Approximation Methods for Hybrid Constitutive Model of Magnetostriction

Abstract

Coupling piezoelectric and magnetoelastic materials together enables the electrical control of magnetism. However, the lack of an accurate non-linear magnetoelastic constitutive model complicates designing macroscale devices with these multiferroic composites. While micromagnetic simulations are the gold standard for mesoscale magnetic modeling, their use is limited to very small systems (i.e., typically below 10s of cubic microns). The focus of this research is on generating an accurate constitutive model for macroscale magnetoelastic materials that is consistent with micromagnetics.Multiple different constitutive models are currently used to describe macroscale magnetoelastic materials [1,2,3]. Two common approaches, polynomial and statistical energy models, each have their own strengths and limitations. For example, series expansion models that use polynomial expansions are straightforward to setup, however they are not capable of capturing the saturating behavior of magnetic materials. Additionally, to obtain nonlinear material properties well-controlled experiments to curve fit the models are required [4], which can be prohibitively challenging, time consuming, and expensive for anisotropic materials. Alternatively, statistical models can overcome the inherent limitations of polynomial expansions. These models utilize Boltzmann statistics to calculate the average magnetization and magnetostriction. These calculations account for the Zeeman, magnetoelastic, and magnetocrystalline anisotropy energies. A challenge of this statistical approach is that the requisite integral expressions do not possess general closed form solutions and therefore require numerical solutions. The first of these integrals is the partition function, which is dependent on the micromagnetic energies of the system u, and the thermal energy. In the title of Figure 1 the partition function includes beta which is used as a nondimensional term representing the magnitude of the micromagnetic energies in u. The other integral to consider is the average magnetization, which is derived from the partition function. While standard numerical integration methods can be utilized, they are computationally expensive, limiting their use in finite element models. In this work a constitutive model utilizing statistical physics to bridge micromagnetics to the macroscale is developed, and the accuracy and computational complexity of several integration methods are analyzed to produce a model suitable for use in high fidelity numerical models.The two methods for approximating the partition function that were examined are a partial series expansion and Laplace’s method. Conceptually a series expansion is asymptotically exact at the point its expanded about, conversely Laplace’s method is asymptotically exact at higher energy values (i.e., the methods are complimentary). The accuracy of each method was compared to an accurate, but computationally expensive numerical integral with relative error set at machine precision. The average magnetization, and therefore the partition function, has a closed form solution when only the Zeeman energy is included. If the additional micromagnetic energies are small compared to the Zeeman energy, they can be expanded in a Maclaurin series, which maintains integrals with exact solutions. Figure 1(a) shows the absolute error of this partial series expansion method as the magnitude of the applied energies (beta) increase. The inset graph also plots the absolute error, but beta 1 represents when only Zeeman energy is applied, and beta 2 shows when only the other micromagnetic energies are applied. Figure 1(b) shows the absolute error of Laplace’s method as the magnitude of the applied energies increase. The inset graph again considers the main contributor to the energy of the system. These two figures show that the two methods are indeed complimentary and so a hybrid model would maintain low relative errors for the entire range of beta values. Additionally, Laplace’s method only requires a small number of points (e.g., between two and six) to approximate integrals, making it more computationally efficient compared to standard integration methods. To conclude, we will use this model to simulate both the magnetization and magnetostriction of Galfenol subjected to magnetic fields up to 20 mT and mechanical stresses between -50 and 50 MPa. The results will be compared to experimental data from the literature [5]. **