# Modeling the viscoelectroelastic behavior of PVDF/PZT-5A composites for sensing and actuation applications

Salah Elbarnaty, Joao Pedro M Correia, Wiyao Azoti, Saïd Ahzi

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Salah Elbarnaty, Joao Pedro M Correia, Wiyao Azoti, Saïd Ahzi. Modeling the viscoelectroelastic behavior of PVDF/PZT-5A composites for sensing and actuation applications. Materials Research Express, 2025, 12 (9), pp.095701. 10.1088/2053-1591/adf959. hal-05280919

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Modeling the viscoelectroelastic behavior of
PVDF/PZT-5A composites for sensing and
actuation applications

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PAPER

# Modeling the viscoelectroelastic behavior of PVDF/PZT-5A composites for sensing and actuation applications

Salah Elbarnaty¹,* , Joao Pedro M Correia¹ , Wiyao Azoti² and Saïd Ahzi¹,³

¹ University of Strasbourg, ICUBE Laboratory-CNRS UMR 7357, Strasbourg, France

² INSA Toulouse, ICA-CNRS UMR 5312, Toulouse, France

³ International University of Rabat, LERMA Laboratory, Sala Al Jadida 11100, Morocco

* Author to whom any correspondence should be addressed.

E-mail: s.elbarnaty@unistra.fr

**Keywords:** viscoelectroelastic composite, PVDF, PZT-5A, frequency analysis, mori-tanaka, hysteresis

## Abstract

Our study presents the modeling of the effective viscoelectroelastic behavior of a composite material, where PVDF is an anistropic piezoelectric matrix and PZT-5A is incorporated in the matrix as piezoelectric long fibers or spheres. Both materials exhibit viscoelectroelastic properties, and their combination offers intriguing multifunctional capabilities. A homogenization scheme (based on the Mori-Tanaka micromechanics modeling) is carried out to understand how both material phases interact within the composite behavior. Frequency analysis is also performed to assess the composite material's response under varying conditions, and time-dependent effective viscoelectroelastic properties are evaluated to simulate its dynamic behavior. The incorporation of PZT-5A inclusions significantly affects the piezoelectric behavior of the composite, especially its spherical-like shape, resulting in an enlarged hysteresis loop. However, the enlargement is not recommended for sensing purposes, as it introduces energy losses, reduces sensitivity, and compromises precision. In contrast, the actuating behavior of the composite remains relatively unaffected by the addition of PZT-5A inclusions, with the hysteresis loop showing no notable changes, ensuring stable and reliable actuation performance. The study also highlights the disappearance of the hysteretic response at relatively high frequencies. Our work provides valuable insights into the interplay of PVDF and PZT-5A in viscoelectroelastic composites, highlighting challenges and opportunities for optimizing their performance. Our findings contribute to the development of advanced materials for applications requiring tailored sensing and actuation behavior, especially in aeronautic applications.

## 1. Introduction

The study of the effective behavior of piezoelectric composites has attracted significant attention due to their wide range of industrial applications, particularly in aerospace [1], aeronautics [2, 3] and automotive engineering [1], where they are used as sensors and actuators. Piezoelectric composites with a polymer matrix exhibit high temperature, frequency and time-dependent properties, [4–7]. The most prevailing piezoelectric polymer to be found is PVDF which is a time, frequency and temperature dependent material. PVDF has been extensively investigated by Vingradov and Halloway [8–10]. It seems to be the best alternative to classical piezoelectric materials [11–13].

The advantages of using PVDF instead of Lead Zirconate Titanate (PZT) or Barium Titanate (BT) stem largely from PVDF's eco-friendliness and its lightweight, flexible nature [11, 14] unlike PZT, which contains lead, and BT, which is relatively heavy and brittle just like PZT [13]. However, PVDF's major disadvantage is that it does not have a strong piezoelectric coefficient compared to the BT or PZT. Many researchers tried to overcome the drawback by introducing nano-inclusions, namely Graphene, Carbon nanotubes (CNT) into a PVDF matrix as in Krishnaswamy et al. [15], but it was not sufficient to compete with conventional piezoelectric
materials (i.e. PZT or BT). Recently, PZT and BT inclusions have been added to PVDF in order to increase its stiffness and piezoelectric behavior [16–18].

Recently, PZT-reinforced PVDF composites have attracted significant research attention due to their potential applications in actuators, sensors, and energy harvesters. Suprapto et al [18] demonstrated a strong correlation between the PZT volume fraction and composite sensitivity, showing that increasing the PZT content enhances the material’s sensing performance. Siponkoski et al [19] investigated the effect of PZT inclusions on the copolymer P(VDF-TrFE) for flexible sensor applications and reported a notable improvement in the strain piezoelectric coefficient $d_{31}$, reaching up to 17 pm V⁻¹ with a 50% PZT volume fraction. Additionally, Yun et al [20] showed that incorporating 20% PZT significantly improves both the tensile and piezoelectric properties of the composite, enhancing its suitability for actuation purposes.

The electromechanical behavior of piezoelectric composites has been widely investigated. The classical form of the electromechanical coupling equations was first introduced in the work of Deeg [21]. The calculations of effective properties of piezoelectric composite materials using a homogenization approach are presented in the work of Dunn and Taya [22, 23]. Various micromechanical models are used in [22] to predict the effective properties of such materials. Among micromechanics models, the Mori-Tanaka (MT) scheme offers the advantages of an explicit scheme, i.e. a closed form solution directly based on the phase properties. In addition, MT is based on Eshelby’s concept of ellipsoidal inclusion, which facilitates the implementation of different inclusion shapes such particles or fibres (short or long) using a single variable called the aspect ratio.

Other micromechanics models such as the iso-field model developed in [24, 25] have been used for estimating the effective properties of piezoelectric structural fibre composites. In the iso-field method, the strain and electric field components parallel to the plane connecting two phases are assumed to be uniform across both phases, and likewise for the stress and electric displacement components normal to the connecting plane [26].

In addition, The effects of phase properties, orientation angles, volume fractions and inclusion aspect ratio on the effective properties of piezoelectric and piezoelectric composite materials have been extensively analyzed by Fakri et al [27, 28]. Odegard [29] introduced a novel modeling approach to predict the electromechanical properties of piezoelectric composite materials. In [30], a comparison between the predictions obtained using the Mori–Tanaka (MT) approach and the Finite Element (FE) method is presented. Li and Dunn [31] studied the viscoelectroelastic behavior of two-phase heterogeneous piezoelectric materials, where lossless piezoelectric inclusions are embedded in a lossy matrix, assuming constant complex coefficients for the matrix properties. The previous work was extended by Azrar et al [4] to include the inclusion coating effects on the effective properties of a viscoelectroelastic composite that consists of non-lossy piezoelectric inclusions embedded in a lossy matrix. The results of Azrar et al [4] are presented in both frequency and time domains. Jiang and Batra [32] used the MT approach to derive closed-form expressions for the effective moduli of viscoelectroelastic composites made of parallel PZT cylinders, with elliptical cross-sections, embedded in a viscoelastic matrix. Jiang and Batra [32] concluded that the viscoelasticity of the matrix has an effect on each effective property of the piezoelectric composite. To our knowledge, the computation of effective properties for viscoelectroelastic composites is a topic that has received limited attention in the literature, especially for PVDF based composites.

In our work, the composites are made of a PVDF matrix reinforced by piezoelectric fillers (PZT-5A) having different shapes. The composites have two phases which exhibit viscoelectroelastic behavior. The modeling framework follows the approach proposed in Azrar et al [4] where the correspondence principle of linear viscoelectroelasticity is combined with the MT micromechanical model. The time and frequency dependent effective properties of viscoelectroelastic composites have been calculated. Subsequently, the overall response of composites has been numerically predicted.

## 1.1. Constitutive laws for linear viscoelectroelastic materials

The mathematical procedures to establish the viscoelectroelastic behavior of a two-phase composite are briefly recalled in the section. More details about the procedures can be found in the works of Azrar et al [4] and Li et Dunn [31]. The constitutive equations for linear time-independent piezoelectric materials are given by:

$$ \begin{aligned} \sigma_{ij} &= C_{ijmn}\varepsilon_{mn} + e_{nij}E_n \\ D_i &= e_{imn}\varepsilon_{mn} - \kappa_{in}E_n \end{aligned} \quad (1) $$

where $\sigma_{ij}$, $\varepsilon_{mn}$, $E_n$ and $D_i$ represent the second-order stress tensor, the second-order strain tensor, the first-order electric field vector and first-order electric displacement vector respectively. In equation (1), $C_{ijmn}$, $e_{ijk}$ and $k_{in}$ are fourth-order elastic stiffness tensor, the third-order piezoelectric tensor and the second-order dielectric permittivity tensor respectively. Equation (1) is usually replaced by the condensed form given by:

$$ \Sigma_{ij} = E_{ijMn}Z_{Mn} \quad (2) $$
with

$$E_{ijkl} = \begin{cases} c_{ijkl} & (J = j = 1, 2, 3; K = k = 1, 2, 3) \\ e_{lij} & (J = 1, 2, 3; K = 4) \\ e_{ikl} & (J = 4; K = 1, 2, 3) \\ -\kappa_{il} & (J = 4; K = 4) \end{cases} \quad (3)$$

In equation (2), $Z_{Mn}$ incorporates the elastic-strain and electric field components and is defined by:

$$Z_{Mn} = \begin{cases} \varepsilon_{mn} & M = 1, 2, 3 \\ -E_n & M = 4 \end{cases} \quad (4)$$

while $\Sigma_{ij}$ incorporates the elastic stress and electric displacement components and is defined by:

$$\Sigma_{ij} = \begin{cases} \sigma_{ij} & J = 1, 2, 3 \\ D_i & J = 4 \end{cases} \quad (5)$$

The constitutive laws written above only describe the time-independent behavior of piezoelectric materials [22, 27, 29]. As for the time-dependent behavior (the present work), the Boltzmann convolution law is adopted in agreement with the work of [4]. Therefore, the constitutive laws, equation (1), can be rewritten for viscoelectroelastic materials as:

$$\begin{aligned} \sigma_{ij}(t) &= \int_{0}^{t} c_{ijkl}(t - \tau) \frac{d\varepsilon_{kl}(\tau)}{d\tau} d\tau + \int_{0}^{t} e_{lij}(t - \tau) \frac{dE_{l}(\tau)}{d\tau} d\tau \\ D_{i}(t) &= \int_{0}^{t} e_{ikl}(t - \tau) \frac{d\varepsilon_{kl}(\tau)}{d\tau} d\tau + \int_{0}^{t} \kappa_{il}(t - \tau) \frac{dE_{l}(\tau)}{d\tau} d\tau \end{aligned} \quad (6)$$

where $c_{ijkl}$, $e_{lij}$ and $\kappa_{il}$ are the elastic, piezoelectric and dielectric relaxation tensors respectively. As the condensed notations are used in our work, the convolutional integral form of equation (2) is written as:

$$\Sigma_{ij}(t) = \int_{0}^{t} E_{ijkl}(t - \tau) \frac{dZ_{kl}(\tau)}{d\tau} d\tau \quad (7)$$

with

$$E_{ijkl}(t) = \begin{cases} c_{ijkl}(t) & (J = j = 1, 2, 3; K = k = 1, 2, 3) \\ e_{lij}(t) & (J = 1, 2, 3; K = 4) \\ e_{ikl}(t) & (J = 4; K = 1, 2, 3) \\ -\kappa_{il}(t) & (J = 4; K = 4) \end{cases} \quad (8)$$

The tensor $E_{ijkl}(t)$, defined in equation (8), is the time-dependent viscoelectroelastic relaxation tensor. In equation (7), the convolution integral provides a compact mathematical expression that links the generalized stress tensor $\Sigma_{ij}$ to the generalized strain tensor $Z_{kl}$ via the viscoelectroelastic generalized relaxation tensor $E_{ijkl}(t)$.

By applying the Carson transformation, the viscoelectroelastic problem can be reduced to an electroelastic one [4, 31, 33]. It is called the correspondence principle, which allows the transition from a viscoelectroelastic problem to an electroelastic one when the time-dependent quantities smoothly change with time [34, 35]. The Carson transformation is given by:

$$\bar{f}(s) = s \int_{0}^{t} f(t) e^{-ts} dt \quad (9)$$

where $s$ is the Carson variable. Using equation (9) with equation (7) leads to:

$$\begin{aligned} \bar{\sigma}_{ij}(s) &= \bar{c}_{ijkl}(s) \bar{\varepsilon}_{kl}(s) + \bar{e}_{lij}(s) \bar{E}_{l}(s) \\ \bar{D}_{i}(s) &= \bar{e}_{ikl}(s) \bar{\varepsilon}_{kl}(s) - \bar{\kappa}_{il}(s) \bar{E}_{l}(s) \end{aligned} \quad (10)$$

with

$$\begin{aligned} \bar{\sigma}(s) &= s\bar{\sigma}(\omega), \quad \bar{c}(s) = s \int_{0}^{t} c(u)e^{-su} du, \quad \bar{e}(s) = s \int_{0}^{t} e(u)e^{-su} du, \\ \bar{E}(s) &= s\bar{E}(\omega), \quad \bar{\varepsilon}(s) = s\bar{\varepsilon}(\omega) \text{ and } \bar{D}(s) = s\bar{D}(\omega). \end{aligned} \quad (11)$$

By analogy with equations (6) and (7), a condensed notation can be applied, and equation (10) can be rewritten as:

$$\bar{\Sigma}_{ij}(s) = \bar{E}_{ijkl}(s) \bar{Z}_{kl}(s) \quad (12)$$

Using the Voigt notation, equation (12) becomes:

$$\bar{\Sigma}_{i}(s) = \bar{E}_{ij}(s) \bar{Z}_{j}(s) \quad (13)$$
or in terms of matrices

$$ \begin{bmatrix} \sigma_1(s) \\ \sigma_2(s) \\ \sigma_3(s) \\ \sigma_4(s) \\ \sigma_5(s) \\ \sigma_6(s) \\ D_1(s) \\ D_2(s) \\ D_3(s) \end{bmatrix} = \begin{bmatrix} C_{11}(s) & C_{12}(s) & C_{13}(s) & 0 & 0 & 0 & 0 & 0 & e_{31}(s) \\ C_{12}(s) & C_{22}(s) & C_{23}(s) & 0 & 0 & 0 & 0 & 0 & e_{32}(s) \\ C_{13}(s) & C_{23}(s) & C_{33}(s) & 0 & 0 & 0 & 0 & 0 & e_{33}(s) \\ 0 & 0 & 0 & C_{44}(s) & 0 & 0 & 0 & e_{15}(s) & 0 \\ 0 & 0 & 0 & 0 & C_{55}(s) & 0 & e_{15}(s) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66}(s) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & e_{15}(s) & 0 & -\kappa_1(s) & 0 & 0 \\ 0 & 0 & 0 & e_{15}(s) & 0 & 0 & 0 & -\kappa_2(s) & 0 \\ e_{31}(s) & e_{32}(s) & e_{33}(s) & 0 & 0 & 0 & 0 & 0 & -\kappa_3(s) \end{bmatrix} \begin{bmatrix} \varepsilon_1(s) \\ \varepsilon_2(s) \\ \varepsilon_3(s) \\ \gamma_4(s) \\ \gamma_5(s) \\ \gamma_6(s) \\ E_1(s) \\ E_2(s) \\ E_3(s) \end{bmatrix} \quad (14) $$

where $2\varepsilon_j = \gamma_j$ for $j = 4,5,6$. From this point onwards, and throughout the subsequent sections of our work, bold symbols will denote 9 × 9 matrix notation.

## 1.2. Homogenization of a two-phase viscoelectroelastic composite: mori-tanaka scheme

The Mori–Tanaka (MT) scheme is a micromechanical homogenization method used to estimate the effective properties of composite material. It is derived from the Eshelby problem [36] and it models a heterogeneous material as a matrix containing inclusions. It takes into account the interaction between the inclusions and the matrix by assuming that each inclusion is embedded in an infinite matrix subjected to the average field of the composite. The MT approach is particularly effective for materials with low to moderate inclusion concentrations and assumes that inclusions are ellipsoidal and aligned [22]. In our work, the MT scheme is applied to a two-phase composite that consists in a PVDF matrix (considered as a lossy piezoelectric matrix) in which PZT-5A inclusions are incorporated. The PZT-5A inclusions can be lossy piezoelectric infinitely long fibers or lossy piezoelectric spheres.

In agreement with [4], the MT scheme of a two-phase viscoelectroelastic in the frequency domain (Carson domain) is given by:

$$ \overline{E^{MT}(s)} = \overline{E_{PVDF}(s)} + c_{PZT-5A} \overline{(\overline{E_{PZT-5A}(s)} - \overline{E_{PVDF}(s)})} \overline{A^{MT}(s)} \quad (15) $$

with

$$ \overline{A^{MT}(s)} = \overline{A^{dil}(s)} [c_{PVDF} \mathbf{I} + c_{PZT-5A} \overline{A^{dil}(s)}]^{-1} \quad (16) $$

and

$$ \overline{A^{dil}(s)} = [\mathbf{I} + \overline{S(s)} \overline{E_{PVDF}^{-1}(s)} (\overline{E_{PZT-5A}(s)} - \overline{E_{PVDF}(s)})]^{-1} \quad (17) $$

where $c_{PVDF}$ and $c_{PZT-5A}$ denote the volume fractions of the matrix (PVDF) and the inclusions (PZT-5A), respectively. $\overline{E_{PVDF}(s)}$ and $\overline{E_{PZT-5A}(s)}$ represent the viscoelectroelastic frequency-dependent matrices of the PVDF and PZT-5A phases respectively. The effective viscoelectroelastic frequency-dependent matrix predicted by the MT method is denoted by $\overline{E^{MT}(s)}$. The matrix $\overline{A^{MT}(s)}$ defined in equation (16) refers to the frequency-dependent strain concentration matrix within the MT framework. The matrix $\overline{A^{dil}(s)}$ defined in equation (17) corresponds to the frequency-dependent strain concentration matrix in the dilute case, where the inclusions are assumed to be non-interacting and embedded in the matrix. In equation (17), $\overline{S(s)}$ is the frequency-dependent piezoelectric Eshelby tensor written in a matrix form. The Eshelby tensor depends on the shape of the PZT-5A inclusions and on the PVDF viscoelectroelastic properties. It must be computed numerically since the PVDF is an orthotropic material [37].

Equation (11) allows us to calculate the viscoelectroelastic effective properties of a two-phase composite using the MT scheme in the frequency domain. A Laplace transformation is used to transform the frequency-dependent properties into time-dependent ones. The time-dependent properties are therefor given by:

$$ \overline{E^{MT}(t)} = \frac{1}{2\pi i} \int_{\delta+i\infty}^{\delta+i\infty} \frac{\overline{E^{MT}(s)}}{s} e^{st} ds \quad (18) $$

It is to be noted that $\overline{E^{MT}(s)}$ in equation (18) depends implicitly on the Carson variable s which requires that the transformation from the frequency domain to the time domain to be evaluated numerically. This is done by using the algorithm proposed by Abate and Valko [38]. The latter was used by many authors to predict time-dependent effective properties and has shown a good performance for viscoelastic [34] and viscoelectroelastic composites [4]. The algorithm used to compute the effective viscoelectroelastic properties in the time domain is given by [38]:
<table><caption>**Table 1.** Properties of PZT-5A: real and imaginary parts of elastic, piezoelectric and dielectric moduli, [31]. Units: $C_{ij}$, GPa; $e_{ij}$, $C^2/m$; $\kappa_0$ (permittivity of free space) = 8.85 × 10⁻¹²$C²/Nm²$.</caption><thead><tr><th>C<sub>11</sub><sup>r</sup></th><th>C<sub>11</sub><sup>i</sup></th><th>C<sub>12</sub><sup>r</sup></th><th>C<sub>12</sub><sup>i</sup></th><th>C<sub>13</sub><sup>r</sup></th><th>C<sub>13</sub><sup>i</sup></th><th>C<sub>33</sub><sup>r</sup></th><th>C<sub>33</sub><sup>i</sup></th><th>C<sub>44</sub><sup>r</sup></th><th>C<sub>44</sub><sup>i</sup></th></tr></thead><tbody><tr><td>121</td><td>0.885</td><td>75.2</td><td>0.061</td><td>75.2</td><td>0.203</td><td>111</td><td>0.710</td><td>21.1</td><td>0.634</td></tr><tr><td>κ<sub>11</sub><sup>r</sup>/κ<sub>0</sub></td><td>κ<sub>11</sub><sup>i</sup>/κ<sub>0</sub></td><td>κ<sub>33</sub><sup>r</sup>/κ<sub>0</sub></td><td>κ<sub>33</sub><sup>i</sup>/κ<sub>0</sub></td><td>e<sub>31</sub><sup>r</sup></td><td>e<sub>31</sub><sup>i</sup></td><td>e<sub>33</sub><sup>r</sup></td><td>e<sub>33</sub><sup>i</sup></td><td>e<sub>15</sub><sup>r</sup></td><td>e<sub>15</sub><sup>i</sup></td></tr><tr><td>916</td><td>-18.3</td><td>830</td><td>-16.6</td><td>-5.4</td><td>0.574</td><td>15.8</td><td>0.27</td><td>12.3</td><td>-0.291</td></tr></tbody></table>

<table><caption>**Table 2.** Properties of PVDF: real part of dielectric and piezoelectric moduli, [29]. Units: $e_{ij}$, $[C^2/m]$; $\kappa_0$ (permittivity of free space) = 8.85 × 10⁻¹²$C²/Nm²$.</caption><thead><tr><th>κ<sub>11</sub><sup>r</sup>/κ<sub>0</sub></th><th>κ<sub>22</sub><sup>r</sup>/κ<sub>0</sub></th><th>κ<sub>33</sub><sup>r</sup>/κ<sub>0</sub></th><th>e<sub>31</sub><sup>r</sup></th><th>e<sub>32</sub><sup>r</sup></th><th>e<sub>33</sub><sup>r</sup></th></tr></thead><tbody><tr><td>7.4</td><td>9.3</td><td>7.6</td><td>0.024</td><td>0.001</td><td>-0.27</td></tr></tbody></table>

$$E^{MT}(t, M) = \frac{r}{M} \left\{ \frac{1}{2} \overline{E^{MT}}(r) \exp(rt) + \sum_{k=1}^{M-1} \text{Re} \left[ \exp(t \cdot s(\theta_k)) \overline{E^{MT}}(s(\theta_k))(1 + i\sigma(\theta_k)) \right] \right\} \quad (19)$$

with $r = \frac{2M}{5t}$, $\theta_k = \frac{k\pi}{M}$, $\sigma(\theta_k) = \theta_k + (\theta_k \cot(\theta_k) - 1) \cot(\theta_k)$ and $s(\theta_k) = r\theta_k(\cot(\theta_k) + i)$. In equation (19), M is the number of terms to be summed and angle $\theta_k$ ranges from $-\pi$ to $\pi$. Finally, equations (15) and (19) allow us to calculate the effective viscoelectroelastic properties in frequency and time domains respectively, using the MT scheme for a two-phase composite.

## 2. Materials properties

Using the above modeling, the effective viscoelectroelastic properties in the time and frequency domains are considered for the PVDF reinforced by both long fibers and spherical PZT-5A inclusions (figure A1). Both PVDF and PZT-5A are materials which exhibit frequency and time dependent elastic behavior [8–10]. However, in terms of piezoelectric and dielectric behavior, only PZT-5A displays constant lossy characteristics [31], while the piezoelectric response of PVDF is frequency and time independent [8]. The properties of both the PVDF and PZT-5A are taken from the literature and are listed in tables 1–2.

The properties of both the PVDF and PZT-5A are taken from the literature. Tables 1–2

The PVDF’s creep compliance in the direction of molecular chain was successfully modeled using the power law- model by Vinogradov and Holloway [10]:

$$M_0(t) = D_0 + D_1 t^n \quad (20)$$

The Carson transformation of the power-law model (equation (20)) gives [4, 34]:

$$M_0(s) = D_0 + \frac{D_1 n!}{s^n} \quad (21)$$

The constants $D_0$, $D_1$ and $n$ are given by table 3. The relaxation modulus is given by $C_0(s) = M_0^{-1}(s)$. for the sack of simplicity the Poisson coefficient $a$ is taken to be constant.

Its noteworthy though, that the main advantage of the Carson transformation is that it allows to preserve the physics of the transformed quantity unlike Laplace transformation [34].

## 3. Results and discussion

### 3.1. Frequency domain analysis

Figures 1 and 2 illustrate the effective viscoelectroelastic properties of the PVDF/PZT-5A composite in the frequency domain for different inclusion shapes and volume fractions. As expected for viscoelastic composites, the real part of effective stiffness $\overline{C}_{11}$ increases as frequency increases, reflecting the stiffer response of the PVDF matrix at high loading rates. The presence of PZT-5A inclusions, especially at higher volume fractions, enhances the stiffening effect, figure 1. Additionally, composites with PZT-5A spherical inclusions exhibit slightly higher moduli compared to those with fiber-like inclusions. It is likely to be due to a more isotropic reinforcement effect. The piezoelectric response remains relatively stable across a wide frequency range, figure 2, with only a
<figure><img src="image_7.png" /><figcaption>**Figure 1.** Complex effective $\overline{C}_{11}$ versus frequency for both spherical and long fiber-like inclusions. (a): $c_{PZT-5A} = 0.1$; (b): $c_{PZT-5A} = 0.4$; (c) $c_{PZT-5A} = 0.8$.</figcaption></figure>

slight decrease at higher frequencies, which can be attributed to the viscoelastic damping in the PVDF matrix. The influence of inclusion shape is also evident in figure 2: fiber-like PZT-5A inclusions lead to a slightly higher piezoelectric response compared to spherical inclusions. It is likely to be due to their more favorable orientation
<figure><img src="image_8.png" /><figcaption>**Figure 2.** Complex effective $\bar{e}_{31}$ versus frequency for both spherical and long fiber-like inclusions. (a): $c_{PZT-5A} = 0.1$; (b): $c_{PZT-5A} = 0.4$; (c) $c_{PZT-5A} = 0.8$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 3.** Effective storage dielectric constant Re (κ₃₃/κ₀) of the viscoelectroelastic composite as a function of PZT-5A volume fraction (Spheroidal shape), with comparison to experimental results for PVDF/PZT composites fabricated via hot-pressing method (120 °C, 50 Hz).</figcaption></figure>

<table><caption>**Table 3.** PVDF viscoelastic properties (power-law model) at room temperature [8, 10].</caption><thead><tr><td>D<sub>0</sub> (GPa<sup>-1</sup>)</td><td>D<sub>1</sub> (GPa<sup>-1</sup> s<sup>-1</sup>)</td><td>n</td><td>a</td></tr></thead><tbody><tr><td>0.3206</td><td>0.005018</td><td>0.107</td><td>0.326</td></tr></tbody></table>

along the loading direction as was reported by Odegard [27, 29]. Figure 3 illustrates the variation of the dielectric constant κ'₃₃/κ₀ for different PZT volume fractions and compares these results with experimental data reported Gowhaman et al [39] for hot press processing method. The comparison demonstrates reasonable agreement between the experimental findings and our model predictions.

Overall, the frequency domain analysis confirms that the mechanical behavior of the composite is significantly frequency-dependent, while the piezoelectric behavior remains more stable. It can be concluded that the composite material studied is suitable for actuation purposes under dynamic conditions.

## 3.2. Hysteresis analysis

The hysteresis phenomenon arises from microscopic mechanisms including internal friction within the PVDF matrix, domain wall movements and dielectric relaxation in PZT-5A, and interfacial stress transfer losses between matrix and inclusions. These contribute to phase lag and energy dissipation during cycling, captured in the enlarged loop areas.

Understanding the hysteresis characteristics under different frequencies and inclusion morphologies allows tailoring composite microstructures for targeted applications. Actuators require minimized hysteresis to ensure repeatability and low energy loss. Sensors may tolerate some hysteresis but should minimize it to maximize sensitivity and resolution, suggesting careful design to balance inclusion shape and volume fraction.

Various parameters can influence the hysteretic behavior of a material. In our work, we solely focus on the effects of frequency and PZT-5A inclusion shape and volume fraction on the hysteretic response of the composite in the case of actuating and sensing applications. We consider that the composite is subjected to a harmonic strain and an electric field solicitation such as:

$$ \varepsilon_{11}(t) = \varepsilon_{11}^{\max} \exp(i\omega t) \qquad (22) $$

$$ E_3(t) = E_3^{\max} \exp(i\omega t) \qquad (23) $$

with ε₁₁max = 1 and E₃max = 200 [kV cm⁻¹]. The sensing and actuating responses under a harmonic strain and an electric field respectively are written as follows [4]:
<figure><img src="image_10.png" /><figcaption>Figure 4. Effective storage strain piezoelectric coefficient $\overline{d}_{31}$ of the viscoelectroelastic composite as a function of PZT-5A volume fraction (Spherical shape), with comparison to experimental results for PVDF/PZT composites fabricated via hot-pressing method (27 kHz).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 5. Strain variation under harmonic electric field of the considered viscoelectroelastic composite with spherical PZT-5A inclusions (volume fraction = 0.4) for different frequencies.</figcaption></figure>

$$D_3(t) = \mathrm{Re}(\overline{e_{31}(\omega)} \varepsilon_{11}^{\max} \exp(i\omega t)) \quad (24)$$

$$\varepsilon_{11}(t) = \mathrm{Re}(\overline{d_{31}(\omega)} \quad E_3^{\max} \exp(i\omega t)) \quad (25)$$

with $\overline{d_{ij}(\omega)} = \overline{e_{ik}(\omega)} \overline{C_{kj}}^{-1}(\omega)$.

It should be noted that the coefficient $d_{31}$ is a key parameter for studying actuators [40–45], as it represents the induced strain or displacement in the longitudinal direction (1-direction) when an electric field is applied in the transverse direction (3-direction). The figure 4 illustrates the variation of the parameter as a function of PZT-5A volume fraction and compares the results with experimental data reported by Suprapto et al [18].

### 3.3. Frequency effect

Figures 5 and 6 present the predicted variation of the strain and the normalized electric displacement under harmonic electric field (actuation) and strain field (sensing) respectively, at different frequencies (0.1, 1 and 50 s⁻¹) and for a volume fraction equal to 0.4 of PZT-5A spherical inclusions. The figures show that the frequency significantly influences the hysteretic response of PVDF/PZT-5A composites. The actuation behavior (figure 5) occurs with minimal changes in hysteresis loop shape at low frequencies but it vanishes at relatively high ones (i.e. 50 s⁻¹). In contrast, in the sensing case (figure 6) the hysteresis loop area expands, indicating greater energy dissipation due to enhanced viscoelastic and dielectric losses. The same predictions were reported in the works
<figure><img src="image_12.png" /><figcaption>**Figure 6.** Normalized electric displacement variation under harmonic strain field of the considered viscoelectroelastic composite with spherical PZT-5A inclusions (volume fraction = 0.4) for different frequencies.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 7.** Strain variation under harmonic electric field of the considered viscoelectroelastic composite for different volume fractions of $c_{PZT-5A}$ (spherical shapes) and at a frequency of $1 s^{-1}$.</figcaption></figure>

of Azrar et al [4]. Such behavior stems from the lossy nature of PZT-5A inclusions, which dominate under dynamic loading. These findings suggest that while high frequencies degrade sensing precision, they have negligible impact on actuation performance, making the composite more suitable for actuator applications under relatively high frequency conditions.

### 3.4. Volume fraction effect of PZT-5A

Figures 7 and 8 present the predicted actuating and sensing behavior of the considered viscoelectroelastic composite with different volume fraction values of PZT-5A inclusions with spherical shape. The predictions are computed for a frequency value of $1 s^{-1}$. The volume fraction of PZT-5A inclusions plays a critical role in determining the composite's hysteretic behavior. In the case of actuation (figure 7), the hysteretic loop seems to slightly change when the PZT-5A volume fraction increases. On the other hand, in the sensing case (figure 8), higher PZT-5A concentrations lead to larger hysteresis loops, particularly at $c_{PZT-5A}$ = 0.4, reflecting increased energy losses from stronger inclusion-matrix interactions. Such drawback matches with experimental results reported by Gowdhaman et al [39] and Suprapto et al [18]. The latter of whom reported that adding PZT fillers
<figure><img src="image_14.png" /><figcaption>**Figure 8.** Normalized electric displacement variation under harmonic strain field of the considered viscoelectroelastic composite for different volume fractions of PZT-5A (spherical shape) and at a frequency of 1 s⁻¹.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 9.** Strain variation under harmonic electric field of the considered viscoelectroelastic composite, at a frequency of 1 s⁻¹, with PZT-5A volume fraction equal to 0.4 and for both long fiber and spherical-like shapes.</figcaption></figure>

increases the composite's viscosity, resulting in more pronounced energy dissipation. In figure 7 we also observe a decrease in strain due to the opposite signs of the piezoelectric constant $d_{31}$ in PVDF and PZT-5A. In fact, the presence of PZT-5A reduces the effective $d_{31}$ until it approaches zero, after which it begins to increase again. A similar behavior was reported by Krishnaswamy et al [16] in their study of PVDF reinforced with piezoceramic barium titanate (BT), which shares a comparable morphology (Perovskite structure [20]) with PZT-5A. Additionally, Suprapto et al [18] experimentally confirmed that incorporating PZT particles into PVDF results in a reduction of the strain piezoelectric coefficient $d_{31}$ of the composite. This decrease is attributed to the complex interplay between the polymer matrix and the ceramic inclusions, where factors such as interfacial bonding, filler dispersion, and volume fraction influence the overall electromechanical coupling [14, 18].
<figure><img src="image_16.png" /><figcaption>**Figure 10.** Normalized electric displacement variation under harmonic strain field of the considered viscoelectroelastic composite, at a frequency of 1 s⁻¹, with PZT-5A volume fraction equal to 0.4 and for both long fiber and spherical-like shapes.</figcaption></figure>

Together, these studies emphasize the critical role of filler selection and composite microstructure in optimizing piezoelectric performance of the composite.

**3.5. Inclusion shape effect of PZT-5A**

Figures 9 and 10 present the predicted actuating and sensing behavior of the considered viscoelectroelastic composite, at a frequency of 1s⁻¹ and with a volume fraction of 0.4 PZT-5A. Both spherical and long fiber-like shapes are considered in figures 9 and 10. The shape of PZT-5A inclusions—whether spherical or fibrous—distinctly affects the composite's hysteretic properties. In sensing applications (figure 10), spherical-shaped inclusions produce more pronounced hysteresis loops compared to fiber-like inclusions. This behavior can be explained by the more isotropic and localized stress concentration around spherical particles, leading to stronger viscoelastic relaxation and dielectric loss mechanisms. Fiber-like inclusions, having a more elongated geometry, tend to distribute stress more evenly along their length, resulting in reduced energy dissipation and narrower hysteresis loops. On the other hand, it is clear that fiber-like inclusions contribute significantly to increasing the output strain due to their alignment with the third principal direction of the composite (figure 9). In contrast, spherical inclusions have a lesser effect because their smaller semi-axis length limits their reinforcement capability compared to the elongated fibers [27]. This suggests that for sensing applications, where minimizing energy loss is critical, spherical inclusions may be preferable. However, for actuation purposes, both inclusion shapes exhibit slightly different hysteresis behaviors, indicating that inclusion geometry has a minor impact on overall performance. Nonetheless, fiber-like inclusions have the advantage of producing greater strain, and thus larger deflection, making them more suitable for actuation.

**4. Conclusion**

In our study, we have presented a micromechanical modeling framework to predict the effective viscoelectroelastic behavior of PVDF/PZT-5A composite, considering both frequency and time effects. Using the Mori–Tanaka homogenization scheme combined with the Carson transformation, we were able to predict the dynamic electro-mechanical behavior of composites incorporating either spherical or fiber-like PZT-5A inclusions within a PVDF matrix.

Our results reveal that the PZT-5A inclusions significantly influence the hysteretic behavior of the composite, especially in sensing applications where an enlarged hysteresis loop indicates higher energy dissipation and reduced sensitivity. The shape and volume fraction of the inclusions play a critical role, with spherical inclusions leading to greater hysteresis loop compared to fiber-like shapes. Conversely, the actuation performance remains relatively unaffected by the inclusion morphology, suggesting that PVDF/PZT-5A composites are well-suited for actuator applications, even under high-frequency solicitations.
Furthermore, our analysis demonstrates that hysteretic effects tend to vanish at relatively high frequencies, which could be valuable for the design of frequency-optimized devices. Our findings contribute to a better understanding of the complex interplay between material microstructure and dynamic functional performance and provide useful guidelines for the design of next-generation smart materials dedicated to sensing and actuation applications, most particularly for aeronautical and aerospace applications.

## Acknowledgments

The authors gratefully acknowledge the funding from both parties: The French ANR ('Agence Nationale de Recherche' project EMBIA - 'Electrical Multiscale Bioinspired Interfaces in Aeronautics', https://anr.fr/Project-ANR-21-CE05-0006) and the European Union's Horizon Europe research and innovation program (HORIZON-EIC-2023-PATHFINDEROPEN-01 under grant agreement No 101 129 952).

## Data availability statement

The data that support the findings of this study are available upon reasonable request from the authors.

## Declaration of competing interest

None.

## Appendix

<figure><img src="image_17.png" /></figure>
## Author contributions

Salah Elbarnaty ID 0009-0008-8616-6381

Conceptualization (lead), Data curation (lead), Formal analysis (lead), Investigation (lead), Methodology (lead), Software (lead), Validation (lead), Visualization (lead), Writing – original draft (lead), Writing – review & editing (lead)

Joao Pedro M Correia ID 0000-0002-4881-8392

Conceptualization (supporting), Writing – review & editing (lead), Methodology (supporting), Supervision (lead)

Wiyao Azoti

Software (supporting), Writing – review & editing (supporting)

Saïd Ahzi

Supervision (supporting)

## References

[1] Aabid A et al 2021 A review of piezoelectric material-based structural control and health monitoring techniques for engineering structures: challenges and opportunities *MDPI AG* **101**

[2] Marouf A, Charbonnier D, Vos J B, El Akoury R, Hoarau Y and Braza M 2024 Morphing of high-lift wing-flap system with cambering and trailing-edge flapping at high reynolds number towards a full airplane application *J. Fluids Struct.* **127** 104111

[3] Abou-Khalil J et al 2024 Aerodynamic performance increase over an A320 morphing wing in transonic regime by numerical simulation at high reynolds number *Int. J. Numer. Methods Heat Fluid Flow* **241** 2709–47

[4] Azrar L, Bakkali A and Aljinaidi A A 2014 Frequency and time viscoelectroelastic effective properties modeling of heterogeneous and multi-coated piezoelectric composite materials *Compos. Struct.* **113** 281–97

[5] Roh Y, Varadan V V and Varadan V K 2002 Characterization of all the elastic, dielectric, and piezoelectric constants of uniaxially oriented poled PVDF films

[6] Zhang T, Li H G and Cai G P 2021 Time delay stability analysis for vibration suppression of a smart cantilever beam with hysteresis property *Journal of Low Frequency Noise Vibration and Active Control* **40** 898–915

[7] Laiarinandrasana L, Besson J, Lafarge M and Hochstetter G 2009 Temperature dependent mechanical behaviour of PVDF: experiments and numerical modelling *Int. J. Plast.* **25** 1301–24

[8] Vinogradov A and Holloway F 1999 Electro-mechanical properties of the piezoelectric polymer PVDF *Ferroelectrics* **226** 169–81

[9] Vinogradov A M and Holloway F 2000 Dynamic mechanical testing of the creep and relaxation properties of polyvinylidene fluoride

[10] Vinogradov A M, Schmidt V H, Tuthill G F and Bohannan G W 2004 Damping and electromechanical energy losses in the piezoelectric polymer PVDF *Mechanics of Materials* **36** 1007–16

[11] Kalimuldina G et al 2020 A review of piezoelectric pvdf film by electrospinning and its applications *Sensors* **20** 5214

[12] Talbourdet A 2018 Structures textiles piézoélectriques à base de PVDF pour la conversion d'énergie mécanique en énergie électrique *Textiles PhD thesis* University of Lille

[13] Defebvin J 2015 Étude des relations structure-propriétés de matériaux hybrides piézoélectriques à base PVDF *PhD thesis* University of Lille 1

[14] Jain A, Prashanth K J, Sharma A K, Jain A and P.n R 2015 Dielectric and piezoelectric properties of PVDF/PZT composites: A review *Polym. Eng. Sci.* **55** 1589–616

[15] Krishnaswamy J A, Buroni F C, García-Macías E, Melnik R, Rodriguez-Templeque L and Saez A 2020 Design of nano-modified PVDF matrices for lead-free piezocomposites: graphene vs carbon nanotube nano-additions *Mech. Mater.* **142** 103275

[16] Krishnaswamy J A, Buroni F C, García-Macías E, Melnik R, Rodriguez-Templeque L and Saez A 2020 Design of lead-free PVDF/CNT/BaTiO₃ piezocomposites for sensing and energy harvesting: the role of polycrystallinity, nanoadditives, and anisotropy *Smart Mater. Struct.* **29** 015021

[17] Krishnaswamy J A, Buroni F C, García-Sanchez F, Melnik R, Rodriguez-Templeque L and Saez A 2019 Improving the performance of lead-free piezoelectric composites by using polycrystalline inclusions and tuning the dielectric matrix environment *Smart Mater. Struct.* **28** 075032

[18] Suprapto et al 2024 Fabrication and characterization of PZT/PVDF composite films for force sensor applications *Polym. Int.* **73** 727–47

[19] Siponkoski T et al 2015 Electromechanical properties of PZT/P(VDF-TrFE) composite ink printed on a flexible organic substrate *Composites B* **80** 217–22

[20] Yun J S et al 2016 The fabrication and characterization of piezoelectric PZT/PVDF electrospun nanofiber composites *Nanomaterials and Nanotechnology* **6**

[21] Deeg W 1980 Dislocation, crack, and inclusion problems in piezoelectric solids (Stanford University)

[22] Dunn M L and Taya M 1993 Micromechanics predictions of the effective electro elastic moduli of piezoelectric composites *Thesis* University of California

[23] Dunn M L 1993 An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities *Proc. R. Soc. Lond.* A **443** 265–87

[24] Lezgy-Nazargah M and Eskandari-Naddaf H 2018 Effective coupled thermo-electro-mechanical properties of piezoelectric structural fiber composites: a micromechanical approach *J. Intell. Mater. Syst. Struct.* **29** 496–513

[25] Lezgy-Nazargah M 2015 A micromechanics model for effective coupled thermo-electro-elastic properties of macro fiber composites with interdigitated electrodes *Journal of Mechanics* **31** 183–99
[26] Kapuria S and Kumari P 2011 Mechanics of materials and structures three-dimensional isofield micromechanics model for effective electrothermoelastic properties of piezoelectric composites msp three-dimensional isofield micromechanics model for effective electrothermoelastic properties of piezoelectric composites santosh kapuria and poonam kumari

[27] Fakri N, Azrar L and El Bakkali L 2003 Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements *Int. J. Solids Struct.* **40** 361–84

[28] Elouafi J, Azrar L and Aljinaidi A A 2015 Closed-form expressions for the effective moduli of heterogeneous piezoelectric materials *Int. J. Solids Struct.* **52** 19–32

[29] Odegard G M 2004 Constitutive modeling of piezoelectric polymer composites *Acta Mater.* **52** 5315–30

[30] Mishra N and Das K 2020 A mori–tanaka based micromechanical model for predicting the effective electroelastic properties of orthotropic piezoelectric composites with spherical inclusions *SN Appl Sci* **2** 1206

[31] Li J Y and Dunn M L 2001 Viscoelectroelastic behavior of heterogeneous piezoelectric solids *J. Appl. Phys.* **89** 2893–903

[32] Jiang B and Batra R C Effective electroelastic properties of a piezocomposite with viscoelastic and dielectric relaxing matrix (10.1106/ CQCM-9WK7-4FVT-5NHP)

[33] Lévesque M, Gilchrist M D, Bouleau N, Derrien K and Baptiste D 2007 Numerical inversion of the Laplace-Carson transform applied to homogenization of randomly reinforced linear viscoelastic media *Comput. Mech.* **40** 771–89

[34] Li K, Gao X L and Roy A K 2006 Micromechanical modeling of viscoelastic properties of carbon nanotube-reinforced polymer composites *Mech. Adv. Mater. Struct.* **13** 317–28

[35] Yancey R N and Pindera M-J 1990 Micromechanical analysis of the creep response of unidirectional composites

[36] Douglas Eshelby J 1957 The determination of the elastic field of an ellipsoidal inclusion, and related problems The determination of the elastic field of an ellip soidal inclusion, and related p roblems *Proc. R. Soc. Lond.* A **241** 376–96

[37] Gavazzi A C and Lagoudas D C 1990 Comput. Mech. On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites

[38] Abate J and Valkó P P 2004 Multi-precision Laplace transform inversion *Int. J. Numer. Methods Eng.* **60** 979–93

[39] Gowdhaman P, Annamalai V and Thakur O P 2016 Piezo, ferro and dielectric properties of ceramic-polymer composites of 0-3 connectivity *Ferroelectrics* **493** 120–9

[40] Smits J G and Choi W-S 1991 The constituent equations of piezoelectric heterogeneous bimorphs

[41] Wang Q-M, Du X-H, Xu B and Cross L E 1999 Electromechanical coupling and output efficiency of piezoelectric bending actuators

[42] El-Sayed A M, Abo-Ismail A, El-Melegy M T, Hamzaid N A and Osman N A A 2013 Development of a micro-gripper using piezoelectric bimorphs *Sensors* **13** 5826–40

[43] Huang C, Lin Y Y and Tang T A 2004 Study on the tip-deflection of a piezoelectric bimorph cantilever in the static state *J. Micromech. Microeng.* **14** 530–4

[44] Khan M U et al 2019 Deflection of coupled elasticity–electrostatic bimorph PVDF material: theoretical, FEM and experimental verification *Microsyst. Technol.* **25** 3235–42

[45] Sharma T, Khurana A and Shukla K K 2018 Analysis of thin walled composite box beam with and without piezoelectric actuators using ABAQUS) View project