# Kramers-Kronig Relations for Nonlinear Rheology: 1. General Expression and
Implications111This paper is dedicated to the memory of Prof. James W. Swan
whose contributions inspired this work.

Sachin Shanbhag<EMAIL_ADDRESS>Department of Scientific Computing, Florida
State University, Tallahassee, FL 32306. USA  Yogesh M. Joshi
<EMAIL_ADDRESS>Department of Chemical Engineering, Indian Institute of
Technology, Kanpur, INDIA

###### Abstract

The principle of causality leads to linear Kramers-Kronig relations (KKR) that
relate the real and imaginary parts of the complex modulus $G^{*}$ through
integral transforms. Using the multiple integral generalization of the
Boltzmann superposition principle for nonlinear rheology, and the principle of
causality, we derived nonlinear KKR, which relate the real and imaginary parts
of the $n^{\text{th}}$ order complex modulus $G_{n}^{*}$. For $n$=3, we
obtained nonlinear KKR for medium amplitude parallel superposition (MAPS)
rheology. A special case of MAPS is medium amplitude oscillatory shear (MAOS);
we obtained MAOS KKR for the third-harmonic MAOS modulus $G_{33}^{*}$;
however, no such KKR exists for the first harmonic MAOS modulus $G_{31}^{*}$.
We verified MAPS and MAOS KKR for the single mode Giesekus model. We also
probed the sensitivity of MAOS KKR when the domain of integration is truncated
to a finite frequency window. We found that that (i) inferring
$G_{33}^{\prime\prime}$ from $G_{33}^{\prime}$ is more reliable than vice-
versa, (ii) predictions over a particular frequency range require
approximately an excess of one decade of data beyond the frequency range of
prediction, and (iii) $G_{33}^{\prime}$ is particularly susceptible to errors
at large frequencies.

integral transform and MAPS and MAOS and causality

##  I Introduction

In the linear viscoelastic (LVE) regime, soft materials are subjected to
infinitesimal deformations so that the mechanical response may be probed
without perturbing the equilibrium microstructure. Linear viscoelasticity is
commonly examined using three types of experiments: (i) step strain, (ii)
creep or step stress, and (iii) oscillatory strain. These experiments lead,
respectively, to the linear response functions: (i) stress relaxation modulus
$G(t)$, (ii) creep compliance $J(t)$, and (iii) complex modulus
$G^{*}(\omega)$. Here, $t$ and $\omega$ represent time and frequency,
respectively. These LVE response functions provide insights into material
microstructure, and are part of standard rheological characterization [1, 2,
3, 4]. These response functions are interrelated. If complete knowledge of any
one property is available, the remaining two can be inferred. The relaxation
modulus $G(t)$ and creep compliance $J(t)$ are related to each other by the
convolution relation. The complex modulus $G^{*}(\omega)$, on the other hand,
is related to the Fourier transform of $G(t)$. The complex modulus has real
and imaginary parts,
$G^{*}(\omega)=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)$, called the
storage (or elastic) and loss (or viscous) modulus, respectively. These moduli
are related to each other via Kramers-Kronig relations (KKR) [5, 6]. They can
be experimentally measured by performing oscillatory shear (OS) experiments.

###  I.1 Linear Kramers-Kronig Relations

Suppose a material is subjected to an arbitrary time-dependent shear strain
$\gamma(t)$. In the LVE regime, the induced shear stress $\sigma(t)$ is
related to the deformation history by the Boltzmann superposition principle
[2, 3, 4],

$\sigma(t)=\int_{-\infty}^{t}G(t-t^{\prime})\dfrac{d\gamma(t^{\prime})}{dt^{\prime}}\,dt^{\prime},$
(1)

where $t$ is the current time, and ${t}^{\prime}$ is the time associated with
application of deformation field. The principle of causality stipulates that
the relaxation modulus $G(t-t^{\prime})=0$ for $t-t^{\prime}<0$. Using this
principle, the complex modulus can be related to the relaxation modulus via a
modified Fourier transform,

$G^{*}(\omega)=i\omega\int_{-\infty}^{\infty}G(t)\,e^{-i\omega
t}\,dt=i\omega\int_{0}^{\infty}G(t)\,e^{-i\omega t}dt.$ (2)

Note that the complex viscosity
$\eta^{*}(\omega)=\eta^{\prime}(\omega)-i\eta^{\prime\prime}(\omega)=G^{*}(\omega)/(i\omega)$
and the relaxation modulus $G(t)$ form a standard Fourier transform pair.

Using the principle of causality again to write $G(t)$ as a sum of an even and
odd function, it can be shown that [3, 4],

$G(t)=\dfrac{2}{\pi}\int_{0}^{\infty}\dfrac{G^{\prime}(\omega)}{\omega}\sin\omega
t\,d\omega=\dfrac{2}{\pi}\int_{0}^{\infty}\dfrac{G^{\prime\prime}(\omega)}{\omega}\cos\omega
t\,d\omega.$ (3)

This constrains the functional forms of $G^{\prime}(\omega)$ and
$G^{\prime\prime}(\omega)$ to be even and odd functions of $\omega$,
respectively. By manipulating equations 2 and 3, we obtain the linear KKR
222For convenience and brevity, we assume that $G^{\prime}(0)=0$ (viscoelastic
liquids), in this work. Generalizations of KKR to include nonzero equilibrium
modulus $G^{\prime}(0)\neq 0$ are straightforward.:

$\displaystyle G^{\prime}(\omega)$
$\displaystyle=-\dfrac{2\omega^{2}}{\pi}\int_{0}^{\infty}\dfrac{G^{\prime\prime}(u)/u}{u^{2}-\omega^{2}}du$
$\displaystyle G^{\prime\prime}(\omega)$
$\displaystyle=\dfrac{2\omega}{\pi}\int_{0}^{\infty}\dfrac{G^{\prime}(u)}{u^{2}-\omega^{2}}du.$
(4)

Since the integrals have a singularity at $u=\omega$, the Cauchy principal
value of the integrals is implied. Originally, KKR were proposed for specific
atomic systems using physical arguments [6, 8, 9], but subsequently
generalized using complex analysis, and assumptions of linearity, causality,
and analyticity [10, 11]. Indeed, KKR can be derived in a succinct form from
the residue theorem in complex analysis with deceptive simplicity as [12],

$\eta^{*}(\omega)=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{\eta^{*}(u)}{u-\omega}du.$
(5)

This equation is equivalent to the two relations in equation 4. Table 1
summarizes different forms in which linear KKR can be expressed 333Note that
$\eta^{*}(\omega)$ is analogous to susceptibility $\chi(\omega)$ in optics.
However, unlike $\eta^{*}=\eta^{\prime}-i\eta^{\prime\prime}$, susceptibility
is defined as $\chi^{*}=\chi^{\prime}+i\chi^{\prime\prime}$. This difference
in definition results in the comparable KKR,
$\chi^{*}(\omega)=1/(i\pi)\int_{-\infty}^{\infty}\dfrac{\chi^{*}(u)}{u-\omega}du.$.

| Complex Form | Pair Form
---|---|---
modulus | $\dfrac{G^{*}(\omega)}{\omega}=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{G^{*}(u)/u}{u-\omega}du.$ | $G^{\prime}(\omega)=-\dfrac{2\omega^{2}}{\pi}\int\limits_{0}^{\infty}\dfrac{G^{\prime\prime}(u)/u}{u^{2}-\omega^{2}}du$
$G^{\prime\prime}(\omega)=\dfrac{2\omega}{\pi}\int\limits_{0}^{\infty}\dfrac{G^{\prime}(u)}{u^{2}-\omega^{2}}du$
viscosity | $\eta^{*}(\omega)=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{\eta^{*}(u)}{u-\omega}du.$ | $\eta^{\prime}(\omega)=\dfrac{2}{\pi}\int\limits_{0}^{\infty}\dfrac{u\eta^{\prime\prime}(u)}{u^{2}-\omega^{2}}\,du$
$\eta^{\prime\prime}(\omega)=\dfrac{2\omega}{\pi}\int\limits_{0}^{\infty}\dfrac{\eta^{\prime}(u)}{\omega^{2}-u^{2}}\,du$
Table 1: Different forms of linear Kramers-Kronig relations.

It is useful to emphasize that the relationship between the real and imaginary
parts of $G^{*}(\omega)$ implied by KKR is underpinned by the principle of
causality. It is merely a mathematical reflection of the physical constraint
$G(t<0)=0$, mediated through the Fourier transform of a real causal function.
Nevertheless, it can be practically useful. For example, in small amplitude
oscillatory shear (SAOS) experiments a sinusoidal strain
$\gamma(t)=\gamma_{0}\sin\omega t$ with amplitude $\gamma_{0}$ and angular
frequency $\omega$ is applied [3, 14]. In the LVE limit, the stress response
is given by,

$\sigma(t)=\sigma_{\text{SAOS}}(t)=\gamma_{0}\left(G^{\prime}(\omega)\sin\omega
t+G^{\prime\prime}(\omega)\cos\omega t\right).$ (6)

Modern rheometers can measure this response, and infer $G^{\prime}(\omega)$
and $G^{\prime\prime}(\omega)$ over a range of frequencies in a typical
frequency sweep experiment. For thermo-rheologically simple materials, this
frequency range can be widened to several decades using the time-temperature
superposition principle [3]. Manual shifting of individual datasets to produce
master-curves of $G^{*}(\omega)$ can result in violation of KKR. Thus, KKR can
be used for data-validation of experimentally measured $G^{*}(\omega)$ [15,
16].

###  I.2 MAOS and MAPS Rheology

In the LVE limit of $\gamma_{0}\rightarrow 0$, stresses induced in a material
are small and harmonic. As the magnitude of the deformation is gradually
increased, nonlinear viscoelastic features start to manifest, and the stress
response becomes non-harmonic. In conventional large amplitude oscillatory
shear (LAOS) experiments, an oscillatory strain field
$\gamma(t)=\gamma_{0}\sin\omega t$ similar to SAOS experiments, is applied
[17, 18, 19]. For large values of amplitude $\gamma_{0}$, higher nonlinear
modes are activated, and the stress response is often represented by a power
series [20],

$\sigma(t)=\sum\limits_{n\in\text{odd}}^{\infty}\sigma_{n}(t)=\sum\limits_{n\in\text{odd}}^{\infty}\sum\limits_{m\in\text{odd}}^{n}\gamma_{0}^{n}\left[G_{nm}^{\prime}(\omega)\sin(m\omega
t)+G_{nm}^{\prime\prime}(\omega)\cos(m\omega t)\right],$ (7)

where the nonlinear complex moduli
$G_{nm}^{*}(\omega)=G_{nm}^{\prime}(\omega)+iG_{nm}^{\prime\prime}(\omega)$
are functions of only frequency. The summations include only odd values of $m$
and $n$ due to the odd symmetry of shear stress with shear strain. In the
medium amplitude oscillatory shear (MAOS) regime, only the weakest nonlinear
modes are activated, and the stress response
$\sigma(t)=\sigma_{1}(t)+\sigma_{3}(t)+\mathcal{O}(\gamma_{0}^{5})$ can be
truncated after the leading nonlinear term $n=3$ in equation 7 as,

$\displaystyle\sigma(t)=$
$\displaystyle\gamma_{0}\left[G_{11}^{\prime}\sin\omega
t+G_{11}^{\prime\prime}\cos\omega t\right]+$
$\displaystyle\gamma_{0}^{3}\left[G_{31}^{\prime}\sin\omega
t+G_{31}^{\prime\prime}\cos\omega t+G_{33}^{\prime}\sin 3\omega
t+G_{33}^{\prime\prime}\cos 3\omega t\right],$ (8)

where $\sigma_{1}=\sigma_{\text{SAOS}}$ and the LVE complex modulus is given
by $G^{*}=G_{11}^{\prime}+iG_{11}^{\prime\prime}$. The stress term
$\sigma_{3}(t)$ associated with cubic power of strain is the MAOS
contribution; the MAOS moduli associated with the first and third harmonic are
$G^{*}_{31}=G^{\prime}_{31}+iG^{\prime\prime}_{31}$, and
$G^{*}_{33}=G^{\prime}_{33}+iG^{\prime\prime}_{33}$, respectively. MAOS
measurements have been used to discriminate between linear and branched
polymers [21, 22, 23], evaluate nanoparticle dispersion quality [24, 25],
droplet size dispersion in polymer blends [26, 27], quantify filler-matrix
interactions in filled rubbers [28, 29] etc.

Experimentally, extraction of the MAOS moduli is indirect, and involves
careful extrapolation. The effort and care required is significantly greater
than that required in the measurement of LVE moduli $G^{\prime}(\omega)$ and
$G^{\prime\prime}(\omega)$, in part, due to the narrow window of suitable
strain amplitudes [30]. If $\gamma_{0}$ is too small, MAOS signals are too
weak and difficult to measure. If $\gamma_{0}$ is too large, the stress
response is contaminated by the contribution of modes higher than the third
harmonic. Furthermore, the optimal range of $\gamma_{0}$ is frequency-
dependent; at low frequencies, higher strain amplitudes are necessary to
ferret out MAOS signatures. In practice, the stress response is measured at
multiple strain-amplitudes in the target zone, and the “true” MAOS moduli are
extracted by extrapolation. Due to the complicated process involved,
validating experimental data before interpretation is paramount. The companion
paper provides a method for efficiently accomplishing this task [31].

Medium amplitude parallel superposition (MAPS) can be seen as a generalization
of the MAOS protocol [32]. Instead of the single-tone sinusoidal strain in
MAOS, the strain waveform in MAPS consists of a superposition of three sine
waves with frequencies $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$,

$\gamma_{\text{MAPS}}(t)=\gamma_{0}\left(\sin(\omega_{1}t)+\sin(\omega_{2}t)+\sin(\omega_{3}t)\right).$
(9)

This perturbation elicits a much richer asymptotic nonlinear response than
MAOS. Indeed, as introduced formally in section II it leads to a strain-
independent third-order complex modulus
$G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ which offers a complete
characterization of the material’s asymptotic nonlinear behavior. By complete,
we mean that using $G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ the nonlinear
response to any arbitrary medium amplitude deformation history can be
predicted via a generalization of Boltzmann superposition principle.

Therefore, MAOS can be thought of as a special, low-dimensional projection of
MAPS: it can be shown that the MAOS moduli, $G_{31}^{*}$ and $G_{33}^{*}$, are
special cases of the third order or MAPS modulus
$G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ [32],

$\displaystyle G_{31}^{*}(\omega)$
$\displaystyle=\dfrac{3}{4}G_{3}^{*}(\omega,-\omega,\omega)$ $\displaystyle
G_{33}^{*}(\omega)$
$\displaystyle=-\dfrac{1}{4}G_{3}^{*}(\omega,\omega,\omega).$ (10)

Due to experimental challenges, characterization of materials using MAPS has
barely started [33, 34, 35]. Regardless, for the purposes of this work, MAPS
provides a convenient general theoretical lens for interpreting MAOS
measurements and KKR.

###  I.3 Organization

This paper is organized as follows: we begin with a generalization of the
Boltzmann superposition principle to nonlinear rheology using a multiple
integral expansion in section II. We highlight similarities and connections
between higher-order terms and their LVE counterparts. We then mathematically
derive a general nonlinear KKR (equation 24).

In section III, we narrow our focus by specializing these general KKR to MAPS
and MAOS rheology. It turns out that for the MAOS modulus $G_{33}^{*}$ we can
formulate a KKR (equation 29); unfortunately no such KKR exists for
$G_{31}^{*}$. Finally, in section IV, we test KKR on the single mode Giesekus
model for which the third-order complex modulus
$G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ is analytically known. In
particular, we verify that the MAPS moduli are consistent with the appropriate
KKR. Finally, we explore the sensitivity of the MAOS KKR when the domain of
integration is limited to a finite frequency window.

##  II Derivation of Kramers-Kronig Relations for Nonlinear Rheology

The Boltzmann superposition principle can be generalized to nonlinear rheology
using a multiple integral expansion [36, 37, 38, 39, 32]. The general
framework for stress induced due to imposed strain can be represented using an
infinite Volterra series,

$\sigma(t)=\sum_{n\in\text{odd}}^{\infty}\,\sigma_{n}(t),$ (11)

where the summation includes only odd values of $n$ due to the odd symmetry
between shear stress and strain, i.e, $\sigma(-\gamma)=-\sigma(\gamma)$. The
contribution of the $n^{\text{th}}$ mode is given by,

$\sigma_{n}(t)=\int_{-\infty}^{t}\cdots\int_{-\infty}^{t}G_{n}(t-t_{1},t-t_{2},\cdots,t-t_{n})\prod_{m=1}^{n}\dot{\gamma}(t_{m})\,dt_{m},$
(12)

where $\dot{\gamma}(t)$ is the shear rate, and
$G_{n}(t-t_{1},t-t_{2},\cdots,t-t_{n})$ is the $n^{\text{th}}$ order
relaxation modulus that generalizes the linear relaxation modulus. The
principle of causality stipulates that
$G_{n}(t-t_{1},t-t_{2},\cdots,t-t_{n})=0$ if $t-t_{i}<0$ for any
$i=1,2,\cdots,n$. The first term in this series,

$\sigma_{1}=\int_{-\infty}^{t}G_{1}(t-t_{1})\,\dot{\gamma}(t_{1})dt_{1}$ (13)

is identical to the Boltzmann superposition principle given by equation 1 with
$G_{1}(t)\equiv G(t)$. Subsequent terms ($n\geq 3$) in equation 12 take into
account stress induced due to the interaction of strains applied at different
times $t_{i}$ and $t_{j}$. In LVE, such cross-effects are negligible.

Nonlinear effects in oscillatory shear flow can evaluated by taking a Fourier
transform (denoted by “hat”) of equation 11,

$\hat{\sigma}(\omega)=\sum_{n\in\text{odd}}^{\infty}\,\hat{\sigma}_{n}(\omega)=\sum_{n\in\text{odd}}^{\infty}\int_{-\infty}^{\infty}\sigma_{n}(t)\,e^{-i\omega
t}dt$ (14)

where the contribution of the $n^{\text{th}}$ mode is,

$\hat{\sigma}_{n}(\omega)=\dfrac{1}{(2\pi)^{n-1}}\int_{-\infty}^{\infty}\overset{n}{\cdots}\,\int_{-\infty}^{\infty}G_{n}^{*}(\omega_{1},\ldots,\omega_{n})\,\delta\left(\omega-\sum_{m=1}^{n}\omega_{m}\right)\left(\prod_{m=1}^{n}\hat{\dot{\gamma}}(\omega_{m})\,d\omega_{m}\right).$
(15)

The nonlinear complex relaxation modulus
$G_{n}^{*}\left({\omega_{1}},{\omega_{2}},\ldots,{\omega_{n}}\right)$ is the
modified Fourier transform of the nonlinear relaxation modulus
$G_{n}(t_{1},\ldots,t_{n})$,

$G_{n}^{*}(\omega_{1},\ldots,\omega_{n})=\left(\prod_{m=1}^{n}i\omega_{m}\right)\int_{0}^{\infty}\overset{n}{\cdots}\,\int_{0}^{\infty}G_{n}(t_{1},\cdots,t_{n})\left(\prod_{m=1}^{n}e^{-i\omega_{m}t_{m}}dt_{m}\right).$
(16)

Note that the first term corresponding to $n=1$ is the usual linear complex
modulus $G_{1}^{*}(\omega)=G^{*}(\omega)$ encountered in the LVE regime as
equation 2. The next odd term corresponding to $n=3$, or
$G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$, represents the leading
nonlinear term that can be experimentally characterized using MAPS rheology.
Due to the Volterra representation, $G_{n}^{*}$ obeys permutation symmetry and
is invariant with respect to the permutation of its arguments. Thus, for
example,
$G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})=G_{3}^{*}(\omega_{2},\omega_{1},\omega_{3})=G_{3}^{*}(\omega_{3},\omega_{2},\omega_{1})$,
etc.

Analogous to the linear complex viscosity
$\eta^{*}(\omega)=G^{*}(\omega)/(i\omega)$, it is convenient to introduce the
$n^{\text{th}}$ order complex viscosity
$\eta_{n}^{*}(\omega_{1},\cdots,\omega_{n})=\eta_{n}^{\prime}(\omega_{1},\cdots,\omega_{n})-i\eta_{n}^{\prime\prime}(\omega_{1},\cdots,\omega_{n})$,
which is related to the $n^{\text{th}}$ order complex modulus $G_{n}^{*}$ via,

$G_{n}^{*}(\omega_{1},\cdots,\omega_{n})=\left(\prod_{m=1}^{n}i\omega_{j}\right)\eta_{n}^{*}(\omega_{1},\cdots,\omega_{n}).$
(17)

Using this definition, we can write equation 16 as,

$\eta^{*}_{n}(\omega_{1},\cdots,\omega_{n})=\int_{0}^{\infty}\overset{n}{\cdots}\,\int_{0}^{\infty}G_{n}(t_{1},\cdots,t_{n})\left(\prod_{m=1}^{n}e^{-i\omega_{m}t_{m}}dt_{m}\right).$
(18)

Now that we have defined all the relevant terms, we can begin deriving a
general form of KKR following the approach of Hutchings et al. [40]. Let $u$
and $\omega$ denote arbitrary frequencies. Consider the following integral
with $p_{i}\geq 0$ for all $i=1,\cdots,n$,

$I=\int_{-\infty}^{\infty}\dfrac{\eta^{*}_{n}(\omega_{1}+p_{1}u,\cdots,\omega_{n}+p_{n}u)}{u-\omega}du$
(19)

Substituting equation 18 for $\eta_{n}^{*}$,

$I=\int_{-\infty}^{\infty}\int_{0}^{\infty}\overset{n}{\cdots}\,\int_{0}^{\infty}G_{n}(t_{1},\cdots,t_{n})\,e^{-i\sum\omega_{m}t_{m}}\cdot\dfrac{e^{-iu\sum
p_{m}t_{m}}}{u-\omega}\,\left(\prod_{m=1}^{n}dt_{m}\right)du,$ (20)

where the summations that occur as arguments to exponential functions,
$\sum\omega_{m}t_{m}$ and $\sum p_{m}t_{m}$, run from $m=1$ to $n$. We can
switch the order of integration to isolate terms that involve $u$ and obtain,

$I=\int_{0}^{\infty}\overset{n}{\cdots}\,\int_{0}^{\infty}G_{n}(t_{1},\cdots,t_{n})\,e^{-i\sum\omega_{m}t_{m}}\left(\prod_{m=1}^{n}dt_{m}\right)\int_{-\infty}^{\infty}\dfrac{e^{-iu\sum
p_{m}t_{m}}}{u-\omega}du$ (21)

We can analytically compute the integral over $u$, by using the following
result for constant $a\neq 0$,

$\int_{-\infty}^{\infty}\dfrac{e^{-iua}}{u-\omega}du=(-i\pi)e^{-i\omega a}.$
(22)

Using equation 22 in equation 21, and invoking equation 18 in the last step,
we can show that,

$\displaystyle I$
$\displaystyle=(-i\pi)\int_{0}^{\infty}\overset{n}{\cdots}\,\int_{0}^{\infty}G_{n}(t_{1},\cdots,t_{n})e^{-i\sum\omega_{m}t_{m}}\cdot
e^{-i\omega\sum p_{m}t_{m}}\left(\prod_{m=1}^{n}dt_{m}\right)$
$\displaystyle=(-i\pi)\int_{0}^{\infty}\overset{n}{\cdots}\,\int_{0}^{\infty}G_{n}(t_{1},\cdots,t_{n})e^{-i\sum(\omega_{m}+p_{m}\omega)t_{m}}\left(\prod_{m=1}^{n}dt_{m}\right)$
$\displaystyle=(-i\pi)\,\eta^{*}_{n}(\omega_{1}+p_{1}\omega,\cdots,\omega_{n}+p_{n}\omega).$
(23)

We can equate the RHS of equations 19 and 23 to obtain a general form of
nonlinear KKR,

$\eta^{*}_{n}(\omega_{1}+p_{1}\omega,\cdots,\omega_{n}+p_{n}\omega)=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{\eta_{n}^{*}(\omega_{1}+p_{1}u,\cdots,\omega_{n}+p_{n}u)}{u-\omega}du.$
(24)

Note that this relation holds when $p_{i}\geq 0$ for all $i=1,\cdots,n$, with
at least one $p_{i}>0$, due to equation 22. This is the most general form of
KKR for nonlinear rheology in this work. Several other useful forms are
special cases of this relation. A particular special case is obtained by
substituting $p_{j}=1$ and $p_{i}=0$ for all $i\neq j$ where $1\leq i,j\leq
n$, $\omega_{j}=0$, $u=\omega_{j}^{\prime}$, and $\omega=\omega_{j}$,

$\eta^{*}_{n}(\omega_{1},\cdots,\omega_{j},\cdots,\omega_{n})=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{\eta^{*}_{n}(\omega_{1},\cdots,\omega_{j}^{\prime},\cdots,\omega_{n})}{\omega_{j}^{\prime}-\omega_{j}}d\omega_{j}^{\prime}.$
(25)

Note that the RHS involves integrating over the $j^{\text{th}}$ input
frequency. For $n=1$, the correspondence with the linear KKR in equation 5 is
obvious. We can obtain an equivalent form in terms of higher order complex
modulus by using equation 17,

$\dfrac{G^{*}_{n}(\omega_{1},\cdots,\omega_{j},\cdots,\omega_{n})}{\omega_{j}}=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{G^{*}_{n}(\omega_{1},\cdots,\omega_{j}^{\prime},\cdots,\omega_{n})/\omega_{j}^{\prime}}{\omega_{j}^{\prime}-\omega_{j}}d\omega_{j}^{\prime}.$
(26)

##  III Kramers-Kronig Relations for MAPS and MAOS

Figure 1: KKR for MAPS relates the third-order modulus at a point (black
circle) to an integral over the first input frequency ($\omega_{1}^{\prime}$)
(see equation 27) illustrated by the dashed red line.

We can specialize the general forms of KKR derived for nonlinear rheology
(equations 24 \- 26) for MAPS and MAOS experiments. For MAPS, two useful forms
follow directly from equations 25 and 26 with $n=3$, and (say) $j=2$:

$\displaystyle\eta^{*}_{3}(\omega_{1},\omega_{2},\omega_{3})$
$\displaystyle=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{\eta^{*}_{3}(\omega_{1},\omega_{2}^{\prime},\omega_{3})}{\omega_{2}^{\prime}-\omega_{2}}d\omega_{2}^{\prime}.$
$\displaystyle\dfrac{G^{*}_{3}(\omega_{1},\omega_{2},\omega_{3})}{\omega_{2}}$
$\displaystyle=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{G^{*}_{3}(\omega_{1},\omega_{2}^{\prime},\omega_{3})/\omega_{j}^{\prime}}{\omega_{2}^{\prime}-\omega_{2}}d\omega_{2}^{\prime}.$
(27)

For concreteness, the RHS of the equations above involve integrating over the
second input frequency. Due to permutation symmetry, equivalent relations can
also be furnished for first and third input frequencies. Figure 1 illustrates
this relation for the case where the integral is expressed over the first
input frequency. This family of KKR is useful for validating MAPS experiments
where $G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ or
$\eta_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ is available.

###  III.1 KKR for MAOS

We can manipulate the general KKR relation (equation 24) to develop KKR that
are useful for relating the real and imaginary parts of the MAOS moduli
$G_{33}^{*}$, where the perturbation is single-tone. With $n=3$, we set
$\omega_{1}=\omega_{2}=\omega_{3}=0$, and $p_{1}=p_{2}=p_{3}=1$ to obtain,

$\eta^{*}_{3}(\omega,\omega,\omega)=\dfrac{i}{\pi}\int_{-\infty}^{\infty}\dfrac{\eta^{*}_{3}(u,u,u)}{u-\omega}du.$
(28)

Using equation 10 and 17, we can rewrite the corresponding KKR in terms of the
modulus,

$G_{33}^{*}(\omega)=\dfrac{i}{\pi}\omega^{3}\int_{-\infty}^{\infty}\dfrac{1}{u^{3}}\dfrac{G_{33}^{*}(u)}{u-\omega}du.$
(29)

We can equate the real and imaginary parts separately to obtain a pair of KKR
relations. Using $G_{33}^{\prime}(-\omega)=G_{33}^{\prime}(\omega)$ and
$G_{33}^{\prime\prime}(-\omega)=-G_{33}^{\prime\prime}(\omega)$, we can
express these KKR for MAOS moduli $G_{33}^{*}$ on a non-negative frequency
domain similar to the linear KKR as,

$\displaystyle G_{33}^{\prime}(\omega)$
$\displaystyle=-\dfrac{2\omega^{4}}{\pi}\int_{0}^{\infty}\dfrac{1}{u^{3}}\dfrac{G_{33}^{\prime\prime}(u)}{\left(u^{2}-\omega^{2}\right)}du$
$\displaystyle G_{33}^{\prime\prime}(\omega)$
$\displaystyle=\dfrac{2\omega^{3}}{\pi}\int_{0}^{\infty}\dfrac{1}{{{u}^{2}}}\dfrac{G_{33}^{\prime}(u)}{\left(u^{2}-\omega^{2}\right)}du.$
(30)

Just like linear KKR, these MAOS KKR can be used for numerically evaluating
one signal from the other, or for data validation. Similar relations for the
third-harmonic are widely used in nonlinear optics [40, 41, 42].

Figure 2: Schematic diagram illustrating KKR MAOS within the context of MAPS:
(A) for $G_{33}^{*}$, application of KKR at a point (blue co-ordinates)
involves an integral (dashed red line) that coincides with $G_{33}^{*}$ at all
frequencies; (B) for $G_{31}^{*}$ the corresponding integral involves a
quantity (dashed red line) that does not coincide with $G_{31}^{*}$.

Interestingly, while we can write specific expressions relating the real and
and imaginary parts of $G_{33}^{*}(\omega)$ (equation 30), MAOS KKR relating
the real and imaginary parts of $G_{31}^{*}(\omega)$ do not exist. In equation
24, since $p_{i}\geq 0$, the integrand on the RHS cannot be expressed as
$\eta^{*}_{3}(u,u,-u)$, which is equal to $\eta_{31}^{*}(u)$. However, by
using $p_{1}=1,p_{2}=0,p_{3}=0$ and
$\omega_{1}=0,\omega_{2}=\omega,\omega_{3}=-\omega$, we obtain
$\eta^{*}_{3}(u,\omega,-\omega)$ with two fixed input frequencies in the
integrand, leading to:

$\displaystyle\eta_{31}^{*}(\omega)$
$\displaystyle=\eta_{3}^{*}(\omega,\omega,-\omega)=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{\eta_{3}^{*}(u,\omega,-\omega)}{u-\omega}du$
$\displaystyle G_{31}^{*}(\omega)$
$\displaystyle=G_{3}^{*}(\omega,\omega,-\omega)=\dfrac{i\omega}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{G_{3}^{*}(u,\omega,-\omega)/u}{u-\omega}du.$
(31)

The same expression can also be obtained by using $\omega_{1}=\omega$ and
$\omega_{3}=-\omega$ in equation 27. This is the closest we can approach a KKR
involving the MAOS modulus $G_{31}^{*}(\omega)$. This situation arises because
MAOS moduli are a projection or subspace of the MAPS modulus
$G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$. As illustrated in figure 2, the
MAOS moduli can be visualized as two particular diagonal vectors (marked by
black lines) in the three-dimensional domain of $G_{3}^{*}$. For $G_{33}^{*}$,
the integrand of the corresponding KKR relation (equation 30) lives in the
same subspace shown by the dashed red line in figure 2A. Unfortunately, the
integrand of equation 31 shown by the dashed red line in figure 2B lives in a
different subspace and does not yield a KKR.

general nonlinear KKR
---
$\eta^{*}_{n}(\omega_{1}+p_{1}\omega,\cdots,\omega_{n}+p_{n}\omega)=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{\eta_{n}^{*}(\omega_{1}+p_{1}u,\cdots,\omega_{n}+p_{n}u)}{u-\omega}du,\quad
p_{i}\geq 0$
general | $\eta_{n}^{*}$ | $\eta^{*}_{n}(\omega_{1},\cdots,\omega_{j},\cdots,\omega_{n})=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{\eta^{*}_{n}(\omega_{1},\cdots,\omega_{j}^{\prime},\cdots,\omega_{n})}{\omega_{j}^{\prime}-\omega_{j}}d\omega_{j}^{\prime}$
$G_{n}^{*}$ | $\dfrac{G^{*}_{n}(\omega_{1},\cdots,\omega_{j},\cdots,\omega_{n})}{\omega_{j}}=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{G^{*}_{n}(\omega_{1},\cdots,\omega_{j}^{\prime},\cdots,\omega_{n})/\omega_{j}^{\prime}}{\omega_{j}^{\prime}-\omega_{j}}d\omega_{j}^{\prime}$
MAPS | $\eta_{3}^{*}$ | $\eta^{*}_{3}(\omega_{1},\omega_{2},\omega_{3})=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{\eta^{*}_{3}(\omega_{1},\omega_{2}^{\prime},\omega_{3})}{\omega_{2}^{\prime}-\omega_{2}}d\omega_{2}^{\prime}$
$G_{3}^{*}$ | $\dfrac{G^{*}_{3}(\omega_{1},\omega_{2},\omega_{3})}{\omega_{2}}=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{G^{*}_{3}(\omega_{1},\omega_{2}^{\prime},\omega_{3})/\omega_{j}^{\prime}}{\omega_{2}^{\prime}-\omega_{2}}d\omega_{2}^{\prime}$
MAOS | $\eta_{33}^{*}$ | $\eta_{33}^{*}(\omega)=\dfrac{i}{\pi}\int\limits_{-\infty}^{\infty}\dfrac{\eta_{33}^{*}(u)}{u-\omega}du$
$G_{33}^{*}$ | $G_{33}^{*}(\omega)=\dfrac{i}{\pi}\omega^{3}\int\limits_{-\infty}^{\infty}\dfrac{1}{u^{3}}\dfrac{G_{33}^{*}(u)}{u-\omega}du$
Table 2: Summary of nonlinear KKR for general ($n^{\text{th}}$ order), MAPS
and MAOS (third order) complex moduli. These equations can be derived from the
general nonlinear KKR listed at the top of the table, which is derived in
section II.

##  IV Validation of KKR for MAPS and MAOS

The various KKR expressions developed hitherto are tabulated in Table 2 for
convenience. In this section, we verify the KKR expressions corresponding to
the MAPS (equation 27) and MAOS moduli (equations 30 and 31) for the single
mode Giesekus model. For this model, analytical expressions for the MAPS and
MAOS moduli are available in the literature [32, 43, 44].

###  IV.1 KKR for MAPS

The single mode Giesekus model has three parameters, the two linear
parameters: modulus $G_{0}$, and the relaxation time $\tau_{0}$, and the
nonlinear parameter $\alpha_{G}$. The zero shear viscosity is related to the
linear parameters via $\eta_{0}=G_{0}\tau_{0}$. Lennon et al. [32] derived the
third-order complex modulus for various constitutive equations including the
single mode Giesekus model, which can be written as,

$\dfrac{\eta_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})}{\eta_{0}\tau_{0}^{2}}=\dfrac{\alpha_{G}\left((3-2\alpha_{G})+i\sum\limits_{j}{{z_{j}}}\right)}{3\left(\prod\limits_{j}{\left(1+iz_{j}\right)}\right)}\dfrac{\left[-3-4i\sum\limits_{j}z_{j}+\sum\limits_{j}z_{j}^{2}+3\sum\limits_{j}\left(\prod\limits_{k\neq
j}z_{k}\right)\right]}{\left[\prod\limits_{j}\left(1+i\sum\limits_{k\neq
j}{{z_{k}}}\right)\right]\left(1+i\sum\limits_{j}z_{j}\right)},$ (32)

using the dimensionless frequency $z_{i}=\omega_{i}\tau_{0}$, with $i=1,2,3$,
for brevity. To validate the MAPS KKR (equation 27), we consider the integral,

$\displaystyle\dfrac{1}{\eta_{0}\tau_{0}^{2}}\int_{-\infty}^{\infty}\dfrac{\eta_{3}^{*}\left(\omega_{1},u,\omega_{3}\right)}{\left(u-{\omega_{2}}\right)}$
$\displaystyle
du=\int_{-\infty}^{\infty}\dfrac{\alpha_{G}}{3(z-z_{2})}\dfrac{\left(3-2\alpha_{G}\right)+i\left({z_{1}}+z+{z_{3}}\right)}{\left(1+iz_{1}\right)\left(1+iz\right)\left(1+iz_{3}\right)}\times$
$\displaystyle\dfrac{-3-4i(z_{1}+z+z_{3})+\left(z_{1}^{2}+{{z}^{2}}+z_{3}^{2}\right)+3\left(z_{1}z+zz_{3}+z_{3}z_{1}\right)}{\left(1+i\left(z_{1}+z\right)\right)\left(1+i\left(z+z_{3}\right)\right)\left(1+i\left(z_{3}+z_{1}\right)\right)\left(1+i\left(z_{1}+z+z_{3}\right)\right)}dz,$
(33)

where $z=\tau_{0}u$. This integral over $z$ can be evaluated analytically. As
expected from the MAPS KKR equation 27, it leads to original expression for
$\eta_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ given by equation 32 with the
appropriate prefactors. It is perhaps useful to point out that in trying to
verify the MAPS KKR, a small typo was discovered in equation 61 and D8 of ref.
[32] (the authors reported missing a factor of $i$ in the numerator of the
first term on the RHS, which is fixed in equation 32). Note that verification
of the MAPS KKR automatically validates equation 31 for $\eta_{31}^{*}$,
which, as alluded to before, is strictly not a KKR as it does not relate the
real and imaginary parts of the same property through an integral transform.

It is known that for inelastic constitutive equations such as generalized
Newtonian fluids, the linear elastic moduli at all frequencies is
${G}^{\prime}=0$. Consequently, generalized Newtonian fluids do not obey the
Fourier transform relation between $G^{\prime\prime}(\omega)$ and $G(t)$ given
by equation 2. They violate the principle of causality, and even linear KKR
given by equation 5 do not apply. For a generalized Newtonian fluid
$\eta_{3}^{*}({\omega_{1}},{\omega_{2}},{\omega_{3}})=\eta^{\prime}(0)$ =
constant [32]. For this case, we get
$\int_{-\infty}^{\infty}\eta_{3}^{*}\left({\omega_{1}},u,{\omega_{3}}\right)/\left(u-{\omega_{2}}\right)du=0$,
and hence MAPS KKR does not hold either. This result is expected because a
generalized Newtonian fluid is an idealization; no real fluid demonstrates
$G^{\prime}(\omega)=0$ at all frequencies.

###  IV.2 KKR for MAOS: Finite Frequency Window

Unlike $\eta_{31}^{*}$, the real and imaginary parts of $\eta_{33}^{*}$ are
related through a KKR (equation 28). As with $\eta_{31}^{*}$, the validity of
the corresponding specialized KKR is automatically implied by the validity of
the MAPS KKR. MAOS KKR (equation 30) may can also be directly verified using
the relations for $G_{33}^{*}$ derived by Gurnon and Wagner [43], and
Bharadwaj and Ewoldt [44]. We obtain the same expressions from the MAPS
relation (equation 32) using $\omega_{1}=\omega_{2}=\omega_{3}=\omega$,

$\displaystyle\dfrac{G_{33}^{\prime}(\omega)}{G_{0}}=-\dfrac{1}{4}\dfrac{G_{3}^{\prime}(\omega,\omega,\omega)}{G_{0}}$
$\displaystyle=\dfrac{\alpha_{G}z^{4}(-21+30z^{2}+51z^{4}+4\alpha_{G}(4-17z^{2}+3z^{4}))}{4(1+z^{2})^{3}(1+4z^{2})(1+9z^{2})}$
$\displaystyle\dfrac{G_{33}^{\prime\prime}(z)}{G_{0}}=-\dfrac{1}{4}\dfrac{G_{3}^{\prime\prime}(\omega,\omega,\omega)}{G_{0}}$
$\displaystyle=\dfrac{\alpha_{G}z^{3}(-3+48z^{2}+33z^{4}-18z^{6}+\alpha_{G}(2-48z^{2}+46z^{4}))}{4(1+z^{2})^{3}(1+4z^{2})(1+9z^{2})},$
(34)

with $z=\omega\tau_{0}$. Note that there is a small typo in Bharadwaj and
Ewoldt [44] in the expression for $G_{33}^{\prime}$ where the term $+30z^{2}$
in the numerator is accidentally replaced by $-30z^{2}$.

Note that MAOS KKR require knowledge of $G_{33}^{*}$ over an infinite
frequency window. In what follows, we explore the sensitivity of MAOS KKR when
experimental data are available only over a limited frequency range,
$\omega_{\min}\leq u\leq\omega_{\max}$. We denote these approximations of MAOS
KKR by decorating the corresponding moduli with a tilde,

Figure 3: Normalized errors for finite-frequency approximations of MAOS KKR
for the single mode Giesekus model with $G_{0}=\tau_{0}=1$ and
$\alpha_{G}=0.2$: (A) $\varepsilon_{33}^{\prime}/\omega^{3}$ and (B)
$\varepsilon_{33}^{\prime\prime}/\omega^{3}$ are plotted as a function of the
parameter $c$, which controls the width of the frequency window via
$\omega_{\min}=\omega/c$ and $\omega_{\max}=c\omega$. Four different values of
the frequency $\omega\tau_{0}$ are selected. When the low frequency correction
dominates the error, we expect $\varepsilon_{33}^{\prime}\propto\omega^{3}/c$
and $\varepsilon_{33}^{\prime\prime}\propto\omega^{4}/c^{3}$, which are
indicated by the dashed gray lines.

$\displaystyle\tilde{G}_{33}^{\prime}(\omega)$
$\displaystyle=-\dfrac{2\omega^{4}}{\pi}\int_{\omega_{\min}}^{\omega_{\max}}\dfrac{1}{u^{3}}\dfrac{G_{33}^{\prime\prime}(u)}{\left(u^{2}-\omega^{2}\right)}du$
$\displaystyle\tilde{G}_{33}^{\prime\prime}(\omega)$
$\displaystyle=\dfrac{2\omega^{3}}{\pi}\int_{\omega_{\min}}^{\omega_{\max}}\dfrac{1}{{{u}^{2}}}\dfrac{G_{33}^{\prime}(u)}{\left(u^{2}-\omega^{2}\right)}du.$
(35)

Fortunately, these integrals can be evaluated analytically for the Giesekus
model, although the resulting expressions are somewhat elaborate. We can
define the absolute error between the true and approximate moduli to quantify
the sensitivity of KKR to truncation of the frequency window as,

$\displaystyle\epsilon_{33}^{\prime}(\omega;\omega_{\min},\omega_{\max})$
$\displaystyle=\left|G_{33}^{\prime}(\omega)-\tilde{G}_{33}^{\prime}(\omega;\omega_{\min},\omega_{\max})\right|$
$\displaystyle\epsilon_{33}^{\prime\prime}(\omega;\omega_{\min},\omega_{\max})$
$\displaystyle=\left|G_{33}^{\prime\prime}(\omega)-\tilde{G}_{33}^{\prime\prime}(\omega;\omega_{\min},\omega_{\max})\right|.$
(36)

The error due to truncation has contributions from the left,
$u\in[0,\omega_{\min}]$, and right, $u\in[\omega_{\max},\infty]$, tails. At
large frequencies $u\rightarrow\infty$, the integrands of both the MAOS KKR
decay rapidly as $1/u^{6}$,

$\lim_{u\rightarrow\infty}\dfrac{1}{u^{3}}\dfrac{G_{33}^{\prime\prime}(u)}{\left(u^{2}-\omega^{2}\right)}\propto\dfrac{1}{u^{6}},\quad\lim_{u\rightarrow\infty}\dfrac{1}{{{u}^{2}}}\dfrac{G_{33}^{\prime}(u)}{\left(u^{2}-\omega^{2}\right)}\propto\dfrac{1}{u^{6}}.$
(37)

Thus, the typical correction due to truncation of the high frequency or right
tail is relatively modest compared to the truncation of the left tail, which
we consider next. At low frequencies,

$\lim_{u\rightarrow
0}\dfrac{1}{u^{3}}\dfrac{G_{33}^{\prime\prime}(u)}{\left(u^{2}-\omega^{2}\right)}\propto\dfrac{u^{0}}{\omega^{2}},\quad\lim_{u\rightarrow
0}\dfrac{1}{{{u}^{2}}}\dfrac{G_{33}^{\prime}(u)}{\left(u^{2}-\omega^{2}\right)}\propto\dfrac{u^{2}}{\omega^{2}}.$
(38)

This left tail contribution to the error in equation 36 can be approximated
as,

$\displaystyle\epsilon_{33}^{\prime}(\omega;\omega_{\min},\infty)$
$\displaystyle=-\dfrac{2\omega^{4}}{\pi}\int_{0}^{\omega_{\min}}\dfrac{1}{u^{3}}\dfrac{G_{33}^{\prime\prime}(u)}{\left(u^{2}-\omega^{2}\right)}du\approx-\dfrac{2\omega^{4}}{\pi}\int_{0}^{\omega_{\min}}\dfrac{u^{0}}{\omega^{2}}du\propto\omega^{2}\omega_{\min}$
$\displaystyle\epsilon_{33}^{\prime\prime}(\omega;\omega_{\min},\infty)$
$\displaystyle=\dfrac{2\omega^{3}}{\pi}\int_{0}^{\omega_{\min}}\dfrac{1}{u^{2}}\dfrac{G_{33}^{\prime}(u)}{\left(u^{2}-\omega^{2}\right)}du\approx\dfrac{2\omega^{3}}{\pi}\int_{0}^{\omega_{\min}}\dfrac{u^{2}}{\omega^{2}}du\propto\omega\omega_{\min}^{3}.$
(39)

Figure 3 depicts the total truncation error (equation 36) at four different
frequencies $\omega\tau_{0}=$ 0.1, 1, 10, and 100 as a function of the width
of the frequency window, which is controlled by the parameter $c$. We set
$\omega_{\min}=\omega/c$ and $\omega_{\max}=c\omega$ for each choice of
$\omega$ and $c$. As $c\rightarrow\infty$, the frequency window becomes
infinite, and the errors $\varepsilon_{33}^{\prime}$ and
$\varepsilon_{33}^{\prime\prime}$ go to zero. Error analysis (equation 39),
which assumes that the left tail is primarily responsible suggests,
$\varepsilon_{33}^{\prime}\propto\omega^{2}(\omega/c)=\omega^{3}/c$. This is
clearly evident in figure 3A at sufficiently large $c$, where the $1/c$
dependence is shown by the dashed line, and the error is normalized by
$\omega^{3}$. Similar analysis suggests that
$\varepsilon_{33}^{\prime\prime}\propto\omega(\omega/c)^{3}=\omega^{4}/c^{3}$,
which is also evident at large $c$ in figure 3B.

At smaller values of $c$ deviations from the asymptotic trendlines are
visible. These deviations surface when the finite frequency window
$[\omega/c,c\omega]$ does not include a sufficient information around the
characteristic relaxation time of the Giesekus model, $\tau_{0}$. That is, it
is important for the finite frequency window to include sufficient information
around the corresponding characteristic frequency, i.e. $\omega_{\min}\ll
2\pi/\tau_{0}\ll\omega_{\max}$. The dataseries corresponding to for
$\omega\tau_{0}=1$ always includes this region for the range of $c$ explored
in 3. Thus, it tracks the asymptotic trendline more faithfully than other
frequencies.

Figure 4: Finite-frequency window approximations of the third harmonic MAOS
moduli $\tilde{G}_{33}^{\prime}(\omega)$ and
$\tilde{G}_{33}^{\prime\prime}(\omega)$ for the single mode Giesekus model
with $G_{0}=\tau_{0}=1$ and $\alpha_{G}=0.2$. Three different widths of the
frequency window where $\omega_{\min}\tau_{0}=1/p$ and
$\omega_{\max}\tau_{0}=p$ are shown. The dashed black lines are the true MAOS
moduli.

Two other practical observations can be made from the figure: (i) at a given
$\omega$ and $c$, the magnitude of the error $\varepsilon_{33}^{\prime\prime}$
is smaller, often much smaller, than the corresponding error
$\varepsilon_{33}^{\prime}$, and (ii) for a fixed but sufficiently large
frequency window, both errors increase rapidly with frequency $\omega$. This
suggests that limiting the frequency window adversely affects high frequency
predictions of $\tilde{G}_{33}^{\prime}(\omega)$ and
$\tilde{G}_{33}^{\prime\prime}(\omega)$.

In practice, standard rheometers have a fixed frequency range, typically
between $10^{-3}$ to $10^{3}$ rad/s. Hence, we can explore the sensitivity of
$\tilde{G}_{33}^{\prime}(\omega)$ and $\tilde{G}_{33}^{\prime\prime}(\omega)$,
when the experimental data are available from a fixed frequency window
$\omega_{\min}=1/p$ and $\omega_{\max}=p$. Figure 4 shows
$\tilde{G}_{33}^{\prime}(\omega)$ for $\omega\tau_{0}\in[0.1,10]$, at three
different values of $p$ = 10, 100, and 1000.

As expected, the agreement between the approximate and true moduli improves as
$p$ increases. Note that when $p=10$, the prediction, particularly for
$\tilde{G}_{33}^{\prime}(\omega)$, is poor. The corresponding prediction for
$\tilde{G}_{33}^{\prime\prime}(\omega)$ is not quite as bad. This seems to be
a general trend: predictions of $G_{33}^{\prime\prime}(\omega)$ using MAOS KKR
on limited frequency data for $G_{33}^{\prime}(\omega)$ are far more reliable
than vice versa. This was foreshadowed by figure 3B, where the error
$\varepsilon^{\prime\prime}<\varepsilon^{\prime}$ at the same value of
$\omega$ and $c$.

However, even for $G_{33}^{\prime\prime}(\omega)$ we need data approximately
one order of magnitude larger ($p=100$ corresponds to $\omega_{\min}=10^{-2}$
and $\omega_{\max}=10^{2}$) than the range of reliable prediction
($\omega\tau_{0}\in[0.1,10]$) shown in figure 4B. Increasing $p$ by another
order of magnitude to 1000, results only in minor improvements in prediction
of $G_{33}^{\prime\prime}(\omega)$. Unlike $G_{33}^{\prime\prime}(\omega)$,
the predictions of $\tilde{G}_{33}^{\prime}(\omega)$ at high frequency, even
at $p=1000$ are quite poor. This is anticipated by equation 39, which
suggested that the error for the storage modulus is proportional to
$\omega^{2}/p$, unlike the loss modulus which is proportional to
$\omega/p^{3}$ and much better behaved.

##  V Summary

Linear KKR are integral transforms that relate the real and imaginary parts of
the complex modulus $G^{*}$ (or complex viscosity $\eta^{*}$). These relations
are a mathematical reflection of the principle of causality which constrains
the linear relaxation modulus ($G(t<0)=0$). We started with a multiple
integral expansion that generalizes the Boltzmann superposition principle to
nonlinear rheology. Nonlinear KKR, similar to their linear counterparts,
arises from the principle of causality which also constrains $n^{\text{th}}$
order relaxation modulus $G_{n}(t_{1},\cdots,t_{n})$.

We derived a general form of nonlinear KKR (equation 24) following the
approach of Hutchings et al. [40]. We specialized this general KKR to MAPS
rheology which relates the real and imaginary parts of the third-order complex
modulus $G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ or complex viscosity
$\eta_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ (equation 27). Recall that
knowledge of $G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$ allows us to
predict the asymptotically nonlinear material response to any arbitrary medium
amplitude deformation history. MAOS rheology can then be considered as a
popular special case of MAPS rheology that is characterized by two moduli
$G_{31}^{*}=(3/4)G_{3}^{*}(\omega,-\omega,\omega)$ and
$G_{33}^{*}=(-1/4)G_{3}^{*}(\omega,\omega,\omega)$. While a MAOS KKR relating
the real and imaginary parts of $G_{33}^{*}$ can be written, no such
expression relating the the real and imaginary parts of $G_{31}^{*}$ exists.

We verified the MAPS KKR relations on the single mode Giesekus model for which
the third-order complex modulus $G_{3}^{*}(\omega_{1},\omega_{2},\omega_{3})$
is analytically known. With practical applications in mind, we investigated
the sensitivity of the MAOS KKR when the domain of integration is truncated
and data is limited to a finite frequency window. We found that: (i) the
truncation error is typically dominated by the low-frequency or left tail,
(ii) inferring $G_{33}^{\prime\prime}$ from $G_{33}^{\prime}$ is more reliable
than vice versa, (iii) making predictions over a particular frequency range
requires approximately an extra decade of data beyond the frequency range of
prediction, and (iv) predictions of $G_{33}^{\prime}$ at large frequencies are
poor, even when two decades of data beyond the prediction range are available.

## Acknowledgments

This work is based in part upon work supported by the National Science
Foundation under grant no. NSF DMR-1727870 (SS). YMJ acknowledges financial
support from Science and Engineering Research Board (SERB), Department of
Science and Technology, Government of India. The authors thank Kyle R. Lennon
and Gareth M. McKinley for help with equation 32, and Ms. Shweta Sharma for
assistance with Mathematica.

## Data Availability Statement

Data that support the findings of this study are available from the
corresponding author upon reasonable request.

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