Abstract
Piezoelectric materials have commonly been used in pressure and stress sensors; however, many designs consist of thin plate structures that produce small voltage signals when they are compressed or extended under a pressure field. This study used finite element methods to design a novel piezoelectric pressure sensor with a C-shaped piezoelectric element and determine if the voltage signal obtained during hydrostatic pressure application was enhanced compared to a standard thin plate piezoelectric element. The results of this study demonstrated how small deformations of this C-shaped sensor produced a large electrical signal output. It was also shown that the location of the electrodes for this sensor needs to be carefully chosen and that the electric potential distribution varies depending on the poling of the piezoelectric element. This study indicated that the utilization of piezoelectric materials of different shapes and geometries embedded in a polymer matrix for sensing applications has several advantages over thin plate solid piezoelectric structures.
1. Introduction
Pressure gauges of different design types (e.g., U-shaped tube gauges, piston gauges, aneroid gauges, Bourdon tube, or diaphragm gauge [1, 2], optical fiber sensors [3, 4], different electronic sensors) are currently used in different applications. Electronic sensors are convenient because they allow for easy, direct integration into electronic control schemes which can be easily miniaturized and have a short response time when used under dynamic conditions. Electronic pressure sensors can be designed by using different electromechanical or magnetomechanical effects. Electronic sensor types include piezoelectric [5], piezoresistive [6], capacitive, magnetic (inductive), potentiometric, resonant, and surface acoustic wave sensors. In the fields of robotics and orthotics the McKibben actuators have been utilized to mimic the behavior of biological muscles. Son and Goulbourne, 2009 [7] showed how the use of the two electrical parameters capacitance or resistance could be used to measure the large strains/pressure of the actuating device. MEMS based on piezoresistive pressure sensors have also been considered for pressure measurement applications; however they posses low sensitivity and suffer thermal drift [6, 8]. Piezoelectric pressure sensors are commonly used in sensor designs due to their high reliability and robustness, large range of measurable pressure, and low sensitivity to the electro-magnetic field. In traditional piezoelectric pressure sensor designs, the piezoelectric element is manufactured as a relatively simple shape such as a disc or plate and is subjected to a simple compressive or tensile load applied by the force collector, such as a diaphragm [5, 9–12]. In more sophisticated designs, the piezoelectric film or thin plate is attached to the substrate and subjected to flexure deformations [13, 14] which produce a combination of bending and stretching deformation in the piezoelectric component and significantly enhance the piezoelectric response [14]. Recently, it has been demonstrated that the bending of piezoelectric nanowires or nanobelts can produce a very strong electric potential output [15–17] which can be used for energy harvesting. One problem which can be faced with practical implementation of piezoelectric materials in sensor or structural applications is low toughness and high probability of brittle fracture. In that respect the use of piezoelectric materials in form of thin fibers or thin plates embedded in polymer matrix is similar to the use of carbon fibers that are brittle as individual component but very resilient when these are embedded as part of a structure in an epoxy matrix. Thus, the widespread use of piezoelectric materials either as structural components or sensors can be resolved by using piezoelectric ceramics in composite materials [18–20].
In the present paper we analyze the possibility of a pressure sensor design in which the shape of the stiff piezoelectric component embedded into the compliant polymer facilitates the bending mode of the piezoelectric sensor deformation when hydrostatic pressure is applied to the composite sensor structure. We consider a relatively simple C-shaped piezoelectric element embedded into a cubic shaped polymer matrix (Figure 1). It is important to note that this study focuses on the piezoelectric element of the pressure sensor which can then be incorporated into the different sensor designs employed to augment the voltage output of the sensing elements as found in the literature [11, 21]. Finite element analyses (FEAs) are used to evaluate the response of this novel sensor design to applied pressure and compare it to the response of a conventional plate-shaped piezoelectric element (Figure 2) subjected to the same pressure conditions.
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2. FEA of a Piezoelectric Sensor
The FEA of an electromechanical system is based on the mechanical and electrical equilibrium equations: where represents the divergence operator, is the stress, represents the body forces, and is the electrical displacement. Material properties are presented in the model by the constitutive equations in the stress-charge form: where is the elastic material matrix, and are the piezoelectric material matrix and its transpose, respectively, is the permittivity matrix, is the strain, and is the electrical field. Material parameters used in (2) can be presented in a matrix form as The finite element formulation for the linear piezoelectric analysis, derived by using the principle of virtual work, is shown in: where is the matrix that maps displacement to strain, is the interpolation matrix, is the displacement and electric potential vector, represents the body forces acting on the system, and is the surface traction vector.
By using (4), an FEA solver has been developed to calculate the displacement and electric potential , which can then be used to compute the strain and electric field [20]. Once the strain and electric field are obtained, the stress and electric displacement can be calculated by using (2). The solver uses the Intel Math Kernel Library Sparse Solver for both direct and indirect solutions. All simulations have been performed on a Windows Workstation with an Intel Core 2 Extreme Processor (2 GHz) and 8 Giga bytes of RAM. All the models have been analyzed using the direct solver. The in-house FEA code makes use of both the 8- and 20-node brick linear elastic and piezoelectric hexahedral elements.
3. FEA Model Parameters
The geometry of the composite sensor with a C-shaped piezoelectric component is shown in Figure 1. The C-shaped piezoelectric element had a thickness of 50 μm and was embedded into a polymer cube of mm. The mesh of the piezoelectric element and the polymer matrix was constructed using 20-node brick piezoelectric elements. However, the material properties of the polymer matrix corresponded to those of the polymer with null piezoelectric constants. The boundary conditions were set to provide 100,000 Pa of pressure uniformly applied to the surfaces of the polymer (hydrostatic pressure). Boundary conditions were applied to prevent rigid body motion and rotation. A single node at the corner of the polymer cube was assigned a zero volt electric potential. The material properties of the piezoelectric component of the sensor were modeled with PZT-5A elements using the stress-charge form of the constitutive equations with the following values: = = N/m2, = = N/m2, = = = = N/m2, = N/m2, = = N/m2, and = N/m2. The piezoelectric properties were modeled as = = −5.351 C/m2, = 15.783 C/m2, and = = 12.295 C/m2. The permittivity was set to F/m and = F/m. In the first model the polarization orientation was set to follow the C-shaped contour of the sensor element, changing discretely between the sections of the sensor element shown with different colors in Figure 1. A second model of the C-shaped embedded sensor with the polarization oriented through the thickness of the sensor was also studied. The properties of the polymer matrix were modeled using the stress-charge form of the constitutive equations with the following values: = = N/m2, = = N/m2, = = = = N/m2, = N/m2, and = = = N/m2. The piezoelectric properties were set to zero. The permittivity was set to F/m.
A thin piezoelectric plate of was also modeled to obtain the reference values of a piezoelectric response as shown in Figure 2. The boundary conditions on the plate were applied to prevent a rigid body motion and rotation. The piezoelectric material was defined using PZT-5A elements and polarized along one side of the thin square plate. A zero volt electric potential was assigned at a single node at the corner of the piezoelectric plate.
4. Results
The C-shaped piezoelectric sensor polarized along the perimeter and embedded in an epoxy matrix under a hydrostatic pressure field was analyzed using FEA. The results of this study demonstrated the formation of an internal bending mode in the piezoelectric C-shaped sensor element. The bending mode is produced in all locations of the C-shaped sensor element as shown in Figure 3(a). The electric potential difference generated in this sensor design was approximately 11.5 V as shown in Figure 3(b). This high potential difference is approximately 11.3 V greater than that of a thin piezoelectric plate of the same thickness polarized along its length under the same hydrostatic pressure (Figure 2). This represents an increase in electric potential difference by a factor of approximately fifty seven times when compared to that of a thin piezoelectric plate. The electric potential distribution on the C-shaped sensor element shown in Figure 3(b) indicates that the electrodes should be placed at the tips of the C-shaped element. However, the ability to polarize a C-shaped sensor element along its length would present a challenge during manufacturing and would require that multiple interdigitated electrodes be placed at every corner of the sensor. Another challenge presented by this type of sensor is its manufacturing. The C-Shape piezoelectric component could be micromachined using lasers as it has been shown in [22]. Other authors have shown the use of etching processes for micromachining of bulk piezoelectric substrates [23]. Although, these etching processes are mostly applied to thin films, it is foreseeable that thin C-shaped film sensors could be manufactured for later encapsulation into an epoxy matrix. Finally, a Ceramic Injection Molding combined with sintering and micromachining could be used to produce the basic shape of the sensor as described in [24]. The electrodes could then be printed on the sensor through deposition technology as it is commonly performed in the semiconductor and electronic industry. The final step of the manufacturing process would be the attachment of miniaturized wires to the sensor electrodes for polarization and signal reception. The sensor itself would then be surrounded by an epoxy matrix to protect the sensor and provide it with resilience of a composite structure as well as the required interaction between the piezoelectric element and the epoxy matrix necessary for the augmentation of the electric potential output described in this study.
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The second set of simulations analyzed the performance of the same C-shaped piezoelectric sensor element with a through-thickness polarization. A through thickness polarization would only require two electrodes placed on the inner and outer surfaces of the C-shaped sensor element and would make the sensor easier to manufacture since miniaturized interdigitized electrodes through the perimeter of the sensor would not be required. The results shown in Figures 4(a)-4(b) indicate that the deformed sensor produces an electric potential of approximately 0.6 V. This electric potential is significantly lower than it would be if the sensor was polarized along its circumference. However, it has also been shown that the electric potential output is still approximately three times greater than that of a single thin piezoelectric plate under the same hydrostatic pressure.
5. Conclusions
The results of this study indicate that an enhanced electric potential output from a composite sensor with a C-shaped piezoelectric element can be obtained. This larger signal output is achieved through a combination of bending deformation of the piezoelectric element and polarization orientation along the piezoelectric material portion of the sensor. This study demonstrates that a through thickness polarization orientation, while presenting benefits from a manufacturing point of view, results in a much lower signal output level. It has been shown that pressure sensors making use of piezoelectric materials can substantially increase their electric potential output by optimizing their deformation mode and material polarization distribution. Although this study has not optimized the shape and size of the sensor, it has been demonstrated that an electric potential signal could be increased by a factor of 3 to 57, based on the sensor shape design and the polarization of the material in comparison to a single flat plate made of the same piezoelectric material.