Microstructured Multiferroic Materials: Modelling Approaches Towards Efficiency and Durability

Abstract

Magnetoelectric composites are investigated by numerical simulation. Nonlinear material models describing the magneto-ferroelectric or electro-ferromagnetic behaviors of the two constituents are presented. The ferroelectric model additionally accounts for damage evolution due to micro crack growth. The constitutive equations and weak forms of balance equations have been implemented within a finite element framework. A so-called condensed approach is also elaborated towards multiferroic compounds. Numerical simulations focus on the prediction of local domain orientation, the overall constitutive behaviors, the calculation of magnetoelectric coupling coefficients, and the investigation of damage processes, predominantly during magneto-electric poling, as well as mutual interactions of these aspects. A particle and a laminated composite are compared as examples.

Appendix

The coefficients of cobalt ferrite (CF) and barium titanate (BT) are listed in Table 2. The coefficient $\mu _{11}$ of BT is determined from the magnetic suszeptibility $\chi _{v}^{\mathrm {CGS}}=-8.47\cdot 10^{-8}$ at 293.8 K [41] according to $\mu _{11}=\mu _{0} \left( \chi _{v}^{\mathrm {SI}}+1\right)$ with $\chi _{v}^{\mathrm {SI}}=4\pi \chi _{v}^{\mathrm {CGS}}$. The coefficient $\mu _{22}$ was estimated, adapting the transversal isotropy of the dielectric constants $\kappa _{ij}$ to the magnetic permeability. In the state of maximum magnetostriction the magnetic suszeptibility of CF is $\chi _{v}^{\mathrm {SI}}=1.16$, providing a permeability of $\mu _{11}=2.71\cdot 10^{-6}\,\mathrm {Ns^{2}/C^{2}}$. The coefficient $\mu _{22}$ is determined based on the same idea as with BT.

Table 2 Material properties of $\mathrm {BaTiO_{3}}$ and $\mathrm {CoFe_{2}O_{4}}$ [22, 29, 41, 43]

Additionally, the quantities in Table 3 have been identified for the nonlinear reversible constitutive model. Due to the fact, that an appropriate value of $\mu _{11}^{r}$ could not be found in the literature, $\mu _{11}^{r}=5$ has been chosen, relying on similar ferromagnetic materials.

Table 3 Parameters of the phenomenological material model adapted to the constitutive behavior of $\mathrm {CoFe_{2} O_{4}}$

Table 4 illustrates the procedure of how to identify the parameters of the constitutive model based on experimental and numerical curves.

Table 4 Identification of the parameters of the phenomenological constitutive model

For a plane stress state, the values of $\sigma _{11}^{s}$ and $\sigma _{22}^{s}$ are calculated as follows:

$$\begin{aligned} \begin{aligned} \sigma _{11}^{s}&= C_{11}\eta _{1} + C_{12} \eta _{2} - \frac{C_{12}}{C_{22}} \left( C_{12} \eta _{1} + C_{23} \eta _{2} \right) \ , \\ \sigma _{22}^{s}&= C_{12}\eta _{1} + C_{22} \eta _{2} - \frac{C_{23-}}{C_{22}} \left( C_{12} \eta _{1} + C_{23} \eta _{2} \right) \ . \end{aligned} \end{aligned}$$

(50)

Moreover, the quantities in Table 5 have been applied for the physically motivated model. Due to the lack of elastic and dielectric constants in literature, the values of Table 2 have been taken for AlNiCo 35/5 as well, assuming the same orders of magnitude.

Table 5 Parameters of the physically based model adapted to the constitutive behavior of $\mathrm {BaTiO_{3}}$ and AlNiCo 35/5 [10, 22, 34, 39] (in [34] $\mu ^{r}$ is denoted as $\mu _{rec}$)