Magneto-deformation and transverse elastic waves in hard-magnetic soft laminates Mechanics of Materials

We study the magneto-mechanical behavior of periodic laminates made of hard -magnetic active elastomers (HMAEs). We formulate the amended free-energy function for HMAEs, and derive an explicit expression for the induced deformation of the HMAE laminate as a function of the applied magnetic field. Next, we employ the “ small-on-large ” framework and examine the small-amplitude shear waves propagating in the finitely deformed HMAE laminate in a magnetic field. We find that the remanent magnetization of HMAE phases can result in compressive deformations (in the direction of the applied magnetic field), as opposed to the induced tensile deformation in previously considered soft -magnetic active laminates. Further, we derive the dispersion relations for the transverse elastic waves propagating in the direction perpendicular to the layers. We use the analytical results to illustrate the tunability of the shear wave band gaps with varying remanent magnetizations of the phases; moreover, the shear wave band gaps can be actively controlled by a remotely applied magnetic field.


Introduction
Magnetoactive elastomers (MAEs) are composite materials consisting of magnetizable particles embedded in a soft matrix. Under an externally applied magnetic field, the magnetized particles interact, resulting in modification of the mechanical properties and leading to deformation occurring mostly in the matrix. The remote and reversible principle of actuation and property tunability makes the materials attractive for the development of variable-stiffness devices (Erb et al., 2012;Ginder et al., 2002), vibration absorbers (Ginder et al., 2001;Li et al., 2014) and isolators (Opie and Yim, 2011), actuators (Kashima et al., 2012;Tang et al., 2018), and sensors (Tian et al., 2011), and biomedical devices (Makarova et al., 2016) among others. Typically, a polymeric matrix material (e.g., silicone rubber) in its liquid state before polymerization, is mixed with magnetizable particles of micro or even nano size. Curing in the presence of magnetic field results in the formation of chainlike structures aligned along the direction of the magnetic field. Through this microstructure modification, different interactions between magnetizable particles are induced, thus, enabling tunability of the overall magneto-mechanical behavior of MAEs.
The foundation of magneto-elasticity (and mathematically analogous electro-elasticity) was developed by Truesdell and Toupin (1960). Since then, the magneto-elastic theory has been further developed by Ogden (2003, 2004), Bustamante et al. (2006), Vu and Steinmann (2007), and Destrade and Ogden (2011). In parallel, significant efforts have been made towards the development and implementation of nonlinear magneto-elasticity into numerical schemes (Haldar et al., 2016;Labusch et al., 2014;Metsch et al., 2016). An analytical approach for estimating the response and effective properties of MAEs with the random distribution of magnetoactive particles has been developed by Ponte Castañeda and Galipeau (2011). Galipeau et al. (2014) showed that MAEs with seemingly similar microstructures might exhibit significantly different magneto-mechanical properties. Rudykh and Bertoldi (2013) employed a micromechanics approach to analyze the macroscopic stability in anisotropic MAEs. Goshkoderia and Rudykh (2017) have investigated the long-wave instability in particulate MAE composites via a numerical homogenization. Goshkoderia et al. (2020) have reported the experimental observations of the instability-induced patterns in soft magneto-sensitive periodic systems. More recently, Pathak et al. (2022) studied the onset of microscopic instabilities in MAEs with bi-phasic layered microstructure exhibiting ferromagnetic behavior.
The aforementioned MAEs belong to so-called soft-magnetic active elastomers, typically based on embedding particles such as iron, soft ferrite, iron-silicon alloys, iron-nickel alloys (Jolly et al., 1996). These soft-magnetic materials have low coercivity and do not retain remanent magnetization once the external magnetic field has been removed (Bertotti, 1998). Recently, particles of high-coercivity ferromagnetic materials or hard-magnetic materials (such as NdFeB, hard ferrite, alnico alloys, samarium-cobalt) have been used to fabricate hard-magnetic active elastomers (HMAEs). These new magnetoactive composites showed programmable and complex shape transformations at low magnetic fields (Kim et al., 2018;Lum et al., 2016;Montgomery et al., 2020;Yan et al., 2021b). Once exposed to a large magnetic field, hard-magnetic materials retain their magnetization even after removing the external field. The high coercivity allows the hard-magnetic materials to sustain their remanent magnetization over a wide range of applied magnetic fields (that are below the coercive field strength). When the applied magnetic field is not aligned with the magnetization direction of the hard-magnetic particles, the induced magnetic torque acts to align the particle's magnetization direction with the applied field, leading to deformation of the HMAE composite. The remote and reversible shape-transformative behavior of HMAEs has enabled functionalities in areas such as soft robotics (Cui et al., 2019;Hu et al., 2018), biomedicine (Wang et al., 2021), mechanical metamaterials (Chen et al., 2021), self-assembly and self-organization (Piranda et al., 2021), and actuation (Bowen et al., 2015;Crivaro et al., 2016). To describe the behavior of the materials, Zhao et al. (2019) proposed a phenomenological model for ideal HMAEs. In the model, the magnetic flux density in the material is assumed to be linear with the external field strength, with the permeability constant equal to that of the vacuum. Yan et al. (2021a) have extended the model to both uniform and non-uniform magnetic fields. Garcia-Gonzalez and Hossain (2021) have proposed a lattice-based model that incorporates information of the particles' distribution into the constitutive formulations. A recent review by Lucarini et al. (2022) summarizes the development in the field of HMAEs.
The material tunability by a remotely magnetic fieldeither through property modification or induced deformationholds the intriguing potential for designing metamaterials for elastic wave manipulation. The foundation of the analysis of infinitesimal motion superimposed on finite deformations can be found in the work of Green et al. (1952), further developed by Ogden (2007), and is widely adopted as "small-on-large" theory (Bertoldi and Boyce, 2008;Guo et al., 2017Guo et al., , 2022Norris and Parnell, 2012;Rudykh and Boyce, 2014). Here, we employ the small-on-large framework to examine the propagation of small-amplitude elastic waves in finitely deformed magnetoelastic materials. Following the work by Maugin (1981), providing the governing equations for magnetoelastic waves in magnetizable deformable materials, the nonlinear surface waves (Abd-Alla and Maugin, 1987;Hefni et al., 1995) and inhomogeneous plane waves (Boulanger, 1989) have been studied. Based on the formulation of Dorfmann and Ogden (2004), Destrade and Ogden (2011) examined the small-amplitude plane harmonic waves in the finitely strained magnetoelastic material. Ogden (2011, 2012) studied the Rayleigh surface and Love waves propagating in a finitely strained isotropic and layered half-space magnetoelastic solid immersed in a magnetic field. Recently, Karami Mohammadi et al. (2019) have investigated transverse elastic wave propagation in finitely deformed bi-phase periodic magnetoelastic layered composites, and illustrated the shear wave bandgap tunability by magnetically induced deformation. However, these aforementioned works considered soft-magnetic active elastomers, while little is known about the wave propagation in HMAE medium.
In this work, we investigate the finite deformation and superimposed small-amplitude transverse elastic waves in bi-phasic layered HMAEs with remanent magnetization. We study the magnetostriction of the periodic layered HMAEs and derive the expression for the induced deformation as a function of the applied magnetic field, mechanical and magnetic properties of the phases, and their volume fractions. Moreover, we analyze the influences of the applied magnetic field and material parametersespecially the remanent magnetizationon the shear wave band gaps in the HMAE laminates.

Nonlinear magneto-elasticity
Consider a magnetoelastic deformable solid occupying domain Ω 0 in the reference configuration where no magnetic and mechanical fields are applied. Under the action of combined magnetic and mechanical loadings, the magnetoelastic solid occupies a deformed configuration denoted as Ω. The deformation is described by a function x = x(X) that maps the material point X from the reference state to the new position x in the deformed state. The deformation gradient is thus defined as F = ∂x/∂X. For incompressible solids, J ≡ detF = 1.
Here, we follow the work by Dorfmann and Ogden (2004) and denote by B, H, and M, respectively, the magnetic induction, the magnetic field, and the magnetization in the deformed configuration. They are related as where the constant μ 0 is the magnetic permeability in a vacuum.
Neglecting electrical, thermal, and relativistic effects, the quasi-static Maxwell equations in terms of the magnetic induction B and the magnetic field H in the deformed state, can be written as divB = 0 and curlH = 0. (2) Here and thereafter, the upper case first letter denotes the differential operator in the reference configuration, whereas the lower case denotes the ones in the deformed configuration. The magnetostatic equations in the reference configuration are where B L = F − 1 ⋅B and H L = F T ⋅H are the Lagrangian counterparts of the magnetic induction and the magnetic field, respectively. Following the work of Kovetz (2000), the magnetization is defined in terms of the specific free-energy density φ (F, B) as where ρ is the density in the deformed configuration. The total Cauchy stress σ can then be written in the form where is the so-called Maxwell stress. The specific free-energy density in the Lagrangian form is defined as Φ(F, B L ) = φ(F, FB L ). In terms of Φ, an "amended" free-energy function has been proposed by Dorfmann and Ogden (2004) where ρ 0 = ρ (J = 1) is the density in the reference configuration. Thus, the total first Piola-Kirchhoff stress tensor P, and Lagrangian magnetic field H L are Here, p is the Lagrange multiplier introduced due to the incompressibility constraint. Correspondingly, the total Cauchy stress tensor is Under the quasi-static condition and in the absence of body forces, the total Cauchy and first Piola-Kirchhoff stresses satisfy the following equilibrium equations divσ = 0 and DivP = 0.

Incremental motions
Next, we consider an infinitesimal incremental motion superimposed on the deformed state. We denote the incremental changes in F, p, P, B L , and H L as Ḟ , ṗ, Ṗ , Ḃ L , and Ḣ L , respectively. According to Eq. (8), the linearized constitutive relations can be written as where The "push-forward" counterparts (in the deformed configuration) of the incremental changes Ṗ , Ḃ L , and Ḣ L are Denoting u =ẋ as the infinitesimal incremental displacement superimposed on the deformed state, the incremental constitutive laws in Eq.
We consider steady-state small amplitude waves propagating in the deformed magnetoelastic medium. Following Destrade and Ogden (2011), the solution for Eq. (17) can be written in the form of plane waves with constant polarization where g and d are constant unit vectors denoting polarizations, f, q, and Π are single-variable functions with respect to the argument n⋅ x − ct, and n is the constant unit vector denoting the direction of propagation. The use of the incompressibility constraint in Eq. (15) and the incremental motion of Eq. (17) 2 yields g ⋅ n = 0 and d⋅n = 0.

Energy density function for HMAEs
Due to the relatively large coercivity, hard-magnetic materials can retain high remanent magnetization even in the absence of an externally applied magnetic field once they are saturated. The high coercivity further allows the magnetic materials to sustain the remanent magnetization over a wide range of applied magnetic fields (below the coercivity level). The magnetic behavior of the HMAEs can be approximated by a linear function when the applied magnetic field is below the coercivity level (Zhao et al., 2019).
As illustrated in Fig. 1a, we assume that the magnetic induction B of the HMAEs is linearly related to the applied magnetic field H when the field strength is small enough relative to the coercivity level (Lovatt and Watterson, 1999), namely where B r is the remanent induction and is related to the remanent magnetization M r by B r = μ 0 M r . The combination of Eqs. (1) and (20) yields where μ r = μ/μ 0 is the relative magnetic permeability. The idealized magnetization loop with the linear region represented by Eq. (21) is illustrated in Fig. 1b. Alternatively, the magnetization can be expressed with respect to the magnetic induction where χ = (μ − μ 0 )/μ is the magnetic susceptibility. Based on Eq. (22), the specific free-energy function can be written as where w e denotes the purely elastic response of the HMAEs. The constructed specific free-energy function satisfies Eq. (4). Therefore, the Lagrangian form of the specific free-energy function is where W e (F) can be defined by the appropriate hyperelastic energy potentials. In Eq. (24), M Lr is the remanent magnetization in the Lagrangian description. It is related to the remanent magnetization in the deformed configuration via RM Lr = M r , where R is the rotational component of the deformation gradient F. Such a definition is adopted since the remanent magnetization M r in the deformed configuration is determined by the rotations of the HMAEs, independently of stretches. This relation has been examined by Mukherjee et al. (2021) through full-field homogenization simulations, showing some similarities with the behavior of the isotropic soft-magnetic MAEs (Danas et al., 2012). Recalling Eq. (7), the Lagrangian amended energy function can be written as Upon substitution of Eq. (25) into Eq. (9), the total Cauchy stress in the HMAEs is evaluated as
We consider the case when the remanent magnetizations in the laminates are in the same direction as the applied field (in the e 2 direction; see Fig. 2b), namely Under the action of the applied field, the laminate stretches along the e 2 direction. In the deformed laminate, the layer thicknesses become where λ (1) 2 and λ (2) 2 are the phase stretch ratios in the e 2 direction and 2 . Here and thereafter, the parameters of the alternating layers are denoted as (•) (1) and (•) (2) , respectively.
The average deformation gradient and Eulerian magnetic induction are defined as and respectively. The displacement continuity condition along the interface between the layers enforces  ( where s is an arbitrary unit vector perpendicular to n (the unit vector denoting the lamination direction; see Fig. 2a). Using Eq. (31) and the symmetry of the deformation gradient defined in Eq. (29) with the phase incompressibility, we obtain λ (2) 2 = λ. Therefore, the deformation gradients in alternating phases are The traction continuity condition across the interface between the layers implies that ( Moreover, in the absence of free current at the interface, the jump conditions for the Eulerian magnetic induction and magnetic field are ( Using the magnetic induction jump condition (34) 1 , for the macroscopically applied magnetic load defined in Eq. (30), we obtain

Magneto-deformation
Consider the laminates consisting of HMAE phases, whose constitutive behavior is governed by the energy function (25); we assume that their purely elastic response is expressed as a function of the first invariant of the right Cauchy-Green deformation tensor where I (ξ) 1 = F (ξ) : F (ξ) . Thus, the stress field in the layer ξ (ξ = 1, 2) is where Ψ We assume that the HMAE laminate is surrounded by a vacuum, and no mechanical loadings are applied. The stress field jump condition across the interface between the laminate and vacuum yields is the Maxwell stress in the vacuum. Once again, the magnetic induction jump condition across the interface between the laminate and vacuum enforces B * = B = Be 2 .
For simplicity, we consider that the purely elastic response of the HMAE phases is dictated by the neo-Hookean model, namely Where G (ξ) is the shear modulus. Using Eqs. (33) and (38)- (41), we obtain . By eliminating the Lagrange multipliers from Eq. (42), we obtain the expression of the induced deformation as a function of the magnetic induction, mechanical and magnetic properties of layers, and their volume fractions, namely where Defining the normalized magnetic induction as B n = B/ ̅̅̅̅̅̅̅̅ ̅ Gμ 0 √ and normalized remanent magnetization as M (43) can be written as Note that for the case M (ξ) n = 0, expression (45) reduces to the result corresponding to the case for the laminate with soft-magnetic active elastomers (Pathak et al., 2022), namely In this case of the soft-magnetic laminate, the application of the external magnetic fields leads to stretching in the e 2 direction. (2) n = +0.6 in Fig. 3b), compressive deformation (i.e., λ < 1) developed under a low magnetic induction level (see the red and blue curves in Fig. 3a and b). From a physical point of view, at a low magnetic induction level, the magnetic stress in the HMAE phases is dominated by − r , which is positive in layer 1 and negative in layer 2. Since (2) n ⃒ ⃒ , the resulting magnetic stress inside the laminate is positive. At a small magnetic induction level, the resulting positive magnetic stress is larger than the Maxwell stress σ * m outside the laminate. To satisfy the mechanical traction-free boundary conditions, the magnetic stress inside the laminate is partially compensated by a compressive (negative) mechanical stress, resulting in the compressive deformation. However, as the magnetic induction further increases, the magnetic stress in the HMAE phases is dominated by B (ξ) ⊗ B (ξ) /μ (ξ) , which is smaller than the Maxwell stress σ * m outside the laminate, thus, tensile (positive) mechanical stress develops inside the laminate and stretch deformation is observed. Note that, for the case μ (1) r = μ (2) r = 1.0 which means the permeability of the HMAE phases equals that of the vacuum surrounding the laminate, only compressive deformation develops (see the black curves in Fig. 3a and b). This is because the magnetic stress inside the laminate is always larger than the Maxwell stress σ *  (2) n = + 1.5, correspondingly. The volume ratio and shear modulus contrast are v (1) = 0.5 and G (1) /G (2) = 15, respectively. laminate (see the black line in Fig. 3c), since the magnetic stress com- (2) r canceled each other, thus, the magnetic stress inside the laminate equals the Maxwell stress σ * m outside the laminate. Fig. 4 shows the dependence of the induced stretch λ on the magnetic induction for laminates in which the remanent magnetizations of both phases are positive (i.e., in the same direction of magnetic induction). The results are given for HMAE laminates with volume ratio v (1) = 0.5, and shear modulus contrast G (1) /G (2) = 15. One can see that when the remanent magnetization M (2) n = + 2.5, as an example: at a relatively small magnetic induction B n , the stretch λ becomes smaller when the relative permeability μ (ξ) r takes a larger value (see Fig. 4a). However, after reaching a transition point at B n = 2.5, the stretch λ increases correspondingly as the relative permeability μ (ξ) Maxwell stress σ * m does not change with HAME's magnetic properties, that is, the total stress inside the laminate also remains constant. Therefore, an increase in magnetic stress is compensated by a decrease in mechanical stress. Thus, the laminate undergoes comparatively smaller deformation as the relative permeability μ (ξ) r increases. By contrast, for B n larger than a critical value, the magnetic stress component B (ξ) ⊗ B (ξ) /μ (ξ) dominates the magnetic stress, thus, the stretch increases with an increment in relative permeability μ (ξ) r . Be noted, due to the presence of remanent magnetization in the HMAE laminate, B n = 0 does not mean that the external applied magnetic field equals zero. The relation between the magnetic induction inside the laminate and the external applied magnetic field is provided in Appendix A.
The layer thicknesses l (ξ) and phase velocity c (ξ) are functions of the stretch ratio as determined by Eq. (28) and Eq. (48), respectively. Note that the obtained dispersion relation for HMAE laminates (66) is identical to that for the purely elastic problem , and soft-magnetic laminates (Karami Mohammadi et al., 2019). However, the dependence of the stretch on the applied magnetic field differs, resulting in different tunability of the dispersion curves for the different material systems. To illustrate the dependence, one should consider a material model with a stiffening effect (see the discussion of the results for neo-Hookean HMAE laminates in Appendix C). To this end, we consider the HMAE laminate with the phases whose elastic response is dictated by the Gent model, namely where J (ξ) m is the locking parameter; here, we assume that both phases are characterized by identical locking parameters J Submitting Eq. (67) into Eq. (48) yields the corresponding phase velocity By making use of Eqs. (28), (66), (68) and (69), band structure diagrams for transverse waves in the HMAE laminate with alternating Gent phases subjected to the magnetic field perpendicular to the layers are constructed.  Fig. 5a displays the dispersion curves at B n = 0. In the reported frequency range, four transverse wave band gaps are observed, as highlighted by the shaded areas. The dependence of the transverse wave band gaps on the magnetic induction B n is shown in shifts the lower boundary of the first bandgap from f n = 0.397 to 0.557 and widens it from △f n = 0.285 to 0.399.

Conclusion
In this paper, we investigated the behavior of hard-magnetic soft laminates with bi-phasic layered microstructure. We considered the HMAE laminates subjected to a magnetic field perpendicular to the direction of the layers. First, we derived the expression for the magnetic field-induced stretch of the HMAE laminates with remanent magnetizations. The induced stretch is expressed as a function of the volume ratio and the magnetoelastic constants (including the remanent magnetizations) of the HMAE phases. Remarkably, the results indicate that the HMAE laminate undergoes compressive deformation along the direction of the magnetic field when the direction of the remanent magnetizations in the phases is opposite to that of the applied magnetic field. This is in contrast to the previously considered laminates made out of soft-magnetic active elastomers, which can only develop tensile strains in the magnetic field direction.
Second, we examined the propagation of shear waves in the HMAE laminates undergoing finite strains in the presence of an external magnetic field. Interestingly, the derived dispersion relation for HMAE laminates is identical to the one for the purely elastic setting and the one for laminates made of soft-magnetic active elastomers. We note that the bandgap of the shear waves propagating in the direction perpendicular to the layers in the HMAE laminate with neo-Hookean phases is independent of magnetic excitations. However, the dependence of the stretch on the applied magnetic field differs, resulting in different tunability of the dispersion curves for the different material systems. We illustrate this dependence by considering a material model with a stiffening effect; namely, we consider laminates with HMAE Gent phases. We show that the width and position of the shear wave band gaps in the HMAE laminates with remanent magnetizations can be tuned by a remotely applied magnetic field.
The results can guide the design of novel materials with potential applications in remotely controlled wave manipulating devices. We note that the derived exact solutions are based on the material and geometry idealization and may not account for material (Hauseux et al., 2017(Hauseux et al., , 2018Rappel et al., 2019) or geometrical Ding et al., 2021;Yu et al., 2022) uncertainties (or imperfections). These uncertainties in material properties and geometrical (the layer thickness and shape) parameters can affect the wave propagation characteristics in the HMAE laminates. Moreover, the imperfections of the interface or interphase between layers may be introduced in the material manufacturing processes (Arora et al., 2019). The influence of such uncertainties can be quantified through multi-field coupled stochastic analyses (Elouneg et al., 2021;Mazier et al., 2022).

Author agreement
The manuscript has been approved by the authors, Dr. Quan Zhang and Dr. Stephan Rudykh. This work has not been published previously, and it is not under consideration for publication elsewhere.

Data availability
No data was used for the research described in the article.

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. (

Appendix B. The magnetoelastic moduli tensors for the HMAE Phases
We consider HMAE laminates with phases defined by the amended energy function in Eq. (25). By recalling Eq. (36), the tensors of magnetoelastic moduli defined in Eq. (16) are where Ψ r .

Appendix C. Band structure for neo-Hookean HMAE laminates
Consider that the purely elastic response of the HMAE phases is described by the neo-Hookean model, submitting Eq. (41) into Eq. (48) yields c (ξ) By making use of Eqs. (28), (45), (66) and (72), the dispersion diagrams for the neo-Hookean HMAE laminate can be constructed. Fig. 6 shows the results for laminates with v (1) = 0.5, ρ (1) /ρ (2) = 1.0, and G (1) /G (2) = 15. The dispersion structure is presented in Fig. 6a, with the band gaps highlighted by the shaded areas. The evolution of the first four band gaps with respect to magnetic induction B n is plotted in Fig. 6b. It can be found that the band gaps are indifferent to the applied magnetic field and induced deformation. The reason can be found by comparing Eq. (28) and Eq. (72), which indicates that the change in the geometry induced by deformation is identical to the change in phase velocity; this implies that the term l (ξ) / c (ξ) in Eq.
(66) keeps constant regardless of the value of the magnetic field and the magnetic parameters (including the relative permeabilities and the remanent magnetizations). Therefore, the band gap structures in Fig. 6 are true for any values in relative permeabilities and remanent magnetizations. To achieve magnetic field-induced tunability of the transverse wave band gaps, one should consider laminates with phases exhibiting stronger stiffening , for example, Arruda-Boyce or Gent phases.