1. Introduction
Due to their electromechanical properties, piezoelectric materials are widely used as sensors and actuators [13]. Under low driving levels, their behavior remains linear and can be described by means of linear constitutive equations. A majority of the transducers is used on these levels. Increasing the levels of electric ﬁeld or stress leads to a depoling that results in the degradation of the dielectric and piezoelectric performances. This latter phenomenon is usually considered to be due to the irreversible motion of the domain walls [411]. The resulting nonlinear and hysteretic nature of piezoelectric materials induces a power limitation for heavy duty transducers or a lack of controllability for positioners. Consequently, a nonlinear modeling including a hysteresis appears to be a key issue in order to obtain a good understanding of transducer behavior and to determine the boundary conditions of use.
Several models have been proposed in the literature found understanding the hysteretic behavior of various materials.12–14. However, a majority of these phenomenological models is purely eclectic, and it is consequently difficult to interpret the results as a function of other parameters (stress and temperature) in order to obtain a clear physical understanding.
2. Stress/electrical scaling in ferroelectrics
2.1. Presentation of the scaling law
In order to determine a scaling law between the electric ﬁeld and the stress, one should start by following piezoelectric constitutive equations restricting them in one dimension.
These equations can be formulated with stress and electric field as independent variables, thus giving
where E, T, and S represent the electric ﬁeld, the mechanical stress, and the strain, respectively. The constants
The coefficients are deﬁned as
From a given P:
It can also be descried as following;
The interrelation between the strain (S) and the spontaneous polarization (P) is estimated using a global electrostrictive relationship, i.e., the strain is an even function of the polarization of the polarization,
Here, n is the polynomial order and α_{x} is the electrostrictive coefﬁcient of order x.
The derivatives of the strain are
Introducing the attest relationship in the previous calculations leads to:
Thus,
Thus, we consider that the term h(P)T plays the same role to the electric field
As illustrated in the figure 1, the scaling law (1) can be used to derive the stress polarization P behavior from the
2.2. Determination of the parameters of the scaling law
Considering physical symmetries in the materials, a similar polarization behavior (P) can be observed during variation of an electric ﬁeld (E) or the mechanical stress (T). Both of these external disturbances are caused by the depoling of the sample. An explication concerning how to apply the scaling law is here given based on the equations developed in Sec. II.Starting from Eq. (7), the entire derivation of strain by polarization can be calculated based on experimental data,
Material  Structure  Function h(P(E,T=0)) 
PMN25PT  Rhomohedral (R) 

PMN40PT  Tetragonal (T) 

P188  MPB (R+T) 

2.3. Veriﬁcation of the scaling law
The viability of the proposed scaling law was explored using two distinct experiments on soft PZT. Starting from the experimental depoling under stress P=f(T), the depoling was plotted as a function of h[P(E=0,T)]T] giving
In a general manner, the experimental and reconstructed cycles demonstrated reasonable agreements, with regard to both increasing and decreasing paths, for the soft PZT.
This good agreement for both P(E) and P(T) cycles thus conﬁrmed the viability of the scaling law for soft PZT. Only one parameter ruled the “scale” of the strain and the scale of the stress effect. This ease of conversion between P(E) and P(T) cycles by such a simple law gives numerous opportunities regarding the use of piezoelectric materials. It is possible to predict the depoling behavior over the entire stress cycle (compressive or tensile)/ﬁeld plane. These results are important to the design and performance of actuators and sonar transducers.The proposed scaling law can be used for several electrical models in order to understand the hysteretic behaviour of piezoelectric materials [1214]. This scaling law is interesting in order to introduce the stress as an equivalent electric ﬁeld; the behavior of ferroelectric materials under a combined electric ﬁeld (E) and stress (T) can thus be determined. It is also interesting to note that for practical use, the maximum stress can be determined from this scaling law. This result is presented in Fig. 3. The small variations of polarization were observed for applied electric ﬁeld lower than E_{M} (here, 0.7 kV/mm). Therefore polarizations undergo a rapid change in polarization. Based on this E_{M} value, the equivalent stress (T_{M}) can be directly obtained (40 MPa). As a consequence, the maximum stress for application can be obtained without stress experiment.
3. Temperature/electric field scaling in ferroelectrics
3.1. Presentation of the scaling law
In order to determine a scaling law between the electric ﬁeld and the temperature, one should start by following the piezoelectric constrictive equations, restricting them in one dimension.
These equations can be formulated with the temperature and the electric ﬁeld as independent variables; thus, giving
The coefﬁcients are deﬁned as:
For a given P:
which can also expressed as:
Here, θ_{0} and E_{0} correspond to room temperature (298 K) and the initial electric field (0 kV/mm), respectively.
From a physical point of view, the entropy cannot depend on the polarization orientation in the ferroelectrics material. It means that the entropy must be an even function of polarization. Limiting the entropy expansion to the second order and ensuring
Here,
The derivatives of the strain can be written as:
Introducing Eq. (19) in the previous calculations leads to:
The function
Thus, Eq. 21 can be written as
According to Fig. 4, for a given value of polarization (P), we can write the following equality:
Thus,
With; and The term
As illustrated in Fig. 4, the scaling law can be used to derive the behavior of the polarization as a function of the temperature P(θ) from P(E) cycle, or reciprocally to drive the polarization behavior versus the electrical ﬁeld, once the P(E) cycle is known.
3.2. Veriﬁcation of the scaling law
The effects of various electric fields and temperatures on the polarization profile are illustrated in Figure 5, where Figure 5(a) represents the polarization variation as a function of the temperature for an electric field E=0 V/mm. It was shown by Hajjaji et al [15] that the depolarization as a function of the temperature was mainly due to the decrease in the dipole moment and the fact that the variation in this dipole moment was reversible. In the vicinity of the ferroelectric to paraelectric transition, the temperature depolarization of the ceramics
was the result of a 0–90° domain switching, whereas a 0–180° domain switching did not occur with temperature. The effects were thus quite obvious. At a fixed
Despite the difference between the mechanisms of depolarization as a function of electric ﬁeld and temperature, we have try determining a law that links the two (electric ﬁeld E and temperature θ) and to identify one from another.
In order to obtain a suitable scaling relation for the ceramic, one can first follow the suggested scaling law given in Eq. (23). This enables a direct determination of the proportionality coefficients α and β from the experimental data. The coefficient α can be determined from the following equation (24)
In addition, the viability of the proposed scaling law was explored by way of two distinct experiments on soft PZT. Starting from the experimental depoling under temperature
The second comparison was helpful in determining the appropriateness of the scaling law for fields close to the coercive field
It is interesting to note that for purely electrical measurements, the presented law rendered it possible to determine the maximum temperature for practical use (cf. Figure 7). Small variations in polarization were observed for an applied electric field lower than E_{M} (here, 150 V/mm), leading to the conclusion that the polarizations underwent a rapid change. Based on the obtained E_{M} value, one can determine the equivalent temperature (θ_{M}) corresponding to the maximum temperature used.
The relationship
4. Temperature/stress scaling in ferroelectrics
4.1. Presentation of the scaling law
In order to determine the general laws between the mechanical stress, electrical field, and the temperature, we are based on previous studies of Guyomar et al [7]. These studies were proposed a scaling effect between electric field and a term composed by the polarization multiplied by the stress:
Where α is the proportionality constant between ΔE and ΔT. Both ΔE and ΔT represent the electric field and the mechanical stress variation. P(E,T_{0}) is the polarization at zero stress(T_{0}=0MPa).
In the other study Hajjaji et al proposed a scaling law between the electrical field and the temperature [16]. This law is expressed by the following expression.
Here,
In most cases the coefficient
With;
According to equations (25) and (27) we find the following expression:
With;
Thus
As illustrated in figure 8, we determine P(T) and P(θ) from P(E) (steps 1 and 3), P(E) and P(θ) from P(T) (steps 1 and 2), and at the end we can determine P(E) and P(T) from P(θ) (steps 2 and 3),.
4.2 Veriﬁcation of the scaling law
Figure 9 shows the relation between ΔT and
The expression (25) allows expressing the mechanical stress as an equivalent electric field and the electric field as an equivalent stress. Thus, a good agreement between electrical field and mechanical stress proved that the proposed scaling law allows predicting the depoling behavior under stress using only purely electrical measurements. Reciprocally, the predictions of the depoling behaviour under an electrical field were permitted using only purely mechanical measurements. It was found that such an approach permitted the prediction of the maximal stress application from purely electrical measurements (i.e., measurements of S(E) and P(E)). The maximal stress for application is the stress that can be applied to materials without they lose their piezoelectric properties.
In the other study we proposed a scaling law between the electrical field and the temperature [16]. This law is expressed by the expression (25) (
Figure 10 shows the relation between
According to this law, it is possible to determine the behavior of the polarization in function of temperature from the electrical measurements. Reciprocally, it is possible to determine the behavior of the polarization in function of the electric field from thermal measurements. It is interesting to note that for purely electrical measurements, the presented law rendered it possible to determine the maximum temperature for practical use. Small variations in polarization were observed for an applied electric field lower than E_{M} (here, 150 V/mm), leading to the conclusion that the polarizations underwent a rapid change. Based on the obtained E_{M} value, one can determine the equivalent temperature (θ_{M}) corresponding to the maximum temperature used.
Considering physical symmetries, similar behaviors can be observed under stress or temperature. Indeed both external disturbances may result in depling of the sample. We consider here a scaling effect that is described with (equation 30):
Where T is the mechanical stress,
In order to confirm these results, we plotted the mechanical stress as a function to the temperature variation. Figure 12 shows the relation between Δθ and
In the literature, a majority of these phenomenological models are purely electric, mechanic, or thermal [1922]. Consequently, it is difficult to interpret the results as a function of the combined to two or three excitations (mechanical stress and temperature for example).
The proposed scaling law can be used for several models have been proposed in the literature for comprehending the hysteretic behavior of various materials, which renders it interesting for introducing the temperature as an equivalent to the mechanical stress, or reciprocally to introducing the mechanical stress as an equivalent to the temperature.
The behavior of ferroelectric materials under a combined mechanical stress (T) and temperature (θ) can thus be determined, which will help in the identification and understanding of the effect of the simultaneous action of temperature and mechanical stress on ceramics.
According to this law, it is possible to determine the behavior of the polarization in function of temperature from the mechanical measurements. Reciprocally, it is possible to determine the behavior of the polarization in function of the mechanical stress from thermal measurements. It is interesting to note that for purely mechanical measurements, the presented law rendered it possible to determine the maximum temperature for practical use, and reciprocally it is possible to determine the maximum stress for practical use from purely thermal measurements.
5. Predictions of material behavior
Due to their electromechanical properties, piezoelectric materials are widely employed as sensors and actuators [1617]. Most of these piezoelectric materials are utilized under different conditions (stress, electrical ﬁeld, and temperature).It would thus be interesting to predict their behaviors under a variety of excitations without having to perform too much experimental work, i.e., just carrying out a single experiment and providing the other experimental values. For example, from a simple measurement of the polarization as a function of the electric ﬁeld, one could predict the behavior of the polarization as a function of temperature negative and positive(step 2) and stress (compressive and tensile stress) (step 1). In conclusion, we could determine P(T) and P(θ) from P(E) (steps 1 and 3),P(E) and P(θ) from P(T) (steps 1 and 2), and ﬁnally P(E) and P(T) from P(θ) (steps 2 and 3),.
6. Relationship between the coefﬁcients d_{33} and ε_{33}
The proposed scaling law can also be applied to the minor cycles. This fact provides a great advantage for the problem of the relation between
It is quite difﬁcult to experimentally obtain a real d33 corresponding to an exact ɛ_{33}. During the experiment, the electrical ﬁeld (E) was stopped at a certain level to obtain a value of the polarization (P), and the permittivity (ɛ_{33}) could thus be deﬁned. When E was stopped, d_{33} was calculated based on the obtained_{33} and did consequently not correspond to thereal d_{33}. As illustrated in Fig. 14, the measuring point for P was not on the main cycle but slightly beside, on the minor cycle. This result was due to the difference of E_{0}+ΔE from ɛ, not corresponding to that of T_{0}+ΔT from d_{33}. Resultantly, the calculation of d_{33} could not be based on an exact value. By using the proposed simple scaling law, on the other hand, it was possible to obtain an exact value for d_{33},
thus,
Figure 15 depicts the prediction of the piezoelectric constant (d_{33}) under a compressive stress. In this case, d_{33} was calculated from the function h[P(E,T=0)] and compared with experimental values. It could be observed that the experimental and calculated piezoelectric constants displayed a similar variation with the compressive stress. Such a good agreement between simulation and experiment proved that the proposed law scaling rendered it possible to predict the piezoelectric constant (d_{33}) under stress using only purely electrical measurements. Reciprocally, predictions of the dielectric constant (ɛ_{33}) under an electrical ﬁeld were permitted using only purely mechanical measurements.
7. Conclusion
The present chapter proposes a three simple scaling laws taking into account the electrical field, the stress, temperature, and the polarization of ferroelectric materials in the form of
The proposed scaling laws can be used for several models have been proposed in the literature for comprehending the hysteretic behavior of various materials, which renders it interesting to interpret the results as a function of the combined to two or three excitations (mechanical stress and temperature for example).The ease of conversion between P(E), P(θ) and P(T) cycles by such a simple laws gives numerous opportunities regarding the use of piezoelectric materials. It was possible to predict the depoling behavior over the entire stress cycle (compressive or tensile), or to predict the depoling behavior over the entire temperature cycle (negative or positive) from to the hysteresis cycle. Thus, one should note that applying a symmetric electrical field cycle leads to a dissymmetric cycle under stress and temperature. Consequently, the polarization behaves differently as a function of compressive as opposed to tensile stresses. Moreover, the polarization behaves also differently as a function of positive and negative temperature.
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