B Ducharne

B Newell

G Sebald

A unique fractional derivative operator to simulate all dynamic piezo ceramic dielectric manifestations: From aging to frequency dependent hysteresis

Keywords: Aging, fractional derivative, frequency dependence, hysteresis, ferroelectricity, piezoelectric materials

Ferroelectric materials are utilized in many applications ranging from capacitors to data storage.

The hysteresis frequency dependence of ferroelectric materials has been well studied. However, the long term dynamic behaviour including aging has not been as well documented due to the long time frame required to gather experimental data, but is critical for understanding the lifespan of these materials in application. Previous work has shown that the hysteresis frequency dependence of the dielectric properties can be accurately modelled in the time domain using fractional derivative operators applied on a large frequency bandwidth. Currently, the lowest frequencies tested have been restrained to the hysteresis cycle quasi-static threshold. Below this threshold, the hysteresis shape remains unchanged. This research expands the current knowledge by validating the use of fractional derivative operators in long term aging models. The model data is experimentally validated using aged piezoelectric samples with over up to 10 7 seconds. These results confirm that the low and high dynamic dielectric material behaviours are linked and can be consequently modelled using fractional derivative operators.

I. INTRODUCTION

Ferroelectricity is a material characteristic defined by a material ability to exhibit spontaneous electric polarization which can be reversed by the application of an external electric field [START_REF] Burfoot | Polar Dielectric and their Applications[END_REF].

Piezoelectricity is the ability of a material to exhibit polarization variations when subjected to a mechanical stress. Piezoelectric materials also show the converse effect, undergoing mechanical deformation under the application of an electric field [START_REF] Jaffe | Piezoelectric Ceramics[END_REF]. These properties lead to numerous applications for piezo ceramics including sensors, actuators, and energy harvesters. However, the ferroelectric and the piezoelectric properties of a piezo ceramic are often highly non-linear, which can cause challenges in calibration and real world applications. As a result, in depth characterization of piezo-ceramics is needed. This includes characterization of piezoelectric polarization, fatigue, aging, mechanical properties, and electrical properties in accordance with frequency dependence. These properties have been investigated and characterized for use in practical applications [START_REF] Damjanovic | Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics[END_REF]. In application, most piezoelectric devices operate under small signal conditions when compared to the initial load (high voltage) applied to pole the samples. The general operating signal is a low voltage electrical input or minor mechanical deformation. As a result of minimal loading conditions, the material's behavior stays within the linear performance range. However, when piezo materials are used as actuators, the excitation levels required are higher and can be comparable to the initial poling excitation. Precision positioning systems and mechanical vibration energy harvesters are examples of such applications [START_REF] Schitter | Identification and Open-Loop Tracking Control of a Piezoelectric Tube Scanner for High-Speed Scanning Probe Microscopy[END_REF] [START_REF] Zhang | Experimental sea wave energy extractor based on piezoelectric Ericsson cycles[END_REF]. These excitations cause reconfiguration of the domain which can lead to domain wall displacement and variation in the polarization field (P) along with non-linear responses.

By plotting these polarization field variations versus the electric field (E) or versus the mechanical stress (T), the dielectric hysteresis cycle P(E,T) can be observed [START_REF] Ducharne | low frequency modelling of hysteresis behaviour and dielectric permittivity in ferroelectric ceramics under electric field[END_REF]. Piezo-electrics have many advantages including their design flexibility, ability to withstand large strains, low cost, low maintenance, their ability to convert vibrations into electrical energy, pollution free energy source, and they are unaffected by external electromagnetic fields. However, there are limitations as well including the requirement of an input signal such as mechanical deformation or electrical supply. These materials are also generally temperature sensitive, and piezoelectrics are prone to cracking under high stress conditions. With the given, pros and cons in many applications, piezo-electrics are a very good solution. In all product applications, the long term performance of the materials must be considered and is generally defined as aging [START_REF] Carl | Electrical after effect in Pb(Ti, Zr)O3 ceramics[END_REF] and fatigue [START_REF] Lupascu | Fatigue in bulk zirconate titanate actuator materials[END_REF]. Aging is defined as the gradual changes in material properties with time under external equilibrium conditions. In piezo materials, aging is the usually detected by logarithmic time variations of key piezo ceramic parameters [START_REF] Sebald | Electromechanical characterization of 0.55Pb([END_REF] [START_REF] Hajjaji | Synthesis and characterization of 0.65Pb(Mg1/3Nb2/3)O3-0.35PbTiO3 fibers with Pt core[END_REF]. Fatigue is characterized by gradual changes in the material properties under loading conditions.

Aging is an inherently time dependent material based characteristic that demonstrates the material's dynamic behavior under extremely weak frequency levels.

Models based on the space charges migration have been proposed to model ferroelectric aging [START_REF] Carl | Electrical after-effects in Pb (Ti, Zr) O3 ceramics[END_REF]- [START_REF] Takahashi | Space charge effect in lead zirconate titanate ceramics caused by the addition of impurities[END_REF], but these models are currently unable to provide short term behavior information including properties such as dielectric hysteresis dependence (P(E) f ϵ [0.01-10 Hz]). In literature [START_REF] Guyomar | Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives[END_REF]- [START_REF] Guyomar | High nonlinearities in Langevin transducer: A comprehensive model[END_REF], it has been demonstrated that by using fractional derivative operators the frequency dependence of a piezo ceramic dielectric hysteresis can be correctly simulated on the full frequency bandwidth of a variety of piezo material, even including bulk ceramics and thin films.

In these sensitive setups, 3 decades can be seen for a bulk-type ceramic and 6 decades for a thin film. Creep phenomena (the piezo ceramic drift of displacement under a constant applied electric field) has also been simulated with success [START_REF] Guyomar | Time fractional derivatives for voltage creep in ferroelectric materials: theory and experiment[END_REF].

Under excitation frequencies larger than 10 mHz, experimental results are easier to test for and model because experimental tests can be performed in a short period of time. However, in all aging experiments environmental conditions including temperature, pressure, and humidity must be carefully maintained and monitored in order to get reliable results.

Lead based Pb[Zr1-xTix]O3 (PZT) ceramics exhibit excellent temperature stability and are highly adaptable for numerous applications because they can be easily doped with additives to adjust their electrical and material properties to meet desired specifications. Because of the wide usage of these materials and their adaptability, this study is limited to these classic PZT ceramic material. It is assumed that the physical mechanisms of this material is similar to other PZTs and therefore the results of these experiments can be adapted for other materials.

II. BACKGROUND

In the first part of this article, we will briefly discuss the classic dynamic observations and characterizations of a piezo ceramic dielectric behavior, with emphasis on the high and weak excitation time slope dynamic behaviors. The current simulation methods used to produce the dynamic results will be discussed along with frequency dependent hysteresis models, scaling laws, and current aging simulations. The fractional derivative operator and its use for simulating these dynamic behaviors will be explained, including distinguishing between high and weak excitation time slope dynamic behaviors. The last part of this article will be devoted to the experimental methods for aging and a comparison/validation of the predicted model using experimental results.

III. CLASSIC EXPERIMENTS FOR DYNAMIC DIELECTRIC PROPERTIES -UNDER HIGH SLOPE EXCITATION FIELD STIMULI (dE/dt, dT/dt >> 1)

A. High amplitude levels: Frequency dependence of the dielectric hysteresis Below the material Curie temperature, at low frequencies (f << 1 Hz) and for high excitation amplitudes, loop-like hysteresis curves can be observed by plotting the spontaneous polarization P, versus electrical field E, or versus an external mechanical stress T. As the excitation frequency increases and exceeds the quasi-static threshold, the piezo ceramic hysteresis cycle starts to show frequency dependence. This dependence can be observed by plotting the frequency variation of the hysteresis area. On a very large frequency bandwidth, this loop area remains constant up to the quasi-static limit, then increases up to a maximum value that depends on the excitation amplitude. For higher frequency levels, the loop area quickly decreases to a value close to zero.

B. Weak amplitude levels: dielectric spectroscopy

Dielectric spectroscopy consists of measuring the dielectric properties of a medium as a function of frequency and under weak amplitude electrical excitation. Dielectric spectroscopy is based on the interaction of the sample electric dipole moment and the external electrical field E excitation (the dielectric permittivity ε). A ferroelectric ceramic frequency dependence is also called dielectric relaxation and it provides interesting information about the piezo ceramic nature of a material and about the physical relations describing the polarization behaviour. This relaxation is usually described in terms of permittivity as a function of frequency.

IV. CLASSIC EXPERIMENTS FOR DYNAMIC DIELECTRIC PROPERTIES -UNDER WEAK SLOPE EXCITATION FIELD STIMULI (dE/dt, dT/dt << 1)

A. Aging The main consequence of aging is a reduction of the remanent polarization for poled samples and a decrease in the global polarization level during bipolar electric cycling. Effects are amplified for donor-doped materials with electric fields larger than the coercive field [START_REF] Lupascu | Fatigue in bulk lead zirconate titanate actuator materials[END_REF]. The microscopic origin of aging in a piezo ceramic has been well studied in the scientific literature. The usual physical interpretation of these effects is that mobile charged defects stabilize a given domain configuration thereby generating an internal bias field that counteracts the polarization reversal when an electric field is applied. Three major mechanisms are predominant in the aging process: _ The volume effect coming from defect dipoles ( acceptor ion plus oxygen vacancy) having an electric dipole moment as well as an elastic dipole moment due to the deformation of the unit cell [START_REF] Carl | Electrical after-effects in Pb(Ti, Zr)O3 ceramics[END_REF] [START_REF] Arlt | Internal bias in ferroelectric ceramics: Origin and time dependence[END_REF].

_ The domain effect which can be defined as a migration of charged defects in the direction of the domain walls creating extra pinning defects [START_REF] Zhang | In situ observation of reversible domain switching in aged Mn-doped BaTiO3 single crystals[END_REF].

_ The interface effect (also called grain boundary effect) which can be explained as the diffusion of ions, vacancies or electrons to grain boundaries creating space-charge layers [START_REF] Eichel | Defect structure of oxide ferroelectrics -valence state, site of incorporation mechanisms of charge compensation and internal bias fields[END_REF].

V. CLASSIC MODELS AND SIMULATION METHODS OF THE DYNAMIC DIELECTRIC PROPERTIES -UNDER HIGH SLOPE EXCITATION FIELD STIMULI (dE/dt, dT/dt >> 1)

A. High amplitude levels: Frequency dependence dielectric hysteresis models

The simulation of high amplitude excitation dynamic dielectric properties (hysteresis frequency dependence) over a large range of excitation frequencies has been widely studied and documented in the scientific literature [START_REF] Hegewald | Efficient modeling of ferroelectric behavior for the analysis of piezo ceramic actuators[END_REF]. Several methods exist. Two or three-dimensional Ising-type models (multi-scale approach) are used [START_REF] Mccoy | The two-dimensional Ising Model[END_REF] [START_REF] Kleeman | Uniaxial relaxor ferroelectrics: the ferroic random-field Ising model materialized at last[END_REF], but are time intensive and variation is often present between simulated and experimental results. Another method is extension of the quasi-static contribution (Preisach model, Jiles-Atherton model, etc) to dynamic behaviour using the first or a second order additional dynamic contribution provides good results but on a very narrow excitation frequency bandwidth. Neural networks [START_REF] Laosiritaworn | Artificial neural network modeling of ferroelectric hysteresis: an application to soft lead zirconate titanate ceramics[END_REF] or phase shifting operators [START_REF] Cruz-Hernandez | Phase control approach to hysteresis reduction[END_REF] exhibit various limitations such as amplitude, frequency, etc and can only be used directly in the field of an associated application.

B. Scaling laws

There are various methods to observe and to model the dynamic dielectric behaviour of a piezo ceramic. One such method is called "scaling laws", and it consists of focusing on the hysteresis area variations only [START_REF] Yimnirun | Scaling behavior of dynamic hysteresis in soft lead zirconate titanate bulk ceramics[END_REF][START_REF] Guyomar | The use of fractional derivation in modeling ferroelectric dynamic hysteresis behavior over large frequency bandwidth[END_REF]. The evolution of the hysteresis area <A> is studied as a function of the excitation amplitude E0 and frequency f. This method is used to support use in industrial applications [START_REF] Bhattacharyya | Scaling behavior of dynamic hysteresis of a deformed helix ferroelectric liquid crystal[END_REF]. <A> is the energy dissipation in one cycle of domain reversal, and it is related to the non-equilibrium first order phase transitions in ferroelectrics. The hysteresis area provides of interesting information about the dynamics of ferroelectric domains including characteristic times of domain nucleation and domain boundary motion during reversal processes. <A> ꝏ f μ1 E0 μ2 (where μ1 and μ2 are real parameters depending on the geometry and the nature of the system) is the classic "scaling law" it has been developed as a first attempt to tackle investigations on ultrahigh-frequency ceramics. Frequency exponent of 0.3 to 0.6 were found for classic piezoceramics. Scaling laws are an interesting way to understand losses in ferroelectrics but they are not useful simulation tools for modelling ferroelectric aging behaviour as time domain variations cannot be considered.

C. Weak amplitude levels: dielectric spectroscopy simulation

Dielectric spectroscopy is classically described in terms of permittivity as a function of frequency:

  * ( ) '( ) ''( ) 1 ( ) i i                   (1)
Here ε∞ is the sample permittivity under high-frequency excitation, Δε = εs -ε∞ where εs is the quasi-static, low-frequency permittivity, and τ is the characteristic relaxation time. ε'(ω) and ε''(ω) are the real and imaginary parts of the permittivity, respectively. For ideal systems α = β = 1, this leads to the Debye complex formula [START_REF] Debye | Zur Theorie der anomalen dispersion im Gebiete der langwelligen Elektrischen Strahlung[END_REF].

In alternate cases such as BaTiO3, a classic piezo ceramic β = 1 but α ≠ 1. Materials of this type use the Cole-Cole model [START_REF] Cole | Dispersion and absorption in dielectrics -I alternating current characteristics[END_REF][START_REF] Cole | Dispersion and absorption in dielectrics -II direct current characteristics[END_REF]. α is linked to the distribution of relaxation time (0 < α < 1). In the case of ferroelectric polymers, optimal results are obtained with α ≠ 1 and β ≠ 1. In these cases, the unsymmetrical characteristics and broadness of the dielectric dispersion curve must be considered and can be done using an extension called the Havriliak-Negami [START_REF] Havriliak | complex plane analysis of α-disperisions in some polymer systems[END_REF] relaxation. constant, and temperature dependency [START_REF] Lohkämper | Internal bias in acceptor doped BaTiO3 ceramics: Numerical evaluation of increase and decrease[END_REF]. Once this bias field is correctly defined and set for a given sample, the classic P(Eeff) simulation scheme can be used for simulation of polarization variations, where Eeff stands for the effective electric field which is the sum of an external electric field and the time dependent internal bias field.

VI. CLASSIC MODELS AND SIMULATION METHODS

( ) ( ) ( ) eff ext bias E t E t E t   (2)

VII. FRACTIONAL DERIVATIVE CONTRIBUTION FOR THE SIMULATION OF THE DYNAMIC PROPERTIES -UNDER HIGH SLOPE EXCITATION FIELD STIMULI (dE/dt, dT/dt >> 1)

A. High amplitude excitation

The fractional derivative operators can be solved in both the time and frequency domains.

Industrial applications of piezo ceramics under high levels of external excitation like micropositioning or mechanical vibration energy harvesters usually require real time control and consequently time variations of the ceramic property must be known [START_REF] Zhang | Experimental sea wave energy extractor based on piezoelectric Ericsson cycles[END_REF]. For these reasons, the numerical scheme we have been developed is a time domain based solution. It relies on the extension of a hysteresis quasi-static contribution to the dynamic behaviour. As explained previously, the classic dynamic contribution is made from the material's density product to the polarization's first order time derivation. Unfortunately, after comparison with experimental results it is easy to see that a first order consideration can only be used with moderate accuracy on a very narrow frequency bandwidth. To overcome this issue, previous research [START_REF] Guyomar | Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives[END_REF]- [START_REF] Guyomar | High nonlinearities in Langevin transducer: A comprehensive model[END_REF] has used fractional derivatives to balance the low frequency and the high frequency components in a different way than the limited straight time derivative. To introduce a fractional time derivative d α B/dt α in the system equation, the Grünwald-Letnikov or the Riemman-Liouville definitions can be applied [START_REF] Ducharne | Dynamics of magnetic field penetration into soft ferromagnets[END_REF]. Both are particular cases of a general fractional order operator named in fractional calculus, the first one represents the α order derivative, while the other one represents the α fold integral. Consequently the class of functions described by the Riemman-Liouville definition is broader (functions must be integratable) than the Grünwald-Letnikov ones.

However, for a Grünwald-Letnikov type function, both definitions are similar. In this study, the Riemman-Liouville form with α ϵ [0,1] has been chosen where Γ(.) represents the Euler gamma function.

( )

1 ( ) ( ) ( ) (1 ) t t d f t d D f t t f d dt dt                 (3)
According to equation 3, the fractional derivative of a function f(t) is actually the convolution of f(t) by t α /Γ(1-α), where α is the order of the fractional derivation. The additional time derivative present in the formula coincides with the positive occurrence of the gamma function, Γ(.), leading to its convergence towards a finite value. Based on this function, fractional derivatives include memory of the previous states. From a spectral point of view, a powerful consequence of a fractional derivative order is that the frequency spectrum f(ω) of f(t) will be multiplied by (jω) α instead of jω for a first order derivation. Consequently, the fractional derivative provides a possibility to satisfy the balance requirements, previously mentioned between it uses the the f(ω) low and high frequency components. As described in [START_REF] Guyomar | Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives[END_REF]- [START_REF] Guyomar | High nonlinearities in Langevin transducer: A comprehensive model[END_REF], the fractional derivative dynamic term is introduced in the lumped quasi-static contribution and is used as a dynamic contribution (Eq. 5). Fig. 1 gives the block-scheme illustration of eq. 4.

TABLE II SIMULATION DYNAMIC PARAMETERS

The dynamic parameters we used for this simulation are given in tab. 2.

The quasi-static contribution of this material has been previously described in past work and these values were used for this analysis [START_REF] Guyomar | Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives[END_REF]. The quasi-static parameters (γ, σ, the spectrum distribution function) for the P188 PZT were taken from literature [START_REF] Guyomar | Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives[END_REF] and are listed here in Table . 3.

B. Under weak amplitude excitation, dielectric spectroscopy simulation

As explained in the dielectric spectroscopy simulation methods section of this article, the Debye equation has been used to simulate the dielectric relaxation response of an ideal, noninteracting population of dipoles with an alternating external electric field E. Unfortunately, for the classic piezo PZT ceramic, the frequency behaviour is far from ideal. The fractional Cole-Cole equation has been utilized to improve the success of simulations. In the Cole-Cole approach, α, the fractional order is linked to the distribution of relaxation time. Assuming E and P harmonictype waveforms, it is possible to establish a link between the dielectric spectroscopy Cole-Cole model and the fractional derivative high amplitude field consideration described previously [START_REF] Zhang | Characterization of fractional order for high-frequency bandwidth model of dielectric ferroelectrics[END_REF] [START_REF] Ducharne | Multiscale characterization and model for the dynamic behavior of ferroelectric materials using fractional operators[END_REF]. After analysis t of the equation an interesting relation between τ, α, ρ, and εs becomes apparent:

. s      (5)
Comparisons between experimental results using the fractional Cole-Cole equation and simulations on Cole-Cole-type plots for classic PZT ceramic are available in literature [START_REF] Ducharne | Multiscale characterization and model for the dynamic behavior of ferroelectric materials using fractional operators[END_REF] and can be used as validation of the fractional used under weak amplitude excitation. Havriliak-Negami advanced theory is sometimes necessary to gain high correlation between experimental and simulated results. Similarly, comparison with high amplitude models can be established but the comparison leads to a series of fractional contributions instead of a single term as in the Cole-

Cole equation:

.

i i i d P dt     (6)
In past work [START_REF] Ducharne | Multiscale characterization and model for the dynamic behavior of ferroelectric materials using fractional operators[END_REF] comparisons between simulations and experimental results are given for property is particularly interesting in our case because it means that even if the electric field excitation becomes null, as it is the case during aging, the polarization value will still evolve because of the memory of the previous states. The aging simulation is consequently based on eq.

4, where the time window is set to a year. E is nonzero during the poling process and null elsewhere. Once poled, simulated time variations of the remanent polarisation are the direct consequence and manifestation of the piezo ceramic aging.

C. Experimental methods and results

The electric current variations due to depolarizing during aging processes are extremely weak.

Such variations are overwhelmed by the surrounding electromagnetic noises and consequently impossible to be measure using standard test methods and laboratory amplifiers. For these reasons, we have used another method to monitor these variations. Remanent polarization of a piezo ceramic after the poling process is useful to monitor for measuring piezo ceramic aging. To measure the remanent polarization, there is no way to avoid a full hysteresis cycle type excitation and consequently a new distribution of the domain which would effectively erase the aging data by resetting the domain. To overcome this issue we have decided to take another experimental approach. As explained, remanent polarization is challenging to monitor during the aging process, but other physical properties such as the permittivity ε or d33 the piezo-electric coefficient can be measured without any alteration of the material or aging process. d33 is a nonlinear but provides a very relevant image of the remanent polarization [START_REF] Sebald | Electromechanical characterization of 0.55Pb([END_REF][START_REF] Hajjaji | Synthesis and characterization of 0.65Pb(Mg1/3Nb2/3)O3-0.35PbTiO3 fibers with Pt core[END_REF]. By knowing d33(Pr)

we can then indirectly follow the Pr evolution during the aging process. To obtain the d33(Pr) characteristic curves, systematic pre-characterisations of all piezo ceramics experimental samples was conducted. In this work, the aging is assumed to correspond to a relaxation of the polarization only, without any degradation or fundamental change in the microstructure, leading to the assumption that the electrostriction is rather constant. Alternate methods exist for We assumed constant temperature and pressure during the process and no additional mechanical constraints were applied to the material. In order to avoid dielectric breakdown, the specimens were electrically insulated with silicon grease. The electric displacement was calculated by charge measurements using a Kistler amplifier 5011, and the polarization field was calculated by integration of the charge measurements. After obtaining baseline values, monitoring the aging process can begin. A final 1,5 period triangular-type waveform, 4kV/mm amplitude electric field E excitation is applied to all test samples in order to reset the domain distribution and to reach remanent polarization. After that, d33 will be monitored regularly over a 200 day duration to observe the aging effects. The first measure is done 75 seconds after polarization. The initial experimental result is depicted in Fig. 5 and it shows the time variation of d33. As envisaged, a quasi-constant negative slope in the logarithmic scale can be observed. It is also noteworthy that a variation occurred at approximately 10 5 seconds. This could be attributed to changes in the laboratory environment or variations in the sample itself. It is note-worthy that the best results when comparing simulations to experimental results

have been obtained with a value of (α = 0.6) which is relatively close to 0.52 corresponding to the fractional order leading to the best matches under P(E) situations.

In our simulation, a fractional derivative order equal to 1 means a quasi-instantaneous dynamic contribution which will stop right after the electric field excitation removal. This comes from the numerical definition of the first order derivative which to be solved relies on the previous time step. Now for lower fractional order, the resolution needs a window of previous time steps to be solved and the lower the coefficient is, the higher the influence of the initial time steps. From the aging point of view it means that the lower α is, the lower the aging rate will be. An interesting observation is that no matter what the order of the fractional derivative is, if the simulation time is long enough the polarization field will always reach a steady state where no more variations can be observed. The value of α is inversely proportional to this transient phase time.

In Figure 9, the aging rates versus time have been plotted on a linear time scale comparing the simulated and experimental results. Since the aging rate stabilizes after 10 4 seconds, an alternate method of looking at this property was used. In Figure 10, aging versus time decades is displayed.

The experimental aging rate is noisy with an average value around 1.7%. Even if none of the simulation results reproduces strictly the experimental ones, α = 0.6 gives an average aging rate (1.98 %) very close to the experimental one (1.72 %).

VIII. CONCLUSION

Dynamic behaviours in ferroelectric materials can be physically explained as a complex combinations of diffusive and dissipative behaviours. The dissipative behaviours are linked to the dynamics of domain wall motion. The diffusive behaviours are linked to mechanical and thermal relaxation. In the past, fractional derivatives have been used with success for the simulation of systems exhibiting dynamic behavior known as anomalous diffusion [START_REF] Chen | Anomalous diffusion modeling by fractal and fractional derivatives[END_REF]- [START_REF] Metzler | fractional model equation for anomalous diffusion[END_REF]. Where first or second order derivation are used in classic diffusion equations, anomalous diffusions are more complex and consequently require alternative mathematical operators to be simulated. It is also used to complex visco-elastic behavior [START_REF] Scott Blair | The role of psychophysics in rheology[END_REF] as an intermediate state between elastic and viscous bodies. The dynamic behavior including the frequency dependence of hysteresis and aging of piezo ceramics is one of those. As discussed in IV-A aging is due to three major mechanisms: the volume effect, the domain effect and the interface effect. These mechanisms can be combined.

All together they lead to a macroscopic behavior which can be assimilated to an anomalous diffusion and simulated using fractional derivative operators. In our simulation itself, we have no way to distinguish the aging mechanisms as the model is a phenomenological lump approach with no direct connection to the ferroelectric sample micro-structure. With this non-predictive but phenomenological approach, we focus on reproducing experimental signals as they are measured, physical interpretations can be made in term of anomalous diffusion but because of the space discretization absence there is no local information and no direct connection to the ceramic micro-structure. There is therefore no way to estimate the activation energy responsible for aging such as to differentiate the contribution of charges to the ionic space charges.

It has already been demonstrated that the frequency dependence of the dielectric properties can be correctly modelled in the time domain thanks to fractional derivative operators on a very large frequency bandwidth of external excitations. Up to now, the lowest frequencies have been limited to the hysteresis cycle quasi-static threshold generally in the range of 10 mHz for classic PZT ceramics. In this article, a simulation technique using the fractional derivative dynamic contribution was validated under a wide frequency range and particularly in terms of long time variations like those experience during aging. The simulation was validated using experimental data from a piezo ceramic material, P188. The fractional parameters from the simulation are conserved through the frequency dependence of dynamic hysteresis cycle and long-time variation of the remanent polarization which can be physically interpreted as a conservation of the physical behaviour even if these parameters are evolving in different scales of time. This work confirms that fractional derivative based simulation is capable of predicting aging in piezo ceramic materials.

Ferroelectric

  ceramics are subject to fatigue due to loads acquired when the materials are used in real world applications along with natural degradation over time, aging. The physical mechanisms of both degradations are similar and linked to pinning defects which are a result of doping and intentionally impurities in the ferroelectric ceramic material. Donor or acceptor dopants, mobile vacancies, and holes are examples of typical defects.

  OF THE DYNAMIC DIELECTRIC PROPERTIES -UNDER WEAK SLOPE EXCITATION FIELD STIMULI (dE/dt, dT/dt << 1);AGING SIMULATIONThe role of space charges of different natures including electronic, ionic, and, mixed has been pointed out in many studies in the context of aging[20][36]. When comparing aging models, several approaches have been taken including, 1-D, 2-D, and 3-D models. Due to geometric considerations, 1D quantitative models generally vary in comparison to experimental samples.2-D and 3-D drift diffusion models provided valuable contributions and predictions because theyprovide information useful to understanding the aging mechanism for both poled and unpoled piezo ceramics. Such models show a time dependent internal bias field characterized by ferroelectric properties including domain width and surface boundary charge, relaxation

Fig. 1 -Fig. 2 -

 12 Fig. 1 -Block diagram illustration of the whole dynamic ferroelectric hysteresis model. In this diagram, P is the polarization field, Edyn represents the electric field excitation, Estat is

Cole-

  Cole type plots (i.e. variations of the imaginary versus real part of the permittivity under frequency sweep variations of the excitation field). High correlations between experimental and simulate data provide validation of the method.As described and explained in the second part of this article describing current research of simulation methods, the time domain resolution of a fractional derivative includes memory of the previous states. A first order derivation of a f(t) function uses only the f(t-dt) value, whereas the fractional derivation needs the full function f history, which numerically speaking means a series of all the time discretized values of f since the very beginning of the simulation. This

  electrode measuring the remanent polarization, but generally require larger number of samples and alternate assumptions must be made using these methods including the same initial remanent polarization state. The test set-up can be seen in fig[START_REF] Damjanovic | Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics[END_REF]. These pre-characterisations consist of exciting the material with a 1.5 period triangular-type waveform electric field E (fig.4 -A) in order to reach remanent polarization while increasing the value of the maximum electric field excitation (fig.4 -B). Once remanent polarization is reached, the d33 coefficient is measured. The pre-characterisations leads to the d33(Pr) characteristic (fig.4 -C) which will be used later as a standard to return the remanent polarisation in the aging experimental process. Cylindrical specimens with diameters of 6.35 mm and heights of 4mm were subjected to the electric field using a Optilas Trek high-voltage supply.

Fig. 3 -

 3 Fig. 3 -Pre-Characterisation experimental setup.

Fig. 3 Fig. 4 -

 34 Fig. 4 -A 1.5 period triangular-type waveform electric field E, 4 -B related hysteresis cycles, 4 -C Pr(E) characteristic curve, 4 -D d33(Pr) characteristic curve, 4 -E d33(E) characteristic curve.

Fig. 5 -

 5 Fig. 5 -d33(t) variations due to aging effect.

Fig. 6 Fig. 6 -

 66 Fig.6shows the remanent polarisation evolution. To get this curve, a 1D interpolation from

Fig. 7 -

 7 Fig. 7 -P(E) under both poling and aging situation.

Fig. 8 Fig. 8 -

 88 Fig. 8 depicts the simulated results and experimental results. Fig 8-A shows Pr(t) variations for

Fig. 9 -Fig. 10 -

 910 Fig. 9 -Linear Time Scale Aging Rates Experimental Versus Simulated.

  Table. 1 depicts the physical properties of the soft PZT compositions used in this study. The specific material used in this study is P188 obtained from Quartz & Silice, France, Navy type II.


Bhaawan Gupta

Benjamin Ducharne

Tetsuya Uchimoto

Gael Sebald
email: gael.sebald@insa-lyon.fr

Takamichi Miyazaki

Toshiyuki Takagi

Non-destructive Testing on Creep Degraded 12% Cr-Mo-W-V Ferritic Test Samples using Barkhausen Noise

Keywords: Magnetic Barkhausen Noise, Electromagnetic Non-destructive testing, Micro-magnetic modeling, high chromium steel, creep degradation, Jiles-Atherton model

Micro-magnetic non-destructive testing helps in evaluating the mechanical and structural integrity of ferromagnetic components (mostly steel) via local magnetic characterization. As a means of revealing the magnetic domain wall movements, the Barkhausen noise measurement method is a micro-magnetic nondestructive testing method of considerable potential and interest. The Magnetic Barkhausen Noise Energy (MBNenergy) method can be used to reconstruct local hysteresis cycles from Barkhausen noise measurements. These cycles constitute good indicators for understanding the magnetization process and the influence of microstructural and mechanical properties. In this article, the MBNenergy method is employed for evaluating the microstructural changes induced by creep/ageing of high chromium steel subjected to different creep test conditions as stress and temperature. The corresponding magnetic parameters are investigated in relation to precipitations and dislocations. After the experimental analysis, simulations are performed based on the Jiles-Atherton theory for the quasi-static magnetic behavior. This simulation approach yields physically meaningful model parameters that can be analyzed and linked to the sample microstructural characteristics, thereby enabling physical interpretation.

Introduction

Micro-magnetic non-destructive testing (NDT) can be used to evaluate the mechanical properties (such as the hardness and ageing level) of materials. The Barkhausen noise is very sensitive to mechanical changes and residual stresses [START_REF] Gauthier | Measurement of residual stress in steel using the magnetic Barkhausen noise technique[END_REF]- [START_REF] Altpeter | Micro-magnetic evaluation of micro residual stresses of the IInd and III order[END_REF]. These mechanical changes may lead to microstructural changes and magnetic-behavior modifications. The microstructural changes modify the domain wall movements and, hence, Barkhausen noise measurements represent an interesting means of investigating load-induced phenomena such as creep [START_REF] Gauthier | Measurement of residual stress in steel using the magnetic Barkhausen noise technique[END_REF][2] [START_REF] Stewart | Magnetic Barkhausen noise analysis of stress in steel[END_REF] [START_REF] Kleber | On the role of residual internal stresses and dislocation on Barkhausen in plastically deformed steel[END_REF]. Studying creep in materials helps to reveal the level of rupture for a material subjected to a given set of conditions [START_REF] Gupta | Magnetic incremental permeability non-destructive evaluation of 12 Cr-Mo-W-V Steel creep test samples with varied ageing levels and termal treatments[END_REF]. Several factors (such as dislocations, recovery of the crystallization stage, and variations in the carbide precipitates) contribute to the creep in materials. Eventually, these factors all lead to the formation of cracks and material failure after a certain threshold. Creep has been a very important matter of concern for the power plant industries [START_REF] Sposito | A review of non-destructive techniques for the detection of creep damage in power plant steels[END_REF] [START_REF] Garofalo | Fundamentals of creep and creep-rupture in metals[END_REF].

Recently, the application of micro-magnetic non-destructive techniques, such as measuring the magnetic Barkhausen noise (MBN), has increased considerably in industrial fields [START_REF] Dobmann | Industrial applications of 3MA -Micromagnetic Multiparameter Microstructure and Stress Analysis[END_REF]. This is due mainly to the improvement of signal processing techniques that have enabled and simplified the online production monitoring [START_REF] Altpeter | Micro-magnetic evaluation of micro residual stresses of the IInd and III order[END_REF]. However, industrial implementation is limited because the reproducibility of the raw signal results mainly from the acquisition process completed by high-order filters and huge-gain amplifier stages. The Magnetic Barkhausen Noise Energy (MBNenergy) method can be used to reconstruct local hysteresis cycles from Barkhausen noise measurements. These hysteresis cycles constitute good indicators for understanding the magnetization process and the influence of microstructural and mechanical properties. The integration step filters the raw signal and provides a stable image of the Barkhausen noise and is therefore a quite efficient tool. In this work, the MBNenergy method is employed for the evaluation of microstructural changes due to creep in 12 % Cr-Mo-W-V creep steel samples. These samples were previously investigated via the Magnetic Incremental Permeability (MIP) technique described in the previous work of the authors. The motivation is to determine the microstructural characteristics from the magnetic parameters derived from the MBN signal. To avoid repeatability issues due to the sensor, lift off, and sensibility (such as the quest of the most sensitive indicator), as per [10] [11], MBNenergy hysteresis cycles will be plotted and evaluated.

             
As in the case of the classical hysteresis cycles considering the average induction field B versus the tangential magnetic excitation H, MBNenergy hysteresis cycles area are related to an energy.

A raw Barkhausen noise signal can be considered an image of the domain wall speed. By integrating the square of the signal (see Equation ( 1)) the area of the resulting hysteresis cycle is obtained as an image of the kinetic energy. This energy is consumed by the domain walls during the magnetization process. After the experimental analysis, the magnetic signals are modeled using the theory of Jiles-Atherton for quasi-static conditions. The model parameters derived in the case of MBN are analyzed and linked to the sample microstructural characteristics, thereby enabling physical interpretation.

Experimental procedure

Fig. 1 shows a schematic of the experimental set-up. The tested samples are magnetically excited using a sinusoidal magnetic excitation field driven with the help of a soft U-shaped magnetic lamination stack.

The samples are all of the same size, for further details see [START_REF] Gupta | Magnetic incremental permeability non-destructive evaluation of 12 Cr-Mo-W-V Steel creep test samples with varied ageing levels and termal treatments[END_REF]. The yoke is fabricated from silicon steel sheets, the leg size is 10 mm x 20 mm, and the inner-distance between legs is 20 mm. The excitation frequency is set to 0.1 Hz after optimization tests, as explained in the previous study [START_REF] Gupta | Magnetic incremental permeability non-destructive evaluation of 12 Cr-Mo-W-V Steel creep test samples with varied ageing levels and termal treatments[END_REF]. The excitation magnetic field is feedback controlled to ensure a 10 kA/m maximum amplitude. Furthermore, the sensor used to pick up the Barkhausen Noise signal consists of two pancake pick-up coils and a Hall Effect sensor located inside. The Hall sensor measures the tangential surface excitation field H, on the surface of the tested sample. The output of the pick-up coils is transmitted to an electronic analog circuit. These coils operate independently, a commune potential is set, and the differential voltage is amplified. This circuit uses a differential amplifier (P-61, NF Corporation) to provide a first amplification (by a factor of 1000) of the differential Barkhausen signal. Once amplified, this signal is fed to a band pass filter (BPF) with a 1-60 kHz cut off frequency. The output of the BPF is amplified again by a factor of 50 prior to the oscilloscope acquisition. In parallel, the output of the Hall sensor is also amplified by a factor of 50 prior to acquisition by the oscilloscope. The raw MBN signal can be measured directly or processed, as suggested in [START_REF] Ducharne | Characterization and modeling of magnetic domain wall dynamics using reconstituted hysteresis loops from Barkhausen noise[END_REF], to obtain the MBNenergy(H) curve.

(

) 1
For this, in a second electronic circuit, the square of the Barkhausen noise is first calculated using an AD633 analog multiplier, followed by a low noise operational amplifier OPA2604 in an integration configuration ensuring the integration of the signal. A small reed relay D31C2100 provides the reset function of the integrator as soon as the acquisition is completed. E-mail address: gael.sebald@insa-lyon.fr (Gael Sebald)

Materials of Interest and Microstructural Evaluation

Sample description

In this study, three different categories of samples are investigated. For each category, every investigated sample is characterized by a different rupture level. The material considered in the present study, i.e., a 12% Cr-Mo-W-V steel, is a representative martensitic stainless steel (the general composition of this alloy is given in [START_REF] Mutoh | Improving Fretting Fatigue Strength at Elevated Temperatures by Shot Peening in Steam Turbine Steel, Standardization of Fretting Fatigue Test Methods and Equipment[END_REF]). Such alloys are used for high-temperature material applications including steam gas turbines and boilers. The percentages of the samples tested in our study may vary slightly, but we assume that changes in the magnetic behavior due to these weak variations will be extremely limited.

Table 1 provides a short description of the sample conditions. The Larson Miller Parameter is a statistical parameter that combines the effect of Temperature T and test time t in a mathematical expression [START_REF] Larson | [END_REF], and is given as follows:

( log ) LMP T C t  (2)
Where, C is a material specific constant, often taken as 20, t is the time in hours, and T is the temperature in Kelvin. The values of LMP for each sample are calculated and presented in Table 1, which is given as follows:

Microstructural analysis

The microstructure of each sample was analyzed via Scanning Electron Microscopy and Electron Backscatter Diffraction (EBSD), which yields the Kernel Average misorientation (KAM) data. The KAM is defined as the average misorientation angle of a given point with all its neighbors [14] [START_REF] Gupta | Physical interpretation of the microstructure for aged 12 Cr-Mo-V-W steel creep test samples based on simulation of magnetic incremental permeability[END_REF]. Further details of the standard procedures are provided in [START_REF] Moussa | About quantitative EBSD analysis of deformation and recovery substructures in pure Tantalum bout quantitative EBSD analysis of deformation and recovery substructures in pure Tantalum[END_REF] [15], with [START_REF] Gupta | Physical interpretation of the microstructure for aged 12 Cr-Mo-V-W steel creep test samples based on simulation of magnetic incremental permeability[END_REF] describing previous work by the authors where the MIP technique is employed. In the present work, we try to relate these data with the Barkhausen Noise signal and then use a modeling technique for further quantification to derive parameters that describe the correlation between the signal and the microstructure. Fig. 3 shows the most relevant data results obtained from the microstructural analysis. Previous work [START_REF] Gupta | Magnetic incremental permeability non-destructive evaluation of 12 Cr-Mo-W-V Steel creep test samples with varied ageing levels and termal treatments[END_REF] [15] on these samples by the current authors has provided a detailed analysis of the results.

Table 1. List of Samples

_______________________________________ *Corresponding author. E-mail address: gael.sebald@insa-lyon.fr (Gael Sebald)

Experimental Results

Hysteresis cycles constructed from magnetic Barkhausen noise raw signals are referred to as MBNenergy(H) cycles. Fig. 4 shows the different steps of this reconstruction. During the first step (b), the square of the raw signal is calculated. The resulting signal is multiplied by sign(dH/dt), which is 1 with increasing H and -1 otherwise. A time integration is performed immediately after, as illustrated in Fig. 4(c), and the offset is removed. Afterward, a normalization step is performed to ensure equality between the absolute value of both the MBNenergy maximum and the minimum.

Experimental Data Analysis of 12%Cr-Mo-W-V Creep Test Samples

Fig. 5(a) shows the raw magnetic Barkhausen Noise signal for the three 550°C samples. Fig. 5(b) shows the comparison of these three different samples via the reconstructed MBNenergy Hysteresis cycles from the raw MBN signal, using the technique presented in Fig. 4. From the raw signal, clear differences are observed between the samples. The amplitude of the raw signal decreases with increasing creep. Clear differences among the saturation points of the three samples are observed and the coercivity of the ruptured sample is observed to be higher than that of Sample 1. This results from the number of precipitates, which is considerably higher than the number occurring in Sample 1, leading to magnetically hard samples as rupture is approached. Fig. 6 shows the comparison of the samples from the 600°C category. As in the case of the MIP results in [START_REF] Gupta | Magnetic incremental permeability non-destructive evaluation of 12 Cr-Mo-W-V Steel creep test samples with varied ageing levels and termal treatments[END_REF], the results differ slightly among the different samples. However, from the MBNenergy hysteresis cycles, the differences in the magnitude of MBNenergy are quite noticeable. The MBN activity in the raw signals decreases over time as the rupture level increases.

Figure 5. Raw MBN signal for Samples 1-3 and their respective reconstructed MBNenergy Hysteresis cycles (Bottom) (a) Raw MBN signals for samples 1-3 (b) Reconstructed MBNenergy cycles for samples 1-3

Fig. 7 shows the comparisons for the 650°C category of samples. In contrast to the 550°C samples, the amplitude of the MBN of these samples increases with increasing rupture level. Moreover, the Barkhausen noise activity decreases with increasing time and when rupture is close. In this case, again in contrast to the 550°C samples, the coercivity of the loops also decreases as the rupture nears.

Comparison among the ruptured samples from the three different categories

Fig. 8 shows the comparisons of the raw magnetic Barkhausen noise signals of the three ruptured From the microstructural analysis, for the 650°C samples, the number of precipitates is lower than in the other samples, thereby favoring easy domain wall movements. This leads to an easier orientation of the domains in the direction of the applied magnetic field, resulting in higher and more rapid magnetizability than in other directions. Furthermore, for samples treated at similar temperatures (Fig. 5), the coercivity increases (in general) with increasing rupture of the 550°C samples, but decreases in the case of the 650°C samples (Fig. 7). This may have resulted from the effect of the precipitates. At 550°C, the number of precipitates is dominant and at 650°C the size of the precipitates is dominant. Fewer misorientations/dislocations (KAM data can be found in section 3) occur, on average, in the higher-temperature-treated samples (than in the lower-temperature-treated samples), thereby resulting in quicker magnetization and higher permeability. Based on Figs. 567, from the MBNenergy curves, parameters such as the coercivity factor, permeability factor (µfactor) corresponding to the slope of the curves, and MBNenergy amplitude are extracted and correlated with the microstructure of the materials. However, when considering the amplitude of the MBNenergy curve, the absolute values for the same material may vary with different experimental set-ups, because of the gain factor, but the relative relation between the samples will always remain constant.

Figure 8. Comparison of ruptured samples from three different temperature categories (b) Reconstructed MBNenergy cycles for ruptured samples (Samples 3, 6, &9) (c) (a) Raw MBN signals for ruptured samples (Samples 3, 6, &9)

_______________________________________ *Corresponding author. E-mail address: gael.sebald@insa-lyon.fr (Gael Sebald)

Magnetic parameters versus the microstructure

Figs. 9-10 show the correlation of magnetic parameters derived from MBNenergy curves with the microstructural characteristics of precipitates and dislocations (Fig. 3). The use of the Barkhausen noise measuring technique also demonstrates the effect of microstructural changes in the materials. The tendency of the parameters is similar to that of the MIP results [START_REF] Gupta | Magnetic incremental permeability non-destructive evaluation of 12 Cr-Mo-W-V Steel creep test samples with varied ageing levels and termal treatments[END_REF]. The higher temperature samples behave in an opposite manner to the lower temperature samples. For example, the coercivity increases (in general) with increasing number of precipitates in the 550°C samples, leading to magnetic hardening of the materials. However, for the 600°C and 650°C samples, the coercivity decreases with decreasing number of precipitates, due to the soft magnetic behavior of the materials. KAM is an important factor from a materials science point of view, but for an understanding of creep, satisfactory correlation between KAM and the magnetic parameters may be lacking. However, a larger variation is observed for the 650°C samples (as in the case of the MIP technique, B(H) technique, and the modeling parameters derived from those techniques [7] [15]) than for the other samples. The average density of dislocations decreases in the high-temperature samples, thereby resulting in higher permeability and magnetic reversibility. When the material is tested via the Barkhausen technique, the coercivity exhibits the strongest correlation with the microstructure, and this is also verified with the model, as demonstrated in Section 5.3.

Figure 9. Magnetic Parameters derived from MBN versus number of precipitates (a) Coercivity versus Precipitations (b) Permeability factor versus Precipitations

Jiles-Atherton hysteresis theory for the simulation of the MBNenergy hysteresis cycles 5.1. Introduction to the classic Jiles-Atherton (J-A) model

Below a frequency threshold, the cumulative periodic value of ferromagnetic losses becomes frequency independent (in the decreasing direction), and this represents the quasi-static state. This behavior is observable by plotting the spontaneous average magnetic induction B versus the tangential surface magnetic excitation field H under weak frequency (<< 1 Hz for a typical soft ferromagnetic). To obtain correct simulation results for this quasi-static hysteresis, the J-A model considers both contributions -the domain wall bending and the domain wall translations [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF]- [START_REF] Zhang | Dynamic magnetic scalar hysteresis lump model based on Jiles-Atherton quasi-static hysteresis model extended with dynamic fractional derivative contribution[END_REF]. Additionally, the:  model requires a fairly small memory storage, where the evolution history information is limited to a single time step before B;  status can be completely described by five physically meaningful and interdependent parameters;  J-A model is easily reversible (H or B can serve as input for the model) [START_REF] Leite | The inverse Jiles-Atherton model parameters identification[END_REF] [START_REF] Leite | Inverse Jiles-Atherton vector hysteresis model[END_REF]. In the scalar J-A model for ferromagnetic materials, the total magnetization M is decomposed into its reversible (Mrev) and irreversible (Mirr) contribution as shown mathematically in equation ( 3). From a physical point of view, Mrev and Mirr are associated with the magnetic domain rotation and the wall displacements, respectively.

rev irr M M M  (3)
The anhysteretic magnetization Manh can be described by the Langevin equation [START_REF] Zwanzig | Nonlinear generalized Langevin equations[END_REF]:

           (4)
or by using a hyperbolic function:

tanh e anh s H MM a     (5)
In both equations ( 4) and ( 5), Manh is the anhysteretic magnetization, He is the effective field, Ms is the saturation magnetization, and a is the parameter describing the magnetic-field-dimension characteristic of the anhysteretic magnetization shape. According to the J-A theory, a = kB.T / μ0.m, where kB is Boltzmann's constant, T is the temperature, and m is the magnetic moment of a typical domain.

. Here, H, M, and α are the applied tangent excitation field, average sample magnetization, and mean field parameter related to the inter-domain coupling, respectively. The anhysteretic, irreversible, and reversible magnetization are related as follows:

( ) rev an irr M c M M  (7)
Where, c is a proportionality coefficient. This coefficient can be determined from the experimental results by calculating the ratio of the initial differential susceptibilities associated with the first and anhysteretic magnetization curves.

The coefficient k is the pinning parameter, which is linked to the amount of energy dissipated, and δ is a directional parameter, which ensures that energy is always lost through dissipation.

1 / 0 1 / 0 if dH dt if dH dt            (9)
Combining the aforementioned equations yields the main equation of the J-A model:

        (10)
Variations in the magnetization M with respect to variations in H can be calculated from equation [START_REF] Ducharne | Phenomenological model of Barkhausen noise under mechanical and magnetic excitations[END_REF]. However, for some applications (e.g., consideration of the dynamic effect), the magnetic induction B is known before H. References [START_REF] Leite | The inverse Jiles-Atherton model parameters identification[END_REF] and [START_REF] Leite | Inverse Jiles-Atherton vector hysteresis model[END_REF] detail the inverse version of the J-A model for ferromagnetic materials, where the model input is the magnetization B. The physical principles of the model are very similar and the main equation of this inverse model for ferromagnetic materials becomes:

0 0 (1 ) 1 (1 )(1 ) (1 )
            (11)

A Jiles-Atherton-type approach for the simulation of the of MBNenergy(H) cycles

This simulation study is aimed at deriving a reliable parameter or a combination of parameters, which can be used to evaluate the microstructure of the material. As detailed in [START_REF] Gupta | Physical interpretation of the microstructure for aged 12 Cr-Mo-V-W steel creep test samples based on simulation of magnetic incremental permeability[END_REF], MIP yields good correlations for the three modeling parameters, α, k, and c. For the B(H) measurements, α provides an interesting correlation [START_REF] Gupta | Physical interpretation of the microstructure for aged 12 Cr-Mo-V-W steel creep test samples based on simulation of magnetic incremental permeability[END_REF]. The parameter/s applicable to measurements performed via the Magnetic Barkhausen Noise technique is/are determined. The simulation process must be run prior to the correlation step. For the MBNenergy hysteresis cycle, this process starts with the approximation of the experimental MBNenergy anhysteretic curve. For this curve, we have opted for a numerical estimation rather than experimental measurements. We proceed in this manner because the experimental procedure is tedious and the estimated anhysteretic curves of almost all the classical soft magnetic materials differ only slightly from the measured curves. Assuming that the major hysteresis cycle is perfectly symmetrical, the MBNenergy anhysteretic curve can be calculated from the increasing and decreasing parts of the cycle equation:

( ) ( ) () 2 inc dec anhyst i i i i i i H B H B H B   (14)
Here, 𝐻 𝑖 𝑖𝑛𝑐 , 𝐻 𝑖 𝑑𝑒𝑐 and 𝐻 𝑖 𝑎𝑛ℎ𝑦𝑠𝑡 represent the H values corresponding to the increasing part and decreasing part of the major hysteresis cycle, and the anhysteretic cycle, respectively. Ms and a, the Jiles-Atherton anhysteretic parameters are calculated by fitting the estimated anhysteretic curve to the simulated curve using Matlab™ curve fitting toolbox. An optimization code based on the minimization of an error function (eq. 15) is used to determine the optimal α, k, and c combination.

Modeling parameters derived from simulation of MBNenergy curves

A simulated and a measured MBNenergy hysteresis cycle for sample 1 are compared in Fig. 11. In Fig. 11(b), the modeling parameter (k in this case) is derived from the fitting of the simulated curve to the experimental curve. An analysis of the simulation parameters reveals that the most relevant correlation occurs for the coefficient k (Fig. 11(b)). This is verified by calculating the Pearson Correlation coefficient, as shown in Fig. 12. k is evaluated with different (mechanical, magnetic, and mathematical) parameters and a correlation is determined using the Pearson correlation coefficient (for k and α), as shown in Fig. 12. The value of the coefficient increases with increasing strength of the correlation (values close to 1/-1 indicate a strong (positive/negative) correlation).

(a) Simulated and Experimental MBNenergy Curves for Sample 1 (b) Evolution of k parameter versus Precipitates (c) Evolution of k parameter versus Average of KAM Figure 11. Comparison of simulated and experimental curves for sample 1 MBNenergy cycle, dependence of the k coefficient on the number of precipitates, dependence of the k coefficient on KAM

_______________________________________ *Corresponding author. E-mail address: gael.sebald@insa-lyon.fr (Gael Sebald)

The J-A parameters are all inter-related, i.e., each parameter changes with changes in the values of the other parameters. Hence, the most stable parameter (k in this case) with variations in other parameters (α, c), which corresponds to the coercivity of the materials in this case, is chosen. The 550°C sample contains more dislocations and precipitates (Microstructural Analysis, section 3) than the other samples and, hence, the k factor of this sample is higher (in general) than those of the other samples (see Fig. 11). A comparison of Fig. 9(a) and Fig. 11(b), which show the coercivity vs. the number of precipitates and k vs. the number of precipitates, respectively, reveals quite similar tendencies. However, some variations are expected since, k is also associated with the effect of the variations in other parameters (α, c). The variation in the k parameter vs. the creep life fraction t/tr (see Fig. 13) confirms that the evolution trend of the 550°C samples is opposite to that of the 650°C samples. However, the rate of change is also an important factor. In the 550°C samples, the change rate of k with increasing rupture time is considerably higher (in the increasing direction) than the quite small change rate in the 650°C samples (in the decreasing direction). From the precipitation point of view, the high-temperature samples (650°C) exhibit only modest variations, but regarding dislocations, these samples exhibit larger variations when k vs. the average of KAM is considered (see Fig. 11(c)). The variation with respect to KAM observed for the magnetic and model parameters corresponding to the higher temperature samples is larger (also the case for MIP [START_REF] Gupta | Physical interpretation of the microstructure for aged 12 Cr-Mo-V-W steel creep test samples based on simulation of magnetic incremental permeability[END_REF]) than that observed for the lower-temperature samples. Moreover, in the case of MBN, other parameters (e.g., α) are sensitive to only lower temperature samples and c is sensitive to only higher temperature samples. α represents the inter-domain coupling (Jiles-Atherton theory), which is higher for lower temperature samples since the precipitates are smaller than those in the higher temperature samples.

Figure 12. Pearson coefficient for comparisons between simulation parameters (α, k) and microstructural parameters (a) Pearson correlation coefficient for α (b) Pearson correlation coefficient for k

_______________________________________ *Corresponding author. E-mail address: gael.sebald@insa-lyon.fr (Gael Sebald) Fig. 13 shows the evolution of the most reliable parameter (k) for the Magnetic Barkhausen noise in relation to the rupture level. Despite minor discrepancies, the overall evolution tendency of this parameter is quite similar to that of the coercivity factor derived from MBNenergy hysteresis cycles.

Conclusion

This work focuses on a micro-magnetic Magnetic Barkhausen Noise method for evaluating the microstructural state of 12 Cr-Mo-W-V steel. Such micro-magnetic analysis is of major interest as the microstructural state is deeply connected to flaw propagation in these structural steels and, if left unchecked, this propagation leads to severe problems. As a means of revealing the magnetic domain wall movements, the Barkhausen noise measurement is a micro-magnetic non-destructive testing method of significant potential and interest. In this study, rather than following the classical approach, which consists of working directly from the raw signal, MBNenergy(H) hysteresis cycles have been reconstructed and simulated. Working with these cycles limits the reproducibility and the sensitivity issues (such as providing a magnetic signature). The amplitude of each MBNenergy curve decreases (in general) with increasing creep level. Furthermore, based on the observed hysteresis, these MBNenergy curves can be easily simulated using the classical hysteresis models, such as the Jiles-Atherton model, which has been used in this study. Experimental tests and simulations are followed by a detailed analysis aimed at determining the correlations between the magnetic parameters (coercive field, permeability factor), J-A simulation parameters, and the microstructural properties. For example, a correlation factor of 0.9 is obtained when the k parameter is considered with respect to the number of precipitates and an α of 0.8.

Fig. 2

 2 shows all the signals acquired by the oscilloscope. Signal 1 (Blue), Signal 2 (Cyan), and Signal 3 (Red) are the output of the Hall sensor, raw amplified magnetic Barkhausen noise, and square of the MBN (Signal 2), respectively. Signal 4 (Green) is the integrated output of Signal 3. A post processing numerical treatment is performed for the drift correction, sign of the MBNenergy time derivation, cycle symmetry, and final plot.

Figure 2 .Figure 1 .

 21 Figure 2. Data Acquisition for the Magnetic Barkhausen NoiseMeasurement system

Figure 3 :

 3 Figure 3: Microstructural Data Evaluation

Figure 4 .

 4 Figure 4. Derivation of MBNenergy cycle from Raw Magnetic Barkhausen Noise

6 Figure 6 .

 66 Figure 6. Raw MBN signal for Samples 4-6 and their respective reconstructed MBNenergyHysteresis cycles (Bottom)

Figure 7 .

 7 Figure 7. Raw MBN signal for Samples 7-9 and their respective reconstructed MBNenergyHysteresis cycles (Bottom).

Figure 10 .*

 10 Figure 10. Magnetic Parameters derived from MBN versus average of KAM (a)Coercivity versus Average of KAM

  Be is the effective magnetic flux density.

Figure 13 .

 13 Figure 13. Evolution of k and coercivity factor with respect to rupture level.

  MBNenergy hysteresis cycles are obtained by plotting the time integration of the squared Barkhausen noise multiplied by the excitation field time derivation sign as a function of H, see equation 1; Bark(t) is the raw MBN signal:

	MBN	energy	( ) t	0 t	2 ( ') . Bark t sign	dH dt	dt	'

Acknowledgment

This work was performed in the framework of the Japan -France International Laboratory (LIA) ELyTGlobal. The authors gratefully acknowledge the French Region Auvergne-Rhône-Alpes, the French project IDEXLYON of the Université de Lyon in the framework of the "Investissements d'Avenir" program (ANR-16-IDEX-0005), and the Institute of Fluid Science (Tohoku University) through the General Collaborative Program projects # J17I065, #J18I055, and the JSPS KAKENHI Grant, #18H01448.

_______________________________________ *Corresponding author. E-mail address: gael.sebald@insa-lyon.fr (Gael Sebald)