SAOS (cf. Fig. 7).

FIG. 8.

Effect of fractional Zener (FZ)43,44,46 LVE relaxation spectra on TSS MAOS signatures: first example of solid-like terminal TSS MAOS behavior and deviations from it. (a) SAOS, Eqs. (42a) and (42b); (b) TSS MAOS, Eqs. (27a)–(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. FZ model with power-law exponents 0γ=β<α1 and ratio between characteristic moduli Gγ/GFM=103, for increasing departure from classical mechanical behavior: {β=γ=0,α=1} (single-mode Zener23,108 model; black solid lines), {β=γ=0.001,α=1} (red dashed-dotted lines), and {β=γ=0.125,α=0.750} (blue dashed lines). The solid-like terminal SAOS behavior G(ω0)=Geq (black solid lines) is manifested in (i) non-vanishing elastic MAOS nonlinearities at low De [ei](ω0)=[ei]eq, per Eqs. (29a) and (29b), and (ii) the appearance of an additional sign change for [e3](ω) at De101 (cf. Fig. 7). The slight deviation from solid-like terminal SAOS and TSS MAOS scaling (red dashed-dotted lines) is detectable in MAOS (all functions except [e3](ω)) at frequencies much larger 104timesorso than in SAOS (cf. Fig. 7).

Close modal

Fractional mechanical models enable the quantitative description of self-similar dynamics over a broad range of time scales with only a few, physically meaningful material parameters.45,46 Self-similar dynamics, characterized by power-law relaxations coupled with subdiffusive behavior,130–132 is the hallmark of complex systems composed of a large diversity of elementary units interacting on a wide range of length and time scales133 and materials that are far from thermodynamic equilibrium.134 Not surprisingly, power-law relaxation patterns are ubiquitous. Within the realm of polymer science, they have been observed in numerous systems,135 including rubbers,136–140 homopolymers undergoing the glass-rubber transition,141,142 linear entangled homopolymers (both monodisperse143,144 and polydisperse145), branched polymers with star-, H-, or comb-like topologies,146–154 ring polymers,155 polymer blends,150,156–158 polymer fractals,159,160 and chemical gels at the gel point.161–167 In the class of physical gels, examples include filled polymers,168,169 associating polymers,149,170,171 crystallizing polymers,172 foams,173 liquid crystalline polymers,174 microgel dispersions,175,176 suspensions,177,178 soft glassy materials,179,180 chewing gum,181 and microphase-segregated block polymers both above182–187 and below46,188–191 the glass transition temperature of the higher Tg block. Being the result of multimodal and cooperative relaxation processes, self-similar dynamics is not even a prerogative of polymeric materials. Rocks, metals, inorganic glasses,192–194 and proteins194,195 have been found to exhibit power-law relaxations.

A simple and powerful approach to developing fractional models of LVE behavior is to consider classical series-parallel23 models and replace the classical mechanical elements (springs and dashpots) with fractional counterparts (spring-pots196). This guarantees that the required thermodynamic constraints are automatically fulfilled.43,197,198 Each spring-pot, equivalent to hierarchical and fractal arrays (infinite “trees” or “ladders”) of springs and dashpots167,199,200 and customarily depicted by an upright triangle symbolizing a “ladder,”43 is fully characterized by a dyad45 (F, m): the quasi-property45,201–203F (with dimensions of Pa sm) and the power-law exponent m (with 0 ≤ m ≤ 1). Thus, the spring-pot interpolates between a spring (m = 0) and a dashpot (m = 1), and the quasi-property F, intimately related to the stochastic behavior on a microscopic level,130–132 reduces to a modulus and a viscosity in the limiting cases. Schiessel et al.43 and Friedrich et al.44 provided good overviews of fractional versions of classical series-parallel models, where orderly arrangements of (a few) fractional elements naturally lend themselves to the description of general relaxation patterns. The series combination of two spring-pots (FM40–46 model) gives rise to a double power-law behavior, i.e., LVE functions undergoing a transition between an initial gradual relaxation (pseudo-plateau region) and a steeper power-law descent to full relaxation. The parallel combination of the FM model with a spring-pot (FZ43,44,46 model) parsimoniously describes the so-called Z- or S-shaped transitions23 between two pseudo-plateaus.

However complex the arrangement of spring-pots, a fractional model can be simply understood from the shape of its relaxation spectrum H(τ). Since the FM model does not possess an equilibrium modulus (Geq,FM = 0), the analytical expression for the relaxation spectrum, HFM(τ), can be determined from its operational relaxance through integral transformation calculus (as described by Tschoegl23). We find46 

(39)

where zτ/τFM is the dimensionless time scale, τFM ≡ (V/W)1/(αβ) is the characteristic relaxation time of the model, GFMV(W/V)α/(αβ)=V(τFM)α=W(τFM)β is a characteristic modulus, while (V, α) and (W, β) are the parameters of the two spring-pots. Without loss of generality, we take 0 ≤ β < α ≤ 1. Clearly, the FM model reduces to the ordinary Maxwell model for {0 = β, α = 1}. The storage modulus, GFM(De), and the loss viscosity, ηFM(De), are45,46

(40a)
(40b)

where ηFMGFMτFM is a characteristic viscosity (numerically equal to the steady-state zero-shear viscosity, η0, when α = 1; for α ≠ 1, η0).

Figure 7 explores the effect of FM relaxation spectra, HFM(τ) [inset of Fig. 7(a)], on TSS MAOS signatures. This provides another example of non-liquid-like terminal TSS MAOS behavior. A single-mode Maxwell model is shown for reference (black solid lines; the same as in Fig. 3). The slight deviation, elicited by {β = 0, α = 0.999}, from liquid-like terminal SAOS G(ω0)M1Gω2,G(ω0)M0Gω and MAOS [ei](ω0)ω4,[vi](ω0)ω2 scaling, is detectable in MAOS at frequencies larger (10 times or so) than in SAOS, especially through [v1](ω) and [v3](ω). The latter exhibits an additional sign change at a frequency, De ≈ 10−2, which would be easily accessible experimentally. In Fig. 7(c), the strain-controlled linear-to-nonlinear elastic and viscous transitions at low De, i.e., [e3](ω → 0)·[e1](ω → 0) > 0 and [v3](ω → 0)·[v1](ω → 0) < 0, are opposite to simple-fluid behavior (cf. Fig. 3). Somewhat surprisingly, at low De, a constant critical strain governs the onset of nonlinearity, rather than a critical strain rate.

Adding a spring-pot with parameters (X, γ) in parallel to the FM model yields the FZ model. Here, we restrict our attention to the parameter range 0 ≤ γ = β < α ≤ 1. It is evident that the FZ model reduces to the ordinary Zener model for {0 = γ = β, α = 1}. For γ > 0, the FZ model does not possess an equilibrium modulus (Geq,FZ = 0). From the combination rules23 (known as equations of congruence, in structural mechanics204), it then follows that the relaxation spectrum is46 

(41)

where the relaxation spectrum originating from the FM subunit, HFM(z), is still given by Eq. (39), the second term in Eq. (41) is the contribution of the γ-spring-pot, and GγX(W/V)γ/(αβ)=X(τFM)γ represents a second characteristic modulus of the model. Notice that the characteristic relaxation time of the FZ model (τFM), and the first characteristic modulus (GFM), only depend on the parameters of the FM subunit (hence, their subscripts). Obviously, the isolated γ-spring-pot dictates the long-time response and precludes the FZ model from exhibiting a finite steady-state zero-shear viscosity (regardless of the parameter range). It is possible to show43,46 that the FZ model with 0 < γ = β < α < 1 and Gγ/GFM ≪ 1 exhibits three distinct (i.e., fully developed) time regimes of power-law behavior. This is the case considered here since we take Gγ/GFM = 10−3. Consistent with the additivity of the relaxances,23 the storage modulus, GFZ(De), and the loss viscosity, ηFZ(De), are46 

(42a)
(42b)

where ηFMGFMτFM is a characteristic viscosity of the FZ model (originating entirely from the FM subunit).

Figure 8 explores the effect of FZ relaxation spectra, HFZ(τ) [inset of Fig. 8(a)], on TSS MAOS signatures. This provides the first example of a solid-like terminal TSS MAOS response and deviations from it. A single-mode Zener model is shown for reference (black solid lines). Several unique features can be observed. The solid-like terminal SAOS behavior G(ω0)=Geq, exemplified by the Zener model, is manifested in (i) non-vanishing elastic MAOS nonlinearities at low De [ei](ω0)=[ei]eq, per Eqs. (29a) and (29b), and (ii) the appearance of an additional sign change for [e3](ω) at De ≈ 10−1 (cf. Fig. 7). The slight deviation, elicited by {β = γ = 0.001, α = 1}, from solid-like terminal SAOS and TSS MAOS scaling, is detectable in MAOS (all functions except [e3](ω)) at frequencies much larger 104timesorso than in SAOS (cf. Fig. 7). It is worth noting that in this case (red dashed-dotted lines) the low-De response for [e1](ω) increases even though the equilibrium modulus disappears (Geq,FZ = 0 for γ > 0); this is the result of the −2Geq term in Eq. (27a). In Fig. 8(c), the strain-controlled linear-to-nonlinear elastic and viscous transitions at low De, i.e., [e3](ω → 0)·[e1](ω → 0) > 0 and [v3](ω → 0)·[v1](ω → 0) < 0, are opposite to simple-fluid behavior (cf. Figs. 3 and 7).

3. Molecular theories: Rouse and Doi-Edwards

We now consider the TSS MAOS response associated with molecular theories for flexible linear polymer chains: the Rouse3,24,47,48 theory for dilute solutions and the Doi-Edwards49–56 theory for entangled melts. Molecular theories of the viscoelastic behavior of polymers predict discrete distributions of relaxation times,24,92 and therefore the results (i.e., material functions) are in the form of sums. Here, the theoretical predictions for the SAOS material functions will be approximated by closed-form expressions, and these will be used in Eqs. (27a)–(27d) to explore the TSS MAOS behavior.

As mentioned at the end of Sec. II B, outside of the Coleman-Noll class (cf. Subsection 1 of the  Appendix), the expression for G″(ω) can include a purely dissipative term, e.g., ωηsol, where ηsol is a solvent viscosity.3,24,27,48,91,92 This is the case, for instance, of polymer solutions, where it is useful to distinguish between the polymer contribution, ηpol, and the solvent contribution, ηsol, to the steady-state zero-shear viscosity, η0 = ηsol + ηpol = limω→0η′(ω). Under dilute, theta, and free-draining conditions, flexible chain dynamics are explained by the Rouse3,24,47,48 theory. The closed form (i.e., continuous limit) of the Rouse theory, upon which the following equations are based, dates back to the work of Blizard,205 Marvin and Oser,206 and Spriggs and Bird.207 When a solvent viscosity ηsol is included, the closed-form expressions for the storage modulus, GR(De), and the loss viscosity, ηR(De), of the Rouse theory can be written as

(43a)
(43b)

where GR = νkBT is the characteristic modulus3,24,47,48 of the theory, ν is the number density of polymer molecules, ηpol = η0ηsol = (π2/6)GRτR, τR is the longest Rouse relaxation time,153,208ξπ(2De)1/2 is a convenient dimensionless variable, De ≡ ωτR, and

(44)

The TSS MAOS response associated with the Rouse3,24,47,48 theory is illustrated in Fig. 9, for increasing solvent viscosity: ηsol/ηpol = 0 (black solid lines), ηsol/ηpol = 0.01 (red dashed-dotted lines), and ηsol/ηpol = 0.1 (blue dashed lines). Interestingly, the form of Eqs. (27a)–(27d) renders the Rouse TSS MAOS behavior independent of ηsol.

FIG. 9.

TSS MAOS response associated with the Rouse3,24,47,48 theory (in closed form3,24,205–207) for monodisperse linear polymer chains in dilute solutions. (a) SAOS, Eqs. (43a) and (43b); (b) TSS MAOS, Eqs. (27a)–(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. SAOS and TSS MAOS signatures for increasing solvent viscosity, ηsol   :ηsol/ηpol=0 (black solid lines), ηsol/ηpol=0.01 (red dashed-dotted lines), and ηsol/ηpol=0.1 (blue dashed lines), where ηpol is the polymer contribution to the steady-state zero-shear viscosity, η0=ηsol+ηpol=limω0η(ω). The form of Eqs. (27a)–(27d) renders the Rouse TSS MAOS behavior independent of ηsol.

FIG. 9.

TSS MAOS response associated with the Rouse3,24,47,48 theory (in closed form3,24,205–207) for monodisperse linear polymer chains in dilute solutions. (a) SAOS, Eqs. (43a) and (43b); (b) TSS MAOS, Eqs. (27a)–(27d); and (c) linear-to-nonlinear elastic and viscous transitions. Gray, thinner lines are used for G(ω) [(a), top left subplot]. SAOS and TSS MAOS signatures for increasing solvent viscosity, ηsol   :ηsol/ηpol=0 (black solid lines), ηsol/ηpol=0.01 (red dashed-dotted lines), and ηsol/ηpol=0.1 (blue dashed lines), where ηpol is the polymer contribution to the steady-state zero-shear viscosity, η0=ηsol+ηpol=limω0η(ω). The form of Eqs. (27a)–(27d) renders the Rouse TSS MAOS behavior independent of ηsol.

Close modal

The power expansion stress coefficients for the original Doi-Edwards49–56 reptation theory were calculated in 198228,104, with the IAA (i.e., DE-IAA49–52,56 constitutive equation) by Pearson and Rochefort28 and without the IAA (i.e., DE53–56 constitutive equation) by Helfand and Pearson.104 The resulting DE-IAA MAOS functions given by Eqs. (A7)–(A11) of Pearson and Rochefort28 are, corrected for the misprints, equivalent to Eqs. (25)–(27) with Geq = 0 and A = −5/21 (Sec. III B). Except for a numerical prefactor,4,71,72 they are also identical to the Curtiss-Bird (ε = 0)3,5,48,67–70 MAOS functions (Sec. I). The closed-form expressions for the storage modulus, GDE-IAA(De), and the loss viscosity, ηDE-IAA(De), of the DE-IAA theory are28 

(45a)
(45b)

where GDE-IAA = (3/5)νkBT is the DE-IAA plateau modulus,27,56ν=ρNav/MeG is the number density of polymer molecules, ρ is the melt density, Nav is Avogadro’s number, MeG is the Graessley-Fetters208–210 entanglement molar mass, η0 = (π2/12)GDE-IAAτd, τd is the disentanglement time,56 ζπ(De/2)1/2 is a convenient dimensionless variable, De ≡ ωτd, and28 

(46)

Without invoking the IAA, the DE SAOS functions are still given by Eqs. (45a) and (45b), but the DE plateau modulus is now27,56GDE = (4/5)νkBT. Since the DE constitutive equation does not belong to the Coleman-Noll class (cf. Secs. II A 1, II A 2, and III B; Subsection 1 of the  Appendix), its TSS MAOS signatures are different from those given by Eqs. (25)–(27). For the DE MAOS functions, we used Eqs. (28)–(37) of Helfand and Pearson.104 

The TSS MAOS response for the Doi-Edwards49–56 theory is shown in Fig. 10, with the IAA (black solid lines) and without the IAA (red dashed-dotted lines). The mathematical simplification granted by the IAA translates into a specific physical mechanism for nonlinear stress relaxation: after chain retraction,26,27 chain ends are considered still associated with the same tube segment (strain reversibility).4,56 The DE signatures thus illustrate the effect of non-affine, irreversible deformation of reptating chains (hence, deviation from the Coleman-Noll class of “simple fluids”;31,37 cf. Subsection 1 of the  Appendix)4 on the SAOS and TSS MAOS responses.

FIG. 10.

TSS MAOS response for the Doi-Edwards49–56 theory (in closed form28,104) for monodisperse linear polymer chains in undiluted and entangled conditions. (a) SAOS, (b) TSS MAOS, and (c) linear-to-nonlinear elastic and viscous transitions: with the independent alignment approximation [DE-IAA; black solid lines; Eqs. (45a) and (45b), and Eqs. (27a)–(27d) with A=5/21, equivalent to Eqs. (A7)–(A11) of Pearson and Rochefort28 corrected for the misprints] and without [DE; red dashed-dotted lines; Eqs. (28)–(37) of Helfand and Pearson104 corrected for the misprint]. A gray, thinner line is used for G(ω) [(a), top left subplot]. The DE signatures illustrate the effect of non-affine, irreversible deformation of reptating chains (hence, deviation from the Coleman-Noll class of “simple fluids”;31,37 cf. Subsection 1 of the  Appendix)4 on the SAOS and TSS MAOS responses. Noteworthy is the distinct high-De behavior for [v1](ω): discussed in detail by Helfand and Pearson104 (see their Fig. 3), it is not the uniform vertical shift shown in Fig. 3 of Wagner et al.30 and Fig. 4 of Bharadwaj and Ewoldt.5 

FIG. 10.

TSS MAOS response for the Doi-Edwards49–56 theory (in closed form28,104) for monodisperse linear polymer chains in undiluted and entangled conditions. (a) SAOS, (b) TSS MAOS, and (c) linear-to-nonlinear elastic and viscous transitions: with the independent alignment approximation [DE-IAA; black solid lines; Eqs. (45a) and (45b), and Eqs. (27a)–(27d) with A=5/21, equivalent to Eqs. (A7)–(A11) of Pearson and Rochefort28 corrected for the misprints] and without [DE; red dashed-dotted lines; Eqs. (28)–(37) of Helfand and Pearson104 corrected for the misprint]. A gray, thinner line is used for G(ω) [(a), top left subplot]. The DE signatures illustrate the effect of non-affine, irreversible deformation of reptating chains (hence, deviation from the Coleman-Noll class of “simple fluids”;31,37 cf. Subsection 1 of the  Appendix)4 on the SAOS and TSS MAOS responses. Noteworthy is the distinct high-De behavior for [v1](ω): discussed in detail by Helfand and Pearson104 (see their Fig. 3), it is not the uniform vertical shift shown in Fig. 3 of Wagner et al.30 and Fig. 4 of Bharadwaj and Ewoldt.5 

Close modal

The SAOS behavior in Fig. 10(a) is unaffected by the IAA (the minor difference in plateau moduli, hence steady-state zero-shear viscosities, is taken into account by the normalization). Because primitive path fluctuations211,212 are not included in the original Doi-Edwards49–56 theory, the high-De asymptotic behavior of the loss modulus, G″(De) ∼ De−1/2, does not exhibit the scaling211,212G″(De) ∼ De−1/4 typically observed experimentally.26,27 As for the TSS MAOS behavior [Fig. 10(b)], DE-IAA and DE signatures exhibit the same frequency dependence and very similar magnitude (to within ≈10%)104 for [e1](ω), [e3](ω), and [v3](ω). Notable exception is the distinct high-De behavior for [v1](ω): discussed in detail by Helfand and Pearson104 (see their Fig. 3), it is not the uniform vertical shift shown in Fig. 3 of Wagner et al.30 and Fig. 4 of Bharadwaj and Ewoldt5 (cf. Secs. II A 1, II A 2, and III B; Subsection 1 of the  Appendix). Through the first-harmonic viscous nonlinearity, weakly nonlinear perturbations from equilibrium at high De are sensitive to the physical mechanism associated with chain retraction, thereby highlighting the difference between the DE-IAA and DE constitutive equations. This is particularly noteworthy considering that the DE-IAA and DE predictions for h(γ0) are nearly indistinguishable26,27,56,72,81 (Sec. II A 1).

We derived equations for the weakly nonlinear medium-amplitude oscillatory shear (MAOS) response of materials and models exhibiting time-strain separability. In doing so, we addressed the question of what is meant by a time-strain separable (TSS) response in MAOS. Contrary to previous belief,5,109 having nonlinear model parameters as front factors in the MAOS analytical solutions is not a sufficient condition for time-strain separability. The frequency dependence of a TSS MAOS response cannot be arbitrary [Eqs. (25)–(27)]: it is uniquely determined by linear viscoelasticity (LVE) (like it is, e.g., for a TSS nonlinear response in step shear). Magnitude and sign of TSS MAOS functions are set by a single nonlinear parameter, termed A, which was defined here to represent the dimensionless magnitude of the leading-order nonlinearity encoded in the small-strain power series expansion of the shear damping function, i.e., h(γ)=1+Aγ2+Oγ4 [Eq. (11)].

We showed that there are three equivalent ways in which TSS MAOS equations can be written: (i) integrals involving the linear memory function m(s) ≡ −∂G(s)/∂s [Eqs. (25a)–(25d)], (ii) integrals involving the linear shear relaxation modulus G(s) [Eqs. (26a)–(26d)], and (iii) linear combinations of SAOS material functions at different integer-multiple frequencies [i.e., the asymptotically nonlinear counterpart in oscillatory shear of the TSS nonlinear response in step-shear deformations; Eqs. (27a)–(27d)]. TSS MAOS equations in terms of the linear memory function m(s) have not been presented earlier. In addition, our derivation left the functional form for m(s) unspecified, and no assumptions were made regarding the long-time (i.e., terminal) behavior of G(s). As a result, our derived TSS MAOS equations are applicable to any LVE model and for both LVE liquidsGeqlimtG(t)=0,M1G0sG(s)ds<+ and LVE solidsGeq>0.

TSS MAOS signatures reveal much more than just the perturbation parameter A. Specifically, the distribution of terminal relaxation times is significantly more apparent in the TSS MAOS functions [e1](ω),[e3](ω),[v1](ω),[v3](ω) than their LVE counterparts G(ω),η(ω). We theoretically showed that this occurs because TSS MAOS functions depend on higher-order moments of the relaxation spectrum H(τ) (relaxation modulus G(s)) than the ones controlling the response in the LVE regime, which led to the definition of MAOS liquids: Geq = 0 and M3G0s3G(s)ds<+. We thus expect our theoretical results to be beneficial in studies of polymeric systems seeking to detect small differences in large-scale structural features such as molar mass distribution and long-chain branching and more generally in studies of complex fluids and soft matter with wide distributions of relaxation times. Furthermore, when time-strain separability applies, one can use the TSS MAOS equations presented herein to improve inference of the LVE relaxation spectrum, especially at long time scales. When time-strain separability does not apply, relaxation spectra can still be inferred from MAOS data via the equations recently proposed by Martinetti et al.91 

The value of our equations was demonstrated by considering the TSS MAOS response associated with a variety of LVE relaxation spectra, originating from (i) simple analytical functions (log-normal23,38 and asymmetric Lorentzian23,39 distributions, Figs. 5 and 6), (ii) fractional mechanical models (Maxwell40–46 and Zener,43,44,46Figs. 7 and 8), and (iii) molecular theories (Rouse3,24,47,48 and Doi-Edwards,49–56Figs. 9 and 10). These examples showed the potential benefits of studying weakly nonlinear oscillatory perturbations from equilibrium for systems dominated by long-range spatial or temporal correlations. In addition, TSS MAOS signatures were revealed (Figs. 6–8) that differ from the liquid-like terminal MAOS behavior predicted by Bharadwaj and Ewoldt,57 even in the presence of liquid-like terminal SAOS behavior. Indeed, the generality of our derived equations allowed us to examine the first solid-like terminal TSS MAOS behavior and deviations from it (Fig. 8).

The TSS MAOS signatures shown in Figs. 1, 5–10 may serve as fingerprints for comparison with experiments and for model selection. Notably, any other LVE model not considered here (i.e., any other LVE relaxation spectrum) could be used in Eqs. (25)–(27) to predict the frequency-dependent behavior of the four TSS MAOS material functions. This of course includes discrete distributions of relaxation times, such as the semi-empirical distribution proposed by Spriggs213 (which yields a Rouse-like spectrum as a special case3). From the practical standpoint, the large diversity of TSS MAOS signatures accessible through these equations provides a means to describe experimental MAOS data. In this case, the choice of LVE model should be guided by the physics of the system at hand (e.g., dilute polymer solution vs. entangled melt) and the observed LVE and MAOS responses, while the nonlinear parameter A can be determined from asymptotically nonlinear perturbations in either MAOS or stress relaxation experiments. Although the value of A could be inferred from stress relaxation data, we expect MAOS data to provide a more accurate estimate due to a better signal-to-noise ratio and the availability of multiple signals (i.e., the four MAOS material functions).

The inferred value of A indicates evidence for the governing constitutive model; for instance, for a single-mode Maxwell LVE behavior, the corotational Maxwell (CM)2–5 model prescribes A = −1/6 ≃ −0.167 (black solid lines in Fig. 1), while the linear molecular stress function (L-MSF)30,100–103 model specifies A = −29/210 ≃ −0.138 (red dashed-dotted lines in Fig. 1). The task of down-selecting from a set of physically plausible TSS MAOS models can benefit from model selection techniques, such as those based on Bayesian credibility criteria.91,214,215

Finally, the frequency dependence of the critical strain for the onset of nonlinearity shown in Figs. 1(c), 3(c), 5(c)–10(c) has practical implications concerning the measurement of MAOS data via the “frequency-sweep MAOS” procedure recently proposed by Singh et al.216 The behavior shown in Figs. 1(c), 3(c), and 5(c), which refers to MAOS liquids, is the frequency-dependent linear-to-nonlinear transition pertaining to the Coleman-Noll class of “simple fluids”31,37 (Fig. 11): strain-rate-controlled at low De (γ̇0γ0ω=const.) and strain-controlled at high De (γ0 = const.). Upon this assumed functional behavior is based the frequency-dependent strain amplitude trajectory, γ0(ω), chosen in 2018 by Singh et al.216 to measure MAOS material functions for a transient polymer network via the experimental protocol termed “frequency-sweep MAOS”.216 In this new experimental procedure,216 frequency sweeps follow a pre-defined strain amplitude trajectory γ0(ω) designed to elicit a nonlinear stress response that is (i) measurable (i.e., that complies with the low and high torque limits of the instrument57,217) and (ii) confined to the MAOS regime where higher-order nonlinearities are negligible. Selecting such a strain amplitude trajectory γ0(ω) is the key step of the “frequency-sweep MAOS”216 protocol; not only does its shape (i.e., frequency dependence) need to be known a priori (e.g., based on the material class) but one also has to take into account that both shape and magnitude can vary greatly among the four MAOS functions [e1](ω),[e3](ω),[v1](ω),[v3](ω). The results presented in Figs. 1(c), 3(c), 5(c)–8(c) are instrumental to assessing the current limits and challenges of the “frequency-sweep MAOS”216 procedure.

FIG. 11.

Pipkin diagram,218 i.e., strain amplitude γ0 as a function of dimensionless frequency (Deωτchar), illustrating the behavior of Coleman-Noll “simple fluids”31,37 of the first, second, and third order. Notice that the boundaries of their respective regions are set by a constant strain rate γ̇0γ0ω at low De (purely viscous behavior) and by a constant strain γ0 at high De (purely elastic behavior). In the low- and high-De limits, some constitutive equations are shown within their region of applicability: Newtonian fluid,3,27,219 generalized Newtonian fluid3,219 (GNF), second-order fluid,3 Hookean solid27,219 (HS), neo-Hookean solid27,219 (NHS), and Rivlin solid.220–222 

FIG. 11.

Pipkin diagram,218 i.e., strain amplitude γ0 as a function of dimensionless frequency (Deωτchar), illustrating the behavior of Coleman-Noll “simple fluids”31,37 of the first, second, and third order. Notice that the boundaries of their respective regions are set by a constant strain rate γ̇0γ0ω at low De (purely viscous behavior) and by a constant strain γ0 at high De (purely elastic behavior). In the low- and high-De limits, some constitutive equations are shown within their region of applicability: Newtonian fluid,3,27,219 generalized Newtonian fluid3,219 (GNF), second-order fluid,3 Hookean solid27,219 (HS), neo-Hookean solid27,219 (NHS), and Rivlin solid.220–222 

Close modal

The frequency dependence of the critical strain illustrated in Figs. 1(c) and 3(c) exemplifies the experimental linear-to-nonlinear transition γ0(ω) considered by Singh et al.:216 simple-fluid-like and characterized by similar first-harmonic elastic ([e1](ω)) and viscous ([v1](ω)) MAOS regimes (third-harmonic MAOS regimes are not shown). In this case, a single γ0(ω) trajectory, defined as discussed by Singh et al.,216 can reasonably capture (simultaneously) the MAOS regimes of the four material functions. Advantages and disadvantages of the “frequency-sweep MAOS”216 protocol under these conditions were recently shown by Martinetti et al.119 

A slightly more complicated case is represented by the blue dashed curves in Fig. 5(c), associated with a wide (PDI ≈ 9.5) log-normal-based distribution of relaxation times. Here, the linear-to-nonlinear transition γ0(ω) is still characteristic of Coleman-Noll “simple fluids”31,37 (MAOS liquids, in particular); however, the magnitude of the low-De MAOS boundary varies substantially between the first-harmonic elastic and viscous perspectives. As a result, [e1](ω) and [v1](ω) could not be measured simultaneously via a “frequency-sweep MAOS”216 protocol with a single γ0(ω) trajectory (the same is true for [e3](ω) and [v3](ω), which are not shown). Even more challenging, as far as the application of the “frequency-sweep MAOS”216 protocol is concerned, is the situation depicted by the red dashed-dotted curves in Fig. 6(c) and the black solid curves in Fig. 8(c), which refer to non-liquid-like MAOS responses. In these examples, not only is the magnitude of the low-De γ0(ω) trajectory significantly different between [e1](ω) and [v1](ω) but also its functional behavior, no longer archetypical of Coleman-Noll “simple fluids,”31,37 is set by the particular LVE model being considered. The method adopted by Singh et al.216 to design the γ0(ω) trajectory does not apply to these types of linear-to-nonlinear transition. In this case, the results shown in Figs. 6(c)–10(c) [and, more generally, Eqs. (25)–(27)] are beneficial in (i) trying to devise appropriate γ0(ω) trajectories to be followed during “frequency-sweep MAOS” measurements and (ii) judging whether the (more accurate, but also more time and material consuming) “strain-sweep MAOS” approach11,57,109 is required instead.

This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG02-07ER46471, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign. We are grateful to Gareth H. McKinley (Massachusetts Institute of Technology) for prompting us to examine the MAOS signatures of the DE-IAA constitutive equation and for helpful discussions.

1. Constitutive equation and mathematical framework

For the Coleman-Noll class of isotropic, incompressible, viscoelastic “simple materials”31–37 of the integral type, the nth-order approximation to the general constitutive equation can be written as a series expansion,

(A1)

In Eq. (A1), the left-hand side is the difference between the extra stress tensor, σ(t), and the “delayed elastic stress,”36,σel(t) ≡ limtσ(t) (with σel(t) = 0 for “simple fluids”;31,37σel(t) ≠ 0 for “simple solids”31,36,37). The right-hand side of Eq. (A1) represents the viscoelastic response of the material, where σ(i)(t) are i-fold multiple integrals containing scalar-valued material functions of i time variables and terms associated with a tensorial measure of deformation. Often called multiple integral expansion4 or memory-integral expansion,3 Eq. (A1) represents a perturbation expansion from the small-strain limit of the Pipkin diagram,218 as shown in Fig. 11. When deformation and stress are related via Eq. (A1), within the mathematical framework proposed by Coleman and Noll,31–37 materials are said to belong to the class of “simple materials”. For a detailed description of all the mathematical requirements, the reader is referred to the original references.

A fundamental assumption is that the stress is a continuous and smooth functional of the strain history.31–37 In addition, the Coleman-Noll mathematical formulation of the principle of fading memory implies that “simple fluids” behave as Newtonian liquids at low De and as Hookean solids at high De.4,31,37 Their linear-to-nonlinear viscoelastic (LVE-to-NLVE) transition is therefore dictated by a constant strain rate γ̇0γ0ω at low De (purely viscous behavior) and by a constant strain γ0 at high De (purely elastic behavior),4,31,37 as shown in Fig. 11. This type of LVE-to-NLVE transition has been experimentally observed for polymer solutions,223 polymer melts,28 and transient networks of associating polymers.216 The high-De elastic limit, which applies to both “simple fluids” and “simple solids,” carries the implications that (i) for sufficiently rapid deformations the stress is independent from the strain rate or strain path and (ii) that even large strains, if removed quickly enough, have no effect on the stress (strain reversibility).4,31,36,37

Within the Coleman-Noll class are the popular corotational and upper-convected Maxwell3,4 models, the Giesekus (ηsol = 0)110–113 model, the Rivlin-Sawyers (RS)84 equation, and the Kaye-Bernstein-Kearsley-Zapas (K-BKZ)85,86 equation. Notable exceptions include constitutive equations that violate, for instance, (i) the principle of fading memory, e.g., models with a power-law distribution of relaxation times [depending on the order n of Eq. (A1)];4 (ii) the assumed high-De purely elastic behavior, e.g., models with a retardation (purely dissipative) term, such as Oldroyd’s fluid B3,4 and the Curtiss-Bird (ε ≠ 0)3,48,67–69 theory, and the Johnson-Segalman4,224 equation for non-limiting cases of the Gordon-Schowalter convected derivative;225 or (iii) the corollary of strain reversibility, e.g., the Doi-Edwards reptation theory without the IAA54–56 (Secs. II A 1 and IV B 3), and Wagner’s damping functional226,227 model.

2. Third-order “simple materials”

The first-order approximation of Eq. (A1), σ(t) − σel(t) = σ(1)(t), is the equation of finite linear viscoelasticity,31–37 and it involves a single material (memory) function. An additional memory function is required in the second-order approximation, σ(t) − σel(t) = σ(1)(t) + σ(2)(t), which is the equation of finite second-order viscoelasticity.31–37 In this work, we consider third-order “simple materials” of the integral type, or finite third-order viscoelasticity (Fig. 11), defined by31–37 

(A2)

and fully characterized by four independent memory functions.

The strain measure in Eqs. (A1) and (A2) is given by the Finger tensor3,4,27,83C−1 (which implies affine deformation)4 or by any symmetric tensor related to C−1 by a smooth one-to-one transformation.33,35,37 When written in terms of the Finger tensor C−1, the single integral σ(1)(t) yields the constitutive equation of the Lodge rubber-like liquid,228,229

(A3)

where mttGtt/t is the linear memory function,3,4,24,26,27Gtt is the linear shear relaxation modulus,3,4,24,26,27s′ ≡ tt′ is the elapsed time, t′ is the past time, and C1t,t is the Finger measure of the strain accumulated between times t′ and t. Consistent with the choice of strain measure in Eq. (A3), the double integral σ(2)(t) and the triple integral σ(3)(t) can be written as3,36,230

(A4)

and

(A5)

where m2s,s, m3s,s,s, and m4s,s,s are time-dependent nonlinear memory functions, and the elapsed times s″ ≡ tt″ and s‴ ≡ tt‴ are defined with respect to the past times t″ and t‴. Alternative representations of Eqs. (A3)–(A5) are possible (see, e.g., the work of Truesdell and Noll,37 Astarita and Marrucci,83 Davis and Macosko,231 and references therein). Note that all memory functions [and therefore all σ(i)(t) in Eq. (A1)] vanish in the long-time t limit.

3. Neglecting interactions of past kinematic events

The often used single-integral RS84 and K-BKZ85,86 equations can be derived from Eq. (A1) under the assumption that the effects on the stress at time t of strains at all different historic times are independent of each other.3,4,83 Neglecting all interactions of past kinematic events is tantamount to forcing all nonlinear memory functions mi2s,s,=0 unless s′ = s″ = ⋯ = s = tt′.4,83 In this case, all i-fold multiple integrals in Eq. (A1) collapse into single integrals. In particular, Eqs. (A4) and (A5) become

(A6)

and

(A7)

A similar reduction applies to all higher-order terms σ(i≥4)(t) in Eq. (A1). The Cayley-Hamilton theorem3,4,83 then permits to reduce the memory-integral expansion, out to any order, to the RS84 equation,

(A8)

where ψ1tt,I1,I2 and ψ2tt,I1,I2 are time- and strain-dependent nonlinear memory functions,3,4,83I1=IC1 and I2=IIC1 are the first and second invariants of C−1, and C is the Cauchy tensor.3,4,27,83 If ψi are the components of an elastic energy gradient, i.e., ψi = V/Ii, the viscoelastic response is that of a Rivlin network220–222 whose structural elements have finite lifetimes, and Eq. (A8) yields the K-BKZ85,86 equation.

While exactly fulfilled for single-step strain histories,4 the assumption that all interactions of past kinematic events can be neglected may lead to unsatisfactory predictions in double-step stress relaxation81,232,233 and LAOS234 experiments. To remedy this, an integral constitutive equation with partial strain-coupling effects was proposed by Vrentas et al.235 

4. Complete time-strain separability

Complete (i.e., out to any order) time-strain separability ensues when the nonlinear memory functions ψi in Eq. (A8) can be written as a product of time- and strain-dependent factors of the form3,4,17–19

(A9)

where mtt is the linear memory function [as in Eq. (A3)], and the strain-dependent functions ϕiI1,I2 satisfy the requirement3 

(A10)

necessary for a correct reduction to the LVE response limit. In this case, Eq. (A8) yields the factorized or time-strain separable Rivlin-Sawyers (TSS-RS)3,4,84 equation, Eq. (4).

In simple shear, the Finger tensor C1t,t and the Cauchy tensor Ct,t are3,4,27,83

(A11)

and

(A12)

where δ is the identity matrix. Thus, the shear stress σ(t) according to Eq. (4) is

(A13)

where σel(t)limtσ(t)=GeqhIγ(t,) (with σel(t) = Geq = 0 for LVE liquids), hIϕ1I1,I2+ϕ2I1,I2, and II1=II2=3+γt,t2.

If the strain history is a step of strain of height γ0, i.e., γ(t) = γ0s(t), then

(A14)

and Eq. (A13) yields

(A15)

From the definition of nonlinear shear relaxation modulus,3,4,24–27G(t, γ0) ≡ σ(t)/γ0, and Eq. (A15), Eq. (1) is thus obtained.

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To whom this article is dedicated on the occasion of his 95th birthday.

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