TY - GEN A1 - Dornisch, Wolfgang A1 - Schrade, David A1 - Xu, Bai-Xiang A1 - Keip, Marc-André A1 - Müller, Ralf T1 - Coupled phase field simulations of ferroelectric and ferromagnetic layers in multiferroic heterostructures T2 - Archive of Applied Mechanics N2 - The combination of materials with either pronounced ferroelectric or ferromagnetic effect characterizes multiferroic heterostructures, whereby the different materials can be arranged in layers, columns or inclusions. The magnetization can be controlled by the application of electrical fields through a purely mechanical coupling at the interfaces between the different materials. Thus, a magneto-electric coupling effect is obtained. Within a continuum mechanics formulation, a phase field is used to describe the polarization and the magnetization in the ferroelectric and ferromagnetic layers, respectively. The coupling between polarization/magnetization and strains within the layers, in combination with the mechanical coupling at the sharp layer interfaces, yields the magneto-electric coupling within the heterostructure. The continuum formulations for both layers are discretized in order to make the differential equations amenable to a numerical solution with the finite element method. A state-of-the-art approach is used for the ferroelectric layer. The material behavior of the ferromagnetic layer is described by a continuum formulation from the literature, which is discretized using a newly proposed approach for the consistent interpolation of the magnetization vector. Four numerical examples are presented which show the applicability of the newly proposed approach for the ferromagnetic layer as well as the possibility to simulate magneto-electric coupling in multiferroic heterostructures. KW - Multiferroic heterostructure KW - Phase field method KW - Ferroelectric material KW - Ferromagnetic material KW - Finite element method KW - Rotation interpolation Y1 - 2019 U6 - https://doi.org/10.1007/s00419-018-1480-9 SN - 0939-1533 SN - 1432-0681 VL - 89 IS - 6 SP - 1031 EP - 1056 ER - TY - GEN A1 - Nadgir, Omkar A1 - Dornisch, Wolfgang A1 - Müller, Ralf A1 - Keip, Marc-André T1 - A phase-field model for transversely isotropic ferroelectrics T2 - Archive of Applied Mechanics N2 - 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. KW - Phase-field modeling KW - Ferroelectrics KW - Ginzburg–Landau equation KW - Transverse isotropy KW - Numerical simulations Y1 - 2019 U6 - https://doi.org/10.1007/s00419-019-01543-y SN - 0939-1533 SN - 1432-0681 VL - 89 IS - 6 SP - 1057 EP - 1068 ER -