This paper demonstrates the importance of three-dimensional (3-D) piezoelectric coupling in the electromechanical behavior of piezoelectric devices using three-dimensional finite element analyses based on weak and strong coupling models for a thin cantilevered piezoelectric bimorph actuator. It is found that there is a significant difference between the strong and weak coupling solutions given by coupling direct and inverse piezoelectric effects (i.e., piezoelectric coupling effect). In addition, there is significant longitudinal bending caused by the constraint of the inverse piezoelectric effect in the width direction at the fixed end (i.e., 3-D effect). Hence, modeling of these effects or 3-D piezoelectric coupling modeling is an electromechanical basis for the piezoelectric devices, which contributes to the accurate prediction of their behavior.

Piezoelectric materials can serve as sensors based on the accumulation of electric charge in response to mechanical strain, called the direct piezoelectric effect (DPE) [

A typical simplified model of these devices is the thin cantilevered piezoelectric bimorph beam. Its electromechanical behavior has been extensively studied using a variety of theoretical approaches [

In general, the configuration of piezoelectric devices is more complicated in terms of geometry, material positioning, and electrode patterns. Numerical approaches are needed to simulate the electromechanical behavior of such systems [

The present study mainly focuses on thin cantilevered piezoelectric bimorph actuators and sensors. Previous studies used various simplified models for this configuration for both analytical and numerical approaches. However, a thin cantilevered piezoelectric bimorph sensor can have a complicated three-dimensional (3-D) distribution of the electric field in the thin piezoelectric continuum [

Similarly, for a thin cantilevered piezoelectric bimorph actuator, cases for which direct modeling of 3-D piezoelectric coupling is important must be determined. Many studies focus on the finite element modeling of adaptive structural elements, namely, solids, shells, plates and beams [

In

The electromechanical behavior of a piezoelectric material can be expressed as
_{ij}_{i}_{i}_{ijkl}_{ij}_{ijk}_{i}_{ij}_{ij}_{i}_{i}_{i}_{i}

The weak forms of _{i}

Substituting _{ij}

Substituting _{i}

By substituting the piezoelectric constitutive equations (

Based on these equations, the spatially discretized equations of motion for linear piezoelectricity in the global coordinate system are derived as
_{uu} is the global mechanical stiffness matrix,

Piezoelectric ceramics such as lead zirconate titanate (PZT), lithium niobate (LiNbO3), and lithium tantalate (LiTaO3) are the most commonly used piezoelectric materials. In general, these poled piezoceramics are transversely isotropic materials [^{−}^{1}. The vectors for the stress

For PZT, which is used in this study, the expanded forms of ^{E}

The focus of this study is the importance of 3-D piezoelectric coupling in the thin cantilevered piezoelectric bimorph actuator. For this purpose, strong and weak coupling methods are developed. These methods were implemented as in-house computer programs.

Various coupling modeling (monolithic, partitioned, or splitting; strong or weak; implicit or explicit) can be seen in previous literatures. The coupling methods can be roughly divided into two approaches. One is the monolithic approach, and the other is the partitioned approach. The monolithic approach gives strong coupling methods, while the partitioned approach gives both strong and weak coupling methods.

The performance comparison of the several coupling methods including the strong coupling method used here, the monolithic method, and the explicit method was conducted in our previous study for the purpose of illustrating the relative merit and demerit of these methods [

We consider a method based on the block Gauss-Seidel (BGS) algorithm, which is a partitioned iterative coupling algorithm. If convergence is achieved in the solution procedure of this method, the solution will exactly satisfy the coupled equation system. Such a method is said to be strongly coupled [

For a given ^{(i−1)}, which is obtained in the previous iteration ^{(i)} is obtained by solving the IPE equation (^{(0)} is set to ^{(i)} is the maximum bending displacement at the

We denote

Then, the displacement

This procedure results in a weak coupling method because it ignores the DPE, and the solution does not exactly satisfy the coupled equation system (

As shown in

As shown in

The piezoceramic layers are poled in the same direction, namely the thickness direction or the

It should be noted here that the solution procedure for the weak coupling method in the previous section is equivalent to that used to obtain the following closed-form solution of the tip bending displacement of the cantilevered bimorph beam [_{31} is the piezoelectric strain constant,

In this section, the effect of piezoelectric coupling on the analysis results is investigated. A thin cantilevered piezoelectric bimorph actuator is analyzed using the theoretical solution, the numerical weak coupling solution, and the numerical strong coupling solution. The ratio of the length to the sectional dimension is 100, which is large enough to sufficiently satisfy the Euler-Bernoulli beam assumption. Hence, the differences in the solutions are caused only by the differences in the coupling models. The weak coupling model, which considers only the IPE, is used for the theoretical and numerical weak coupling formulations, and the strong coupling model, which considers both the DPE and the IPE (i.e., piezoelectric coupling), is used for the numerical strong coupling formulation.

The piezoelectric actuator model presented in

Material^{*} |
P-10 | P-11 | P-12 | P-15 | P-17 | P-24 | P-31 | P-34 | P-37 | |

_{11}^{E} |
[10^{10} N/m^{2}] |
12.1 | 11.8 | 12.3 | 13.7 | 12.8 | 18.4 | 16.3 | 17.0 | 19.7 |

_{12}^{E} |
7.75 | 7.19 | 7.73 | 8.81 | 7.51 | 11.31 | 10.01 | 10.11 | 11.18 | |

_{13}^{E} |
7.65 | 7.53 | 7.65 | 8.83 | 7.81 | 11.49 | 10.32 | 10.57 | 12.38 | |

_{33}^{E} |
10.7 | 11.2 | 11.1 | 12.0 | 10.7 | 17.1 | 14.2 | 16.8 | 20.8 | |

_{44}^{E} |
2.24 | 2.35 | 2.55 | 1.71 | 2.53 | 3.47 | 2.61 | 3.52 | 4.42 | |

_{66}^{E} |
2.19 | 2.30 | 2.27 | 2.44 | 2.66 | 3.53 | 3.11 | 3.43 | 4.24 | |

_{31} |
[c/m^{2}] |
−4.38 | −5.67 | −7.75 | −10.8 | 2.73 | 2.82 | −3.39 | −3.39 | −2.61 |

_{33} |
18.6 | 19.2 | 23.1 | 8.7 | 35.6 | 21.3 | 18.7 | 14.4 | 11.0 | |

_{15} |
13.3 | 13.2 | 15.5 | 14.4 | 21.5 | 10.7 | 13.6 | 7.9 | 4.6 | |

_{11} |
[10^{−8} F/m] |
1.72 | 1.77 | 2.37 | 1.90 | 4.03 | 1.04 | 1.49 | 0.908 | 0.525 |

_{33} |
1.88 | 2.07 | 2.98 | 1.24 | 3.99 | 1.14 | 1.42 | 1.08 | 0.576 | |

_{31} |
[%] | 34.8 | 35.5 | 35.8 | 41.2 | 33.4 | 23.0 | 35.6 | 26.4 | 10.7 |

Note: ^{*} PZT materials listed here were fabricated by FDK corporation (

The typical finite element mesh is shown in

In order to check the mesh convergence, the bimorph with the P-15 material in

Mesh type | Number of divisions^{*1} |
Number of nodes | Number of elements | Tip deflection [μm] | Relative difference^{ *2} [%] |
||
---|---|---|---|---|---|---|---|

Finer mesh-1 | 100 | 1 | 4 | 3323 | 400 | 0.727245 | 5.40561 × 10^{−3} |

Finer mesh-2 | 100 | 2 | 4 | 5237 | 800 | 0.727284 | NA |

Coarser mesh-1 | 50 | 1 | 4 | 1673 | 200 | 0.727437 | 2.09589 × 10^{−2} |

Coarser mesh-2 | 100 | 1 | 2 | 1913 | 200 | 0.727000 | 3.90079 × 10^{−2} |

Notes: *1 Each mesh is uniformly divided in all directions.

*2 Relative difference between the tip deflection for each mesh and that for the finer mesh-2.

The strong and weak coupling methods are used to analyze the problem described in _{ij}_{ij}_{31} is used here, which is given as [_{31} is the piezoelectric strain constant, _{33} is the dielectric constant, and ^{E}_{11} is the elastic compliance constant. The values of _{31} for the PZT materials are summarized in

_{31} and the relative difference between the numerical solution and the theoretical solution for the tip displacement. The relative difference between the numerical weak coupling solution and the theoretical solution (white circles) ranges from + 0.028% to + 0.053% and is almost independent of _{31}. The Euler-Bernoulli beam assumption is sufficiently satisfied for the present cantilever and the same weak coupling model for piezoelectricity is used for the theoretical formulation in _{31}. For the tested materials shown in _{31} value led to a smaller relative difference between the solutions. For example, the relative difference is −0.68% for _{31 }= 10.7% (the lowest value of _{31} for the tested materials, i.e., PZT material P-37 in _{31 }= 41.2% (the highest value of _{31} for the tested materials, i.e., PZT material P-15 in

To provide more direct evidence of the importance of the DPE, the electric potential distributions in the thin piezoelectric continuum at the free and fixed ends are plotted along the thickness direction in _{31 }= 10.7% is distributed linearly along the thickness direction, as assumed in the theoretical formulation. In contrast, the electric potential for _{31 }= 41.2% (PZT material P-15 in

In this section, the importance of the 3-D piezoelectric effect in the analysis is investigated. First, a thin cantilevered piezoelectric bimorph actuator is analyzed using the theoretical solution and the numerical weak coupling solution. This structure is sufficiently thin but wider than the beam used in the previous section. The dimensions used here are based on an actual actuator. Roller support is used at the fixed end, as shown in

The piezoelectric actuator model presented in _{31} in

In order to check the mesh convergence, the bimorph with the P-15 material in

Mesh |
Number of divisions^{*1} |
Number of nodes | Number of elements | Tip deflection [μm] | Relative^{*2} [%] |
||
---|---|---|---|---|---|---|---|

(1) | 40 | 2 | 4 | 2117 | 320 | 0.870520 | N/A |

(2) | 20 | 2 | 4 | 1077 | 160 | 0.871462 | 0.108204 |

(3) | 40 | 2 | 2 | 1221 | 160 | 0.870573 | 0.00608226 |

(4) | 40 | 1 | 4 | 1343 | 160 | 0.871239 | 0.0826322 |

Notes: *1 Each mesh is uniformly divided in all directions.

*2 Relative difference between the tip deflection for each mesh and that for the mesh (1).

The weak coupling analysis using the 3-D finite element method described in ^{T}

First, we set the 32-component _{32} of the piezoelectric strain coefficient matrix

The piezoelectric strain constant

Piezoelectric strain constants [×10^{−12} m/V] |
Piezoelectric stress constants [c/m^{2}] |
||||||||

_{31} |
_{32} |
_{33} |
_{15} |
_{24} |
_{31} |
_{32} |
_{33} |
_{15} |
_{24} |

−180 | 0 | 337 | 845 | 845 | 5.10 | 13.9 | 24.5 | 14.4 | 14.4 |

Next, we set the 31-component _{31} of the piezoelectric strain coefficient matrix

The piezoelectric strain constant

Piezoelectric strain constants [×10^{−12} m/V] |
Piezoelectric stress constants [c/m^{2}] |
||||||||

_{31} |
_{32} |
_{33} |
_{15} |
_{24} |
_{31} |
_{32} |
_{33} |
_{15} |
_{24} |

0 | −180 | 337 | 845 | 845 | 13.9 | 5.10 | 24.5 | 14.4 | 14.4 |

The superposition of the numerical solution with the IPE in the length direction and the numerical solution with the IPE in the width direction is very close to the numerical solution with the 3-D IPE (the difference is approximately 0.04%). These results indicate that the difference between the weak coupling solution with the 3-D IPE and the theoretical solution is caused by the IPE in the width direction.

In the previous section, we demonstrated that the IPE in the width direction causes longitudinal bending deformation. In this section, this mechanism is discussed. For a width of

In the first step, the deformation from the initial position in Case B occurs due to the IPE in the width direction. In this case, as shown in

To confirm this interpretation, we consider a problem similar to the second step of this process, as shown in

Tip bending displacements are given by theoretical, numerical weak coupling (NWC) and numerical strong coupling (NSC) solutions for the materials P-15, P-24, and P-37 (see

Material | Theoretical and numerical solutions [μm] | Relative difference^{*3} [%] |
|||||

Theory | NWC^{*1} |
NSC^{*2} |
NWC^{*1} |
NSC^{*2} |
NSC–NWC | ||
---|---|---|---|---|---|---|---|

3 | P-15 | 0.86775 | 0.88585 | 0.79706 | +2.085 | −8.147 | −10.232 |

P-24 | 0.38659 | 0.39569 | 0.38377 | +2.354 | −0.728 | −3.082 | |

P-37 | 0.11707 | 0.11963 | 0.11877 | +2.185 | +1.459 | −0.726 | |

7 | P-15 | Equal to the case of |
0.91852 | 0.83013 | +5.851 | −4.336 | −10.187 |

P-24 | 0.41107 | 0.39927 | +6.332 | +3.281 | −3.051 | ||

P-37 | 0.12382 | 0.12298 | +5.766 | +5.052 | −0.714 |

Note: *1 Numerical weak coupling. *2 Numerical strong coupling. *3 Relative difference of numerical solution compared to theoretical solution.

_{A} and the tip displacement along the _{A} is defined as the ratio of the length to the width of the bimorph. In this figure, the numerical and theoretical solutions are distinguishable from each other for _{A} smaller than 5. Furthermore, in this range of _{A}, the numerical solution is larger than the theoretical solution, and their difference increases as _{A} decreases. In an experimental and theoretical study of the quasistatic response of the piezoelectric bimorph [

_{A} and the bowing at the tip of the bimorph, where the bending is the bending deformation along the width direction, the magnitude of the bowing is defined as the relative difference of the corner and midpoint displacements along the _{A }= 5 in the present numerical result and the previous experimental result [_{A} smaller than 5, and it becomes significant as _{A} decreases. This change of the bowing for _{A} correlates highly with the above-mentioned characteristic behaviors in the present computation and the previous experiment [

This study demonstrated the importance of 3-D piezoelectric coupling modeling for accurate predictions of the electromechanical behavior of piezoelectric devices. 3-D finite element analysis methods based on weak and strong coupling models were compared using a thin cantilevered piezoelectric bimorph actuator. An important difference between the strong coupling model for the numerical analysis and the weak coupling model for the numerical and theoretical analyses was that the former solution included the coupling of the DPE and the IPE (i.e., the piezoelectric coupling) accurately, whereas the latter solutions included only the IPE. Furthermore, an important difference between the numerical weak coupling solution and the theoretical solution was that the former included the 3-D IPE, whereas the latter included only the longitudinal IPE.

The present results can be summarized as follows. For a thin cantilevered piezoelectric bimorph actuator, the difference between the strong and weak coupling solutions given by piezoelectric coupling is almost proportional to the electromechanical coupling coefficient. This difference is significant for actual materials. In addition, the longitudinal bending caused by the constraint of the IPE in the width direction at the fixed end is almost proportional to the width dimension, and its contribution to the total longitudinal bending cannot be ignored even for a geometrical configuration where the Euler-Bernoulli beam assumption holds for the mechanical constitutive relationship. The results show that 3-D piezoelectric coupling modeling is important for the accurate analysis of typical piezoelectric devices such as a thin cantilevered piezoelectric bimorph actuator. In addition, the strength of the piezoelectric coupling can be quantitatively measured using the electromechanical coupling coefficient.

This research was supported by the

The authors declare that they have no conflicts of interest to report regarding the present study.