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A phase-field model for transversely isotropic ferroelectrics

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Abstract

We propose an electro-mechanically coupled phase-field model for ferroelectric materials that show cubic–tetragonal phase transition. The cubic phase is idealized by an isotropic formulation, and the tetragonal phase is idealized by a transversely isotropic formulation. We consider a classical phase-field model with Ginzburg–Landau-type evolution of the order parameter. The order parameter drives the transition of all involved moduli tensors such as elastic, dielectric and piezoelectric moduli, which in turn maintain their typical features and stability as a result of a selected phase-transition function. The model is described in coordinate-invariant form and implemented into a finite element framework with implicit time integration of the evolution equation. Representative numerical examples in two and three dimensions demonstrate the main features of the constitutive model and the numerical stability of the formulation.

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Notes

  1. For brevity, we neglect all volumetric sources such as mechanical body forces and free electric charges.

  2. Please note that the realistic grain size of ferroelectrics is much larger. We have selected such small dimensions for the purpose of demonstration only. With the given dimensions, the material is provided with enough freedom to find some suitable domain configuration (the domain-wall width of barium titanate is approximately 1.5 nm).

  3. Note that in this example, we deliberately assume a smooth transition of the spontaneous polarization across grain boundaries. A computational analysis of associated implications is given in [31].

  4. We note that depending on the size of the sample, the proposed model predicts domain configurations with more complex patterns. As such phenomena seem rather unusual in experiments and computations [43,44,45,46], the associated implications are a part of the current research. For the phase-field analysis of size effects in ferroelectrics, we refer to [47,48,49].

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Acknowledgements

O.N. and M.A.K. acknowledge the financial support of the German Research Foundation (DFG) within the Cluster of Excellence in Simulation Technology (EXC 310). W.D. and R.M. were partially supported by the DFG within Research Group FOR 1509 (MU 1370/8-2).

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Authors and Affiliations

  1. Institute of Applied Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    O. Nadgir & M.-A. Keip

  2. Institute of Applied Mechanics, Technische Universität Kaiserslautern, Gottlieb-Daimler-Straße, 67653, Kaiserslautern, Germany

    W. Dornisch & R. Müller

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  1. O. Nadgir
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Correspondence to M.-A. Keip.

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Nadgir, O., Dornisch, W., Müller, R. et al. A phase-field model for transversely isotropic ferroelectrics. Arch Appl Mech 89, 1057–1068 (2019). https://doi.org/10.1007/s00419-019-01543-y

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