[Paper Review] Generalization of Stoney's equation for flexoelectric thin films on elastic substrates

Swarnava Ghosh|arXiv (Cornell University)|Jan 14, 2026

Nonlocal and gradient elasticity in micro/nano structures0 citations

TL;DR

The paper generalizes Stoney’s equation to account for flexoelectric and piezoelectric coupling in thin films on elastic substrates, deriving curvature and stretching strain expressions under open/closed circuit conditions and both uniform and non-uniform film properties.

ABSTRACT

When a thin film is deposited on an incompatible elastic substrate, the film develops an elastic mismatch strain, causing the film-substrate system to bend. Stoney's equation relates the curvature of the bent film-substrate system with the residual stress developed in the film, and can be used to infer film properties from curvature measurements. Certain materials exhibit electromechanical coupling, such as piezoelectricity and flexoelectricity, which can alter the curvature and strains. In this work, we generalize Stoney's equation to include flexoelectric and piezoelectric effects in the film. Considering both open and closed circuit configurations, as well as uniform and non-uniform film properties, we compare different cases of electromechanical coupling and discuss their influence on curvature, strains, and electric polarization in the film.

Motivation & Objective

• Motivate accurate estimation of thin-film properties when electromechanical coupling is present.
• Extend Stoney’s framework to include flexoelectric and piezoelectric effects in the film.
• Derive analytical expressions for substrate curvature and film stretching strain under various electrical boundary conditions.
• Analyze how uniform versus non-uniform film properties influence curvature, strains, and polarization.

Proposed method

• Start from a constitutive framework that includes elastic, dielectric, piezoelectric, and flexoelectric contributions.
• Formulate the bilayer film-substrate geometry with axial symmetry and small-strain assumptions.
• Solve the electrostatic problem under open- and closed-circuit configurations to obtain E-field distributions.
• Compute the total enthalpy by integrating elastic, dielectric, piezoelectric, and flexoelectric densities across the film volume.
• Minimize the total enthalpy with respect to mid-plane stretching and curvature to obtain epsilon0 and kappa expressions.
• Present closed-form expressions for uniform-film cases and extend to non-uniform film properties.

Experimental results

Research questions

• RQ1 How do flexoelectric and piezoelectric couplings modify the classical Stoney curvature relation for thin-film/substrate systems?
• RQ2 What are the analytical expressions for mid-plane stretching and curvature under open- and closed-circuit electrostatic boundary conditions?
• RQ3 How do uniform versus non-uniform film properties affect curvature, strain, and polarization in the bilayer system?

Key findings

• A generalized Stoney framework incorporating elastic, dielectric, piezoelectric, and flexoelectric effects is derived for thin-film on elastic substrates.
• Analytical expressions are obtained for the stretching strain and curvature under both converse (closed circuit) and direct (open circuit) flexoelectric configurations.
• The results show how curvature and strains depend on film/substrate thicknesses, bending moduli, electromechanical constants (e31, k33, mu), and the elastic mismatch εm.
• Explicit formulas are provided for cases with both piezoelectric and flexoelectric effects present, under uniform film properties.
• The analysis is extended to non-uniform film properties, enabling broader applicability to real materials.