Influence of different embedding methods on flexural and actuation performance of piezoelectric intelligent structures

MFC is a kind of piezoelectric fiber sheet with excellent performance, and its essence is a composite material. When MFC modeling, it is often established for the core actuation area without considering its non-actuation area. For the surface-bonded piezoelectric intelligent structure, the influence of the non-actuating region can be ignored in the establishment of the MFC piezoelectric simulation model. The way can simplify the model and increase the calculation efficiency. However, embedded structure needs to consider the influence of the non-actuating area. This paper mainly studies the P1 type MFC. In the process of research, MFC is divided into five layers according to literature [34], as shown in figure 1, namely the kapton layer, the electrode layer, the actuator layer, the electrode layer, and the kapton layer. The thickness of each layer is 0.04 mm, 0.018 mm, 0.18 mm, 0.018 mm, and 0.04 mm, respectively.

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The MFC-d₃₃ type applies an inverse voltage on the adjacent electrodes of the same layer and the same voltage on the symmetrical interdigital electrode layer, so that a relatively evenly distributed electric field load is generated in the fiber direction of the active layer, and the distance between the adjacent electrodes is 0.5 mm. In this paper, the equivalent parameter modeling method is used to establish the model of the MFC actuation region, and the model parameters refer to [34].

There are two embedding methods of MFC, namely forming embedding and cutting embedding, as shown in figure 2. In order to obtain an accurate finite element model, the sections of two kinds of piezoelectric curved shells can be observed through an optical microscope, and the size parameters of the finite element model can be determined through analysis. According to the observation, the end wrinkling area in forming embedding is more obvious than that in cutting embedding mode. The profile under two embedding modes is obtained by fitting.

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The specimen is prepared by the vacuum assisted resin infusion process (VARI). Because MFC is embedded in the composite curved shell, its preparation process is more complex. The preparation diagram is shown in figure 3.

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The specimen preparation process is as follows:

• (1)  
  Layering. In order to facilitate the release, the release agent should be repeatedly smeared on the bending shell mold before layering. In order to ensure a better release effect, it is necessary to apply the release agent twice, once to volatilize the release agent and then once to apply the release agent, and then the layering starts after the release agent volatilizes again. In view of the complex and variable layering angles of composite laminates, which have a significant impact on the overall performance of the structure, only orthogonal layering is considered in this paper, that is, only 0° and 90°. Since the thickness of the MFC is close to that of the single fiber layer, a single layer of fiber cloth can be cut when the cutting embedding method is used. At the same time, it is necessary to arrange the MFC wires reasonably to avoid the influence of excessive embedding of wires on the mechanical properties of the piezoelectric curved shell.
• (2)  
  Vacuum bag package. After the laying of the carbon fiber cloth is completed, a flexible vacuum bag is used to package the carbon fiber cloth, and a layer of demoulding cloth is placed on the top carbon fiber cloth to separate the laminate from the mold after forming. According to the requirements of the parts, size, and shape to determine the resin flow into the channel and out of the channel. The resin flow was guided by the flow guide network, which was divided into two sections when laying the flow guide network. The resin flowed slowly in the middle section without the flow guide network, so that the resin was fully infiltrated into the carbon fiber cloth.
• (3)  
  Resin configuration. According to the size and thickness of the parts, the required amount of resin is calculated. The epoxy resin glue and curing agent are prepared according to the ratio of 10:3, and the glass rod is used to stir until the mixture is evenly mixed. The bubbles in the mixture are removed by a vacuum pumping press, lasting for 45 min.
• (4)  
  Resin injection. When the vacuum degree of the pressure gauge on the vacuum pumping machine reaches above 0.95 MPa, turn off the vacuum pumping machine. Check that the pressure gauge remains unchanged after 15 min before resin suction. When the other end of the rubber mouth overflows resin, the vacuum tube of the rubber mouth of the first layer of vacuum bag is clamped with a clip, so that the resin in the second layer of vacuum bag pressure under the action of 15 min continues pumping, the air bubbles in the plate from the rubber mouth.
• (5)  
  Heat curing. The mold was put into the oven for heating and curing, and the oven temperature was set to 80 °C for 16 h. During the heating and curing process, the working state of the air drying oven was checked regularly to ensure the normal curing process.
• (6)  
  Demoulding. After the setting time of the oven is over, turn off the power supply and wait for the temperature in the oven to drop below 40 °C. Open the chamber and take out the mold for demolding. The piezoelectric curved shell obtained by demoulding is shown in in figure 4.
• (7)  
  Composite laminates cutting. High pressure water cutting quality is stable, high efficiency, and mechanical cutting not only incision chip accumulation phenomenon, and fiber and resin matrix interface will be destroyed. Considering the above factors and the existing conditions of the laboratory, this test decided to adopt the high-pressure water cutting method.

Unconstrained support was used in the bending performance test. The indenter made the specimen reach the failure state or the predetermined deflection value at a constant loading speed, and then the applied force and the deflection at the corresponding time were obtained. Referring to ASTM D7264, the specimen for the three-point bending test is designed, and the span thickness ratio is set to 32:1. T700/5015 is selected for CFRP. The models are divided into two types, that is, forming embedding and cutting embedding. The finite element models for the three-point bending test are established respectively. The model of the three-point bending test is shown in figure 5. The CFRP layer is designed to be six layers. The layers are [0/90/0]₂, [0/90]₃, [0]₆ and [90]₆. The thickness of the specimen is related to the embedding method. The thickness of the non-embedded MFC is 1.5 mm, the thickness of the forming embedding is 1.7 mm, and the thickness of the cutting embedding is 1.6 mm. In order to obtain better test results, the standard MFC with model M2503-P1 is selected as the embedded element to complete the three-point bending test of the embedded piezoelectric structure. The detailed geometric parameters are shown in table 1.

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Combined with relevant papers, manufacturers and some experiments, the mechanical parameters of relevant materials are obtained as shown in table 2.

Table 2. Mechanical property parameters.

Material	Property	Value
	tensile modulus	${E}_{11}=108.31\,{\rm{GPa}},\,{E}_{22}={E}_{33}=8.91\,{\rm{GPa}}$
	shear modulus	${G}_{12}={G}_{13}=5.56\,{\rm{GPa}},{G}_{23}=3.55\,{\rm{GPa}}$
	Poisson's ratio	${\mu }_{12}={\mu }_{13}=0.3,{\mu }_{23}=0.381$
T700/5015	tensile strength	${X}_{T}=1154.8\,{\rm{MPa}},{Y}_{T}=31\,{\rm{MPa}}$
	compressive strength	${X}_{C}=1175.6\,{\rm{MPa}},{Y}_{C}=189\,{\rm{MPa}}$
	shear strength	${S}_{12}={S}_{23}=80\,{\rm{MPa}}$
	density	ρ = 1.59 × 10−9 T mm−3
Epoxy resin	modulus	$E=2.5\,{\rm{GPa}}$
	Poisson's ratio	$\mu =0.35$
	density	ρ = 1.2 × 10−9 T mm−3
MFC Actuation area	tensile modulus	${E}_{11}=29.4\,{\rm{GPa}},{E}_{22}={E}_{33}=15.2\,{\rm{GPa}}$
	shear modulus	${G}_{12}={G}_{13}=6.06\,{\rm{GPa}},{G}_{23}=5.79\,{\rm{GPa}}$
	Poisson's ratio	${\mu }_{12}={\mu }_{23}={\mu }_{13}=0.312$
	density	ρ = 5.44 × 10−9 T mm−3
MFC Non actuation area (Kapton)	modulus	$E=2.5\,{\rm{GPa}}$
	Poisson's ratio	$\mu =0.34$
	density	ρ = 1.42 × 10−9 T mm−3
Cohesive	initial stiffness of interface	${K}_{{\rm{I}}}={{\rm{K}}}_{{\rm{II}}}={K}_{{\rm{III}}}\,=$ 106 N mm−1
	interface peak strength	σmax,I = 50 MPa, σmax,II = 88 MPa, σmax,III = 88 MPa
	interlaminar fracture toughness	GI = 0.55 kJ mm−2, GII = 0.65 kJ mm−2, GIII = 0.65 kJ mm−2
	density	ρ = 1.4 × 10−9 T mm−3

For carbon fiber reinforced composites, the element control of the grid adopts a sweep mode. At the same time, it should ensure that the scanning direction is consistent with the stacking direction of the material attribute setting, that is, it should be the thickness direction, and the element is set as SC8R. For the non-actuating area and resin aggregation area of piezoelectric materials, the structure unit control mode is adopted, and the unit is set as C3D8R. For the actuating area of piezoelectric materials, the sweep unit control mode is adopted. At the same time, it is also necessary to ensure that the scanning direction is consistent with the stacking direction set by the material properties, that is, they are all in the thickness direction, and the unit is set as C3D8E.

Theoretically, the smaller the mesh of the finite element model is divided, the higher the calculation accuracy is. However, when the model is large, as the grid size becomes smaller and the grid density becomes larger, the amount of calculation of the model will increase significantly, so the calculation time will. At the same time, local grid refinement is carried out for the areas with complex stress changes, such as the resin aggregation area. In the model, the single-layer thickness direction of the composite is set as a fixed grid, that is, the grid size is 0.25. The thickness of the actuating region and non-actuating region in the thickness direction of the piezoelectric material is set as a three-layer grid. The size of the grid in the length and width directions needs to be carefully considered.

In this paper, the convergence criterion is used to plan the grid size reasonably, and the influence on the simulation results when the grid size decreases and the number of elements increases is considered. In order to avoid stress singularity, stress is not used as the judgment target of convergence, but the maximum displacement at the end is used as the judgment target. In the simulation process, the size of the grid is gradually reduced and the number of elements is increased from 1056 to 312620. It can be seen from figure 6 that when the grid size is less than 1 mm, the calculation accuracy is significantly improved. However, the improvement is no longer obvious as the size decreases, and the calculation time increases sharply, so the final grid size is 1 mm.

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The three-point bending of curved beam without embedded MFC, cutting embedding curved beam and forming embedding curved beam are simulated by finite element method, respectively. The results of the force varying with displacement are shown in figure 7. It can be seen that under the two ply angles, the force of the punch is the largest under the conditions of forming embedding, and the force is small and close under the conditions of non-embedded MFC and cutting embedding.

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The bending performance of the model is analyzed, and the maximum stress appears in the middle of the convex surface. The calculation formula for bending strength is as follows:

$$\begin{eqnarray}&&{\sigma }_{f}=\displaystyle \frac{3P\cdot l}{2b\cdot {h}^{2}}\end{eqnarray} \tag{ 1 }$$

In the formula, ${\sigma }_{f}$ is the stress on the outer surface of the midpoint of the span, MPa; P is the force, N; l is the span, mm; b and h are the width and thickness, mm.

The results calculated by the formula (1) are shown in table 3. It can be seen that both embedding methods have influence on the bending strength of the piezoelectric curved shell.

In order to further explore the influence of the two embedding methods on the bending performance of the structure, the changes in the bending strength of the two embedding methods under different layers are analyzed, as shown in table 4. The results show that the bending strength caused by embedding MFC will also change with the ply direction, but the forming embedding always has better results. The forming embedding of MFC will cause resin aggregation in the surrounding area, and local fiber ply will wrinkle. The cutting embedding will destroy the continuity of the fiber and reduce its bending strength.

The actuation model is set as a cantilever curved shell, and the M8557-P1 MFC is embedded as an actuator. When building the model, MFC is also divided into cutting embedding and forming embedding, and the actuation performance is compared in the end. The cantilever curved shell model is shown in figure 9, and its geometric parameters are shown in table 7, where the radius of the middle surface of the column shell is R, the thickness is t, the width is w, and the length is L. The distance between the embedded MFC position and the cantilever end is L₁.

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CFRP also adopts the ply angles of [0/90/0]₂ and [0/90]₃, and the thickness of a single layer is 0.25 mm. MFC is embedded in the second layer. The piezoelectric performance parameters of MFC are shown in table 8.

Table 8. Piezoelectric property parameters in the MFC actuation area.

Performance	Value	Performance	Value
d11	$467\times {10}^{-12}\,{\rm{m}}\,{{\rm{V}}}^{-1}$	Dielectric coefficient	$1.43\times {10}^{-8}\,{\rm{F}}\,{{\rm{m}}}^{-1}$
d12	$-210\times {10}^{-12}\,{\rm{m}}\,{{\rm{V}}}^{-1}$	Electrode spacing	$0.5\,{\rm{mm}}$
Voltage range	−500 ∼ 1500V	Thickness	$0.3\,{\rm{mm}}$

The piezoelectric curved shell is arranged as a cantilever curved shell structure. The voltage is applied to the MFC. Considering that the electrode inside MFC-d₃₃ is a flexible interdigital electrode and the direction of the electric field is consistent with the direction of fiber actuation, the voltage is applied by equivalent voltage. The voltage applied to end face A is 0 ∼ 1500 V, and the voltage applied to end face B is 0 V. The relationship between equivalent voltage and applied voltage is as follows:

$$\begin{eqnarray}&&{U}_{e}=\displaystyle \frac{U\times L}{{h}_{e}}\end{eqnarray} \tag{ 2 }$$

In the formula, $U$ is the applied voltage; $L$ is the effective actuation length; and ${h}_{e}$ is the distance between adjacent electrodes.

Figure 10 shows the strain distribution, displacement response and MFC stress distribution of cantilever curved piezoelectric shell under different embedding modes at 1000V voltage. When a voltage is applied to the MFC, the deformation of the MFC will cause the deformation of the structure. Due to the action of the resin aggregation area, the normal stress of the MFC actuating end face is weakened and the shear stress becomes the dominant stress. The normal stress caused by the end face of MFC will directly act on the cut fiber layer, causing the fiber layers on both sides of MFC to produce strain, and greater deformation can be obtained under the action of shear stress. When MFC is embedded in cutting mode, the cutting layer is subject to compression deformation under the action of the MFC, and shear deformation occurs between layers, as shown in figures 10(a) and (b). In contrast, in the way of forming embedding, the large strain is concentrated near the action area of the MFC. As shown in figures 10(c) and (d), the piezoelectric curved shell obtains greater deformation by cutting embedding.

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Figure 11 shows the displacement response of the end intermediate node of the structure with the increase of applied voltage under different MFC embedding modes. The results show that the curved shell of MFC by forming embedding has better displacement response under the same voltage, and the difference in end displacement response gradually increases with the increase of voltage.

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Figure 12 shows the change curve of the MFC end face actuation force of the end intermediate node of the structure under different embedding modes with the increase of applied voltage. The results show that the cutting embedding has greater end face actuation force than the forming embedding. Similarly, the difference of actuation force gradually increases with the increase of voltage.

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