Flexoelectricity (FE) refers to the two-way coupling between strain gradients and the electric field in dielectric materials, and is universal compared to piezoelectricity, which is restricted to dielectrics with noncentralsymmetric crystalline structure. Involving strain gradients makes the phenomenon of flexoelectricity size dependent and more important for nanoscale applications. However, strain gradients involve higher order spatial derivate of displacements and bring difficulties to the solution of flexoelectric problems. This dilemma impedes the application of such universal phenomenon in multiple fields, such as sensors, actuators, and nanogenerators. In this study, we develop a mixed finite element method (FEM) for the study of problems with both strain gradient elasticity (SGE) and flexoelectricity being taken into account. To use C⁰ continuous elements in mixed FEM, the kinematic relationship between displacement field and its gradient is enforced by Lagrangian multipliers. Besides, four types of 2D mixed finite elements are developed to study the flexoelectric effect. Verification as well as validation of the present mixed FEM is performed through comparing numerical results with analytical solutions for an infinite tube problem. Finally, mixed FEM is used to simulate the electromechanical behavior of a 2D block subjected to concentrated force or voltage. This study proves that the present mixed FEM is an effective tool to explore the electromechanical behaviors of materials with the consideration of flexoelectricity.

Because of the flexoelectric effect, dielectric materials usually polarize in response to a strain gradient. Soft materials are good candidates for developing a large strain gradient because of their good deformability. However, they always suffer from lower flexoelectric coefficients compared to ceramics. In this work, a flexoelectriclike effect is introduced to enhance the effective flexoelectricity of a polydimethylsiloxane bar. The flexoelectriclike effect is realized by depositing a layer of net charges on the middle plane of the bar to form an electret. Experiments show that the enhancement of flexoelectricity depends on the density of inserted net charges. It is found that a charged layer with surface potential of $-5723\mathrm{V}$ results in a 100 times increase of the material’s flexoelectric coefficient. We also show that the enhancement is proportional to the thickness of electrets. This work provides a new way of enhancing flexoelectricity in soft materials and further prompts the application of soft materials in electromechanical transducers.

Magnetically driven small‐scale soft robots are promising for applications in biomedicine, due to their fast, programmable deformation, and remote, untethered actuation to accomplish complicated tasks. Although diverse materials and designs have been proposed for magnetic soft robots with programmable shape transformation, it is still challenging to produce strong actuation by a small magnetic field. Inspired by arthropod species, magnetic soft millirobots with joint structures by 3D printing hydrogels have been developed. The joints can turn the bending deformation into the folding deformation, with the jointed region deforming locally. Different from homogeneous bending deformation, such local deformation allows larger motions of robots and reduces the overall energy consumption at the same time. Through experiments and numerical simulations, it is shown that the magnetic arthropod millirobots are capable of performing multimodal locomotion and programmed shape transformation, such as move, flip, catch, carry, and release. Finally, ex vivo experiments of removing a foreign object from porcine organs (e.g., aorta, stomach, and intestine) are presented to demonstrate the potential surgery application of magnetic arthropod millirobots.

In dielectric materials, due to the flexoelectric effect, a large inhomogeneous deformation can break or significantly alter the inversion symmetry of the material and induce net polarization. Such a phenomenon is distinctive comparing to other forms of electromechanical coupling, such as piezoelectricity and electrostriction, in that it is an intrinsic and universal effect whose strength scales up with the decrease in the sample size. Research works in the past decade indicate that the emergence of flexoelectricity not only provides a new way of realizing electromechanical coupling, but also has impact on multiple fields including smart materials, devices, and even physics. Our objective of this perspective is to briefly introduce the field of flexoelectricity and discuss on its future development. Also, current challenges and ambiguities that may impede the further development of the field are pointed out. Furthermore, our opinions on why flexoelectricity matters and how can we use it in novel ways are presented.

Flexoelectric effect is a universal and size-dependent electromechanical coupling between the strain gradient and electric field. The mathematical framework for flexoelectricity, which involves higher-order gradients of field quantities, is difficult to handle using traditional finite element method (FEM). Thus, it is important to develop an effective numerical method for flexoelectricity. In this paper, we develop a three-dimensional (3D) mixed finite element considering both flexoelectricity and strain gradient elasticity. To validate the developed element, we simulate the electromechanical behavior of a flexoelectric spherical shell subjected to inner pressure and compare the numerical results to analytical results. Their excellent agreement shows the reliability of the proposed FEM. The developed finite element is also used to simulate the electromechanical behavior of a nanometer-sized flexoelectric truncated pyramid. By decreasing the sample size, we observed the increase of its effective piezoelectricity. However, due to the effects of strain gradient elasticity and the influence of flexoelectricity on stiffness, the dependency of effective piezoelectricity on the sample size is not trivial. Numerical results indicate that, when the sample size is smaller than a certain value, the increase of effective piezoelectricity slows down. This finding also shows the importance of a numerical tool for the study of flexoelectric problems.

In this paper, the effects of flexoelectricity and strain gradient elasticity on Lamb waves propagating in an infinite piezoelectric nanoplate are analytically investigated. The dispersion equations are obtained and numerically solved for both the electrically open and short cases. A detailed discussion about the dispersion relations of the lowest mode is subsequently presented. Numerical results indicate that flexoelectricity and strain gradient elasticity have a substantial effect on Lamb wave propagation. Compared with the phase velocity of classical piezoelectric Lamb waves, flexoelectricity could decrease the phase velocity, while strain gradient elasticity could increase it. The effects of flexoelectricity and strain gradient elasticity largely depend on the wave number, material properties and nanoplate thickness. In infinite PZT-5H ceramic nanoplates, the effects of flexoelectricity and strain gradient elasticity on the phase velocity are negligible for wave numbers less than 10 ⁶ m ⁻¹ . However, for larger wave numbers, not only the phase velocity but also the dispersion relations can be significantly changed with the introduction of flexoelectricity and strain gradient elasticity. In addition, the effects of flexoelectricity and strain gradient elasticity on the electromechanical coupling coefficient are significant and complicated. Since the flexoelectric effect and the strain gradient elasticity effect exist in almost all dielectric materials, the conclusions obtained in this paper are general and can be applied to the analysis and design of the acoustic devices based on Lamb wave propagation.