However, polarization mechanisms in ferroelectrics, at the origin of piezoelectricity, show time-dependent effects that are not related by the original model described above. Particular relaxation-like phenomena may yield effects such as stabilization and accommodation, creep, or change in the hysteresis loop with the frequency for instance. This study aims at enriching the previously exposed approach by including dynamical effects. In particular, two phenomena will be under focus. First, transient regime, resulting from depolarization for instance and yielding bias in strain even at rest (zero or constant applied voltage) or with time constants not directly related to the driving voltage frequency, will be under consideration. Such an effect can be seen as a change in the initial condition of the actuator. Therefore, during the first cycles upon the voltage application, a drift in the strain response may be observed, which is not related by the initial model exposed in

Section 2.1. This effect can physically be partly explained by irreversible polarization and domain rearrangement. Additionally to this transient regime, frequency-dependent behavior can be observed [

23], with the hysteresis shape and thus strain response changing as the voltage changes in frequency (but not in magnitude). The strategy adopted in the present study consists in treating such behaviors as particular relaxation phenomena. Furthermore, as these effects may be related to the electrical polarization state of the material, it will be assumed that these relaxations are applied to the driving voltage only (

Figure 2).

These transient and harmonic effects are besides related to the whole history of the actuator. Therefore, these relaxation-related effects cannot be taken into account as a classical, Debye-like relaxation model. Instead, non-integer (i.e., fractional) derivatives are good tools for relating these particular features. Indeed, such operators take into account the whole history, with an unlimited memory, of signals and are therefore well adapted to ferroelectrics. On a physical aspect, they are between energy storage terms (potential or kinetic-i.e., zero and second derivation orders of typical quantities such as strain or voltage for instance) and pure losses (first derivation order of typical quantities), therefore allowing taking into account complex phenomena that occur at the material or system level. An example of such an approach is the well-known Cole–Cole relaxation model in dielectrics [

32,

33], that is, however, classically only considered in the frequency domain (and without hysteresis as well). It could also be noted that, compared to the present approach, the Cole–Cole representation assesses the material behavior at low-field excitation (dielectric spectroscopy for instance), and cannot represent the hysteretic behavior at a system-level as the presently proposed model.

Mathematically, fractional derivatives arise from pseudo-differential operators, and correspond to the fact that, to obtain a classical first-order derivation, a

n-fractional derivative operator

${D}^{n}$ (with

n the possibly non-integer derivation order) has to be applied

n times. For example, for

$n=1/3$, the fractional derivation operator should be applied three times to obtain the first-order derivative of a function

f:

with ∘ the composition operator. Consistently with Laplace and Fourier transforms, these fractional operators in the time-domain (as physical systems are ruled by—potentially non-integer—partial differential equations in time-domain) yield a multiplication by

${s}^{n}$ and

${\left(j\omega \right)}^{n}$ (respectively, in Laplace and Fourier spaces), with

s the Laplace variable,

$\omega $ the angular frequency and

j the unit imaginary number. As an illustration,

Figure 3 and

Figure 4 show the frequency response and transfer characteristics of the fractional order unit relaxor

H defined in the Laplace domain as:

where

s is recalled as the Laplace variable and

n is the fractional order (

$n\in \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}$).

$in\left(s\right)$ and

$out\left(s\right)$, respectively, denote the input and output signals of the system represented by the transfer function

$H\left(s\right)$. These illustrative results have been obtained using the FOMCON (Fractional-order Modeling and Control - version 1.22.0.0, Tallinn University of Technology, Tallinn, Estonia) toolbox for MATLAB

^{®} (version R2020a, Mathworks, MA, USA) developed by Tepljakov et al. [

34,

35]. Therefore, it can be seen from

Figure 3 that the frequency response can be finely tuned both in terms of magnitude response and phase, allowing changing the attenuation slope and bandwidth in a precise way. Phase tuning through the fractional order

n also permits shaping differently the elliptical input-output characteristics and its frequency behavior as shown in

Figure 4. Moreover, a remarkable property when considering unipolar input (

Figure 4b) is the ability of relating an almost constant shift after a transient stage, while the entire derivative order

$n=1$ yields offsets that significantly change as the frequency increases. Finally, the possibility of finely controlling the system dynamics can also be observed from the step response depicted in

Figure 5, showing that low fractional order expands the settling time and thus well relates long-term phenomena.

Therefore, in order to take into account transient and harmonic effects into the static model and according to

Figure 2, the driving voltage

V is changed into an equivalent static model input voltage

${V}^{*}$ that is obtained after two filtering processes: one representing the transient effect whose transfer function is denoted

${G}_{t}$ and the other harmonic phenomena with an associated transfer function

${G}_{d}$. This, therefore, yields in the Laplace domain:

with

${G}_{t}\left(s\right)$ and

${G}_{d}\left(s\right)$, respectively, defined as

with

${\tau}_{i}$ and

${\lambda}_{i}$ (

$i=t,d$) the time constant and fractional order associated to each effect, respectively. Obviously, as the transient effect is expected to have a response spanning a wider time period compared to harmonic losses and conversion, one can expect that

${\tau}_{d}^{{\lambda}_{d}}.{s}^{{\lambda}_{d}}<{\tau}_{t}^{{\lambda}_{t}}.{s}^{{\lambda}_{t}}$ for relatively small value of

s and with

${\lambda}_{t}<{\lambda}_{d}$.

Alternatively, instead of computing the voltage response in Laplace domain to go back in time domain (with time variable

t) for applying the hysteresis model, it is possible to obtain the equivalent voltage by convolution of the impulse responses

${g}_{t}\left(t\right)$ and

${g}_{d}\left(t\right)$, respectively associated with

${G}_{t}\left(s\right)$ and

${G}_{d}\left(s\right)$:

where ∗ is the convolution operator, and

${g}_{t}\left(t\right)$ and

${g}_{d}\left(t\right)$ are defined as [

36]

with

${E}_{\alpha ,\beta}\left(z\right)$ the Mittag–Leffler function: