Magnetoelastic instabilities in soft laminates with ferromagnetic hyperelastic phases

We investigate the microscopic and macroscopic instabilities developing in magnetoactive elastomer (MAE) composites undergoing large deformations in the presence of an external magnetic field. In particular, we consider the MAEs with bi-phasic layered microstructure, with phases exhibiting ferromagnetic behavior. We derive an explicit expression for the magnetic field-induced deformation of MAEs with hyperelastic phases. To perform the magnetoelastic instability analysis, we employ the small-amplitude perturbations superimposed on finite deformations in the presence of the magnetic field. We examine the interplay between the macroscopic and microscopic instabilities. We find that the layered MAEs can develop microscopic instability with antisymmetric buckling modes, in addition to the classical symmetric mode. Notably, the antisymmetric microscopic instability mode does not appear in a purely mechanical scenario (when a magnetic field is absent). Furthermore, our analysis reveals that the wavelength of buckling patterns is highly tunable by the applied magnetic field, and by the properties and volume fractions of the phases. Our findings provide the information for designing materials with reconfigurable microstructures. This material ability can be used to actively tune the behavior of materials by a remotely applied magnetic field. The results can be utilized in designing tunable acoustic metamaterials, soft actuators, sensors, and shape morphing devices.

In principle, MAEs are composite materials consisting of magnetizable particles (for example, carbonyl iron, nickel, or Terfenol-D) embedded in an elastomeric matrix material (such as silicone rubber, polyurethane) [18]. The magnetizable particles (from micro-to nano-size) are added into the matrix material in its liquid state. Upon polymerization, the MAEs with randomly distributed magnetizable particles are produced. Curing in the presence of a magnetic field, however, results in the alignment of magnetizable particles into chain-like structures (for a detailed description of the MAE synthesis, interested readers are referred to the review article by Bastola and Hossain [19]).
There is a significant body of studies concerning the magnetomechanical characterization of MAEs with different microstructures " random and chain-like" are present in the literature. Jolly et al. [18] and Danas et al. [20] studied the shear response of chain-structured MAEs, showing, for example, that the effective shear modulus increases in the presence of a magnetic field. The effective moduli of MAEs are also reported to be increased by the applied magnetic field under uniaxial compression [21] and tensile tests [22]. The magnetostriction of MAEs with randomly distributed magnetizable particles under a very high magnetic field is analyzed by Bednarek [23]. Ginder et al. [2] and Guan et al. [24] determined the magnetostriction of random and chain-structured MAEs. The effect of particle rotation on the effective magnetization of MAEs is investigated by Lanotte et al. [25]. Moreno et al. [26] provided a comprehensive experimental characterization of MAEs with a special focus on the material response under various strain rates. Dargahi et al. [27] performed the dynamic characterization of MAEs subjected to a wide range of excitation frequencies and magnetic flux densities. In these studies, the magnetizable particles are effectively rigid as compared to the elastomer matrix. The magneto-mechanical coupling observed in these MAEs is therefore majorly governed by the two underlying mechanisms, namely, magnetic torques and magnetic interaction between the particles.
The pioneering works of Brown [28], Maugin and Eringen [29], Tiersten [30], Toupin [31], Truesdell and Toupin [32] laid the foundation for the theory of magnetoelastic (and mathematically analogous electroelastic) behavior of continuum, which has been reformulated and further developed [33][34][35]. In parallel, a number of microstructural-based magneto-elastic constitutive models are also developed, for example the lattice model [18,36,37]. Additionally, significant efforts have been made to implement the non-linear magnetoelastic framework into numerical schemes [38][39][40]. Castañeda and Galipeau [41] proposed an analytical approach to estimate the effective behavior of MAEs with the random distribution of magnetoactive particles. In particular, they developed a finite strain nonlinear homogenization framework to determine the total magnetoelastic stress in MAEs under the combined mechanical and magnetic loading. By employing this framework, Galipeau and Castañeda [42] studied the effects of randomly distributed magnetizable particle shape, distribution, and concentration on the effective properties of MAEs. Moreover, Galipeau et al. [43] investigated the behavior of MAEs with periodic arrangements of circular and elliptical fibers, showing that by tailoring the periodic microstructure of MAEs, their magneto-mechanical behavior could be highly tuned. We note that these systems share some similarities with their mathematically analogous dielectric elastomer composites [44][45][46].
While the heterogeneity provides access to the tailored and enhanced coupled behavior, it is also a source for the development of microstructural instabilities. The instability phenomenon historically has been considered as a failure mode, which is to be predicted and avoided. This motivated the investigation of instabilities in composites subjected to purely mechanical loading [47][48][49][50][51][52][53][54][55][56]. Recently, the elastic instability phenomenon has been embraced to design materials with unusual properties and switchable functionalities [57,58]. Examples include instability-induced elastic wave band gaps [59,60], auxetic behavior [61][62][63], and photonic switches [64]. The possibility of controlling the instability development via magnetic field can provide the opportunity to activate these functionalities remotely.
Extending the instability analysis for the coupled magnetomechanical case, Ottenio et al. [65] studied the onset of magneto-mechanical instabilities in isotropic MAEs with a focus on surface instabilities of a homogeneous magnetoactive half-space. Kankanala and Triantafyllidis [66] investigated the failure modes of a rectangular MAE subjected to plane-strain loading conditions in the presence of a magnetic field. Rudykh and Bertoldi [67] analyzed the onset of macroscopic instabilities in anisotropic MAEs by deriving the exact solution for MAEs with layered microstructure. Danas and Triantafyllidis [68] studied the finite-wavelength instability modes occurring in an MAE substrate/layer system under a transverse magnetic field. Recently, Goshkoderia and Rudykh [69] employed a numerical finite element-based code to investigate macroscopic instabilities in MAEs with circular and elliptical inclusions. Very recently, Goshkoderia et al. [70] experimentally illustrated that the instability pattern can be tailored by the application of magnetic field in particulate magnetoactive composites.
Motivated by recent experimental studies showing the tunability of finite-wavelength instabilities via magnetic field [70,71], in this work, we study the onset of microscopic instabilities and associated buckling patterns in MAEs. In particular, we consider the MAEs with bi-phasic layered microstructure exhibiting ferromagnetic behavior. To investigate the onset of instabilities, we consider the small-amplitude perturbations superimposed on finite deformations in the presence of magnetic field to perform the microscopic instability analysis. Moreover, we analyze the limit corresponding to the long-wave or macroscopic instability. We examine the influences of the applied magnetic field and material parameters on MAE's stability. Additionally, we study the magnetostriction of layered MAEs and derive an explicit expression for the induced stretch as a function of the applied magnetic field, mechanical and magnetic properties of layers, and their volume fractions. This paper is structured as follows: Section 2 presents the theoretical background on magneto-elastostatics and incremental analysis together with the constitutive laws for magnetically linear and ferromagnetic behavior. Section 3 is concerned with the analysis of the magnetodeformation and determination of the microscopic and macroscopic instabilities in layered MAEs. Section 4 illustrates the results with examples of magnetostriction and magnetoelastic instabilities in MAEs with various morphologies and material properties. Section 5 concludes the paper with a summary and a discussion.

Magneto-elastostatics
Consider a magnetoelastic deformable body that occupies a region B 0 in the undeformed configuration. Under the action of a combination of mechanical loading and magnetic field, the body deforms into the current configuration B. The deformation is described by the function x that maps the reference point X in B 0 to its deformed position x = x(X) in B. The deformation gradient is thus defined as F = ∂x/∂X with its In this work, we consider quasi-static deformation in the absence of an electric field, electrical charges, or electrical currents within the material. Consequently, the magnetic induction B and magnetic intensity H (in the current configuration) satisfy the following field equations divB = 0 and curlH = 0, where the div and curl are the differential operators defined with respect to x. In the Lagrangian form, Eq. (1) can be written as [33] DivB 0 = 0 and CurlH 0 = 0, where the Div and Curl operators are defined with respect to X, B 0 = JF − 1 B and H 0 = F T H are the Lagrangian counterparts of the magnetic fields. Moreover, in a heterogeneous body, magnetic fields satisfy the jump conditions across the interface or alternatively, where N and N 0 denote the normal to the interface in the deformed and reference configurations, respectively. The jump operator [[•]] ≡ (•) + − (•) − is defined such that N and N 0 are pointing towards phase (•) − . The magnetization is customarily defined as where μ 0 is the vacuum magnetic permeability.
Following the works of Brown [28], Coleman and Noll [72], Kovetz [73], the magnetization is constitutively defined in terms of free-energy function ϕ(F, B) as where ρ is the material density in the current configuration. The total Cauchy stress σ is given by In terms of these relations, the energy-density function ϕ fully characterizes the behavior of magneto-active elastomers. Note that in the absence of material (or vacuum), the stress tensor (7) is still non-zero and depends on the magnetic field. The corresponding stress tensor is also referred to as Maxwell stress. The free energy in Lagrangian form is defined as Φ(F,B 0 ) = ϕ(F,J − 1 FB 0 ). In terms of Φ, a Lagrangian amended energy function can be constructed as [33] W ( F, B 0 ) where ρ 0 = ρJ is the material density in the reference configuration.
Then, the corresponding Lagrangian variables are given by where P is the 1st Piola Kirchhoff stress tensor. Eqs. (7)-(9) hold true for compressible hyperelastic materials. For incompressible materials (J = 1), however, the total Cauchy stress tensor is where p is the Lagrange multiplier associated with the incompressibility constraint.
In the absence of body forces, the total Cauchy and 1st Piola-Kirchhoff stress tensors satisfy the equilibrium conditions divσ = 0 and DivP = 0.
The corresponding jump conditions at the interface are

Incremental equations
Here, following the approach commonly used to study instabilities [65,67,69,74,75], we define the governing equations for the incremental deformation superimposed on finite deformation in the presence of a magnetic field. The incremental governing equations are DivṖ = 0, DivḂ 0 = 0 and CurlḢ 0 = 0, where Ṗ , Ḃ 0 , and Ḣ 0 are the incremental changes in P, B 0 , and H 0 , respectively. Under the assumption that the incremental quantities are sufficiently small, the linearized constitutive relations can be expressed using the Einstein summation notation as where For an incompressible material, Eq. (14) 1 modifies to where ṗ is the incremental change in p.

Magnetic energy functions
In this work, we assume the magnetoactive elastomers to be magnetically soft, so that the hysteresis effects can be neglected. Moreover, we consider the magnetic particles to be isotropic and superparamagnetic, i.e., demagnetization effects are neglected. On the basis of whether the materials show saturation effects, they can be constitutively defined either by linear or ferromagnetic material models.

Linear magnetic materials
Linear magnetic materials show a linear dependence of magnetization on the magnetic induction B, namely, μ 0 M = χB, (23) where χ is the magnetic susceptibility. 1 Alternatively, the constitutive relation can be written as B = μH, (24) where μ = μ 0 /(1 − χ) is the magnetic permeability. The corresponding magnetic energy is

Ferromagnetic materials
For ferromagnetic materials, the magnetization reaches a saturation state at sufficiently high magnetic fields, beyond which there is no further increase in magnetization. Assuming the soft ferromagnetic behavior and magnetic particles being large compared to the typical domain size, the material behavior can be idealized as having a singlevalued constitutive behavior. Although other models can be used, we use the isotropic Langevin model to define the ferromagnetic behavior in the forthcoming analysis. For this model, the magnetization is defined by the following relation 1 In this work, we use the magnetic susceptibility, χ, defined via magnetic induction as μ 0 M = χB. Note the alternative definition of magnetic susceptibility in terms of magnetic intensity is M = χ H H. These susceptibilities are related as where m s is the saturation magnetization and B is the magnitude of the magnetic induction vector B, i.e., B = |B|. Alternatively, the constitutive relation can also be expressed as where The corresponding magnetic energy is . As expected, the B − H curve for linear magnetic material shows a linear response. However, for ferromagnetic materials, the dependence is nonlinear, specifically at small magnetic fields. However, once the saturation limit of magnetization is achieved at a relatively high magnetic field, they show the linear relation in H and B. Fig. 1 b shows the normalized magnitude of magnetization M /m s as the function of normalized magnetic induction B/(μ 0 m s ). The solid curves represent the response of the ferromagnetic materials, whereas the dash-dotted curves correspond to the linear magnetic materials. We consider the materials with three initial susceptibilities: χ = 0.9 (black curves), χ = 0.7 (green curves), and χ = 0.5 (red curves). As expected, the linear magnetic materials show the linear dependence of magnetization on magnetic induction, with slopes proportional to their corresponding magnetic susceptibilities χ. Ferromagnetic materials also show the linear response, however, only at small magnetic fields. At relatively high magnetic induction magnitudes, the magnetization in these materials approaches the saturation values, M/m s →1 (see the solid curves). In ferromagnetic materials with higher initial susceptibilities, the saturation magnetization values are achieved at comparatively smaller magnetic induction magnitudes.

Analysis and Results
We examine incompressible magnetoactive elastomers with bilayer microstructure (schematically shown in Fig. 2) having lamination direction L. The volume fraction of the matrix phase is c (m) , and that of the stiff layer is c (f) = 1 − c (m) . Here and thereafter, we denote the parameters and fields corresponding to the matrix and stiff layers as (•) (m) and (•) (f) , respectively. The average deformation gradient F and magnetic induction B are defined as In this work, we investigate the magneto-mechanical loading defined as where λ is the stretch along the direction of layers, and λ 2 = λ − 1 for incompressible MAEs. Note that we consider an idealization of the periodic microstructure unit cells (schematically shown in Fig. 2b) situated far from the specimen boundaries. Under the assumed separation of length scales, the mechanical and magnetic fields can be considered to be homogeneous in each layer of the laminate and are determined by the appropriate jump conditions. The displacement continuity condition at the layer interface implies ( where s is a unit vector perpendicular to the lamination direction L. Using Eq. (32), for the deformation gradient F (31) 1 with incompressible phases, we can write In the deformed configuration, the thicknesses of the matrix and stiff layers are L (m) = c (m) L and L (f) = c (f) L, respectively, where L is the period of the layered material in the current state. Moreover, using the magnetic induction jump condition (3) at the interface for the current magnetic loading (31) 2 , we obtain We consider the laminates with the isotropic layers, with each layer (r) ∈ {m, f} defined by the following amended energy function where W (r) e is the elastic part and W (r) m is the magnetic part. Although the analysis presented here is general, we consider the elastic part of both phases to adopt the neoHookean material model for simplicity. The corresponding energy function is where G (r) is the shear modulus of the phase (r) and I (r) is the right Cauchy-Green deformation tensor, and F (r) is the deformation gradient. The magnetic part of the amended energy function W (r) m for each layer is defined as where the term ρ (r) ϕ (r) m can be defined either using the expression (25) or (29), according to the magnetic behavior of the layer; B (r) denotes the magnetic induction vector. Note that the second term, B (r) ⋅B (r) /(2μ 0 ), is independent of material constants; therefore, the magnetic energy is non-zero in the free space or in a non-magnetic material. The non-zero components of the corresponding magnetoelastic moduli tensors A ijkl , M ijk , and H ij for each phase are provided in Appendix A, separately for both types of magnetic behaviorslinear and ferromagnetic.

Magnetostriction
Here, we evaluate the deformation of the magnetoactive laminates with the application of magnetic field (31) 2 , without any mechanical traction. In particular, we study the homogenized response of the periodic unit cell shown in Fig. 2b. Using Eq. (10), the stress field inside an incompressible layer (r), with the amended energy function given by Eqs. (35)-(37) can be written as where the magnetic permeability μ (r) can either be constant or a function of B (28) depending on the choice of the energy function.
The stress field jump condition across the interface L = e 2 yields σ (m)

22
= σ (f) 22 . We assume that the finite MAE specimen (Fig. 2a) is surrounded by a vacuum. Using the mechanical traction-free boundary conditions and the stress field jump condition, we obtain where σ * m is the Maxwell stress tensor defined as Then, the stress components become By eliminating the Lagrange multipliers p (m) and p (f) from Eq. (41), the relation between the applied magnetic field and induced stretch is obtained as where μ r is the weighted harmonic mean of relative magnetic permeabilities, defined as Here, μ (r) r = μ (r) /μ 0 is the relative magnetic permeability of phase (r). In the case of the linear magnetic layer, μ (r) r is a constant. However, for the ferromagnetic layer, μ (r) r can be expressed as a function of B, in terms of layer's magnetic saturation value m (r) s and the initial magnetic susceptibility χ (r) (see Eq. (28)). Hence, expression (42) is applicable for MAEs having layers with any type of magnetic behaviorlinear or ferromagnetic. Equation (42) further simplifies to yield an explicit expression for λ, namely, For magnetoactive layers (μ r > 1), the application of magnetic field results in contraction along the layer direction, λ < 1 (or λ 2 > 1). We note that certain magneto-mechanical loading conditions can lead to the development of magnetoelastic instabilities [66,67]; the analysis of the magnetoelastic instabilities is provided in the next subsection.

Magnetoelastic instabilities
The onset of instabilities in MAE with bilayer microstructure is determined as follows. In each layer, we seek a solution for Eq. (22) of the form where k 1 is the wavenumber along the e 1 -direction. The incompressibility constraint implies where In terms of the non-zero components of magnetoelastic tensors, the incremental governing Eq. (22) can now be written as Eqs. (47), (48), and (49) provide a set of six linear homogeneous firstorder differential equations that depend on the vector of six unknown The equations can be written together as where the non-zero components of the matrix R are given in Appendix B.
The solution to Eq. (50) can be determined in the form where W is the eigenvector matrix of R, Z( Here, z is the eigenvalue vector of matrix R; u 0 is an arbitrary constant vector which will be determined using the continuity and quasi-periodicity conditions of the unit cell. For the periodic unit cell of the layered composite (as shown in Fig. 2), the quasi-periodic boundary conditions are where k 2 ∈ [0, 2π /L) is the Floquet parameter. In the domain 0 < x 2 < L + L (m) , solution (51) takes the form On substituting Eq. (53) into (52), we obtain The set of interface jump conditions for the incremental fields are Using Eqs. (46), (47), and (48), the jump conditions (55) can be rewritten in terms of the components of ũ as Eq. (56) Finally, by using Eq. (53) we obtain Combining Eqs. (54) and (58) results in the following condition for the existence of a non-trivial solution where Thus, if the condition (59) is satisfied for a combination of mechanical and magnetic loads, a solution of the form (46) exists. The solution ũ lies in the real space, and from Eq. (52), exp(ik 2 L) is also realvalued; hence, non-trivial solutions are exp(ik 2 L) = ±1.The instability criterion (59) is evaluated with scanning over the values of k 1 at different deformation levels for a given magnetic field until the eigenvalue with |exp(ik 2 L)| = 1 is obtained. Once the condition is satisfied, the corresponding stretch along the direction of layers (e 1 ) that separates the unstable and stable domain (illustratively shown in Fig. 6a) is termed as the critical stretch λ cr and the corresponding wavenumber is the critical wavenumber k 1cr .
Based on the buckling pattern wavelength, we distinguish the macroscopic (or long-wave) and microscopic instabilities. Macroscopic instability is characterized by the critical wavelength significantly larger than the characteristic microstructure (k 1cr →0), whereas microscopic instability may lead to the formation of a new periodicity of the order of the initial microstructure (see [48] for the purely mechanical case). Furthermore, depending on the two possible values of eigenvalue: exp(ik 2 L) = 1 and − 1, the buckling modes can be classified as symmetric for k 2 L = 2nπ and antisymmetric for k 2 L = (2n − 1)π (with n being an integer). For illustrating these instability modes in the plots, hereafter, we use k 2 L = 2π and k 2 L = π (with n = 1) to represent the symmetric and antisymmetric modes, respectively. These buckling modes are schematically shown in Fig. 3.

Examples
In this section, we illustrate the analysis through the examples for the laminate MAEs with magnetically inactive matrix (i.e., χ (m) = 0), and different magnetic behaviors of the stiffer active layer. In the discussion hereafter, we denote the magnetic parameters corresponding to the stiffer active layer without the superscript (f).

Magnetostriction in layered MAEs
In this subsection, we analyze the magnetic field induced deformation in the layered MAEs. In Fig. 4, we plot the field-induced stretch as Clearly, the magnetic field induced stretch λ mag increases with an increase in the applied magnetic field for both the MAEs with linear magnetic and ferromagnetic behaviors. We observe that MAE with the linear magnetic behavior undergoes larger deformations as compared to those with the ferromagnetic behavior. For instance, at B m = 7, the induced stretch corresponding to linear magnetic MAE is λ mag = 2.01, whereas in MAE with m s μ 0 = 2 T it is λ mag = 1.66. Moreover, among the MAEs with ferromagnetic behavior, the stretch λ mag decreases with a decrease in magnetic saturation value. For example, at B m = 10, the magnetic field-induced stretch decreases from λ mag = 2.74 to λ mag = 2.03 as magnetic saturation decreases from m s μ 0 = 10 T to m s μ 0 = 2 T (see green and red curves).
The observed dependence of magneto-deformation on m s μ 0 values is due to the variation in MAE's effective magnetic permeability. In particular, with the decrease in the magnetic saturation values, the effective magnetic permeability also decreases (28), leading to an increase in the contribution of magnetic stress into the total stress, Eq. (38). However, Maxwell's stress σ * m does not change with MAE's magnetic properties, and to satisfy the mechanical traction-free boundary conditions, the total stress inside the MAE also remains constant, Eq. (39). Therefore, an increase in magnetic stress is compensated by a decrease in mechanical stress. Thus, the MAE undergoes comparatively smaller deformations as the active layer's magnetic saturation value decreases.
Next, in  Fig. 5b. We observe that the field-induced stretch monotonically increases with an increase in c (f) , regardless of MAE's magnetic behavior (see Fig. 5a) and shear modulus contrast (see Fig. 5b). This is because, only the stiff layer contributes to the response of MAEs under the applied magnetic field. Similar to the observations in Fig. 4, the induced stretch is higher for linear magnetic behavior, and λ mag decreases with a decrease in m s μ 0 . As expected, the magnetic field induced deformation decreases with an increase in shear modulus contrast (see Fig. 5b).

Magnetoelastic instabilities in layered MAEs
In this subsection, we analyze the magnetoelastic instabilities in MAEs with bilayered microstructure. First, we investigate the effect of the applied magnetic field B m on the critical stretch λ cr and wavenumber k cr , and related instability modes. Here, λ cr denotes the critical stretch value (λ 1 along the direction of layers e 1 ) corresponding to the onset of instability. In the second part of this subsection, we examine the role of phase volume fraction in the development of instabilities in MAEs with different magnetic behaviors. In the following examples, we consider the MAEs with initial shear modulus contrast G (f) /G (m) = 10.

Effect of magnetic field on magnetoelastic instabilities
We start by illustrating the influence of the applied magnetic field on the stability of MAEs with linear magnetic behavior. Fig. 6 shows the critical stretch (a) and normalized critical wavenumbers: k * 1 and k * 2 (b) as the functions of normalized magnetic induction B m . The wavenumbers are normalized with respect to the period length L in the current configuration (see Fig. 2) as k * 1 = k 1 L and k * 2 = k 2 L. We consider the MAEs with stiff layer volume fraction c (f) = 0.6 and initial magnetic susceptibility χ = 0.95. Here and thereafter, we use solid and dotted curves for macroscopic and microscopic instabilities, respectively (see Fig. 6a). Furthermore, solid and dash-dotted curves denote the critical wavenumbers k * 1 and k * 2 , respectively (see Fig. 6b). We find that the critical stretch increases with an increase in the applied magnetic field. Furthermore, we observe that when the MAE is subjected to a smaller magnetic field (B m ≤ 2.5), it develops instabilities  under compressive strains (λ cr < 1). Interestingly, at higher magnetic fields, MAE is unstable even under tensile strains. For example, the MAE is unstable for λ < 1.59, when subjected to B m = 7. Moreover, we find that the instability mode switches at a certain threshold magnitude of magnetic induction B th m . In particular, macroscopic instability appears for B m < B th m , whereas microscopic instability emerges for B m > B th m . For the considered MAE, the threshold value is B th m = 4.1. The transition in the instability mode is also evident from the evolution of the critical wavenumbers (k * 1 and k * 2 ) with the magnetic field (see Fig. 6b). For B m > B th m , the wavenumbers have finite non-zero values, hence, representing the microscopic instability. In particular, the MAEs develop an antisymmetric mode of microscopic instability, as the critical wavenumber k * 2 = π, when subjected to this range of magnetic field values (see. Fig. 6b). Moreover, we find that the wavenumber k * 1 monotonically decreases with an increase in B m , hence, showing the tunability of buckling patterns with an applied magnetic field. For magnetic induction magnitudes smaller than B th m , both the critical wavenumbers (k * 1 and k * 2 ) approach zero, k cr →0, showing the long-wave or macroscopic loss of stability.
Next, we investigate the development of magnetoelastic instabilities in MAEs with ferromagnetic behavior. Fig. 7 shows the critical stretch Similar to MAEs with linear magnetic behavior, the MAEs with ferromagnetic behavior also develop instabilities at higher stretches when subjected to higher magnetic fields. However, we observe that the critical stretch, at a particular magnetic induction magnitude, decreases with a decrease in the MAE magnetic saturation value.  Fig. 7a and c). These observations hold regardless of the volume fraction of phases.
The effect of the applied magnetic field on the buckling patterns and instability modes strongly depends on the stiff layer volume fraction and its magnetic behavior. First, consider the MAEs with a high stiff layer volume fraction, c (f) = 0.4. We observe that in these MAEs, the  behaviors. However, the wavelength of the buckling pattern is smaller (higher k * 1 ) in MAEs with the ferromagnetic behavior as compared to linear ones, when they are to develop microscopic instabilities (see the black and blue curves in Fig. 7b).
The MAEs with smaller volume fraction, c (f) = 0.05, develop microscopic instabilities when subjected to smaller magnitudes of the magnetic field. The instability mode switches to macroscopic at magnitudes B m > B th m (see Fig. 7c and d). Moreover, the threshold magnitude B th m increases with a decrease in m s μ 0 (see inset in Fig. 7c). However, for smaller magnetic saturation values, m s μ 0 = 5 T and m s μ 0 = 2 T, the transition in the instability mode does not occur in the considered range of the applied magnetic field. Hence, as opposed to MAEs with high volume fractions (c (f) = 0.4), in MAEs with c (f) = 0.05, a decrease in magnetic saturation values promotes microscopic (or finite-wavelength) instabilities. Moreover, these MAEs develop the symmetric mode of microscopic instability, with the critical wavenumber k * 2 = 2π (see the dash-dotted curves in Fig. 7d).
The results indicate that in addition to the influence of the applied magnetic field and phase magnetic behavior, the instability development and associated buckling patterns significantly depend on the volume fraction of layers. A detailed analysis of the effect of phase volume fractions is provided in Section 4.2.2.
Next, we study the influence of initial magnetic susceptibility on the magnetoelastic instabilities in the ferromagnetic layered MAEs. To this end, in Fig. 8 The critical stretch decreases with a decrease in the initial magnetic susceptibility; this is observed for all magnetic saturation values. For instance, the critical stretch at B m = 10 corresponding to MAEs with m s μ 0 = 10 T decreases from λ cr = 1.69 to λ cr = 1.10 as susceptibility varies from χ = 0.95 to χ = 0.375 (compare the blue curves in Fig. 8a and e). Moreover, the critical wavenumber k * 1 increases with a decrease in χ, in the MAEs that develop microscopic instabilities. We note that the effect of initial magnetic susceptibility on the critical parameters, λ cr and k cr , is similar to that observed in the case of magnetic saturation values (see Fig. 7a and b). This is because a decrease in magnetic saturation and/or initial magnetic susceptibility values leads to a decrease in MAE's magnetization and vice-versa, at a given magnetic field

Effect of volume fraction of phases on magnetoelastic instabilities
Here, we study the effect of the phase volume fraction on the magnetoelastic instabilities. First, we examine the linear magnetic MAEs with χ = 0.95. In Fig. 9, we plot the critical stretch (a) and wavenumber (b) as the functions of stiff layer volume fraction c (f) . We consider the MAEs subjected to B m = 1 (blue curves), B m = 5 (green curves), and B m = 10 (red curves). For the sake of convenient discussion, in Fig. 9a, we have marked the first and second instability mode transition points as 'S and 'A, respectively. In particular, 'S represents the switch from symmetric microscopic instability mode to macroscopic, whereas 'A denotes the transition from macroscopic to antisymmetric microscopic instability, with an increase in c (f) .
For the MAEs subjected to smaller magnetic field levels, for example, B m = 1, the critical stretch increases with an increase in c (f) up to a certain value; beyond that volume fraction value, the critical stretch decreases with a further increase in the volume fraction. Moreover, when the stiff layer volume fraction is smaller than a particular threshold value, c (f) th , the MAEs develop symmetric microscopic buckling modes (k * 2 = 2π). However, at higher values of c (f) , a macroscopic loss of stability occurs. We also observe that the wavelength of the buckling pattern increases (k * 1 decreases) with an increase in c (f) , and it approaches the long-wave limit (k * 1 →0) for c (f) ≥ c (f) th . The corresponding threshold value is c (f) th = 0.07, which is marked as 'S on the blue curve (see Fig. 9a). We note that similar variation of critical parameters with stiff layer volume fraction has also been reported for layered composites subjected to purely mechanical loadings [55].
However, MAEs subjected to higher magnetic induction values show contrastingly different instability mode transitions and highly nonmonotonous variation of critical parameters. For example, consider the MAEs under B m = 5; these MAEs, similar to MAE under B m = 1, also show the first transition 'S in the instability mode. Interestingly, these MAEs undergo an additional transition, back to microscopic instability, at higher values of c (f) ; this shift in the instability mode is marked as 'A on the green curve in Fig. 9a. Both transitions are also evident from the evolution of critical wavenumbers with stiff layer volume fraction (see the green curves in Fig. 9b). Furthermore, we observe that in the MAEs developing microscopic instabilities, the critical wavelength significantly varies with the volume fraction. This high tunability of wavelength (or k * 1 ) is very pronounced in the vicinity of the extreme volume fraction values, i.e., c (f) →0 and c (f) →1 (see Fig. 9b).
Remarkably, the morphologies of MAEs that are to develop micro-scopic instabilities, can exhibit antisymmetric and symmetric instability modes with distinct values for critical wavenumber k * 2 , dictated by the stiff layer volume fraction. In particular, MAEs with c (f) smaller than that corresponding to first transition point 'S, i.e., c (f) < c (f) thI , has k * 2 = 2π. However, for stiff layer volume fraction higher than that of, c (f) > c We observe that the instability in ferromagnetic MAEs develops at smaller stretches than in their linear magnetic counterparts. Among the ferromagnetic MAEs, the lesser the magnetic saturation value, the smaller is the critical stretch. Moreover, the critical wavelength (wavenumber) decreases (increases) with a decrease in m s μ 0 . These findings are consistent with the previous observations in Fig. 7. Similar to the linear magnetic MAEs, ferromagnetic MAEs also offer a high tunability of the critical wavenumber k * 1 , especially in the vicinity of the extreme volume fraction values (see Fig. 10b).
Comparing the critical parameters of ferromagnetic MAEs (Fig. 10) with those of linear magnetic MAEs (in Fig. 9), we find that a decrease in the magnetic field magnitude (in linear MAEs) has a similar influence as decreasing the magnetic saturation value (in ferromagnetic MAEs under a constant magnetic field). This occurs because of the magnetic saturation effect present in the ferromagnetic MAEs. In particular, the saturation effect takes place at smaller magnetic fields in MAEs with small saturation values. Therefore, when subjected to higher values of B m , the influence of the applied magnetic field on the magnetoelastic tensors of ferromagnetic MAEs is significantly weaker than that in their magnetically linear counterparts.
The transition of instability modes also demonstrates the behavior resembling that in Fig. 9. Finally, we illustrate the influence of the initial magnetic susceptibilities on the critical parameters and instability mode transition with phase volume fraction. Fig. 11 shows the critical parameters for linear magnetic MAEs with χ = 0.95 (black curves), χ = 0.80 (blue curves), and χ = 0.375 (red curves). We consider the MAEs subjected to magnetic inductions B m = 1 ( Fig. 11a and b), B m = 5 ( Fig. 11c and d), and B m = 10 ( Fig. 11e and f).
Consistent with the findings in Fig. 8, we observe that the MAEs with lower values of χ develop instabilities at smaller stretch levels. Moreover, the critical wavenumber k * 1 decreases with an increase in χ, in the MAEs that develop microscopic instabilities. This holds independent of the magnitude of magnetic induction. The interplay between the instability modes is also dictated by the magnetic susceptibility of the MAEs. In particular, we observe that the threshold value corresponding to the transition point 'S increases with a decrease in χ, irrespective of the magnetic fields magnitude.

Conclusion
In this paper, we investigated the behavior of MAEs with bi-phasic layered microstructure with ferromagnetic hyperelastic phases. We considered the MAE laminates subjected to a magnetic field perpendicular to the direction of layers. First, we derived an explicit expression for the field-induced stretch in response to the remotely applied magnetic field. Second, we performed the magnetoelastic instability analysis for layered MAEs, by employing the small amplitude perturbation superimposed on finite deformations in the presence of a magnetic field. While the formulation developed here is generalfor any magnetic behavior of phases, the results are presented for the special class of MAEs with magnetically inactive matrix and active stiff layer phase.
We found that the layered MAEs experience tension along the direction of the magnetic field, and the induced stretch increases with an increase in the applied magnetic field. However, because of the magnetic saturation effect, the MAEs with smaller saturation values attain smaller deformation levels. We also showed that the MAEs with higher volume fractions of the active phase develop large deformations, irrespective of shear modulus contrast between the phases.
The layered MAEs, when subjected to higher magnitudes of the magnetic field, develop instabilities at higher stretches along the direction of layers (perpendicular to the magnetic field). In fact, MAEs are observed to be unstable even under tensile strains in the presence of a strong magnetic field. The magnetic saturation effect, however, results in a decrease of critical stretch levels. Moreover, the wavelength of buckling patterns is shown to be highly tunable by the applied magnetic field. The comparison of critical parametersfor MAEs with various morphologiesshows that a decrease in magnetic susceptibility and/or magnetic saturation values (at a given magnetic field magnitude) has a similar response as reducing the applied magnetic field magnitude. The instability mode and their transitions in layered MAEs are strongly dictated by the volume fraction of phases together with the applied magnetic field. In the presence of a weak magnetic field, similar to the purely mechanical case of layered composites, the layered MAEs also show the transition in instability modes once, with the change in volume fraction. Thus, the symmetric microscopic instability occurs at small volume fractions of the active stiff phase, whereas macroscopic loss of stability occurs at higher volume fractions. Under higher magnetic fields, however, the MAE laminates show two transitions with three distinct instability modes at different active phase volume fractions. First, the symmetric microscopic instability is detected at smaller volume fractions. Second, at moderate volume fractions, the long-wave instability emerges. Interestingly, the MAEs with higher volume fractions develop microscopic instability with antisymmetric buckling patterns. We found that under stronger magnetic fields, the range of active stiff phase volume fractions, at which the antisymmetric mode is attained, further increases. Hence, the application of magnetic field promotes the development of the antisymmetric buckling patterns. It is worth noting that the antisymmetric microscopic instability mode is inadmissible in the purely mechanical setting (without a magnetic field).
The presented results can help widen the design space for novel materials with switchable functionalities with potential applications in remotely controlled soft microactuators and sensors. Moreover, the theoretically predicted antisymmetric buckling mode can motivate further experimental studies of the microstructured MAEs. In the study, we have considered the MAEs subjected to quasi-static loading; therefore, the viscous and inertial effects have not been considered. However, for the dynamic loading, these effects can influence the material stability, as observed, for example, in the soft laminates [76]. To investigate of the influence of time-dependent magneto-mechanical loading on the instability development in MAEs, one should account for the phase rate-dependent behavior and inertia in the modeling. Additionally, the understanding of the material behavior can benefit from the implementation of multiscale modeling that could more accurately capture the global finite size effects, as well as smaller length-scale effects (such as, for example, dipole-dipole interactions).

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.