Insights Into Flexoelectric Solids From Strain-Gradient Elasticity

2014 ◽  
Vol 81 (8) ◽  
Author(s):  
Sheng Mao ◽  
Prashant K. Purohit
Keyword(s):  
Strain Gradient ◽  
Gradient Elasticity ◽  
Small Deformation ◽  
Material Behavior ◽  
Reciprocal Theorem ◽  
Isotropic Case ◽  
Strain Gradient Elasticity ◽  
Size Dependent ◽  
Hard Materials ◽  
Navier Equation

A material is said to be flexoelectric when it polarizes in response to strain gradients. The phenomenon is well known in liquid crystals and biomembranes but has received less attention in hard materials such as ceramics. Here we derive the governing equations for a flexoelectric solid under small deformation. We assume a linear constitutive relation and use it to prove a reciprocal theorem for flexoelectric materials as well as to obtain a higher-order Navier equation in the isotropic case. The Navier equation is similar to that in Mindlin's theory of strain-gradient elasticity. We also provide analytical solutions to several boundary value problems. We predict size-dependent electromechanical properties and flexoelectric modulation of material behavior. Our results can be used to interpret experiments on flexoelectric materials which are becoming increasingly sophisticated due to the advent of nanoscale probes.

Size Dependent Vibration Behavior of an AFM with Sidewall and Top-Surface Probes Based on the Strain Gradient Elasticity Theory

2015 ◽  
Vol 07 (03) ◽  
pp. 1550046 ◽  
Author(s):  
Mohammad Abbasi
Keyword(s):  
Elasticity Theory ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Strain Gradient Theory ◽  
Strain Gradient Elasticity ◽  
Size Dependent ◽  
Vibration Behavior ◽  
Vertical Extension ◽  
Strain Gradient Elasticity Theory ◽  
Gradient Elasticity Theory

In this paper, the size-dependent vibration behavior of an atomic force microscope with assembled cantilever probe (ACP) is analyzed utilizing the modified strain gradient elasticity theory. The proposed ACP comprises a horizontal cantilever, a vertical extension and two tips located at the free ends of the cantilever and extension. Because the vertical extension is located between the clamped and free ends of the microcantilever, the cantilever is modeled as two beams. The results of the current model are compared to those evaluated by both modified couple stress and classical beam theories. The results indicate that the resonant frequency and sensitivity of the proposed ACP is strongly size-dependent especially when the contact stiffness is very low or it is very high. The results also declare that utilizing the strain gradient theory is essential in the analysis of the vibration behavior of the proposed ACP.

Bernoulli–Euler Dielectric Beam Model Based on Strain-Gradient Effect

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Xu Liang ◽  
Shuling Hu ◽  
Shengping Shen
Keyword(s):  
Electric Field ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Beam Theory ◽  
Field Strain ◽  
Beam Model ◽  
Strain Gradient Elasticity ◽  
Cantilever Beams ◽  
Size Dependent ◽  
Classical Beam Theory

The theoretical investigation of the size dependent behavior of a Bernoulli–Euler dielectric nanobeam based on the strain gradient elasticity theory is presented in this paper. The variational principle is utilized to derive the governing equations and boundary conditions, in which the coupling between strain and electric field, strain gradient and electric field, and strain gradient and strain gradient are taken into account. Different from the classical beam theory, the size dependent behaviors of dielectric nanobeams can be described. The static bending problems of elastic, pure dielectric (nonpiezoelectric), and piezoelectric cantilever beams are solved to show the effects of the electric field-strain gradient coupling and the strain gradient elasticity. Comparisons between the classical beam theory and the strain gradient beam theory are given in this study. It is found that the beam deflection predicted by the strain gradient beam theory is smaller than that by the classical beam theory when the beam thickness is comparable to the internal length scale parameters and the external applied voltage obviously affects the deflection of the dielectric and piezoelectric nanobeam. The presented model is very useful for understanding the electromechanical coupling in nanoscale dielectric structures and is very helpful for designing devices based on cantilever beams.

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