Two-Dimensional Free Vibration Analysis of Axially Functionally Graded Beams Integrated with Piezoelectric Layers: An Piezoelasticity Approach

Abstract

<jats:p> For the first time, a two-dimensional (2D) piezoelasticity-based analytical solution is developed for free vibration analysis of axially functionally graded (AFG) beams integrated with piezoelectric layers and subjected to arbitrary supported boundary conditions. The material properties of the elastic layers are considered to vary linearly along the axial ([Formula: see text]) direction of the beam. Modified Hamiltons principle is applied to derive the weak form of coupled governing equations in which, stresses, displacements and electric field variables acting as primary variables. Further, the extended Kantorovich method is employed to reduce the governing equation into sets of ordinary differential equations (ODEs) along the axial ([Formula: see text]) and thickness ([Formula: see text]) directions. The ODEs along the [Formula: see text]-direction have constant coefficients, where the ODEs along [Formula: see text]-direction have variable coefficients. These sets of ODEs are solved analytically, which ensures the same order of accuracy for all the variables by satisfying the boundary and continuity conditions in exact pointwise manner. New benchmark numerical results are presented for a single layer AFG beam and AFG beams integrated with piezoelectric layers. The influence of the axial gradation, aspect ratio and boundary conditions on the natural frequencies of the beam are also investigated. These numerical results can be used for assessing 1D beam theories and numerical techniques. </jats:p>

2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

“…It should be emphasized that the current results are in good agreement with the 3‐D FE except near to the edge. This disagreement at the very clamped support is well‐reported and discussed in literatures [34–36, 45].…”

“…The Reissner‐type variational principle for cylindrical bending analysis of the panel, without any body force source and unit width along y ‐direction, can be expressed as [36] $$$$\begin{eqnarray} && \int _{V} [\delta u (\sigma _{x,x}+\tau _{xz,z})+\delta v (\tau _{xy,x}+\tau _{yz,z})+\delta w (\tau _{zx,x}+\sigma _{z,z}) +\delta {\sigma _{x}}(\epsilon _{x}-u_{,x})+{\delta \sigma _{z}}({\epsilon _z-w}_{,z})+{\delta \tau _{yz}}(\gamma _{yz}-v_{,z}) \nonumber\\ &&\quad +\,{\delta \tau _{zx}}(\gamma _{zx}-u_{,z}-w_{,x})+{\delta \tau _{xy}}(\gamma _{xy}-v_{,x})]\,dV=0,\quad \forall \delta u_{i}, \delta v_{i}, \delta w_{i}, \delta \sigma _{i}, \delta \tau _{ij} \end{eqnarray}$$$$where V stands for the volume of panel having unit width along the y ‐direction. Now, the expressions of strain components …”

“…Earlier, most of the developed solutions were using the two-dimensional theories of elasticity but, later Kapuria and Kumari [34] extended it to develop the solutions based on three-dimensional elasticity. Further, a technique was applied to analyze the stress behavior of the piezoelectric plates [35] and functionally graded plates [36]. Singh et al [37] have presented an analytical solution for the bending analysis of axially functionally graded panels.…”

On the Vibration Analysis of Rotating Piezoelectric Functionally Graded Beams Resting on Elastic Foundation with a Higher-Order Theory

“…Nan et al [7] explored the static bending and free vibration analysis of porous functionally graded piezoelectric nanobeams utilizing the electric enthalpy variation and Hamilton's principle, as well as differential equations for regulating the bending and free vibration. Using an analytical method, Singh and Kumari [8] investigated the free vibration analysis of axially functionally graded beams that were integrated with piezoelectric layers and subjected to arbitrarily supported boundary conditions in a free-falling state. Larkin and colleagues [9] used the modified couple stress theory to investigate the impact of small-scale phenomena on the natural frequencies and power density of macro-to nanoscale functionally graded energy harvesters with beam lengths ranging from 62.5 mm to 6.25 m. They found that the natural frequencies and power density of the harvesters were affected by small-scale phenomena.…”

Free vibration of piezoelectric FGM sandwich beam with porous core embedded in thermal environment and elastic foundation

“…In this work, the analytical solution in conjunction with Reddy's beam theory was used. Singh and Kumari [16] presented an analytical solution based on piezoelectricity for two-dimensional free vibration analysis of axially FGM beam integrated with piezoelectric layers.…”