This work presents the development of an electromechanical size-dependent theory for flexoelectric curved beams. A two-way coupled linear 3D electroelastic theory is presented for flexoelectric materials, considering both direct and converse flexoelectric effects. The direct flexoelectric effect is defined as the coupling of electric field with strain gradient, while the converse effect is described by the coupling between electric field gradient and strain. In the literature, most of the flexoelectric theories and models consider free energy as a function of polarization and polarization gradient [1-2]. However, electric field is easier to control and measure in practical applications. Thus, the present formulation is developed considering a free energy function dependent on infinitesimal strain, strain gradient, electric field, and electric field gradient. The model also incorporates the converse flexoelectric effect, which is often ignored in the literature.Along with the electromechanical coupling, the flexoelectric effect shows a strong dependency on the size of the structure as the size effects due to strain and electric field gradients are prominent at the micro scale. The modified strain gradient theory developed by Lam et al. [3] is incorporated in the present formulation to consider mechanical size effects using three mechanical length scale parameters for isotropic materials. Two additional electrical length scale parameters are introduced to consider the electrical size effects. Symmetry of electric field gradient tensors allows the representation of higher-order dielectric coefficients in terms of two length scale parameters.The generalized theory is specialized to an Euler-Bernoulli curved beam and solved as a boundary value problem. A quadratic variation of electrostatic potential is assumed across the thickness of the flexoelectric layer. The 1D governing equations and boundary conditions are derived using the variational formulation. The governing equations are solved analytically using Fourier series analysis for simply-supported boundary conditions. An electromechanically coupled C2 continuous finite element formulation is also developed to analyze the boundary value problem for different boundary and loading conditions. Finite element results are compared with analytical results for the actuator and sensor response of the flexoelectric curved beams, and various parametric studies are performed to analyze the effect of radius of curvature, layer thickness, mechanical and electrical size effects on the responses of the flexoelectric beam. The results conclude that the developed flexoelectric model is able to capture electrical and mechanical size effects at lower scales. Specifically, it is observed that the effective stiffness and electric permittivity of the material increases at lower scales. These scale effects are also observed in various experimental studies by the researchers [3-4]. The present finite element model can be used as a computational design tool for designing flexoelectric beams as sensors and actuators in various device applications.References1.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Deng, Qian, Liping Liu, and Pradeep Sharma. "A Continuum Theory of Flexoelectricity." In&nbsp;Flexoelectricity in Solids: From Theory to Applications, pp. 111-167. 2017.2.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Zhao, Xie, Shijie Zheng, and Zongjun Li. "Effects of porosity and flexoelectricity on static bending and free vibration of AFG piezoelectric nanobeams."&nbsp;Thin-Walled Structures&nbsp;151 (2020): 106754.3.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Lam, David CC, Fan Yang, A. C. M. Chong, Jianxun Wang, and Pin Tong. "Experiments and theory in strain gradient elasticity."&nbsp;Journal of the Mechanics and Physics of Solids&nbsp;51, no. 8 (2003): 1477-1508.4.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Ren, He, and Wei-Feng Sun. "Characterizing Dielectric Permittivity of Nanoscale Dielectric Films by Electrostatic Micro-Probe Technology: Finite Element Simulations."&nbsp;Sensors&nbsp;19, no. 24 (2019): 5405.

Analytical and Finite Element Modeling of Flexoelectric Curved Beams

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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