# Crack Protective Layered Architecture of Lead-Free Piezoelectric Energy Harvester in Bistable Configuration

^{*}

## Abstract

**:**

_{3}ceramics. This material is very brittle and might be cracked in small amplitudes of oscillations. However, the main aim of our development is the design of a crack protective layered architecture that protects an energy harvesting device in very high amplitudes of oscillations. This architecture is described and optimized for chosen geometry and the resulted one degree of freedom coupled electromechanical model is derived. This model could be used in bistable configuration and the model is extended about the nonlinear stiffness produced by auxiliary magnets. The complex bistable vibration energy harvester is simulated to predict operation in a wide range of frequency excitation. It should demonstrate typical operation of designed beam and a stress intensity factor was calculated for layers. The whole system, without presence of cracks, was simulated with an excitation acceleration of amplitude up to 1g. The maximal obtained power was around 2 mW at the frequency around 40 Hz with a maximal tip displacement 7.5 mm. The maximal operating amplitude of this novel design was calculated around 10 mm which is 10-times higher than without protective layers.

## 1. Introduction

_{3}, and like polyvinylidene fluoride (PVDF). The recent development is forced to focus on lead free ceramic materials and composites for energy harvesting devices [3]. Mainly, barium titanite ceramics are discussed in the case of lead-free applications [4]. Piezoelectric properties of the lead-free ceramic materials, e.g., BCZT ceramics [5], are sensitive to fabrication and processing methods. Alternatives to ceramic materials are provided by polymers like polyvinylidene fluoride (PVDF) [6].

_{31}coefficient which is a key for an efficient energy harvesting; in case of lead-free solution BaTiO

_{3}this d

_{31}coefficient is more than two times higher than the one of PVDF.

_{3}piezoceramic beam with protective layers for bistable operation of piezoelectric energy harvesters, which provide a maximal peak to peak displacement around 20 mm without a brittle fracture. This large oscillation range of bistable energy harvester allows to generate a much higher output power.

## 2. Bimorph Cantilever Beam Design

_{t}= 5 g. The piezoelectric layers are considered to be made from BaTiO

_{3}and are electrically connected in parallel. The substrate together with protective layers are considered to be made from ZrO

_{2}and ATZ (Alumina Toughened Zirconia). These materials were chosen due to their relatively close values of thermal expansion coefficient α and, as a consequence, moderate levels of thermal residual stresses σ

_{res}are induced within the considered multilayer structure upon its fabrication. Material properties of the considered materials are listed in Table 1.

#### 2.1. Optimization of the Layer Configuration within the Proposed Multilayer Structure

_{3}layer, and thus causing a malfunction of the system. At the same time, BaTiO

_{3}layers should be as thick as possible to minimise their capacitance so that the amount of generated electrical power is not suppressed.

_{res}within the layers and at the same time to obtain the highest possible resistance to surface crack propagation quantified with the so-called apparent fracture toughness [30]. The apparent fracture can be effectively calculated by employing the weight function method described in [31], evaluating the apparent fracture toughness using the distribution of thermal residual stresses within the layers and a so-called weight function. Upon calculation, the crack path is assumed to not be affected by thermal residual stresses (i.e., the crack is assumed to grow perpendicular to the structure’s surface). Thermal residual stresses σ

_{res}, which are required for determining the apparent fracture toughness, can be quantified within i-th layer using relations from classical laminate theory [32] as:

_{i}, ν

_{i}, and α

_{i}is the elastic modulus, Poisson’s ratio and thermal expansion coefficient of the i-th layer respectively and ΔT is the temperature difference between the room and the zero-strain (reference) temperature. The term $\overline{\alpha}$ represents the apparent thermal expansion coefficient of the whole laminate and h

_{i}is the thickness of the i-th layer. The distribution of σ

_{res}is assumed to not be affected by piezoelectric properties of BaTiO

_{3}layers. The apparent fracture toughness K

_{R,eff}can then be calculated using the weight function approach [31] as:

_{c,}

_{0}is the intrinsic fracture toughness of current layer, a is the crack length, and h(z, a) is a weight function defined, e.g., in [33].

_{R,eff}in the protective layers and at the same time keep the values of σ

_{res}within the layers at a reasonably low level by changing thicknesses and material order of the substrate and protective layers without changing the total thickness of the structure. The optimization process was split into two phases. During the first phase which is schematically shown in Figure 3, the individual materials within outer protective layers and the substrate were changed from ATZ to ZrO

_{2}. The thickness ratio h

_{1}/h

_{2}of the outer protective layers was changed in such a manner that either both layers were equally thick or one of them was significantly thicker. The thickness of BaTiO

_{3}layers h

_{p}could take values from a discrete set {0.1, 0.15} mm and the thickness of substrate h

_{s}could take values from a discrete set {0.2, 0.3, 0.4} mm. The whole structure was subjected to a temperature change ΔT = −1430 °C. This value represents a common temperature difference between the zero-strain temperature and the room temperature [34]. The first phase showed that the best results which are presented in Figure 4, i.e., reasonably low level of thermal residual stresses and sufficiently high apparent fracture toughness in outer protective layers, are achieved when both outer protective layers are equally thick (h

_{1}/h

_{2}= 1), BaTiO

_{3}layers being 0.15 mm thick and ZrO

_{2}substrate being 0.4 mm thick. Other combinations lead either to high thermal residual stresses within ATZ and BaTiO

_{3}layers or no increase in apparent fracture toughness (K

_{R,eff}was not higher than the intrinsic fracture toughness K

_{c,}

_{0}of a particular material) of outer protective layers.

_{i}of used materials were changed so that the BaTiO

_{3}layers were as thick as possible in order to reduce their capacitance. The resulting, optimised configuration is shown in Figure 5. ZrO

_{2}layers including the substrate were made thinner to allow for thicker BaTiO

_{3}layers due to their similar thermal expansion coefficients α.

_{3}layers and high compressive stresses of −433 MPa in ATZ layers. The Figure 6b shows a significant increase of apparent fracture toughness in ATZ protective layers, compared with the intrinsic value of 3.2, the actual value of 11.73 means almost four-times improved resistance to unstable surface crack propagation. Note that the increase in thickness of BaTiO

_{3}layers led to a much higher apparent fracture toughness in ATZ protective layers and substantially lower tensile residual stresses within BaTiO

_{3}layers compared with the results from the first optimization phase (Figure 4).

## 3. 1DOF Model of Multilayer Piezoelectric Harvester

_{r}, its transverse displacement relative to the base w can be represented by its first, mass normalised, mode shape ϕ

_{1}and a modal coordinate η as:

_{r}can be extracted from the following transcendental equation from [35]:

_{t}is the tip mass and m is mass of the composite beam per unit of its length which is simply defined as:

_{i}is a density of the i-th layer. λ

_{1}is then defined as:

_{i}and cross-sectional moment of inertia J

_{i}of i-th layer referenced to the geometrical centre of the beam’s cross section. ϕ

_{1}can then be extracted from:

_{1}is defined as:

_{1}in (7) represents a modal amplitude constant which should be evaluated from (9) so that the first mode shape is mass normalised:

_{r}is a damping ratio, κ is the modal electromechanical coupling term, U is the voltage drop generated in piezoelectric layers, f is the modal forcing function, C

_{p}is the capacitance of piezoelectric layers, and R

_{l}is connected resistive load. κ is in [35] defined as:

_{31}is piezoelectric modulus of BaTiO

_{3}– ${e}_{31}=E{d}_{31}$, h

_{p}is the thickness of BaTiO

_{3}layers, and h

_{s}is the thickness of the substrate. Next, f is defined as:

_{0}is the acceleration amplitude of kinematic excitation. C

_{p}is simply defined as:

_{3}measured at constant mechanical strain and defined as ${\u03f5}_{33}^{S}={\u03f5}_{33}^{T}-{d}_{31}{e}_{31}$.

## 4. Design of Auxiliary Magnetic Spring and Model of Bistable Energy Harvester

_{3}piezoceramic layers (Section 2.1) and the auxiliary magnetic system which are analysed and designed for maximal effectivity of harvested power, exactly for the presented BaTiO

_{3}architecture.

_{eff}, it is summed with the nonlinear force from magnets, resulting in the total force. This resultant force affects the vibrations of the considered energy harvester and allow for calculation of oscillator’s potential energy via this resultant force.

_{beam}= k

_{eff}.q). The total force is the main characteristic that might be used to estimate the behaviour of the energy harvester. It could provide stable behaviour with a hardening or softening characteristic, or bistable behaviour.

_{mag}is the force from auxiliary magnets, the same as in Figure 11.

## 5. Simulation Results

## 6. Estimation of Critical Tip Displacement Amplitude

_{3}layer, and thus causes a malfunction of the whole multilayer structure. To determine critical displacement amplitude of the harvester’s free end, the stress intensity factor K

_{appl}at the tip of the crack within ATZ protective layers upon vibrations must not be higher than 11.7 MPa·m

^{0.5}(see the end of Section 2.1). The stress intensity factor K

_{appl}is defined through the weight function method as:

_{x}

_{0}(z) is the amplitude of bending stress distribution within the multilayer structure. The highest values of bending stress upon vibrations consisting solely of the first mode shape are found in the vicinity of the clamping at x = 0, which is, therefore, the most suitable location for potential surface cracks. The amplitude of bending stress distribution σ

_{x}

_{0}(z) is, in the case of heterogenous multilayer structure, a layer-wise function and near the clamping at x = 0, it can be expressed in individual layers as:

_{x}

_{0}(z) and stress intensity factor K

_{appl}are assumed to vary linearly with the beam tip displacement, one can easily estimate the critical tip displacement amplitude using a safety factor k

_{BF}defined as:

_{ATZ}is the location of ATZ/BaTiO

_{3}interface. If the calculated safety factor k

_{BF}is equal or lower than 1, then a potential surface crack will propagate through both protective layers and cause a brittle fracture of the whole multilayer structure.

_{appl}along with the apparent fracture toughness K

_{R,eff}calculated in Section 2.1. The stress intensity factor K

_{appl}reaches the value of 7MPa·m

^{0.5}at ATZ/BaTiO

_{3}interface which gives a safety factor of 1.67. Therefore, the critical amplitude of the free end’s displacement upon which a brittle fracture occurs is approximately 10 mm. This means a significant increase in resistance against the surface crack propagation compared to a typical harvester from [10] whose critical amplitude of the free end displacement is about 3 mm.

## 7. Discussion

#### 7.1. Effect of the Protective Layers

_{3}beam without the protective layer might have critical tip displacement around 1 mm, which means about 10 times higher crack resistivity with layered architecture than without it.

#### 7.2. PZT Solution vs. Proposed Lead-Free Design

#### 7.3. Effect of the Auxiliary Magnets Producing Nonlinear Behaviour

## 8. Conclusions

_{3}bimorph cantilever can only operate in low amplitudes of the excited oscillations due to a very brittle behaviour. The proposed design of a lead-free cantilever with protective layers allows to use this design for operation in an extended bistable system with significantly higher amplitudes. Due to the implementation of the proposed cantilever design, the auxiliary magnets were used to produce the energy harvesting system with two extended equilibrium distance of bistable system. Furthermore, the stress intensity factor at the tip of potential surface crack in the system, calculated for the maximal displacement demonstrates that the designed architecture is more resistant to surface crack propagation in the simulated operational range.

_{3}material is lead-free, which is very important recently. Moreover, the proposed design is improved by the protective layers allowing higher excitation amplitudes for which the power is just 50% lower than at commercial PZT solutions.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**A ZrO

_{2}–ATZ (Alumina Toughened Zirconia)–BaTiO

_{3}composition after the first phase optimization: (

**a**) distribution of thermal residual stresses within the layers and (

**b**) the apparent fracture toughness.

**Figure 5.**The second phase of optimization showing the input configuration from the first optimization phase (

**top**) and the resulting ideal layer composition for the multilayer piezoelectric harvester (

**bottom**).

**Figure 6.**The calculated outcome for the ideal layer configuration loaded with ΔT = −1430 °C: (

**a**) Distribution of thermal residual stresses within the layers. (

**b**) The apparent fracture toughness of the structure.

**Figure 7.**Topology of used auxiliary magnets fixed on the novel multilayer piezoelectric cantilever beam.

**Figure 9.**The magnetic force and potential energy in bistable oscillator with (

**a**) huge barrier and (

**b**) low barrier between two stable positions.

**Figure 10.**Final design of the Auxiliary Magnetic System; dimensions are in (mm). The depth is 10 mm.

**Figure 13.**Sweep up with amplitude 0.5 g. (

**a**) Linear, (

**b**) nonlinear oscillator, and (

**c**) detail of nonlinear oscillatations.

**Figure 14.**Sweep up with amplitude 0.9 g. (

**a**) Linear, (

**b**) nonlinear oscillator, and (

**c**) detail of nonlinear oscillations.

**Figure 15.**(

**a**) Amplitude of the bending stress within the laminate kinematically excited with an amplitude of 0.9 g. (

**b**) Stress intensity factor K

_{appl}(blue solid curve) corresponding to excitation amplitude 0.9 g and the apparent fracture toughness K

_{R,eff}(red dashed curve); brittle fracture of the whole structure occurs if K

_{appl}> K

_{R,eff}in ATZ layer, i.e., if the blue solid curve appears in the red-filled area.

Material | ρ (kg/m ^{3}) | E (GPa) | ν (–) | α (ppm/K) | K_{c,0} (MPa·m^{0.5}) | d_{31}(C/N) | ${\mathit{\u03f5}}_{33}^{\mathit{T}}/{\mathit{\u03f5}}_{0}$ (–) | |
---|---|---|---|---|---|---|---|---|

ATZ | 4050 | 390 | 0.22 | 9.8 | 3.2 | – | – | |

ZrO_{2} | 5680 | 210 | 0.31 | 10.3 | 3 | – | – | |

BaTiO_{3} | 6020 | 70 | 0.22 | 11.5 | 0.7 | –58·10^{-12} | 1250 |

m_{eff}(g) | b_{eff}(Ns/m) | k_{eff}(N/m) | m_{acc}(g) | C_{p}(nF) | θ (μN/V) | R_{l}(kΩ) |
---|---|---|---|---|---|---|

6.3 | 3.75·10^{-2} | 555.8 | 7.1 | 75.6 | 245 | 45 |

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**MDPI and ACS Style**

Rubes, O.; Machu, Z.; Sevecek, O.; Hadas, Z.
Crack Protective Layered Architecture of Lead-Free Piezoelectric Energy Harvester in Bistable Configuration. *Sensors* **2020**, *20*, 5808.
https://doi.org/10.3390/s20205808

**AMA Style**

Rubes O, Machu Z, Sevecek O, Hadas Z.
Crack Protective Layered Architecture of Lead-Free Piezoelectric Energy Harvester in Bistable Configuration. *Sensors*. 2020; 20(20):5808.
https://doi.org/10.3390/s20205808

**Chicago/Turabian Style**

Rubes, Ondrej, Zdenek Machu, Oldrich Sevecek, and Zdenek Hadas.
2020. "Crack Protective Layered Architecture of Lead-Free Piezoelectric Energy Harvester in Bistable Configuration" *Sensors* 20, no. 20: 5808.
https://doi.org/10.3390/s20205808