Coupled flexoelectric-photovoltaic phenomena in nanoscale materials: Variational modeling and analytical solutions for advanced energy harvesting

This work presents a comprehensive variational framework for modeling coupled flexoelectric and photovoltaic phenomena in nanoscale materials, with a focus on BaTiO 3 thin plates. By integrating mechanical, electrical, and photonic interactions, the model captures the complex interaction between strain gradients, electric polarization, and light-induced photocurrents. The internal energy formulation incorporates higher-order gradient terms, with rigorously derived tensor symmetries using continuum mechanics and Noether’s theorem. Specializing the theory to thin plates via Kirchhoff–Love kinematics yields reduced governing equations that account for curvature-induced flexoelectric and photovoltaic effects, including dynamic terms such as inertial contributions. Closed-form solutions for clamped square and circular plates under uniform transverse and oscillatory loadings reveal that flexoelectric and photonic couplings reduce the effective bending modulus while enhancing curvature-driven electric polarization and photocurrent. For instance, under oscillatory loading <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1000</mml:mn> <mml:mspace width="0.25em"/> <mml:mi mathvariant="normal">N</mml:mi> <mml:mo>∕</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi mathvariant="normal">sin</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>100</mml:mn> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , the model predicts a time-averaged energy output of approximately <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>.</mml:mo> <mml:mn>7</mml:mn> <mml:mo>×</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> J for a 10-mm circular plate, while a more optimized configuration achieves up to 0.011 J, underscoring the potential of such systems for energy harvesting. Analytical predictions show strong agreement with experimental data, establishing both the theoretical rigor and practical relevance of the framework for the design of multifunctional microelectromechanical systems–based sensors and nanoscale energy harvesters.