Constitutive Modeling of Oriented and Non-oriented Magnetostrictive Particulate Composites

Abstract

This study presents a mathematical framework for two-phase magnetostrictive composites composed of oriented and non-oriented magnetostrictive Terfenol-D particles embedded in passive polymer matrices. The phase constitutive behavior of the monolithic Terfenol-D with arbitrary crystal orientations is represented by a recently developed discrete energy averaged model. This unique Terfenol-D constitutive model results in close-form and linear algebraic equations accurately describing the nonlinear magnetostriction and magnetization in magnetostrictive composites subjected to a given loading or magnetic field increment. The effectiveness of this new mathematical framework in capturing magnetostrictive particle size orientation, phase volume fractions, mechanical loading conditions, and magnetic field excitations are validated using a series of experimental data available in literature. Compared to existing models that prevalently addressed particle orientation in composite constitutive level, the model framework in this study directly handles particle orientation in the phase constitutive level, and therefore achieves enhanced efficiency while maintaining comparable accuracy.

1. Introduction

Magnetostrictive materials change in shape in response to a magnetic field. Conversely, their magnetization status varies with respect to mechanical excitations. All ferromagnetic materials are magnetostrictive. Currently, terbium-iron-dysprosium (Tb_0.3Dy_0.7Fe_1.92) alloy, also known as Terfenol-D, provides the largest magnetically activated strain (~1400 ppm), or magnetostriction, at room temperature (Moffett et al., 1991). The magnetostrictive response is rapid (~ microseconds) and varies with respect to the applied field strength. Therefore, Terfenol-D is a promising material candidate for compact and broadband ultrasonic transducers (Li et al., 2021). One of the main challenges in Terfenol-D deployment is its low tensile strength (<40 MPa) (Peterson et al., 1989). Complex protection mechanisms, such as flextensional cages, are required to protect the Terfenol-D from cracking (Yan et al., 2015). Terfenol-D also exhibit severe eddy currents because of its low electrical resistivity (~60×10⁻⁸ Ω/m) (Sandlund et al., 1994). To resolve the brittleness and eddy current loss in monolithic Terfenol-D for high frequency applications, previous studies have developed particulate magnetostrictive composites consisting of Terfenol-D micro powders dispersed in a polymer matrix (Sandlund et al., 1994; Duenas and Carman, 2000; McKnight and Carman, 2002). It is noted that the monolithic Terfenol-D means a solid Terfenol-D sample that could either be single crystal or polycrystalline. The development of particulate Terfenol-D composites has recently facilitated the additive manufacturing of magnetostrictive materials and led to innovations in magnetically activated bioscaffolds for tissue engineering (Ribeiro et al., 2016). To further enhance the magnitude of magnetostriction, researchers have investigated various particle geometries (e.g., needles, spheres), particle concentrations, and post-processing methods (e.g., magnetically annealing) for Terfenol-D composite fabrication (Or et al., 2003; McKnight and Carman, 2002; Or and Carman, 2005). Elhajjar et al. (2018) have systematically reviewed the microstructures, material properties, potential applications, and model frameworks of magnetostrictive composites.

Constitutive models for magnetostrictive composites need to accurately reproduce the magneto-mechanical coupling in Terfenol-D particles, which exhibit significant hysteresis, anisotropy, and saturation. For small perturbations, the magneto-mechanical coupling in Terfenol-D particles have been approximated by either a linear function (Herbst et al. (1997), Nan (1998), Nan and Weng (1999), Huang et al. (2002), Yan et al. (2003), Guan et al. (2009), Shen and Lin (2021), and Lin and Lin (2021a)) or a quadratic function of the applied magnetic field (Zhou and Shin (2005), Altin et al. (2007), Zhong et al. (2013), Aboudi et al. (2014), Elhajjar and Law (2015), and Zhan and Lin (2021)). Using a phenomenological Preisach operator, researchers have developed constitutive models accounting for all nonlinearities. However, these models require many empirical parameters that have no theoretical justification. Kamlah (2001) discovered that significant improvements in material modeling can be achieved through physical-based approach. Recent studies have constructed physical-based constitutive models for monolithic Terfenol-D based on the classical thermodynamics theory of reversible systems, in which the magneto-mechanical coupling is represented by Gibbs free energy density (Smith, 2005). However, these models exclude material hysteresis and require empirical shape functions. By merging the thermodynamic theory with the Weiss theory of ferromagnetism, Jiles and Atherton have developed a constitutive model, in which the material hysteresis originates from the irreversible magnetic domain rotation (Jiles, 1995). The Jiles-Atherton model has accurately recreated the hysteretic magnetostrictive response using incremental analytical equations. However, it still requires empirical shape functions to describe material saturation and anisotropy. Recent studies have incorporated statistical mechanics theory to the Jiles-Atherton model to eliminate the empirical shape functions. These new models assume that macroscopic magnetization and magnetostriction in magnetostrictive materials are weighted sums of local magnetization and magnetostriction associated with individual magnetic domains (Armstrong, 1997; Smith et al., 2003). Based on the magnetic domain theory, the local responses were calculated analytically based on magnetic domain orientations. To further reduce the computational expense, Evans and Dapino (2010) further proposed a discrete energy averaged (DEA) model based on the assumption that the magnetic domains are oriented along the local minima of the Gibbs energy density equation. Deng (2017) has applied the same DEA model concept to simulate the magnetostrictive responses in monolithic Terfenol-D and developed a coordinate transformation algorithm to account for the misalignment between crystalline coordinates and sample coordinates.

An ideal constitutive model for particulate Terfenol-D composites should not only account for the magneto-mechanical coupling in the Terfenol-D particles, but also capture the particle-matrix interaction, including particle geometry, concentration, distribution, and orientations. The Mori-Tanaka micromechanics model converting particulate composites into statistically homogeneous media has been employed to simulate the particle-matrix interaction. This model has found widespread applications in predicting the magneto-mechanical (Huang et al., 2002; Yan et al., 2003; Guan et al., 2009; Zhan and Lin, 2021; Lin and Lin, 2021a) and magneto-electro-mechanical (Lin and Lin, 2021b; Lin and Liu, 2022) couplings in magnetostrictive composites. It should be emphasized, however, that the constitutive model used by these authors are restricted to magnetically-annealed composites, in which all Terfenol-D particles are oriented along a certain crystal direction. Consequently, these micromechanics models cannot mimic the behavior of non-oriented magnetostrictive composites with randomly distributed Terfenol-D particles.

This study aims at presenting an accurate and efficient mathematical framework to model magnetostrictive Terfenol-D composite with arbitrary particle orientations by nesting the DEA constitutive model developed by Deng (2017) within the Mori-Tanaka micromechanics model. This article is organized in the following structure. After the introduction, Section 2 summarizes the DEA constitutive law for monolithic Terfenol-D material. An analytical Jacobian matrix is derived here in order to facilitate the further integration with the micromechanics model. Section 3 presents a Terfenol-D composite constitutive model based on a micromechanics formulation to simulate constitutive behaviors for oriented and non-oriented Terfenol-D composites, respectively. Section 4 provides extensive comparisons with experimental data from McKnight and Carman (2002) to validate the reliability of the present mathematical framework. Key takeaways and future work are summarized in Section 5.

2. Monolithic Terfenol-D Constitutive Model

This study first modified the explicit and efficient DEA model of monolithic Terfenol-D developed by Deng (2017) to enhance its accuracy under large magnetic fields. An analytical Jacobian matrix and a coordinate transformation algorithm are then presented to facilitate the Mori-Tanaka model development.

2.1. Magneto-mechanical coupling

Terfenol-D particles are randomly distributed in non-oriented magnetostrictive composites, or composites that are cured under no magnetic field, as shown in Fig. 1.

Figure 1.

Schematic of non-oriented Terfenol-D composite cured under no magnetic field. The local coordinates (e₁, e₂, e₃) are defined for each Terfenol-D particle while the global coordinate (e′₁, e′₂, e′₃) is used for the magnetostrictive polymer matrix composite.

According to McKnight and Carman (2002), the composite samples are based on commercial Terfenol-D micropowders. Those micropowders are synthesized by milling highly-textured <112>-oriented polycrystalline monolithic Terfenol-D. Therefore, they are multigrained and multidomain. Since the feedstock material is highly-textured, it is also reasonable to assume that the individual polycrystal micro-powder could be approximated by a single domain particle in theoretical modeling. To capture the misalignment between the Terfenol-D crystal structure and the global sample coordinate ($e_1^{\prime }$, $e_2^{\prime }$, $e_3^{\prime }$), local coordinates (e₁, e₂, e₃), which are along their <100> directions, are defined for each particle. Based on the Weiss theory of ferromagnetism, the macroscopic magnetization M and magnetostriction S in Terfenol-D are weighted sums of local magnetization $M_m^k$ and magnetostriction $S_m^k$ associated with individual magnetic domains given as

$$M=\sum _{k=1}^{\pi }{M}^k{\xi }^k\text{ and }S=\sum _{k=1}^n{S}_m^k{\xi }^k,$$ (1)

where ξ^k is the volume fraction of the magnetic domains aligned along the kth potential orientation and n is the total number of potential orientations. For pseudo cubic symmetric Terfenol-D, n =8 (see a discussion in Deng (2017)) and the magnetic domains are oriented near the <111> directions of its crystal structure, or the easy axes c^k. Local responses of a single magnetic domain are

$$M_m^k=m^k{M}_s,$$ (2)

$$S_m^k={\left\{\frac{3}{2}{\lambda }_{100}{\left(m_1^k\right)}^2\frac{3}{2}{\lambda }_{100}{\left(m_2^k\right)}^2\frac{3}{2}{\lambda }_{100}{\left(m_3^k\right)}^{2}3{\lambda }_{11}{m}_2^k{m}_3^{k}3{\lambda }_{111}{m}_1^k{m}_3^{k}3{\lambda }_{111}{m}_2^k{m}_2^k\right\}}^{\top },$$ (3)

where $m^k={\left\{\begin{array}{c}{m}_1^k{m}_2^k{m}_3^k\end{array}\right\}}^{\top }$ is the kth orientation of the magnetic domain, λ₁₀₀ is the maximum magnetostriction along the <100> direction, λ₁₁₁ is the maximum magnetostriction along the <111> direction, and M_s is the saturation magnetization.

Following the discrete energy average modeling concept that has been validated by Evans and Dapino (2010), the Gibbs energy density around c^k is defined locally as

$$G^k=\frac{1}{2}K{\left|m^k-c^k\right|}^2-S_m^k\cdot T-{\mu }_0{M}_{\text{s}}{m}^k\cdot H+K_a^k,$$ (4)

where K is the magnetic anisotropy constant, μ₀ is the magnetic permeability in free space, T is the stress tensor in Voigt notation, H is the magnetic field vector, and $K_a^k$ is a constant accounting for the material anisotropy. The magnetic domain orientation m^k is calculated such that G^k is minimized. Based on the assumption that the magnetic domain orientations are near the easy axes or m^k ·m^k = 1 ≈ m^k ·c^k, analytical solution for m^k have been found in literature by solving ∂G^k ∂m^k = 0 (Evans and Dapino, 2010). As the magnitude of H increases, m^k may no longer stay near the easy orientation c^k. Therefore, a normalized magnetic domain orientation ${\widehat{m}}^k$ is used to compensate for any over-prediction in magnetization. Similar strategies have been validated for iron-gallium alloy modeling (Chakrabarti and Dapino, 2011). Analytical solution of ${\widehat{m}}^k$ is presented in Appendix 1.

Assume the material hysteresis originates from the irreversible magnetic domain rotation, the volume fraction change is defined incrementally as

$$\Delta {\xi }^k=\left(1-c_{\text{an}}\right)\Delta {\xi }_{\text{irr}}^k+c_{\text{an}}\Delta {\xi }_{\text{an}}^k,$$ (5)

where c_an describes the percentage of reversible magnetic domain rotation. $\Delta {\xi }_{\text{irr}}^k$ and $\Delta {\xi }_{\text{an}}^k$ are the volume fraction variations due to irreversible and reversible kth magnetic domain rotation, respectively. Based on the Boltzmann distribution, ${\xi }_{\text{an}}^k$ is expressed as

$${\xi }_{\text{an}}^k=\frac{\text{exp}\left(-G^k/\Omega \right)}{{\sum }_{k=1}^n\text{exp}\left(-G^k/\Omega \right)},$$ (6)

where Ω is a smoothing factor. Therefore, $\Delta {\xi }_{\text{an}}^k$ can be calculated analytically as

$$\Delta {\xi }_{\text{an}}^k=\sum _{i=1}^6\left(\frac{\partial {\xi }_{\text{an}}^k}{\partial T_i}\Delta T_i\right)+\sum _{i=1}^3\left(\frac{\partial {\xi }_{\text{an}}^k}{\partial H_i}\Delta H_i\right),$$ (7)

where ΔT_i and ΔH_i are the increments in stress T_i and magnetic field H_i components, respectively. Detailed derivation for $\partial {\xi }_{\text{an}}^k/\partial T_i$ and $\partial {\xi }_{\text{an}}^k/\partial H_i$ are presented in Appendix 2. Following the Jiles-Atherton model, $\Delta {\xi }_{\text{irr}}^k$ can be written as

$$\Delta {\xi }_{\text{irr }}^k=\frac{1}{K_{\text{p}}}\left({\xi }_{\text{an}}^k-{\xi }_0^k\right)[{\mu }_0{M}_{\text{s}}\left(\left|\Delta H_1\right|+\left|\Delta H_2\right|+\left|\Delta H_3\right|\right)+\frac{3}{2}{\lambda }_{100}\left(\left|\Delta T_1\right|+\left|\Delta T_2\right|+\left|\Delta T_3\right|\right)+3{\lambda }_{111}\left(\left|\Delta T_4\right|+\left|\Delta T_5\right|+\left|\Delta T_6\right|\right)],$$ (8)

where K_p is the pinning site density in magnetostrictive materials and ${\xi }_0^k$ is the initial volume fraction.

2.2. Jacobian matrix

The Jacobian matrix in particle coordinates is

$$J=\left[\begin{array}{cc}\frac{\partial M}{\partial H}& \frac{\partial M}{\partial T}\\ \frac{\partial \mathbf{\lambda }}{\partial H}& \frac{\partial \mathbf{\lambda }}{\partial T}\end{array}\right].$$ (9)

The 2nd-order tensor ∂M∂H is a 3×3 matrix and the ith column is

$$\frac{\partial M}{\partial H_i}=M_{\text{s}}\sum _{k=1}^n\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}{\xi }^k+{\widehat{m}}^k\frac{\partial {\xi }^k}{\partial H_i}\right).$$ (10)

The 3rd-order tensor ∂M∂T is arranged in a 3×6 matrix and the ith column is

$$\frac{\partial M}{\partial T_i}=M_{\text{s}}\sum _{k=1}^n\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}{\xi }^k+{\widehat{m}}^k\frac{\partial {\xi }^k}{\partial T_i}\right).$$ (11)

Similarly, the 3rd-order tensor ∂λ∂H is arranged in a 6×3 matrix and the ith column is

$$\frac{\partial \mathbf{\lambda }}{\partial H_i}=M_{\text{s}}\sum _{k=1}^n\left(\frac{\partial S^k}{\partial H_i}{\xi }^k+S^k\frac{\partial {\xi }^k}{\partial H_i}\right).$$ (12)

The 4th-order tensor ∂λ∂T is written in a 6×6 matrix and the ith column is

$$\frac{\partial \mathbf{\lambda }}{\partial T_i}=M_{\text{s}}\sum _{k=1}^n\left(\frac{\partial S^k}{\partial T_i}{\xi }^k+S^k\frac{\partial {\xi }^k}{\partial T_i}\right).$$ (13)

Analytical expressions of the partial derivatives ∂ξ^k ∂H_i, ∂ξ^k ∂T_i, $\partial S_m^k/\partial H_i$ and $\partial S_m^k/\partial T_i$ in Eqs. (10)–(13) are presented in Appendix 3.

2.3. Coordinate Transformation

The macroscopic response M and derived in Section 2.1 are in the local coordinate.

Therefore, coordinate transformation is required to calculate the macroscopic magnetization M′ and magnetostriction λ′ in the global coordinate system. The magnetization is the 1st-order tensor and the transformation follows

$$M^{\prime }=QM\text{ and }M=Q^{\top }{M}^{\prime }\text{. }$$ (14)

The magnetostriction is the 2nd-order tensor and the transformation follows

$${\lambda }^{\prime }=X\lambda \text{ and }\lambda =X^{-1}{\lambda }^{\prime }\text{. }$$ (15)

Similarly, the magnetic field H′ and mechanical stress T′ applied in the global coordinate can be transformed into the local coordinate using

$$H=Q^{\top }{H}^{\prime }.\text{ and }\lambda =X^{-1}{\lambda }^{\prime }.$$ (16)

The coordinate transformation matrices Q and X in Eqs. (14)–(16) are derived in Appendix 4.

The outputs of the micromechanics model in this study are flux density B′ and strain S′, which are correlated with magnetization and magnetostriction as

$$B^{\prime }={\mu }_{0}I{H}^{\prime }+M^{\prime },$$ (17)

$$S^{\prime }=s{T}^{\prime }+{\lambda }^{\prime },$$ (18)

where I is an identity matrix. The compliance s in Eq. (18) is expressed as

$$s=\frac{E_{\text{s}}}{(1+v)(1-2v)}\left[\begin{array}{cccccc}1-v& v& v& 0& 0& 0\\ v& 1-v& v& 0& 0& 0\\ v& v& 1-v& 0& 0& 0\\ 0& 0& 0& 0.5-v& 0& 0\\ 0& 0& 0& 0& 0.5-v& 0\\ 0& 0& 0& 0& 0& 0.5-v\end{array}\right].$$ (19)

Consequently, the Jacobian matrix in the global coordinate is defined as

$$J^{\prime }=\left[\begin{array}{cc}\frac{\partial B^{\prime }}{\partial H^{\prime }}& \frac{\partial B^{\prime }}{\partial T^{\prime }}\\ \frac{\partial S^{\prime }}{\partial H^{\prime }}& \frac{\partial S^{\prime }}{\partial T^{\prime }}\end{array}\right]=\left[\begin{array}{cc}{\mu }_{0}I+Q\frac{\partial M}{\partial H}{Q}^{\top }& Q\frac{\partial M}{\partial T}{X}^{-1}\\ X^{-\top }\frac{\partial \mathbf{\lambda }}{\partial H}{Q}^{\top }& s+X^{-\top }\frac{\partial \mathbf{\lambda }}{\partial T}{X}^{-1}\end{array}\right].$$ (20)

Details of Jacobian matrix coordinate transformation are also presented in Appendix 4.

For micromechanics modeling of magnetostrictive composite materials, it is convenient to define the following vectors:

$$\Delta Z^{\prime }={\left\{\Delta B_1^{\prime }\Delta B_2^{\prime }\Delta B_3^{\prime }\Delta S_{11}^{\prime }\Delta S_{22}^{\prime }\Delta S_{33}^{\prime }2\Delta S_{23}^{\prime }2\Delta S_{13}^{\prime }2\Delta S_{12}^{\prime }\right\}}^{\top },$$ (21)

$$\Delta Y^{\prime }={\left\{\Delta H_1^{\prime }\Delta H_2^{\prime }\Delta H_3^{\prime }\Delta T_{11}^{\prime }\Delta T_{22}^{\prime }\Delta T_{33}^{\prime }\Delta T_{23}^{\prime }\Delta T_{13}^{\prime }\Delta T_{12}^{\prime }\right\}}^{\top },$$ (22)

Eqs. (20)–(22) can be finally written in the compact form given as

$$\Delta Z^{\prime }=J^{\prime }\Delta Y^{\prime },$$ (23)

where Jacobian matrix J′ represents the consistent tangent material matrix, which is 9 × 9. This incremental matrix formulation will greatly simplify the subsequent composite constitutive model development.

3. Terfenol-D composite constitutive model

This section presents a model framework that applies to both oriented and non-oriented particulate Terfenol-D composites. The constitutive model is first established by the use of a micromechanics method for oriented Terfenol-D composites and then a coordinate transformation procedure is employed to address the non-oriented Terfenol-D composites.

3.1. Oriented Terfenol-D composites

Mori-Tanaka micromechanics approach which was previously developed for linearly elastic composites has been generalized for the prediction of the behavior of nonlinearly magnetostrictive composites. This method, proposed by Mori and Tanaka (1973) and normalized by Benveniste (1987), assumes that the composite microstructure of a magnetostrictive particle composite can be modeled as a particulate inhomogeneity placed into a uniform field that is equal to its average over the matrix part of the composite. Through the use of the average matrix field, the Mori-Tanaka method is able to address the interaction between particles. As a result, modeling of a particulate composite is equivalent to solving the interaction between a single ellipsoidal particle and an unbounded domain. Under uniform exterior loadings, the problem of a single inclusion embedded in an infinite matrix can be solved by the elegant Eshelby (1957) formalism that is based on eigenstrain concepts. Following the volume average scheme, the increments of the independent and dependent fields of a composite are written as, respectively:

$$\Delta {\overline{Y}}^{\prime }=\sum _{r=1}^2{c}^{(r)}\Delta Y^{\prime (r)},$$ (24)

$$\Delta {\overline{Z}}^{\prime }=\sum _{r=1}^2{c}^{(r)}\Delta {Z^{\prime }}^{(r)},$$ (25)

where the superscript (r) denotes the phase. Let r = 1 be the matrix phase and r = 2 be the particle phase. c^(r) the volume fraction of the rth phase in the composite. In average, the increments in the effective magnetic flux density and strain of the composite are given as

$$\Delta {\overline{Z}}^{\prime }=J^{\prime }\Delta {\overline{Y}}^{\prime }.$$ (26)

Constitutive relation of the rth phase in the composite is cast into an incremental form as

$$\Delta {Z^{\prime }}^{(r)}={J^{\prime }}^{(r)}\Delta {Y^{\prime }}^{(r)}.$$ (27)

According to Hill (1965), one can define a concentration-factor tensor to bridge the field variables between phase and composite level, given as

$$\Delta {Y^{\prime }}^{(r)}=B^{(r)}\Delta {\overline{Y}}^{\prime }.$$ (28)

Substituting Eq. (28) into Eq. (27) to replace ΔY’^(r) yields

$$\Delta {Z^{\prime }}^{(r)}=J^{\prime (r)}{B}^{(r)}\Delta {\overline{Y}}^{\prime }.$$ (29)

Substituting Eq. (29) into Eq. (25) to eliminate ΔZ’^(r) results in effective incremental magnetic flux density and strain:

$$\Delta {\overline{Z}}^{\prime }=\sum _{r=1}^2{c}^{(r)}{J^{\prime }}^{(r)}{B}^{(r)}\Delta {\overline{Y}}^{\prime }.$$ (30)

By comparing Eq. (26) and Eq. (30), the effective consistent tangent material matrix of the composite is derived as:

$${\overline{J}}^{\prime }=\sum _{r=1}^2{c}^{(r)}{J}^{\prime (r)}{B}^{(r)}.$$ (31)

Up to this point, the concentration-factor tensor B^(r) is the only unknown. Mori-Tanaka method provides a unique means to determine B^(r). Details of the Mori-Tanaka mean-field approach can be found in Weng (1984) and Benveniste (1987), which focused on linear elastic medium only. Here, we generalize this approach for nonlinear magnetostrictive materials. A quick derivation for the Mori-Tanaka concentration-factor tensor presented in an incremental manner is demonstrated as follows.

Fig. 2 shows a representative unit of particulate composites consisting of the rth homogeneous ellipsoidal inclusion with a generalized compliance J′^(r) (the domain Θ) embedded into an unbounded homogeneous matrix with the generalized compliance J′⁽¹⁾ (the domain D − Θ).

Figure 2.

An ellipsoidal subdomain Θ embedded in the infinite domain D.

Without the inclusion, the increments of the effective generalized fields, $\Delta {\overline{Y}}^{\prime }$ and $\Delta {\overline{Z}}^{\prime }$, is dominated by the incrementally homogeneous boundary conditions in the domain D, ΔY′⁰ or ΔZ′⁰. Therefore,

$$\Delta {\overline{Y}}^{\prime }=\Delta Y^{\prime 0}\text{ and }\Delta {\overline{Z}}^{\prime }=\Delta Z^{\prime 0}\text{ in }D.$$ (32)

The presence of the rth inclusion in the domain D results in an increment of the generalized independent field inside its domain Θ as

$$\Delta {Y^{\prime }}^{(r)}=\Delta {\overline{Y}}^{\prime }+\Delta Y^{\prime pt},\text{ in }\Theta ,$$ (33)

where the incrementally generalized perturbation field ΔY’^pt is owing to the deviating material properties of the rth inclusion. Therefore, the incrementally generalized dependent field of the rth inclusion is given by the constitutive relation in Eq. (27) as

$$\Delta Z^{\prime (r)}={J^{\prime }}^{(r)}\left(\Delta {\overline{Y}}^{\prime }+\Delta Y^{\prime pt}\right),\text{ in }\Theta .$$ (34)

The rth inclusion can be replaced by the matrix material subjected to a incrementally generalized “eigen-field” ΔY’* (i.e. magnetic flux density-free and strain-free fields), which results in no increments of the generalized magnetic flux density and strain fields of the rth inclusion inside the domain Θ. Therefore, Eq. (34) can be rewritten as

$$\Delta {Z^{\prime }}^{(r)}={J^{\prime }}^{(1)}\left(\Delta {\overline{Y}}^{\prime }+\Delta Y^{\prime pt}-\Delta Y^{{\prime }^{*}}\right),\text{ in }\Theta .$$ (35)

The relation between the incrementally generalized perturbation field ΔY’^pt and the incrementally generalized eigen-field ΔY’* is initially introduced by Eshelby (1957) for linear elastic solids and is generalized here for nonlinear magneto-elastic materials as following

$$\Delta {Y^{\prime }}^{pt}={\mathbb{S}}^{Y^{\prime }}\Delta Y^{{\prime }^{*}},$$ (36)

where ${\mathbb{S}}^{Y^{\prime }}$ is the so-called conjugate Eshelby tensor, and it should not be confused with the strain S. The superscript Y′ indicates that the conjugate Eshelby tensor is formulated by using (H′, T′) as an independent field variable pair. This is different from the generalized Eshelby tensor ${\mathbb{S}}^{Z^{\prime }}$ that employs (H′, S′) as an independent field variable pair. To evaluation of ${\mathbb{S}}^{Z^{\prime }}$ can be found and Zhan and Lin (2021), and a useful conversion between ${\mathbb{S}}^{Z^{\prime }}$ and ${\mathbb{S}}^{Y^{\prime }}$ was provided by Li and Dunn (2001). The equivalence of Eqs. (34) with (35) is the well-known equivalent inclusion equation given as

$$J^{\prime (r)}\left(\Delta {\overline{Y}}^{\prime }+\Delta Y^{\prime pt}\right)=J^{\prime (1)}\left(\Delta {\overline{Y}}^{\prime }+\Delta Y^{\prime pt}-\Delta Y^{\prime *}\right),\text{ in }\Theta .$$ (37)

By substituting Eq. (36) into (37), we replace ΔY’*and then eliminate ΔY’^pt by utilizing Eq. (33), the following equation is obtained

$$\Delta Y^{\prime (r)}={\left[\mathbb{I}+{\mathbb{S}}^{Y^{\prime },(r)}{\left(J^{\prime (1)}\right)}^{-1}\left(J^{\prime (r)}-J^{\prime (1)}\right)\right]}^{-1}\Delta {\overline{Y}}^{\prime },$$ (38)

where $\mathbb{I}$ is the fourth-order identity tensor and it should not be confused with the identity matrix I in Eq. (17). By comparing Eq. (38) with (28), a concentration-factor tensor is derived as

$$B^{\text{dil},(r)}={\left[\mathbb{I}+{\mathbb{S}}^{Y^{\prime },(r)}{\left({J^{\prime }}^{(1)}\right)}^{-1}\left(J^{\prime (r)}-{J^{\prime }}^{(1)}\right)\right]}^{-1}.$$ (39)

B^dil,(r) is the concentration-factor tensor of dilute micromechanics model (Eshelby, 1957) because it is derived by considering only a single rth inclusion embedded in an infinite matrix which is suitable for a very low concentration of inhomogeneities.

If inhomogeneities in a composite increase, the generalized fields of a typical rth inclusion will be affected by the average fields of its surrounding matrix instead of by the mean fields of the overall composite as does in the dilute micromechanics model. The effects of the presence of other inhomogeneities on the rth inclusion should be incorporated in the surrounding matrix, which is done by replacing the increment of the generalized independent field of the composite $\Delta {\overline{Y}}^{\prime }$ with the one of the generalized independent field of the matrix ΔY′⁽¹⁾ in the Eq. (28) and then yields

$$\Delta Y^{\prime (r)}=B^{\text{dil},(r)}\Delta Y^{\prime (1)}.$$ (40)

By substituting Eq. (40) into (24), the incremental field for the rth inhomogeneity is given as

$$\Delta Y^{(r)}=B^{\text{dil, }(r)}{\left[c^{(1)}\mathbb{I}+\sum _{n=2}^N{c}^{(n)}{B}^{\text{dil},(n)}\right]}^{-1}\Delta {\overline{Y}}^{\prime }.$$ (41)

Similarly, by comparing Eq. (41) with (28), another concentration-factor tensor is derived as

$$B^{(r)}=B^{\text{dil},(r)}{\left[c^{(1)}\mathbb{I}+\sum _{n=2}^N{c}^{(n)}{B}^{\text{dil},(n)}\right]}^{-1}.$$ (42)

B^(r) is the concentration-factor tensor of Mori-Tanaka micromechanics model (Mori and Tanaka, 1973) because it accounts for the interactions between inhomogeneities. Here, we had derived the Mori-Tanaka concentration-factor tensor for magneto-elastic medium.

However, due to the nonlinear constitutive behavior of the Terfenol-D constituent, the linearized constitutive relation (in an incremental form) would lead to an error estimation of the magnetic field and stress in each phase. This error is defined by a residual, R, in the following form:

$$R=\left\{\begin{array}{l}\Delta Y^{(1)}-B^{(1)}\Delta {\overline{Y}}^{\prime }\hfill \\ \Delta Y^{(2)}-B^{(2)}\Delta {\overline{Y}}^{\prime }\hfill \end{array}\right\}.$$ (43)

Once R is minimized (by numerical iterations), the concentration-factor tensor B^(r) can be obtained. Consequently, the effective incremental magnetic flux density and strain $\Delta {\overline{Z}}^{\prime }$ and effective consistent tangent material matrix $\overline{J^{\prime }}$ can be determined by using Eqs. (30) and (31), respectively. Finally, the effective total magnetic flux density and strain can be calculated by adding consecutive increment of $\Delta {\overline{Z}}^{\prime }$. To this end, we have reformulated the Mori-Tanaka model with the magnetic field and stress as independent variables to simulate the nonlinear macroscopic behavior of oriented Terfenol-D composites.

3.2. Non-oriented Terfenol-D composites

Non-oriented Terfenol-D composites that are cured without a magnetic field result in randomly-distributed Terfenol-D particles in the polymer matrix. Therefore, constitutive models for non-oriented Terfenol-D composites need to account for the contribution from each potential particle orientations. As shown in Fig. 1, a local cartesian coordinate system V that aligns with the cubic Terfenol-D crystal is defined as

$$V=\left\{\begin{array}{lll}{e}_1\hfill & e_2\hfill & e_3\hfill \end{array}\right\}=\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right].$$ (44)

A global cartesian coordinate systemU is defined to describe the orientation of the macroscopic Terfenol-D composite. A matrix U_k that describes the coordinates of U in the V of the kth Terfenol-D particle is defined as

$$U_k=\left\{\begin{array}{c}{e}_{1,k}^{\prime }{e}_{2,k}^{\prime }{e}_{3,k}^{\prime }\end{array}\right\},$$ (45)

where $e_{3,k}^{\prime }$ aligns along the axial direction of the Terfenol-D composite rod. To recreate the randomly-distributed particle orientation, a group of $e_{3,k}^{\prime }$ that are evenly-distributed in three-dimensional space are created using a spherical coordinate system below

$$e_{3,k}^{\prime }={\left\{\text{sin}\left({\theta }_i\right)\text{cos}\left({\varphi }_j\right)\text{ sin}\left({\theta }_i\right)\text{sin}\left({\varphi }_j\right)\text{ cos}\left({\theta }_i\right)\right\}}^{\text{T}},$$ (46)

where θ_i and ϕ_j are linearly space arrays within [0, π] and [0, 2π], respectively. The total number of θ_i and ϕ_j are N_θ and N_ϕ, respectively. Therefore, the total potential $e_{3,k}^{\prime }$ is N_e = N_θN_ϕ. The selection of $e_{1,k}^{\prime }$ and $e_{2,k}^{\prime }$ is not unique for a given $e_{3,k}^{\prime }$. In this study, we assume U_k = V when $e_3^{\prime }=e_{3,k}^{\prime }$. In other words, U_k is created by rotating the V coordinate system around R_axis,k by β. Here,

$$R_{\text{axis, }k}={\left\{\text{sin}\left({\theta }_i\right)\text{cos}\left({\varphi }_j\right)-\text{sin}\left({\theta }_i\right)\text{sin}\left({\varphi }_j\right)0\right\}}^{\text{T}}.$$ (47)

This coordinate transformation ensures that the rotation axis R_axis,k sits in the e₁ −e₂ plane while being perpendicular to the projection of $e_{3,k}^{\prime }$ in the e₁ − e₂ plane. The impact of $e_{1,k}^{\prime }$ and $e_{2,k}^{\prime }$ selection becomes less significant as N_θ and N_ϕ increase.

For each input ${\overline{Y}}^{\prime }$, the composite response Z_k is calculated for each set of U_k, or each potential orientation of Terfenol-D particle. Finally, the macroscopic response ${\overline{Z}}^{\prime }$ for non-oriented Terfenol-D composite follows a weighted summation, given as

$${\overline{Z}}^{\prime }=\frac{1}{N_{\text{e}}}\sum _{k=1}^{N_e}{Z}_k.$$ (48)

4. Model Validation

McKnight and Carman (2002) have characterized the magnetization and magnetostriction of bulk Terfenol-D and particulate Terfenol-D composites. Using these experimental results, this study validates the effectiveness of the new model framework in capturing particle concentration, particle orientation, and material nonlinearities.

4.1. Monolithic Terfenol-D model

Materials properties for the DEA model are first calibrated based on the characteristic curves of a monolithic <112>-oriented Terfenol-D rod. The applied prestresses and magnetic field, as well as the resulting magnetization and magnetostriction, are all along the Terfenol-D rod’s axis, namely the direction in <112>. Table 1 lists the resulting material properties via curve fitting (Nas et al., 2017). Fig. 3 shows the comparison of the DEA modeling result (solid lines) and the experimental data (circles) under various prestresses, −6, −12, −16 and −24 MPa. Table 2 summarizes the relative modeling error at each prestress.

Table 1.

Material properties of Terfenol-D particles (Deng, 2017 and Nas et al. 2017)

Property	Terfenol-D
K (×105 J/m3)	1.37
$K_0^1$ (×104 J/m3)	14.12
$K_0^2$ (×104 J/m3)	10.40
$K_0^3$ (×104 J/m3)	9.48
$K_0^4$ (×104 J/m3)	9.78
μ0Ms (T)	1.03
λ000 (×10−4)	19.42
λ111 (×10−4)	10.04
Ω (×104 J/m3)	0.55
Esa (GPa)	110.00
v a	0.3
Kp (x103 J/m3)	18.83
c an	1 for anhysteretic while 0.18 for hysteretic

^aE_s and v are directly obtained from literature (McKnight and Carman, 2002).

Figure 3.

Comparisons of the modeling result (solid lines) to the experimental data (circles) from McKnight and Carman (2002) on the magnetostriction for a bulk <112>-oriented Terfenol-D rod under −6, −2, −16 and −24 MPa prestresses at room temperature.

Table 2.

Average errors between the DEA model predictions and the experimental results of the monolithic Terfenol-D

Model╲Prestress (MPa)	− 6	− 12	− 16	− 24
DEA model	2.2%	0.5%	1.6%	3.2%

4.2. Anhysteretic Terfenol-D composite model

The accuracy of the proposed model framework in predicting the constitutive behavior of particulate Terfenol-D composites is evaluated using both oriented particle composite (OPC) and non-oriented particle composite (NOPC). Material hysteresis is first ignored (c = 1) to facilitate model-experiment comparison. Other DEA model parameters are the same as those shown in Table 1. The epoxy matrix represented by an isotropic elastic medium and its properties are listed in Table 3. Based on the microstructural images presented by McKnight and Carman (2002), the aspect ratios a₃a₁ of the Terfenol-D particles are 3.7 and 2.5 for OPC and NOPC, respectively. The upper and lower bound estimations based on a₃/a₁ = 1 (spherical particle) and a₃/a₁ = ∞ (fiber), respectively, are also simulated. To recreate the random particle orientations in NOPC, N_θ and N_ϕ used in this simulation are 10 and 10, respectively, corresponding to a total of 100 evenly-distributed particle orientation in three-dimensional space.

Table 3.

Material properties of epoxy compound

Property	Epoxy
E (GPa)	3
v	0.3

The effective Young’s modulus as a function of particle volume fraction predicted by the new model framework is first compared with experimental results in Fig. 4. The micromechanics prediction (solid line) is in good agreement with the experimental results (circles) in a wide range of particle volume fraction. The mean absolute percentage errors for OPC and NOPC are 16.3% and 16.5%, respectively.

Figure 4.

Comparisons of the modeling and experimental results (McKnight and Carman 2002) on the effective Young’s modulus ${\overline{E}}_3$ for (a) oriented and (b) non-oriented Terfenol-D composites at room temperature.

Both the micromechanics prediction and the experimental measurements fall within the bounds and Young’s modulus increases monotonically with respect to particle volume fraction. Moreover, NOPC exhibits smaller Young’s modulus than that of the OPC with a similar particle volume fraction. This is phenomenon has been observed by McKnight and Carman (2002). This is because the Terfenol-D particles in an OPC form chain-like structures and consequently yields much higher mechanical stiffness than a NOPC whose microstructure is like particle-reinforced composite structures. According to Fig. 4, the effects of particle alignment and volume fraction can both be properly addressed by the present mathematical framework.

The performance of the present mathematical framework on predicting Terfenol-D composite magnetostriction is then examined using magnetostriction data from McKnight and Carman (2002). The particle aspect ratios are still 3.7 and 2.5 for OPC and NOPC, respectively. The particle volume fractions are 0.36 and 0.30 for OPC and NOPC, respectively. Both OPC and NOPC are subjected to an ascending magnetic field and a constant compressive stress of −8 MPa. The stress, magnetic field, and the resulting magnetostriction are all along the rod sample’s axis. Fig. 5 confirms the close match between experimental results (circles) and the micromechanics predictions (solid lines) along with the corresponding bounds (denoted by dashed lines).

Figure 5.

Comparisons of the modeling and experimental results (McKnight and Carman 2002) on the macroscopic magnetostriction ${\overline{\lambda }}_{33}^{\prime }$ for (a) oriented and (b) non-oriented Terfenol-D composites at room temperature.

Further model validation is conducted using magnetostriction data collected under various prestresses of −4, −8, −12 and −16 MPa. The Terfenol-D particle aspect ratios remain the same; while the particle volume fractions are 0.49 and 0.30 for OPC and NOPC, respectively. Figs. 6 and 7 confirm that the present model framework can accurately capture the magnetostriction in oriented and non-oriented Terfenol-D composites under various stress conditions. The model accurately predicts the phenomenon that the prestress would increase the magnetostrictions regardless of the alignment of the Terfenol-D particles.

Figure 6.

Comparisons of the modeling and experimental results (McKnight and Carman 2002) on the effective magnetostriction ${\overline{\lambda }}_{33}^{\prime }$ of Terfenol-D OPC (with a particle volume fraction of 0.49) undergoing various prestresses of (a) −4 MPa, (b) −8 MPa, (c) −12 and (d) −16 MPa.

Figure 7.

Comparisons of the modeling and experimental results (McKnight and Carman 2002) on the effective magnetostriction ${\overline{\lambda }}_{33}^{\prime }$ of Terfenol-D NOPC (with a particle volume fraction of 0.30) undergoing various prestresses of (a) −4 MPa, (b) −8 MPa, (c) −12 and (d) −16 MPa.

McKnight and Carman (2002) also characterized the magnetostriction of Terfenol-D OPC with various particle volume fractions of 0.22, 0.36 and 0.49 under −8 and −16 MPa prestresses. The simulation results shown in Fig. 8 accurately captured the trend that the magnetostriction increases monotonically to the Terfenol-D particle volume fraction regardless of the intensity of a prestress.

Figure 8.

Comparisons of the modeling and experimental results (McKnight and Carman 2002) on the effective magnetostriction ${\overline{\lambda }}_{33}^{\prime }$ of Terfenol-D OPC (with a particle volume fraction VF of 0.49, 0.36 and 0.22) undergoing a prestress of (a) −8 MPa and (b) −16 MPa.

4.3. Hysteretic Terfenol-D composite model

Eventually, the DEA parameter is set back to 0.18 to demonstrate the new model’s c capability in predicting the hysteretic responses of Terfenol-D OPC. The particle aspect ratio is 3.7 and the prestress is −16 MPa. Fig. 9 shows the hysteretic magnetostriction loops with particle volume fraction of 1 (bulk materials), 0.22, 0.36, and 0.49, respectively. Again, the model predictions are in close agreement with the experimental data. Based on the model-experiment comparisons presented in Figs 3 – 9, the present mathematical framework coupling DEA constitutive model and Mori-Tanaka micromechanics model can accurately capture the effects of particle volume fraction, particle aspect ratio, particle orientation, magnetic field, and prestress on Terfenol-D composite responses.

Figure 9.

Comparisons of the modeling and experimental results (McKnight and Carman 2002) on the effective magnetostriction ${\overline{\lambda }}_{33}^{\prime }$ of Terfenol-D OPC under a prestress of −16 MPa. The particle volume fractions VF are (a) 1 (bulk material), (b) 0.22, (c) 0.36, and (d) 0.49.

5. Conclusions and Discussion

By nesting the magnetostrictive DEA model within the Mori-Tanaka micromechanics model, a closed-form, incremental, and nonlinear constitutive model that can simulate the hysteretic magneto-elastic behavior of oriented and non-oriented particulate Terfenol-D composites has been established. A variety of mechanisms, such as particle aggregation, distribution of particles, residual stress, and magnetic loading intensity, would affect the overall response of a Terfenol-D polymer matrix composite. In this study, we assumed the Terfenol-D particle reinforcements in the composites are in an ideal ellipsoidal shape and the Terfenol-D particle distribution is uniform. Four additional assumptions are:

1. Particles are perfectly bonded with the polymer matrix, i.e., that is no any significant contents of pores and defects are presented between the reinforcement phase and the matrix phase. To release assumption, future researchers could introduce a third phase (i.e., adhesive layer) between the reinforcement and the matrix through the Mori-Tanaka model, as discussed by Gao and Zhang (2011).

2. Terfenol-D particles do not deform, dislocate, rotate, or aggerate to form a chain (i.e., long-range connectivity) in the polymer matrix under either mechanical or magnetic stimuli. This assumption could be lifted by considering how the particles displace/rotates due to the magnetic force, as discussed by Zhao et al. (2019).

3. A Terfenol-D polymer matrix composite does not contain any residual stress before loadings. Thermal stress may be built up during the curing process due to thermal expansion mismatch. However, the curing temperature for the samples used in this study is 100 °C and the matrix is stabilized during the 8-hour curing procedure. As a result, the thermal stress would be minimum.

4. The polymer matrix is assumed to be elastic. The polymer matrix could have significant influence on the overall responses of a Terfenol-D polymer matrix composite. For example, a soft matrix may be not able to provide enough stiffness to sustain a complete composite microstructure when the composite is subjected to magnetic fields. Ze et al. (2020) observed this phenomenon in magnetic shape memory polymer matrix composites. Similarly, the viscoelasticity of the polymer matrix could lead to time-dependent composite response, as discussed by Shen and Lin (2021) and Lin and Liu (2022) and Lin and Hung (2023). However, the Terfenol-D composite investigated in this study has a relative stiff matrix and both the microstructure change and viscoelasticity are negligible.

The accuracy and reliability of the present model framework have been validated using comprehensive experimental data reported in McKnight and Carman (2002). According to the model-experiment comparison, the new model framework can accurately predict the effective Young’s modulus and magnetostriction that are strongly dependent on the orientations of Terfenol-D particles, phase volume fractions, prestresses, and magnetic field excitations. Therefore, the aforementioned assumptions are reasonable.

For the OPC composites, the particle crystalline structure is well known (the long axis is along [112]). The OPC composites were also annealed in a magnetic field, which forces the long axis of the particles to be aligned along the sample axis. Based on these results, a micromechanics model with reliable crystalline orientations can be constructed. The crystalline structure of NOPC composites, on the other hand, is not well investigated in literature. In this study, the cubic or equiaxial Terfenol-D particles are assumed to remain highly-textured. However, the mechanical energy applied to the particles during milling may have changed the particle crystalline structure. In addition, due to the lack of magnetic annealing, the Terfenol-D particles are assumed to be evenly distributed in free 3D space. Both assumptions on NOPC composites may have reduced the model accuracy. As a result, the discrepancy between the modeling and experimental results is more significant in the NOPC case than that in the OPC case, as presented in Fig. 6 (OPC case) and Fig. 7 (NOPC case).

Terfenol-D micropowders are polycrystalline and anisotropic, as the magnetic domains prefer the <111> family, or the local minima of the Gibbs energy expression. To account for the crystal orientations and material anisotropy, traditional micromechanics models targeted modifications in the composite constitutive level. The individual Terfenol-D particle is assumed to be isotropic, which results in an orientation-independent stiffness or even a constant stiffness in the phase constitutive level. The material anisotropy is later considered using a phase volume averaging process during the Mori-Tanaka micromechanics model development, in which an orientation distribution function is used to approximate randomly-distributed Terfenol-D particles. Examples of such conventional approaches are the works of Srinivas and Li (2005) and Kuo and Ling (2020). The model presented in this study is more advantageous because the DEA model can directly link the Terfenol-D anisotropy with crystal orientation and researchers can directly address the Terfenol-D material anisotropy in the phase constitutive model level. Hence, more mathematically accurate macroscopic predictions on effective composite properties are expected. Furthermore, the new mathematical framework bypasses the use of the orientation distribution function and the orientation averaging process, and therefore result in enhanced model efficiency.

The present model is unable to address the effects of charge rate, percolation, or eddy current loss in Terfenol-D due to the limitation of the DEA model. Future studies could target incorporating a rate-dependent DEA model (Deng, 2017) with the Mori-Tanaka framework. Because theories and equations for ellipsoidal particles are well-established by Eshelby (1957) and Mori and Tanaka (1973), this study only focused on ellipsoidal Terfenol-D particles. Future studies could also investigate different particle morphologies, such as sphere, elliptic cylinder, penny shape, oblate spheroid, and prolate spheroid. Modifications based on particle morphology and actual SEM images will further enhanced model accuracy.

In conclusion, the information gained from this study would benefit the design of Terfenol-D composite and guide the Terfenol-D composite in development for additive manufacturing. The contributions of this work provide a new avenue to address Terfenol-D particle orientation in a composite material, which has potential applications on modeling of engineered magnetostrictive particulate composites. Future studies will include nesting the proposed mathematical framework into a finite element model and predicting the behavior of magnetostrictive devices (such as sensors and actuators) based on Terfenol-D composites.

Appendix 1

This appendix presents an analytical expression of the normalized kth orientation of the magnetic domain, ${\widehat{m}}^k$. The Gibbs energy density can be rewritten as

$$G^k=\frac{1}{2}{m}^k\cdot K{m}^k-m^k\cdot B^k+K_a^k,$$ (A1)

where

$$K=\left[\begin{array}{ccc}K& 0& 0\\ 0& K& 0\\ 0& 0& K\end{array}\right]-3\left[\begin{array}{ccc}{\lambda }_{100}{T}_1& {\lambda }_{111}{T}_4& {\lambda }_{111}{T}_6\\ {\lambda }_{111}{T}_4& {\lambda }_{100}{T}_2& {\lambda }_{111}{T}_5\\ {\lambda }_{111}{T}_6& {\lambda }_{111}{T}_5& {\lambda }_{100}{T}_3\end{array}\right],$$ (A2)

and

$$B^k=K{c}^k+{\mu }_0{M}_{s}H.$$ (A3)

Due to the symmetry in Terfenol-D crystal structure given as $K_a^1=K_a^2=K_0^1$, $K_a^3=K_a^4=K_0^2$, $K_a^5=K_a^6=K_0^3$ and $K_a^7=K_a^8=K_0^4$, the eight easy axes are defined as

$$c=\left\{\begin{array}{c}{c}^1{c}^2{c}^3{c}^4{c}^5{c}^6{c}^7{c}^8\end{array}\right\}=\frac{1}{\sqrt{3}}\left[\begin{array}{cccccccc}1& -1& -1& 1& 1& -1& 1& -1\\ 1& -1& 1& -1& -1& 1& 1& -1\\ -1& 1& 1& -1& 1& -1& 1& -1\end{array}\right].$$ (A4)

Consequently, an analytical expression of the normalized magnetic domain orientation can be determined by

$${\widehat{m}}^k=\frac{m^k}{\Vert m^k\Vert },$$ (A5)

where

$$m^k=K^{-1}\left(B^k+\frac{1-c^k\cdot K^{-1}{B}^k}{c^k\cdot K^{-1}{c}^{-1}}{c}^k\right).$$ (A6)

Appendix 2

This appendix presents two derivations for $\partial {\xi }_{\text{an}}^k/\partial T_i$ and $\partial {\xi }_{\text{an}}^k/\partial H_i$ that are required for evaluation of reversible volume fraction increment $\Delta {\xi }_{\text{an}}^k$. The derivatives of reversible volume fraction with respect to magnetic field and mechanical stress excitations are

$$\frac{\partial {\xi }_{\text{an}}^k}{\partial H_i}=-\frac{1}{\Omega }{\xi }_{\text{an}}^k\frac{\partial G^k}{\partial H_i}+\frac{1}{\Omega }{\xi }_{\text{an}}^k\sum _{i=1}^8\left({\xi }_{\text{an}}^i\frac{\partial G^i}{\partial H_i}\right),$$ (A7)

$$\frac{\partial {\xi }_{\text{an}}^k}{\partial T_i}=-\frac{1}{\Omega }{\xi }_{\text{an}}^k\frac{\partial G^k}{\partial T_i}+\frac{1}{\Omega }{\xi }_{\text{an}}^k\sum _{i=1}^8\left({\xi }_{\text{an}}^i\frac{\partial G^i}{\partial T_i}\right),$$ (A8)

where the derivatives of Gibbs energy density are calculated using the normalized magnetic domain orientation ${\widehat{m}}^k$; that is

$$\frac{\partial G^k}{\partial H_i}={\left({\widehat{m}}^k\right)}^{\top }K\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)-{\left({\widehat{m}}^k\right)}^{\top }\frac{\partial B^k}{\partial H_i}-{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}^{\top }{B}^k,$$ (A9)

$$\frac{\partial G^k}{\partial T_i}={\left({\widehat{m}}^k\right)}^{\top }K\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)+\frac{1}{2}{\left({\widehat{m}}^k\right)}^{\top }\frac{\partial K}{\partial T_i}-{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}^{\top }{B}^k.$$ (A10)

In Eqs. (A9) and (A10), the derivatives of the normalized kth magnetic domain orientation are

$$\frac{\partial {\widehat{m}}^k}{\partial H_i}=\frac{1}{\Vert m^k\Vert }\frac{\partial m^k}{\partial H_i}-\frac{m^k}{{\Vert m^k\Vert }^3}\left[{\left({\widehat{m}}^k\right)}^{\top }\frac{\partial m^k}{\partial H_i}\right],$$ (A11)

$$\frac{\partial {\widehat{m}}^k}{\partial T_i}=\frac{1}{\Vert m^k\Vert }\frac{\partial m^k}{\partial T_i}-\frac{m^k}{{\Vert m^k\Vert }^3}\left[{\left({\widehat{m}}^k\right)}^{\top }\frac{\partial m^k}{\partial T_i}\right],$$ (A12)

where the derivatives of the non-normalized kth magnetic domain orientation are

$$\frac{\partial m^k}{\partial H_i}=K^{-1}\left[\frac{\partial B^k}{\partial H_i}-\frac{{\left(c^k\right)}^{\top }\left(K^{-1}\frac{\partial B^k}{\partial H_i}\right)}{{\left(c^k\right)}^{\top }\left(K^{-1}{c}^k\right)}{c}^k\right],$$ (A13)

$$\frac{\partial m^k}{\partial T_i}=K^{-1}\left[-\frac{\partial K}{\partial T_i}{m}^k+\frac{{\left(c^k\right)}^{\top }\left(K^{-1}\frac{\partial B^k}{\partial H_i}{m}^k\right)}{{\left(c^k\right)}^{\top }\left(K^{-1}{c}^k\right)}{c}^k\right].$$ (A14)

In Eqs. (A13) and (A14), the derivatives of B^k with respect to magnetic field H_i are

$$\frac{\partial B^k}{\partial H_1}=\left\{\begin{array}{c}{\mu }_0{M}_{\text{s}}\\ 0\\ 0\end{array}\right\},$$ (A15)

$$\frac{\partial B^k}{\partial H_2}=\left\{\begin{array}{c}0\\ {\mu }_0{M}_{\text{s}}\\ 0\end{array}\right\},$$ (A16)

$$\frac{\partial B^k}{\partial H_3}=\left\{\begin{array}{c}0\\ 0\\ {\mu }_0{M}_{\text{s}}\end{array}\right\}.$$ (A17)

Moreover, the derivatives of K with respect to tensile stresses are

$$\frac{\partial K}{\partial T_1}=-3{\lambda }_{100}\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right],$$ (A18)

$$\frac{\partial K}{\partial T_2}=-3{\lambda }_{100}\left[\begin{array}{lll}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right],$$ (A19)

$$\frac{\partial K}{\partial T_3}=-3{\lambda }_{100}\left[\begin{array}{lll}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right].$$ (A20)

Similarly, the derivatives of K with respect to shear stresses are

$$\frac{\partial K}{\partial T_4}=-3{\lambda }_{111}\left[\begin{array}{lll}0\hfill & 1\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right],$$ (A21)

$$\frac{\partial K}{\partial T_5}=-3{\lambda }_{111}\left[\begin{array}{lll}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill \end{array}\right],$$ (A22)

$$\frac{\partial K}{\partial T_6}=-3{\lambda }_{11}\left[\begin{array}{lll}0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill \end{array}\right].$$ (A23)

Appendix 3

This appendix presents analytical expressions of ∂ξ^k/∂H_i, ∂ξ^k ∂T_i, $\partial S_m^k/\partial H_i$ and $\partial S_m^k/\partial T_i$ that are required for determination of ∂λ∂H_i and ∂λ/∂T_i. The derivatives of the kth volume fraction with respect to magnetic field H_i and stress T_i are, receptively,

$$\frac{\partial {\xi }^k}{\partial H_i}=c_{\text{an}}\frac{\partial {\xi }_{\text{an}}^k}{\partial H_i}-\left(1-c_{\text{an}}\right)\frac{\partial {\xi }_{\text{ir}}^k}{\partial H_i},$$ (A24)

$$\frac{\partial {\xi }^k}{\partial T_i}=c_{\text{an}}\frac{\partial {\xi }_{\text{an}}^k}{\partial T_i}-\left(1-c_{\text{an}}\right)\frac{\partial {\xi }_{\text{ir}}^k}{\partial T_i}.$$ (A25)

The derivatives of the kth irreversible volume fraction with respect to magnetic field H_i and stresses T_i are, receptively,

$$\frac{\partial {\xi }_{\text{irr}}^k}{\partial H_i}=k_p^{-1}\left({\xi }_{\text{an}}^k-{\xi }_0^k\right){\mu }_0{M}_{\text{s}},$$ (A26)

$$\frac{\partial {\xi }_{\text{irr}}^k}{\partial T_i}=\frac{3}{2}{k}_p^{-1}\left({\xi }_{\text{an}}^k-{\xi }_0^k\right){\lambda }_{100},\text{ where }i=1,2,3,$$ (A27)

$$\frac{\partial {\xi }_{\text{irr }}^k}{\partial T_i}=3k_p^{-1}\left({\xi }_{\text{an }}^k-{\xi }_0^k\right){\lambda }_{111},\text{ where }i=4,5,6.$$ (A28)

The derivatives of the kth local magnetostriction are with respect to magnetic field H_i and stress T_i are, receptively,

$$\frac{\partial S_{\text{m}}^k}{\partial H_i}=3\left\{\begin{array}{c}{\lambda }_{100}{\left({\widehat{m}}^k\right)}_1{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_1\\ {\lambda }_{100}{\left({\widehat{m}}^k\right)}_2{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_2\\ {\lambda }_{100}{\left({\widehat{m}}^k\right)}_3{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_3\\ {\lambda }_{111}\left[{\left({\widehat{m}}^k\right)}_2{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_3+{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_2{\left({\widehat{m}}^k\right)}_3\right]\\ {\lambda }_{111}\left[{\left({\widehat{m}}^k\right)}_1{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_3+{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_1{\left({\widehat{m}}^k\right)}_3\right]\\ {\lambda }_{111}\left[{\left({\widehat{m}}^k\right)}_1{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_2+{\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_1{\left({\widehat{m}}^k\right)}_2\right]\end{array}\right\},$$ (A29)

$$\frac{\partial S_{\text{m}}^k}{\partial T_i}=3\left\{\begin{array}{c}{\lambda }_{100}{\left({\widehat{m}}^k\right)}_1{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_1\\ {\lambda }_{100}{\left({\widehat{m}}^k\right)}_2{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_2\\ {\lambda }_{100}{\left({\widehat{m}}^k\right)}_3{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_3\\ {\lambda }_{111}\left[{\left({\widehat{m}}^k\right)}_2{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_3+{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_2{\left({\widehat{m}}^k\right)}_3\right]\\ {\lambda }_{111}\left[{\left({\widehat{m}}^k\right)}_1{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_3+{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_1{\left({\widehat{m}}^k\right)}_3\right]\\ {\lambda }_{111}\left[{\left({\widehat{m}}^k\right)}_1{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_2+{\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_1{\left({\widehat{m}}^k\right)}_2\right]\end{array}\right\},$$ (A30)

where ${\left({\widehat{m}}^k\right)}_j$, ${\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_j$, and ${\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_j$ are the jth entity in column vectors. Analytical expressions of the partial derivatives ${\left(\frac{\partial {\widehat{m}}^k}{\partial H_i}\right)}_j$ and ${\left(\frac{\partial {\widehat{m}}^k}{\partial T_i}\right)}_j$ are in Eqs. (A13) and (A14), respectively.

Appendix 4

This appendix presents several analytical expressions that are required for evaluation of Jacobian matrix coordinate transformation. Coordinate transformation algorithms were developed to transfer data between the local coordinates and the global coordinates. In this study, the prime symbol indicates variables in the global coordinate while the unprimed symbol of course denotes variables in the local coordinate. The cartesian coordinate defined for the Terfenol-D particles is

$$V=\left\{\begin{array}{lll}{e}_1\hfill & e_2\hfill & e_3\hfill \end{array}\right\}=\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right].$$ (A31)

The global coordinate defined in relative to the local coordinates is

$$U=\left\{\begin{array}{c}{e}_1^{\prime }{e}_2^{\prime }{e}_3^{\prime }\end{array}\right\}.$$ (A32)

For the first order tensor O_i, such as magnetic field, magnetization and magnetic flux density, the coordinate transformation rule follows

$$O_i^{\prime }=Q_{ij}{O}_j,$$ (A33)

where Q_ij = e′_i ·e_j or Q=U^TV. Notice that Q^T = Q⁻¹, this coordinate transformation can be written in matrix form as

$$O=Q^{\top }{O}^{\prime }\text{ and }{O}^{\prime }=QO.$$ (A34)

For the second order tensor P_ij, such as strain and stress, the coordinate transformation rule follows

$$P_{ij}^{\prime }=Q_{ip}{Q}_{jq}{P}_{pq}.$$ (A35)

In this study, the second order tensor is described in Voigt notation given as

$$P={\left\{\begin{array}{c}{P}_{11}{P}_{22}{P}_{33}{P}_{23}{P}_{13}{P}_{12}\end{array}\right\}}^{\text{T}}.$$ (A36)

Therefore, the coordinate transformation ruler can be written as

$$P=X^{-1}{P}^{\prime }\text{ and }{P}^{\prime }=XP,$$ (A37)

where

$$X=\left[\begin{array}{cccccc}{Q}_{11}^2& Q_{12}^2& Q_{13}^2& 2Q_{12}{Q}_{13}& 2Q_{11}{Q}_{13}& 2Q_{11}{Q}_{12}\\ Q_{21}^2& Q_{22}^2& Q_{23}^2& 2Q_{22}{Q}_{23}& 2Q_{21}{Q}_{23}& 2Q_{21}{Q}_{22}\\ Q_{31}^2& Q_{32}^2& Q_{33}^2& 2Q_{32}{Q}_{33}& 2Q_{32}{Q}_{33}& 2Q_{31}{Q}_{32}\\ Q_{21}{Q}_{31}& Q_{22}{Q}_{32}& Q_{23}{Q}_{33}& Q_{22}{Q}_{33}+Q_{23}{Q}_{32}& Q_{21}{Q}_{33}+Q_{23}{Q}_{31}& Q_{21}{Q}_{32}+Q_{22}{Q}_{31}\\ Q_{11}{Q}_{31}& Q_{12}{Q}_{32}& Q_{13}{Q}_{33}& Q_{12}{Q}_{33}+Q_{13}{Q}_{32}& Q_{11}{Q}_{33}+Q_{13}{Q}_{31}& Q_{11}{Q}_{32}+Q_{12}{Q}_{31}\\ Q_{11}{Q}_{21}& Q_{12}{Q}_{22}& Q_{13}{Q}_{23}& Q_{12}{Q}_{23}+Q_{13}{Q}_{22}& Q_{11}{Q}_{23}+Q_{13}{Q}_{21}& Q_{11}{Q}_{22}+Q_{12}{Q}_{21}\end{array}\right].$$ (A38)

For the fourth order tensor R_ijkl, such as compliance, the coordinate transformation rule follows

$$R_{\mathit{\text{ijkl}}}^{\prime }=Q_{ip}{Q}_{jq}{Q}_{kr}{Q}_{ls}{R}_{\mathit{\text{pqrs}}}\text{ and }{R}_{\mathit{\text{pqrs}}}=Q_{ip}{Q}_{jq}{Q}_{kr}{Q}_{ls}{R}_{\mathit{\text{ijkl}}}^{\prime }.$$ (A39)

Since the compliance in this study is also described in Voigt notation, the coordinate transformation rule in matrix form is

$$R^{\prime }=X^{-\top }R{X}^{-1}.$$ (A40)

The coordinate transformation rule for the third order tensor W_ijk can be written as

$$W_{ijk}^{\prime }=Q_{ip}{Q}_{jq}{Q}_{kr}{W}_{pqr}\text{ and }{W}_{pqr}=Q_{ip}{Q}_{jq}{Q}_{kr}{W}_{ijk}^{\prime }\text{. }$$ (A41)

Due to Voigt notation implemented in this study, the coordinate transformation rule follows

$$\frac{\partial M^{\prime }}{\partial T^{\prime }}=Q\frac{\partial M}{\partial T}{X}^{-1}\text{ and }\frac{\partial {\lambda }^{\prime }}{\partial T^{\prime }}=Q\frac{\partial \lambda }{\partial T}{X}^{-1}.$$ (A42)