Modelling of precipitate hardening in ferroelectric materials

Abstract This investigation presents a mechanical model to determine energetically favourable particle shapes, motivated by precipitation hardening in ferroelectric materials. We examine Li-doped $$\text {NaNbO}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>NaNbO</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> </mml:math> and propose a system of equations to derive the correct material constants for this material system. To model the elongated precipitate shape we introduce two approaches: An elliptic shape and a shape based on cubic B-spline-curves. We subject both definitions to an optimisation routine that yields equilibrium shapes. We further use the spline approach to derive configurational forces on the interface.