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\DOI{10.5802/crmeca.283}
\datereceived{2024-07-02}
\daterevised{2024-11-05}
\dateaccepted{2025-01-06}
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\dateposted{2025-01-27}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Nomograms of solid linear viscoelastic materials in time and
frequency domains}

\alttitle{Nomogrammes des mat\'{e}riaux visco\'{e}lastiques
lin\'{e}aires solides dans les domaines temporel et fr\'{e}quentiel}

\author{\firstname{Tiago Lima} \lastname{de Sousa}\CDRorcid{0000-0002-8992-6293}}
\address{Department of Mechanical Engineering, Federal University of
Parana, Curitiba, 81530-000, Brazil}
\address{Department of Mechanical Engineering, Federal University of
Pernambuco, Recife, 50740-550, Brazil}
\email[T. L. de Sousa]{tiago.sousa.eng@gmail.com}

\author{\firstname{J\'{e}derson} \lastname{da Silva}\CDRorcid{0000-0002-2714-3330}\IsCorresp}
\addressSameAs{1}{Department of Mechanical Engineering, Federal University of
Parana, Curitiba, 81530-000, Brazil}
\address{Department of Mechanical Engineering, Federal University of
Technology -- Parana, Londrina, 86036-370, Brazil}
\email[J. da Silva]{jedersonsilva@utfpr.edu.br}

\author{\firstname{Juc\'{e}lio Tom\'{a}s} \lastname{Pereira}\CDRorcid{0000-0002-2483-4339}}
\addressSameAs{1}{Department of Mechanical Engineering, Federal University of
Parana, Curitiba, 81530-000, Brazil}
\email[J. T. Pereira]{jucelio.tomas@ufpr.br}

\author{\firstname{Carolina Mocelin Gomes} \lastname{Pires}\CDRorcid{0000-0002-5171-3217}}
\addressSameAs{1}{Department of Mechanical Engineering, Federal University of
Parana, Curitiba, 81530-000, Brazil}
\email[C. M. G. Pires]{carolinamocelin@ufpr.br}

\begin{abstract}
The mechanical behavior of isotropic solid viscoelastic material (VEM)
can be described both in time or frequency domain considering
temperature effects. Thus, one can make use of viscoelastic functions
such as Young's and/or shear moduli and the Poisson's ratio. The
viscoelastic dynamic behaviors in different temperature and frequency
or time ranges can be grouped into a single graph, named nomogram. The
present work proposes a method for constructing nomograms for
viscoelastic functions, Young's and shear relaxation moduli, and
Poisson's ratio, defined in the time domain. It also proposed a
nomogram for the complex Poisson's ratio in the frequency domain.
\end{abstract}

\begin{altabstract}
Le comportement m\'{e}canique des mat\'{e}riaux visco\'{e}lastiques
solides isotropes (VEM) peut \^{e}tre d\'{e}crit \`{a} la fois dans les
domaines temporel et fr\'{e}quentiel, en prenant en compte les effets
de la temp\'{e}rature. Ainsi, il est possible d'utiliser des
fonctions visco\'{e}lastiques telles que les modules de Young et/ou de
cisaillement, ainsi que le coefficient de Poisson. Les comportements
dynamiques visco\'{e}lastiques dans diff\'{e}rentes plages de
temp\'{e}rature, de fr\'{e}quence ou de temps peuvent \^{e}tre
regroup\'{e}s en un seul graphique, appel\'{e} nomogramme. Ce travail
propose une m\'{e}thode pour construire des nomogrammes pour les
fonctions visco\'{e}lastiques, les modules de relaxation de Young et de
cisaillement, et le coefficient de Poisson, d\'{e}finis dans le domaine
temporel. Il propose \'{e}galement un nomogramme pour le coefficient de
Poisson complexe dans le domaine fr\'{e}quentiel.
\end{altabstract}

\shortrunauthors

\keywords{\kwd{Viscoelastic behavior}
          \kwd{Viscoelastic functions}
          \kwd{Complex Poisson's ratio}
          \kwd{Complex Young's modulus}
          \kwd{Complex shear modulus}}

\altkeywords{\kwd{Comportement visco\'{e}lastique}
          \kwd{Fonctions visco\'{e}lastiques}
          \kwd{Coefficient de Poisson complexe}
          \kwd{Module de Young complexe}
          \kwd{Module de cisaillement complexe}}

\thanks{State of Amazonas-BR, FAPEAM}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

In recent decades, the use of viscoelastic materials (VEMs) has been
increasing in several engineering areas, such as civil construction,
the automotive and aerospace industries, wind energy, transportation,
and the biomedicine. Examples of VEMs include asphalt mixtures, pads,
bumpers, water piping, packaging, household utensils, gastrointestinal
probes, and syringes. To estimate the mechanical behavior of these
materials, it is necessary to characterize the \mbox{parameters} of
viscoelastic rheological models, which can be done in frequency or in
the time domain~\cite{1,2,3}.

Considering problems described solely in the frequency domain, several
studies have been conducted in recent decades aimed at identifying the
complex Young's and/or shear moduli functions. There are various
methodologies proposed with this focus~\cite{2,4,5,6,7}. Similarly,
considering the time domain, some
works~\cite{8,9,10,11,12,13,14,15,16,17} present methodologies for
identifying the relaxation and/or creep moduli (in traction or shear).
Another important viscoelastic function for predicting VEMs behavior is
Poisson's ratio, which can also be defined in time or in frequency
domain. Aiming at identifying this function, several works have been
published~\cite{18,19,20,21,22,23,24}. According to
authors~\cite{3,4,5,25}, the dynamic properties of complex viscoelastic
functions can be expressed in terms of a loss factor and the dynamic
modulus, over large ranges of temperature and frequency. On the other
hand, those functions can be grouped into a single graph, named a
nomogram, which results in a compact, simple, and fast way of obtaining
the properties of the material under\break analysis.

In this context, the main contribution of the present work is to
propose the construction of a nomogram for the complex Poisson's ratio
in the frequency domain. Constructions of nomograms for the relaxation
moduli (Young's and shear), defined in the time domain, are also
proposed. It is important to highlight that in all the mentioned
methodologies, the VEMs are considered to be isotropic, linear, and
thermorheologically simple. Among numerous engineering applications,
these nomograms can be used to obtain the materials' viscoelastic
properties for use in both the frequency and time domains to control
vibrations in structures involving rubbery materials, as presented in
the works~\cite{26,27,28}.

\section{Theoretical concepts}\label{sec2}

VEMs are materials defined by their elastic and viscous behaviors
simultaneously. Such mechanical behavior can be described through
rheological models involving springs and dampers, where the spring
represents the elastic part and the damper represents the viscous part.
Thus, for any association, whether in series, in parallel, or in a
combination of those arrangements involving springs and Scott-Blair's
fractional dampers, fractional viscoelastic rheological models are
obtained~\cite{29}. According to~\cite{30}, for one-dimensional
problems, the modeling of those systems results in fractional
differential equations such as
{\begin{equation}\label{eq1}
\left[1+\sum_{{r}=1}^{{n}}{a}_{{r}}\frac{\mathrm{d}^{{\beta _{{r}}}}}
{\mathrm{d}t^{{\beta _{{r}}}}}\right]\sigma ({t})=\left[{k}+
\sum_{{r}=1}^{{n}}{b}_{{r}}\frac{\mathrm{d}^{{\beta _{{r}}}}}
{\mathrm{d}t^{{\beta _{{r}}}}}\right]\varepsilon ({t}).
\end{equation}}\unskip

In this equation, $n$, $k$, $a_r$, and $b_r$ ($r = 1,\ldots, n$) are
parameters associated with the chosen rheological model,  $\sigma
({t})$ is the stress function, $\varepsilon ({t})$ is the strain
function, $t$ is time, and $\mathrm{d}^{{\beta _{{r}}}}(\cdot)/
\mathrm{d}t^{{\beta _{{r}}}}$ represents a differential operator of
non-integer order ${\beta}_{r}$, which is given by $\beta_r = r +
\beta - 1$, where  $0< \beta < 1$~\cite{29}. In the present study,
Riemann--Liouville's definition of the fractional derivative are the
most appropriate, since it is assumed that the structural system is
initially at rest, and there is no need to treat the information that
occurs for a time $t < 0$~\cite{29}. Thus, considering a function $f(t)$,
the Riemann--Liouville's definition of fractional derivative on the
left, with an order $\beta $, is given by
{\begin{eqnarray}\label{eq2}
\frac{\mathrm{d}^{\beta}{f}({t})}{\mathrm{d}t^{\beta }}=
{}_0D_{t}^{\beta }
[{f}({t})]=\frac{1}{\Gamma ({m}-\beta)}
\frac{\mathrm{d}^{{m}}}{\mathrm{d}t^{{m}}}{\int }_{0}^{{t}}
\frac{{f}(\tau)}{({t}-\tau)^{1+\alpha -{m}}}
\mathrm{d}\tau ,{\quad}\mbox{where }{m}-1< \beta < {m},
\end{eqnarray}}\unskip
in which $\beta $ is a positive real number, m is a positive integer
number and $\Gamma (\cdot)$ is the Euler's Gamma function~\cite{29}.

According to~\cite{29,31,32}, there are several constitutive models
involving fractional derivatives, such as Maxwell, Kelvin-Voigt, and
Zener. Pritz~\cite{33} and Ciniello et~al.~\cite{34} claim that the
Fractional Zener Model (Figure~\ref{fig1}) has shown to be efficient in
predicting the behavior of linear VEMs in time and/or frequency
domains.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Fractional Zener model illustration. The
parameters are related to uniaxial and shear tests.}
\end{figure}

Considering that the mechanical behavior of VEMs can be described by
the rheological model presented in Figure~\ref{fig1}, the differential
equation that relates stress and strain can be shown as:
{\begin{equation}\label{eq3}
\left(1+\frac{\rmmu _{{E}}}{{E}_{1}+{E}_{2}}
\frac{\mathrm{d}^{\beta }}{\mathrm{d}t^{\beta }}\right)
\sigma ({t})=\left(\frac{{E}_{1}{E}_{2}}{{E}_{1}+
{E}_{2}}+\frac{{E}_{2}\rmmu _{{E}}}{{E}_{1}+
{E}_{2}}\frac{\mathrm{d}^{\beta }}{\mathrm{d}t^{\beta }}\right)
\varepsilon ({t}),
\end{equation}}\unskip
where $E_{1}$ and $E_{2}$ are stiffness moduli of the elastic elements,
and $\rmmu _{{E}}$ is the intensity of the Scott-Blair
element~\cite{29,31,32,34}.

Defining the auxiliary parameters ${E}_{\infty}={E}_{1}
{E}_{2}/({E}_{1}+{E}_{2})$, ${E}_{0}={E}_{\infty }{r}_{{E}}$,
${r}_{{E}}=({E}_{1}+{E}_{2})/{E}_{1}$, and ${\tau }_{{E}}^{\beta}=
\rmmu _{{E}}/({E}_{1}+{E}_{2})$. it is possible to rewrite
Equation~(\ref{eq3}) as
{\begin{equation}\label{eq4}
\left(1+{\tau }_{{E}}^{\beta }\frac{\mathrm{d}^{\beta }}
{\mathrm{d}t^{\beta }}\right)\sigma ({t})=
\left({E}_{\infty }+{E}_{0}{\tau }_{{E}}^{\beta }
\frac{\mathrm{d}^{\beta }}{\mathrm{d}t^{\beta }}\right)
\varepsilon ({t}).
\end{equation}}\unskip

Equation~(\ref{eq4}) is written for a uniaxial tension test. This
equation also has a corresponding form when considering shear tests. In
this case, the stress--strain relation in shear test can be placed as:
{\begin{equation}\label{eq5}
\left(1+{\tau }_{G}^{\beta }\frac{\mathrm{d}^{\beta }}
{\mathrm{d}t^{\beta }}\right)\tau ({t})=\left({G}_{\infty }+
{G}_{0}{\tau }_{G}^{\beta }\frac{\mathrm{d}^{\beta }}
{\mathrm{d}t^{\beta }}\right)\gamma ({t}),
\end{equation}}\unskip
where ${\tau}_{G}^{\beta }=\rmmu _{G}/({G}_{1}+{G}_{2})$, ${G}_{\infty
}={G}_{1}{G}_{2}/({G}_{1}+{G}_{2})$, ${G}_{0}={G}_{\infty }{r}_{G}$ and
${r}_{G}=({G}_{1}+{G}_{2})/{G}_{1}${. The $G_{1}$ and $G_{2}$
parameters are the stiffness moduli of the elastic elements, and $\rmmu
_{G}$ is the intensity of Scott-Blair element for shear constitutive
model.

According to Tschoegl~\cite{30} and demonstrated by Sousa 
et~al.~\cite{24}, applying the Laplace transform to all terms of
Equation~(\ref{eq4}), for ${t}> 0$, and considering a steady-state
sinusoidal excitation of axial frequency, the steady-state response can
be expressed as
{\begin{equation}\label{eq6}
\sigma ({s})+{\tau }_{{E}}^{\beta }{s}^{\beta }\sigma ({s})=
{E}_{\infty }\varepsilon ({s})+{E}_{0}{\tau }_{{E}}^{\beta }
{s}^{\beta }\varepsilon ({s}).
\end{equation}}\unskip

Defining $\tilde{E}(s)=\sigma ({s})/\varepsilon (s)$, where
$\tilde{E}(s)$ is the function named Young's operational
modulus~\cite{30}, one obtains
{\begin{equation}\label{eq7}
\tilde{E}(s)=\frac{{E}_{\infty }+{E}_{0}{\tau }_{{E}}^{\beta }
{s}^{\beta }}{1+{\tau }_{{E}}^{\beta }{s}^{\beta }}.
\end{equation}}\unskip

Analogously, applying Laplace transform to Equation~(\ref{eq5}) one can
obtain
{\begin{equation}\label{eq8}
\left(1+{\tau }_{G}^{\beta }{s}^{\beta }\right)
\tau ({s})=\left({G}_{\infty }+{G}_{0}{\tau }_{G}^{\beta }
{s}^{\beta }\right)\gamma ({s}).
\end{equation}}\unskip

So, the shear operational modulus is obtained, defined as
$\tilde{{G}}(s)=\tau (s)/\gamma (s)${, produces
{\begin{equation}\label{eq9}
\tilde{{G}}(s)=\frac{{G}_{\infty }+{G}_{0}{\tau }_{G}^{\beta }
{s}^{\beta }}{1+{\tau }_{G}^{\beta }{s}^{\beta }}.
\end{equation}}\unskip

According to Tschoegl~\cite{30}, from the operational moduli (Young's
and shear), one can obtain the Poisson's operational ratio, denoted by
$\tilde{\upsilon }(s)$, given by
{\begin{equation}\label{eq10}
\tilde{\upsilon }(s)=\frac{\tilde{E}(s)}{2\tilde{{G}}(s)}-1.
\end{equation}}\unskip

Thus, from Equations~(\ref{eq7}), (\ref{eq9}), and (\ref{eq10}), one
can obtain the Poisson's operational ratio, as
{\begin{equation}\label{eq11}
\tilde{\upsilon }(s)=\frac{{E}_{\infty }}{2{G}_{\infty }}
\left[\frac{\left({r}_{{E}}{\tau }_{{E}}^{\beta }
{\tau }_{G}^{\beta }\right){s}^{2\beta }+
\left({r}_{{E}}{\tau }_{{E}}^{\beta }+{\tau }_{G}^{\beta }\right)
{s}^{\beta }+1}{\left({\tau }_{{E}}^{\beta }{s}^{\beta }+
1\right)\left({r}_{G}{\tau }_{G}^{\beta }
{s}^{\beta }+1\right)}\right]-1.
\end{equation}}\unskip


From the viscoelastic operational functions described in this section,
the viscoelastic responses defined in the time domain can be obtained
by applying the Laplace's inverse transform. These functions can also
be described in the frequency domain for experiments in steady-state,
where ${s}=\mathrm{i}\Omega$.

\subsection{The influence of temperature in the mechanical behavior of
VEMs}\label{sec21}

Temperature is a variable that directly influences the behavior of
VEMs. According to Lakes~\cite{1} and Ferry~\cite{35}, the viscoelastic
response can be described by using the time-temperature superposition
principle, which proposed a shift in the time (or frequency) scale of
the viscoelastic response. This means that each time ($t$) or frequency 
$(\Omega)$ record at a temperature~($T$) can be shifted to a reference
temperature ($T_0$) according to
{\begin{equation}\label{eq12}
{t}_{\mathrm{R}}={t}/\alpha _{T}\left({T},\,{T}_{0}\right)
\quad \mbox{and/or}{\quad}\Omega _{\mathrm{R}}=\alpha _{T}
({T},\,{T}_{0})\Omega,
\end{equation}}\unskip
where ${t}_{\mathrm{R}}$ and $\Omega _{\mathrm{R}}$ are named `reduced time' and
`reduced frequency', respectively~\cite{35,36,37}. In this case, 
$\alpha _{T}({T},\,{T}_{0})$ is a function that represents the
temperature shift factor.

The literature presents several models that describe the shift factor,
including those by Goldstein, Bestul-Chang, Arrhenius, and
Williams--Landel--Ferry. The most widely used in scientific works is the
mathematical model proposed by Williams, Landel, and Ferry~\cite{38},
named the WLF model, which is given by
{\begin{equation}\label{eq13}
\log\alpha _{T}({T},\,{T}_{0})=\frac{-{C}_{1}^{T}
({T}-{T}_{0})}{{C}_{2}^{T}+({T}-{T}_{0})},
\end{equation}}\unskip
where ${C}_{1}^{T}$ and ${C}_{2}^{T}$ are intrinsic constants of the
material.

\subsection{Viscoelastic functions in the frequency domain}\label{sec22}

According to Tschoegl~\cite{30} and Park and Schapery~\cite{39},
Young's modulus in the frequency domain, considering the influence of
temperature, can be obtained by combining \pagebreak Equations~(\ref{eq7})
and~(\ref{eq12}), resulting in
{\begin{equation}\label{eq14}
{E}^{*}(\Omega _{\mathrm{R}})=\frac{\overline{{E}}_{0}+
\overline{{E}}_{\infty }{\tau }_{{E}}^{\beta }
\left(\mathrm{i}\Omega _{\mathrm{R}}\right)^{\beta }}
{1+{\tau }_{{E}}^{\beta }(\mathrm{i}\Omega _{\mathrm{R}})^{\beta }},
\end{equation}}\unskip
where $\overline{{E}}_{0}={E}_{\infty}$, and 
$\overline{{E}}_{\infty}={E}_{0}$. In addition, the complex shear
modulus, ${G}^{*}(\Omega _{\mathrm{R}})$, can be obtained similarly, relating
Equations~(\ref{eq9}) and  (\ref{eq12}), resulting in
{\begin{equation}\label{eq15}
{G}^{*}(\Omega _{\mathrm{R}})=\frac{\overline{{G}}_{0}+
\overline{{G}}_{\infty }{\tau }_{G}^{\beta }
\left(\mathrm{i}\Omega _{\mathrm{R}}\right)^{\beta }}
{1+{\tau }_{G}^{\beta}\left(\mathrm{i}\Omega _{\mathrm{R}}\right)^{\beta }},
\end{equation}}\unskip
where $\overline{{G}}_{0}={G}_{\infty} $, and 
$\overline{{G}}_{\infty}={G}_{0}$. This model, Equation~(\ref{eq14}) or
Equation~(\ref{eq15}), is known in the literature as a
four-parameter fractional constitutive model~\cite{29,31,32}.

Based on Equation~(\ref{eq11}), the complex Poisson's ratio can be
posed in its final form as
{\begin{equation}\label{eq16}
\upsilon ^{*}(\Omega _{\mathrm{R}})=\frac{\overline{{E}}_{0}}
{2\overline{{G}}_{0}}\left[\frac{\left({r}_{{E}}{\tau }_{{E}}^{\beta }
{\tau }_{G}^{\beta }\right)
(\mathrm{i}\Omega _{\mathrm{R}})^{2\beta }+\left({r}_{{E}}{\tau }_{{E}}^{\beta }+
{\tau }_{G}^{\beta }\right)(\mathrm{i}\Omega _{\mathrm{R}})^{\beta }+1}
{\left({\tau }_{{E}}^{\beta }
(\mathrm{i}\Omega_{\mathrm{R}})^{\beta }+1\right)\left({r}_{G}{\tau }_{G}^{\beta }
\left(\mathrm{i}\Omega _{\mathrm{R}}\right)^{\beta }+1\right)}\right]-1.
\end{equation}}\unskip

This complex function can be rewritten in its general form as given by}
$\upsilon ^{*}(\Omega _{\mathrm{R}})=\upsilon '(\Omega _{\mathrm{R}})-
\mathrm{i}\upsilon''(\Omega _{\mathrm{R}}),$ where $\upsilon '(\Omega _{\mathrm{R}})$ and
$\upsilon ''(\Omega_{\mathrm{R}})$  correspond to the real and imaginary parts,
representing the dynamic modulus and the loss modulus of the complex
Poisson's ratio, respectively. Thus, by relating these parts, one can
obtain the loss factor of the complex Poisson's ratio, denoted as} $\eta
_{\upsilon }(\Omega _{\mathrm{R}})$, mathematically defined as
{\begin{equation}\label{eq17}
\eta _{\upsilon }(\Omega _{\mathrm{R}})=\upsilon ''(\Omega _{\mathrm{R}})/\upsilon '(\Omega _{\mathrm{R}}).
\end{equation}}\unskip

Therefore, based on the viscoelastic parameters, one can construct
several curves associated with the complex viscoelastic functions
presented in the current section. Hence, nomograms in the frequency
domain can be obtained.

\subsection{Viscoelastic functions in time domain}\label{sec23}

According to Ciniello et~al.~\cite{34}, the Young's relaxation modulus
in the time domain, considering the influence of temperature for the
fractional Zener model, can be obtained by applying the Laplace inverse
transform to Equation~(\ref{eq6}), resulting in
{\begin{equation}\label{eq18}
{E}({t}_{\mathrm{R}})={E}_{\infty }\left[1+{r}_{{E}}{E}_{\beta }
\left(-\left(\sfrac{{t}}{\left(\tau _{{E}}\alpha _{T}
({T},{T}_{0})\right)}\right)^{\beta }\right)\right],
\end{equation}}\unskip
where ${E}_{\beta }(\cdot)$ is the Mittag-Leffler function of order
$\beta $~\cite{40}.

Similarly, starting from Equation~(\ref{eq9}), the relaxation in shear
modulus function is obtained considering temperature, which can be
expressed as
{\begin{equation}\label{eq19}
{G}({t}_{\mathrm{R}})={G}_{\infty }\left[1+{r}_{G}{E}_{\beta }
\left(-\left(\sfrac{{t}}{\tau _{G}\alpha _{T}({T},{T}_{0})}
\right)^{\beta }\right)\right].
\end{equation}}\unskip

According to Sousa~\cite{41}, the Poisson's ratio function, defined in
the time domain, considering the effect of temperature, can be written
as
{\begin{eqnarray}
\upsilon ({t}_{\mathrm{R}})&=&\frac{{E}_{\infty }}{2{G}_{\infty }}
\left[1+\left(\frac{{\tau }_{G}^{\beta }{\tau }_{{E}}^{\beta }{r}_{{E}}
-{\tau }_{{E}}^{2\beta }{r}_{{E}}-{\tau }_{G}^{\beta }{\tau }_{{E}}^{\beta }
+{\tau }_{{E}}^{2\beta }}{\left({\tau }_{G}^{\beta }{r}_{G}
-{\tau }_{{E}}^{\beta }\right){\tau }_{{E}}^{\beta }}\right)
\left({E}_{\beta }\left(\,-{{t}}_{\mathrm{R}}^{\beta }/{\tau }_{{E}}^{\beta }\right)\right)+\right.\cdots 
\nonumber\\&&
\left.  +\left(\frac{-{\tau }_{G}^{\beta }\left({\tau }_{G}^{\beta }{{r}}_{G}^{2}
-{\tau }_{{E}}^{\beta }{r}_{{E}}{r}_{G}-{\tau }_{G}^{\beta }{r}_{G}+{\tau }_{{E}}^{\beta }
{r}_{{E}}\right)}{\left({\tau }_{G}^{\beta }{r}_{G}-{\tau }_{{E}}^{\beta }\right)
{r}_{G}{\tau }_{G}^{\beta }}\right)\left(\,{E}_{\beta }\left(\,-{{t}}_{\mathrm{R}}^{\beta }/
{r}_{G}{\tau }_{G}^{\beta }\right)\right)\right]-1.
\label{eq20}
\end{eqnarray}}\unskip

The viscoelastic functions presented in this section constitute a basis
on which it is possible to predict the mechanical behavior of linear,
isotropic, and thermorheologically simple VEMs in the time domain,
considering the effect of temperature. Thus, from the material
parameters identified by Sousa et~al.~\cite{24} or Sousa~\cite{41}, and
considering the viscoelastic functions presented here, nomograms are
constructed in the time and frequency domains.

\section{Methodology for construction and evaluating of nomograms in
time and frequency domains}\label{sec3}

For the frequency domain, the dynamic properties of the complex
viscoelastic functions, at several temperatures, can be presented in a
single graph, called a nomogram. Similarly, in the time domain, the
nomograms can be constructed for viscoelastic functions like the
relaxation in traction/shear modulus and the Poisson's ratio.

Table~\ref{tab1} schematically presents a method for constructing
nomograms in frequency or in time domain. The methodology consists of a
pseudo-algorithm that describes a computational procedure used for
construct nomograms of viscoelastic functions step-by-step.

\begin{table}
\caption{\label{tab1}Pseudo-algorithm implemented for constructing
nomograms in the time and frequency domains}
\begin{tabular}{ll}
\tbody
{Step 1:} & \parbox[t]{12cm}{{Obtain the mechanical properties of VEMs considering the influence of temperature.}}\vspace*{2pt} \\ 
{Step 2:} & \parbox[t]{12cm}{{Define the viscoelastic function to construct its nomogram.}} \\ 
{Step 3:} & \parbox[t]{12cm}{{Specify the reference temperature and the frequency/reduced time interval.}} \\ 
{Step 4:} & \parbox[t]{12cm}{{Plot the viscoelastic function(s) at the reference temperature within the defined interval for reduced frequency. The abscissa axis represents the reduced frequency (or reduced time), and the left ordinate axis shows the values of the viscoelastic functions.}}\vspace*{2pt}\\ 
{Step 5:} & \parbox[t]{12cm}{{Create a right ordinate axis while keeping the abscissa axis fixed.}} \\ 
{Step 6:} & \parbox[t]{12cm}{{Construct another graph for different temperatures on the same axis: reduced frequency versus frequency. The resulting curves are straight.}}\vspace*{2pt}
\botline
\end{tabular}
\end{table}

\subsection{Obtaining dynamical properties from nomograms}\label{sec31}


Figure~\ref{fig2} presents a nomogram constructed at a reference
temperature of 5~\textdegree C and includes a time interval from 
0.1~s to a value around 2 years (ordinate axis on the right) for
graphical analysis. The method for obtaining the dynamic property
corresponding to the relaxation modulus is described in the
pseudo-algorithm presented in Table~\ref{tab2}. Similarly, others
viscoelastic functions can be constructed and evaluated---defined in
time or frequency domain---to determine their dynamic properties.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Nomogram in time domain for the Young's relaxation
modulus.}
{\vspace*{6pt}}
\end{figure}


\begin{table}
\caption{\label{tab2}Pseudo-algorithm to obtain dynamic properties of
VEMs, defined in time and frequency domains}
\begin{tabular}{ll}
\tbody
{Step 1:} & \parbox[t]{12cm}{{Define the time and temperature to obtain the relaxation modulus (for example, 2 years and 30~\textdegree C).}}\vspace*{2pt} \\ 
{Step 2:} & \parbox[t]{12cm}{{From the specified time (e.g., 2 years), draw a horizontal line up to the desired temperature (e.g., 30~\textdegree C), as illustrated in Figure~\ref{fig2}.}}\vspace*{2pt} \\ 
{Step 3:} & \parbox[t]{12cm}{{Draw a vertical line up to the relaxation curve, which corresponds the reference temperature.}}\vspace*{2pt} \\ 
{Step 4:} & \parbox[t]{12cm}{{Draw a horizontal line from the intersecting point of the relaxation curve up to the left ordinate axis.}}\vspace*{2pt} \\ 
{Step 5:} & \parbox[t]{12cm}{{Obtain the value of the relaxation modulus for the specified time and temperature.}}
\botline
\end{tabular}
{\vspace*{6pt}}
\end{table}

\subsection{Viscoelastic parameters}\label{sec32}

The viscoelastic functions can be constructed by simply replacing the
viscoelastic parameters in the models discussed in Section~\ref{tab2}. The
adopted viscoelastic parameters (Table~\ref{tab3}) were determined by
Sousa~\cite{41} through an optimization procedure employing a hybrid
method that integrates Genetic Algorithms and nonlinear programming.
This approach utilized experimental data from the
EAR{\textregistered}-C1002 material, as reported by Jones~\cite{42}.
The minimized objective function was the mean squared error between the
empirical data and the theoretical fractional Zener viscoelastic
models, characterized by four parameters. These models are formulated
in the frequency domain and incorporate temperature effects.
\vspace*{-3pt}

\begin{table}
\caption{\label{tab3}Viscoelastic parameters obtained from Sousa~\cite{41}}
\begin{tabular}{cc}
\thead
Viscoelastic parameters & Numerical values \\
\endthead
${E}_{0}~(\mathrm{MPa})$ & $2.128667556644\times 10^{6}$ \\
${E}_{\infty }~(\mathrm{MPa})$ & $1.723757249223\times 10^{9}$ \\
${\tau}_{{E}}^{\beta}$ & $1.279369260811\times 10^{-2}$\vspace*{2pt} \\
${G}_{0}~(\mathrm{MPa})$ & $7.119033966574\times 10^{5}$\vspace*{2pt} \\
${G}_{\infty}~(\mathrm{MPa})$ & $6.202192164498\times 10^{8}$ \vspace*{2pt}\\
${\tau }_{G}^{\beta }$ & $1.189166029556\times 10^{-2}$ \vspace*{2pt}\\
$\beta $ & $4.524176145415\times 10^{-1}$\vspace*{2pt} \\
${C}_{1}^{T}$ & $1.433136320936\times 10^{1}$ \vspace*{2pt}\\
${C}_{2}^{T}$~(\textdegree C) & $1.031149458767\times 10^{2}$
\botline
\end{tabular}
\end{table}

\section{Results and discussion}\label{sec4}

The current section presents the nomograms for the complex moduli
(Young's and shear) and the proposed nomogram for the complex Poisson's
ratio. Subsequently, the nomograms proposed for the relaxation moduli
in tension and shear are shown, as well as the Poisson's ratio defined
in the time domain.
\vspace*{-3pt}

\subsection{Nomograms of Young's and shear complex moduli, and complex
Poisson's ratio: frequency domain}\label{sec41}

Using the methodology presented for constructing nomograms and the
mechanical properties (Table~\ref{tab3}),
nomograms are created for the complex viscoelastic functions (Young's
and shear complex moduli, and complex Poisson's ratio). In all cases,
the mechanical properties can be obtained over a frequency interval
from $10^{-2}$~Hz to $10^{6}$~Hz for the
following temperatures: ${-}$30~\textdegree C, ${-}$20~\textdegree C,
${-}$10~\textdegree C, 5~\textdegree C, 30~\textdegree C, 50~\textdegree
C, and 80~\textdegree C. However, other temperatures can also be
selected to be included in the nomogram.

Figure~\ref{fig3} presents the nomograms constructed for the Young's
and shear complex moduli. In these graphs, the storage moduli and the
loss factor are depicted together. It should be noted that the loss
factor values shown in the graph need to be divided by $10^{3}$. For
instance, at a frequency of approximately 1~Hz and a temperature of
${-}$20~\textdegree C on the nomogram corresponding to the Young's
complex modulus (Figure~\ref{fig3}, left),
the dashed line indicates a loss
factor close to 0.1. Additionally, the value of Young's dynamic modulus
is approximately 2000 MPa.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Nomogram of complex Young's (left) and shear
(right) moduli.}
\end{figure}

The complex Poisson's ratio can be obtained by inserting the parameters
from Table~\ref{tab3} into
Equation~(\ref{eq16}). Hence, Figure~\ref{fig4} proposes a nomogram for
the complex Poisson's ratio, which presents its dynamic modulus and
loss factor. These properties can be obtained in a similar manner. For
example, at a frequency near 1 Hz and a temperature of 5~\textdegree C,
the value of the dynamic modulus and that of the Poisson's loss factor
are approximately 0.49 and 0.002, respectively.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Nomograms of complex Poisson's ratio: dynamic
modulus and its loss factor.}
\end{figure}

The nomograms presented in this section establish a basis that can be
used to quickly and approximately obtain the mechanical properties of
VEMs. Furthermore, the methodology discussed in this paper enables the
determination of the dynamic properties of the viscoelastic functions
under analysis.

\subsection{Nomogram of the shear traction relaxation modulus and the
Poisson's ratio: Time domain}\label{sec42}

Using the parameters presented in Table~\ref{tab3} to consolidate the
viscoelastic functions for the time domain, nomograms are proposed for
the traction and shear relaxation moduli, as shown in
Figure~\ref{fig5}. Subsequently, a nomogram for the Poisson's ratio in
the time domain is presented in Figure~\ref{fig6}. In all cases, these
properties are observed over a time interval ranging from 1~s and
$10^{8}$~s (close to two years) at temperatures including:
${-}$45~\textdegree C, ${-}$40~\textdegree C, ${-}$35~\textdegree C,
${-}$25~\textdegree C, ${-}$10~\textdegree C, 5~\textdegree C,
30~\textdegree C, and 50~\textdegree C. However, other temperatures can
also be selected for inclusion in the nomogram.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Nomograms in time domain: Young's (left) and shear
(right) relaxation moduli.}
\end{figure}

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Nomogram of Poisson's ratio, defined in time domain.}
\end{figure}

One observes that the graphs of the nomograms in the time domain are
inverse forms of the corresponding nomograms in the frequency domain.
For example, the relaxation modulus is a monotonically decreasing curve
over time, whereas its counterpart in the frequency domain, the dynamic
Young's modulus, exhibits an increasing trend with higher frequencies.
Conversely, in the time domain, the Poisson's ratio shows a monotonic
increase. However, the dynamic Poisson's modulus displays a decreasing
trend as frequency increases. These results ensure a coherent physical
interpretation of the material behavior~\cite{21,41,43},
demonstrating the methodology's effectiveness and its potential for
application across various engineering fields.

\section{Final remarks}\label{sec5}

The present work describes and implements a methodology for
constructing viscoelastic functions in both the time and frequency
domains. Initially, in the frequency domain, nomograms were presented
for the complex Young's and shear moduli. Subsequently, a nomogram was
proposed for the complex Poisson's ratio, incorporating two
viscoelastic functions: the dynamic modulus and the Poisson's loss
factor. These properties' values can be easily obtained for any desired
frequency and temperature.

Nomograms of viscoelastic functions were proposed, defined in the time
domain. Using the presented methodology, nomograms were created for the
Young's and shear relaxation moduli, and subsequently, a nomogram for
the Poisson's ratio was proposed in the time domain. The graphs
obtained for these nomograms describe the inverse behavior of the
corresponding nomograms in the frequency domain. These results ensure a
coherent physical understanding of the material's behavior.

The use of a viscoelastic model based on a parametric constitutive
model (here, the fractional Zener model) enables the derivation of two
algebraic expressions for Poisson's ratio, one in the time-domain and
the other in the frequency-domain. Accordingly, the methodology applied
in this work is suitable whenever the constitutive model used allows
the conversion between the time-frequency domains and the obtaining of
the expression for the Poisson's ratio. This, in turn, enables the
construction of a nomogram for this crucial material property.

An important point to highlight is that the proposed model can be
applied in a finite element analysis involving VEMs and requires only
the input of material properties. For finite element analyses in the
time-domain, the mechanical properties of the medium are evaluated
incrementally at each time interval. Thus, Poisson's ratio is updated
at each time increment. For finite element analyses in the
frequency-domain, such as harmonic analysis, the structure's response
depends on the properties at each frequency. Therefore, the physical
parameters (including Poisson's ratio) must be evaluated at each
frequency value.

Note that in a time-domain FEM analysis, for very short times,
Poisson's ratio has a value close to 0.39 (Figure~\ref{fig6}). This
corresponds to an analysis in the frequency-domain for very high
frequencies (Figure~\ref{fig4}). This behavior is close to that of a
solid with low viscosity. On the other hand, for very long times, the
value of Poisson's ratio is close to 0.49 (Figure~\ref{fig6}). This
behavior corresponds to an analysis in the frequency-domain for very
low values of this variable (Figure~\ref{fig4}), characterizing a
behavior close to that of a nearly incompressible fluid.

As its main contribution to established knowledge, the present work
proposes the construction of new nomograms of viscoelastic functions in
the time and frequency domains, which allow for prompt and precise
visualization of the dynamic behaviors of the VEM under analysis.
Additionally, it is important to note that the methodology outlined for
constructing nomograms can be extended to other viscoelastic functions
(e.g., bulk modulus), various VEMs, and diverse combinations of
constitutive models and shift factors.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.


\section*{Acknowledgments}

TLdS thanks the Foundation for Research Support of the State of
Amazonas-BR, FAPEAM, for the financial support granted in the form of a
scholarship, for doctoral degree, without which it would not have been
possible to develop the present work.

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