The active manipulation of phononic bandgaps has been a topic of great interest in the recent past. Phononic crystals, or periodic composite structures built from soft elastomers, offer the potential for reversible manipulation of their phononic bandgaps through finite deformation of the periodic composite. By using hard-magnetic soft materials, which undergo reversible, finite deformations when subjected to an applied magnetic flux density, it is possible to tune the frequency ranges of elastic wave bandgaps or generate new bandgaps through magnetic stimuli. Here, we present a theoretical model for the analysis of large magneto-deformation and the anti-plane shear wave bandgaps in an infinite 2D periodic two-phase hard magnetic soft composite structure subjected to magnetic stimuli. The constitutive behavior of the phases in the hard-magnetic soft composite is described using the incompressible Gent model. To solve the incremental anti-plane wave equations, the finite element method and the Floquet-Bloch theorem for periodic medium are utilized. Using the developed framework, we numerically study the dependency of the bandgap width and their location on the direction and magnitude of applied magnetic flux density vector, material parameter contrasts, and geometry and volume fraction of the inclusion phase. The numerical results reveal that significant tunability of the bandgap is achieved when the applied magnetic flux density direction is along the residual magnetic flux density direction. Also, it is seen that the geometry of the inclusion has significant effect on the bandgap width. Keywords: Phononic crystals, Wave propagation, Tunable band gaps, Soft materials, Hard-magnetic soft materials, Finite element method. ∗Corresponding author Email address: atulksharma@iitj.ac.in (Atul Kumar Sharma ) 1Contributed equally May 7, 2023 1. Introduction 1 Phononic Crystals (PnCs) have received growing attention since their discovery more than a 2 decade ago. PnCs are engineered composite materials or structures designed to tailor the transmission 3 of elastic waves, such as sound and vibrations, by creating periodic variations in the elastic properties 4 of the constituent materials (Sigalas & Economou, 1992; Kim & Yang, 2014; Xia et al., 2019; 5 Zhang & Gao, 2020; Vasileiadis et al., 2021; Chen et al., 2022; Liu et al., 2023). One of the 6 key features of PnCs is the existence of band gaps (Kushwaha et al., 1993), which are frequency 7 ranges where elastic/acoustic waves cannot propagate. The existence of band gaps in PnCs is 8 attributed to Bragg scattering (Kushwaha et al., 1994) and local resonance (Raghavan & Phani, 9 2013). PnCs have a wide range of potential applications, including acoustic filters (Zhang & To, 10 2013), waveguides (Khelif et al., 2004), sensors (Gharibi & Mehaney, 2021), acoustic cloaking 11 (Zheng et al., 2014), noise suppressors (Badreddine Assouar et al., 2012), multiplexing devices 12 (Moradi & Bahrami, 2019), superlenses (Dubus et al., 2011) and many more. 13 One of the key challenges in design and development of PnCs is that the phononic bandgaps are 14 typically fixed once the phononic composite/structure is manufactured. In contrast, there may be 15 wide range of applications in which it is desirable to be able to actively tune the band gaps after the 16 composite has been manufactured. To overcome this challenge of tunability, there have been several 17 efforts in the recent past (Wang et al., 2008; Bou Matar et al., 2012; Nimmagadda & Matlack, 2019; 18 Bertoldi & Boyce, 2008; Gei et al., 2011). In the context of PnCs made up of stiff/hard materials, 19 tunability of band gaps has been explored using piezoelectrics (Wang et al., 2008, 2010; Vatanabe 20 et al., 2014), magneo-elastics (Bou Matar et al., 2012; Hu et al., 2022), temperature variations 21 Xia et al. (2016); Nimmagadda & Matlack (2019), etc, however, due to small-deformation of the 22 structure tunability is limited. Recently, there has been growing interest in utilizing soft active 23 materials, such as soft elastomers (Wang & Bertoldi, 2012; Bertoldi & Boyce, 2008; Rudykh & 24 Boyce, 2014; Shim et al., 2015; Shmuel & Band, 2016; Chen et al., 2019), dielectric elastomers 25 (Gei et al., 2011; Shmuel & deBotton, 2012; Shmuel, 2013; Getz et al., 2017; Jandron & Henann, 26 2018; Alam & Sharma, 2022; Zhao et al., 2023), and magneto-active elastomers (Pierce et al., 2020; 27 Karami Mohammadi et al., 2019) which undergo large deformation when subjected to external 28 stimuli to achieve significant tunability of phononic band gaps, i.e., shifting of band gaps. 29 2 In the present work, we consider phononic composites composed of hard-magnetic soft materials 30 (HMSMs) that possess elastic wave band gaps. The size and location of these band gaps can be 31 adjusted dynamically by applying an external magnetic flux density in real-time. HMSMs are a 32 new family of soft magneto active materials that can undergo large, reversible deformations when 33 actuated by magnetic field (Zhao et al., 2019; Moreno-Mateos et al., 2023; Rahmati et al., 2023b; 34 Wang et al., 2020; Yan et al., 2023; Rahmati et al., 2023a). HMSMs differ from soft magneto- 35 active elastomers in that they are manufactured by incorporating hard-magnetic particles with high- 36 coercivity in the soft elastomer matrix (Lum et al., 2016; Kim et al., 2018; Wu et al., 2020a; Sano 37 et al., 2022; Zhao & Zhang, 2022), while soft magneto active elastomers are manufactured by 38 incorporating soft magnetic particles with low-coercivity in the elastomer matrix (Wu et al., 2020b; 39 Garcia-Gonzalez et al., 2021). Further, HMSMs exhibit high remnant properties, allowing them to 40 retain high residual magnetic flux density even when the external magnetic stimuli are removed. 41 Owing to this property, significant efforts have gone into using HMSMs in a variety of engineering 42 applications such as soft transducers (Lee et al., 2020; Nagal et al., 2022; Nandan et al., 2023), 43 soft/flexible robotics (Kim et al., 2019; Wang et al., 2021), Metamaterials (Zhang et al., 2023), and 44 many more. A recent review on the various applications of HMSMs may be found in Lucarini et al. 45 (2022). 46 Using HMSMs as a route for obtaining magnetically tunable, soft PnCs was considered firstly 47 by Zhang & Rudykh (2022) who developed a theoretical model for investigating the propagation 48 of transverse elastic waves in 1D two-phase hard-magnetic soft laminates. They demonstrated the 49 active manipulation of the band gaps in the shear mode with varying remanent magnetizations of the 50 laminate constituents and found that a remotely applied magnetic field can significantly control the 51 shear wave band gaps. Further, Alam et al. (2023) reported a 1D finite element model to investigate 52 the longitudinal wave band gaps in hard-magnetic soft laminates. They demonstrated that the 53 application of a magnetic flux density in the opposite direction to the residual magnetic flux density 54 yields positive effects on the band gap properties of hard-magnetic soft laminates. In particular, 55 increasing the applied magnetic flux density results in the widening of the band gap width and a 56 shift towards higher frequencies. Li et al. (2022) investigated the active tuning of the band gaps in 57 programmable hard-magnetic soft PnCs with three distinct magnetic anisotropy encoding modes 58 through finite element simulation under the influence of an applied magnetic field. To the best of 59 3 authors’ knowledge, the propagation of elastic waves and tunable band gap characteristics of 2D 60 two-phase hard-magnetic soft composites (HMSCs) has not been hitherto reported. To this end, 61 we develop a numerical finite element framework for analyzing the propagation of incremental 62 anti-plane waves superimposed on the magnetic field induced large deformation of the infinite 2D 63 periodic two-phase HMSCs. Further, the developed numerical framework is utilized to demonstrate: 64 (I) the magnetic tunability, (II) the influence of material parameter contrasts and volume fraction of 65 composite phases, and (III) the influence of the geometry of the inclusion phase on the anti-plane 66 wave band gap characteristics of the HMSC. 67 The remainder of manuscript is divided into five sections. Section 2 provides a concise overview 68 of the fundamental formulation of the nonlinear field theory of HMSMs, along with its linearized 69 incremental wave propagation theory. Section 3 presents a theoretical framework for analyzing 70 the finite deformation behavior of a two-phase HMSC under a magnetic field in the quasi-static 71 regime. In Section 4, we employ finite element method and Floquet-Bloch theorem to obtain the 72 incremental anti-plane wave band gaps in an infinite 2D periodic two-phase HMSC. Section 5 73 presents the numerical results, which show how the applied external magnetic loading, volume 74 fraction of phases, geometry of the inclusions, and material contrast parameters affect the band 75 gap characteristics of an infinite 2D periodic HMSC. Finally, in Section 6, we summarize the 76 conclusions drawn from the current study and also provide further outlook. 77 2. Dynamics of hard-magnetic soft composites 78 In this section, we provide a concise review of the governing equations pertaining to the dynamics 79 of deformable HMSCs following the the nonlinear field theory of HMSMs (Zhao et al., 2019) and 80 the related linearized incremental theory (Dorfmann & Ogden, 2014). 81 Consider an arbitrary incompressible, soft, deformable, hard-magnetic, soft-composite body 82 made up of two phases aandb. This body occupies a domain Ωa R∪Ωb R= Ω R⊂R3with 83 the boundary ∂ΩRin the reference configuration. When actuated by biasing magnetic fields, 84 the composite body undergoes deformation and occupies a domain Ωa∪Ωb= Ω⊂R3in the 85 current configuration. The boundary of the body in the current configuration is labeled by ∂Ω. The 86 deformation χ: ΩR→Ωmaps material particles Xin the reference configuration to the spatial 87 points xin the current configuration. In the following discussion, the differential operators ’Grad’, 88 4 ’Div’, and ’Curl’ are used to denote the gradient, divergence, and curl in the reference configuration, 89 respectively, while ’grad’, ’div’, and ’curl’ are used to denote the same operators in the current 90 configuration. For defining different relevant fields in the current and reference configurations, we 91 introduce the deformation gradient tensor F, Jacobian J, and the left Cauchy-Green strain tensor 92 b. 93 F= Grad χ, J = det F>0, b=FFT. (1) Neglecting the mechanical body forces, the total Cauchy stress tensor σsatisfies the balance of 94 linear momentum equation in the current configuration as follows: 95 divσ=ρa (2) where ρdenotes the material mass density, which remains constant as the material is incompressible 96 J= 1, anda=χ,ttrepresents the acceleration. 97 In the case of an ideal hard-magnetic soft material (with no free current) and neglecting the 98 dynamic coupling between electro-magnetic fields, the Maxwell equations in the current configuration 99 can be expressed as follows 100 divB= 0, curlH=0 (3) here,HandBare the magnetic field and the magnetic flux density in the current configuration, 101 respectively. 102 Jump boundary conditions between interfaces separating phases aandbare: 103 [ [σ] ]·n=0, n·[ [B] ] = 0 , n×[ [H] ] =0 (4) here, [ [•] ] = (•)a−(•)brepresents the jump operator, and ndenotes the unit normal vector to the 104 surface element in the deformed configuration. 105 The constitutive relations for an incompressible hyperelastic hard-magnetic material are obtained 106 from the nominal Helmholtz free energy density function Ψ(F,B0)(per unit volume in the undeformed 107 configuration) (Zhao et al., 2019) as follows: 108 5 P=∂Ψ ∂F+γF−T, H0=∂Ψ ∂B0(5) where P=JσF−T,H0=FTH, andB0=JF−1Bare the totol first Piola-Kirchhoff stress tensor, 109 Lagrangian magnetic field and magnetic flux density vectors, respectively, and γis an arbitrary 110 scalar accounting for the incompressibility constraint. 111 Following Dorfmann & Ogden (2014), we consider a time-dependent small amplitude incremental 112 wave motion ˙x=˙χ(X, t), along with an incremental change ˙B0(X, t)in magnetic flux density, 113 superimposed on the quasi-static large magnetodeformation of the hard magnetic soft body Ω(χ). 114 In the sequel, to distinguish the incremental quantities from their equilibrium counterparts, we use 115 a dot above them. 116 In the updated Lagrangian form, the governing equations for the incremental motion and magnetic 117 field are written as: 118 divΣ=ρ˙x,tt, divˇB= 0, curlˇH=0 (6) where Σ=J−1˙PFT,ˇB=J−1F˙B0, and ˇH=F−T˙H0are the push-forwards of increments in 119 Lagrangian quantities ˙P,˙B0, and ˙H0, respectively. 120 For an incompressible hard-magnetic soft material, the linearized incremental constitutive equations 121 are written as 122 Σ=Ch−γhT+ ˙γI+BˇB, ˇH=BTh+AˇB, (7) whereC,B, andAare the constitutive tensors and their components are written as 123 Cijkl=1 JFjI∂2Ψ ∂FiI∂FkJFlJ,Bijk=FjI∂2Ψ ∂FiI∂B0JF−1 Jk,Aij=JF−1 Ii∂2Ψ ∂B0I∂B0JF−1 Jj(8) and˙γis the increment in γ, and the incremental displacement gradient tensor h= grad ˙xsatisfies 124 the material incompressibility constraint trh= div ˙x= 0. 125 6 3. Nonlinear Magnetodeformation of two-phase hard-magnetic soft composites 126 We consider a 2-D periodic two-phase composite consisting of square-shaped inclusions (incompressible 127 hard-magnetic soft phase a) and embedded in a different incompressible hard-magnetic soft material 128 (phase b) which is infinitely large in the (x1, x3)plane. The composite is infinitely long along x2 129 direction and the different magneto-mechanical fields are assumed to be invariant with respect to x2 130 direction. In the undeformed configuration, the unit cell for this 2-D periodic two-phase composite 131 is assumed to be a square of length Las shown in Fig. 1(a). Hereafter, the superscript (•)pis used 132 to denote the physical quantities of the composite phases (p=a, b). 133 To describe the constitutive response of the HMSC, we consider that the hard-magnetic soft 134 phases are modeled by the incompressible Gent model of hyperelasticity (Gent, 1996) in conjunction 135 with the ideal hard-magnetic soft material model (Zhao et al., 2019). The expression for the 136 Helmholtz free energy of the ideal hard-magnetic soft phase is given by 137 Ψp=−GpJp lim 2ln 1−trbp−3 Jp lim −1 µ0FpBp r0·Bappliedp(9) where Gpis the shear modulus, Jp limis the limiting stretch parameter, µ0is the vacuum permeability, 138 Bp r0is the residual magnetic flux density vector in the reference configuration, and Bappliedpis the 139 applied magnetic flux density vector in the current configuration. Utilizing Eqs. (5) (9), the total 140 Cauchy stress generated in each phase is obtained as 141 b Bapplied Br a x3 l = λL x2 x1 L b a X3 X1 X2 Br b Bapplied a x3 x2 x1 l = λL Br (a) (b) (c) Figure 1: Schematic description of a two-phase HMSC with square shape inclusion in (a) reference configuration, (b) current configuration with residual magnetic flux density aligned with the direction of applied magnetic flux density, and (c) current configuration with residual magnetic flux density opposing the direction of applied magnetic flux density. 7 σp=Gp  1−trbp−3 Jp limbp−1 µ0Bappliedp⊗FpBp r0+γpI. (10) For known values of residual and applied magnetic flux density vectors, respectively, the total 142 Cauchy stress tensor σpis a function of the deformation gradient tensor Fonly. Further, if in the 143 current configuration, the direction of Bappliedpis parallel to the direction of Bp r0, the total Cauchy 144 stress becomes symmetric. However, for more general cases, the total Cauchy stress tensor is 145 asymmetric Zhao et al. (2019); Zhang et al. (2023). 146 We consider that the composite phases are perfectly bonded and is subjected to the magnetic 147 magnetic flux density in the x2direction (parallel to the direction of residual magnetic flux density 148 vector) only, as shown in Fig. 1. The presence of perfect bonding between phases leads to 149 homogeneous deformation and magnetic flux density in each phase, which can be described as 150 Fp= λ0 0 0λ−20 0 0 λ ,Bappliedp= 0 B2 0 (11) where λdenotes the in-plane stretch ratio and B2is the component of the magnetic flux density 151 vector along x2direction. Substituting Eq. (11) into Eq. (10), the expressions for the the nonzero 152 components of total Cauchy stress tensor in terms B2andλare obtained as 153 σp 11=σp 33=Gpλ2  1−2λ2+λ−4−3 Jp lim+γp, σp 22=Gpλ−4  1−2λ2+λ−4−3 Jp lim+γp−B2Bp r02 µ0λ2,(12) where Bp r02is the component of residual magnetic flux density vector for pthphase. 154 For fulfilling the traction free boundary conditions, the components of total Cauchy stress 155 tensors must satisfy 156 σp 11=σp 33= 0, νaσa 22+νbσb 22= 0 (13) where νaandνb= 1−νaare the volume fractions of the phases a, and b, respectively. Utilizing 157 8 the Eqs 13 and 12 and assuming limiting stretch parameter Ja lim=Jb lim=Jlim, we obtain the 158 nonlinear relationship between the applied magnetic flux density B2, and the in-plane stretch ratio 159 λas 160 λ−2−λ4  1−2λ2+λ−4−3 Jlim=¯Br02B2 ¯Gµ0, (14) where ¯G=Gaνa+Gbνb, and ¯Br02=Ba r02νa+Bb r02νb. 161 According to Eq. (14), the in-plane stretch ratio λcan be found once the material parameters 162 and the applied magnetic flux density B2are prescribed. 163 4. Incremental anti-plane waves in a 2D infinite periodic two-phase hard-magnetic soft composite 164 Based on the magnetically induced nonlinear finite deformation obtained in the aforementioned 165 section, we now study the superimposed incremental anti-plane elastic shear waves propagating in 166 the plane (x1, x2)of the HMSC. We assume that the anti-plane strain condition holds ( ˙x1= ˙x3= 0) 167 and the incremental anti-plane displacement field ˙x2depends on coordinates (x1, x3)and time t. 168 For propagation of anti-plane shear waves in HMSC, the governing incremental equation of motion 169 (Eq. 6) takes the following form 170 ∂Σ21(x1, x3, t) ∂x1+∂Σ23(x1, x3, t) ∂x3=ρ(x1, x3)∂2˙x2(x1, x3, t) ∂t2. (15) Using incompressibility constraint (h22= 0) and incremental constitutive relation (Eq. 7), the 171 components Σ21andΣ23are obtained as 172 Σ21=C2121h21+C2123h23,Σ23=C2323h23+C2321h21 (16) where,C2121=C2323=G(x1, x3)λ2  1−2λ2+λ−4−3 Jlim=˜GandC2123=C2321= 0. 173 Assuming incremental anti-plane displacement field to be time-harmonic as ˙x2(x1, x3, t) = 174 u2(x1, x3) exp(−iωt),and inserting the components Σ21andΣ23from Eq. (16) into Eq. (15), we 175 obtain the incremental anti-plane wave equation in the frequency domain as 176 9 ∂ ∂x1 ˜G∂u2(x1, x3) ∂x1 +∂ ∂x3 ˜G∂u2(x1, x3) ∂x3 =−ω2ρ(x1, x3)u2(x1, x3) (17) where u2(x1, x3)is the incremental anti-plane displacement field that is spatially dependent, and ω 177 is the angular frequency associated with the anti-plane wave motion. 178 Next, we employ standard displacement based finite element approach to solve Eq. (17). The 179 weak form of Eq. (17) is obtained by multiplying it with test function δw, and then integrating over 180 the entire computational domain (2D HMSC unit cell depicted in Fig. 2a) as 181 ˆ Ωt˜G∂u2 ∂x1∂δw ∂x1+∂u2 ∂x3∂δw ∂x3 dx1dx3=ˆ Ωtω2ρu2δwdx 1dx3. (18) The computational domain is divided into a set of 4-noded quadrilateral element finite elements 182 Ωt=nel∪ e=1Ωe t, and the incremental displacement field u2and test function δware interpolated 183 element-wise in terms of nodal quantities using nodal shape functions as 184 u2=NIuI 2, δw =NIδwI, (19) where NIrepresents the nodal shape function with Idenoting the number of nodes in a finite 185 2π/l x3 x1 a b l l k3 k1 Г X M 2π/l (a) (b) Figure 2: Schematic of the deformed unit cell in the plane of periodicity (x1, x3), and (b) the corresponding first Brillouin zone. 10 element, and uI 2is the incremental anti-plane displacement values at node I. Upon substituting the 186 finite approximations from Eq. (19) into Eq. (18), we obtain the eigenvalue problem as 187 nel∪ e=1KeuJ 2=ω2nel∪ e=1MeuJ 2 (20) where 188 Ke IJ=ˆ Ωe t˜G∂NI ∂xj∂NJ ∂xjdx1dx3,Me IJ=ˆ Ωe tρNINJdx1dx3, (21) are the components of elemental level stiffness Keand mass Mematrices, respectively, and index 189 j= 1,3. 190 Next, we employ Bloch-Floquet theorem (Kittel et al., 1996) for obtaining the anti-plane wave 191 band structure of an infinite periodic two-phase HMSC. According to Bloch-Floquet theorem, the 192 incremental anti-plane displacement field must satisfy 193 u2(xΓ2) = exp( ik1l)u2(xΓ1), u 2(xΓ4) = exp( ik3l)u2(xΓ3), (22) in which k1andk3denote the components of 2D Bloch wave vector k=k1e1+k2e2, varying along 194 the edges [Γ−X−M−Γ]of the deformed first irreducible Brillouin zone for the square unit cell 195 (Wang et al., 2007) as depicted in Fig. 2b and Γ1,Γ2,Γ3, and Γ4are the boundaries of the unit cell 196 as shown in Fig. 2a. We implemented the Bloch-Floquet complex-valued boundary conditions (Eq. 197 22) using augmented penalty method (Alam et al., 2023; Felippa, 2001). By solving the eigenvalue 198 problem (Eq. 20) in conjunction with the complex-valued boundary conditions (Eq. 22), we obtain 199 the band structure. The aforementioned numerical framework of extracting the anti-plane wave 200 band structure is implemented through a MATLAB code developed in-house. 201 5. Numerical results and discussion 202 In this section, we present the numerical results to investigate the effect of applied magnetic 203 loading, inclusion geometry and volume fraction (Phase b), and material parameter contrasts on the 204 tunability of the elastic anti-plane shear wave bandgap structure of the two-phase HMSC. 205 In the numerical simulations, we discretize the undeformed unit cell, with geometric parameter 206 11 (a) (b) (c) (d) (e) (f) Figure 3: (a) Nonlinear finite mgnetodeformation response of the composite. (b-f) Band structures of the two- phase HMSC subjected to the different levels of normalized applied magnetic flux density. The normalized frequencies ˜ωare plotted as functions of the reduced wave vector kalong Γ−X−M−Γ. 12 L= 5 mm , using 4-noded quadrilateral elements with a single degree of freedom, i.e., the anti- 207 plane displacement field. We examine a HMSC with Phase bpossessing the following material 208 properties: b:Gb= 303 kPa ,ρb= 2434 kg /m3,Br02b= 0.143 T , and extract the band structure 209 for representative values of phase amaterial properties defined using the shear contrast parameter 210 α=Ga Gb, magnetic contrast parameter β=Ba r02 Bb r02, and volume fraction νa= 1−νbat different 211 values of the normalized applied magnetic flux density, denoted as ˜B2=B2p (νaGa+νbGb)µ0. 212 We assume that the mass density of phase ais the same as that of Phase b, i.e., ρb= 2434 kg /m3. 213 We take the limiting stretch parameter to be Jlim= 10 . Note that in the initial numerical results 214 presented herein to investigate the influence of external magnetic loading, volume fraction of the 215 inclusion, and material parameter contrasts on the bandgap characteristics, we consider an inclusion 216 with a square cross-section as shown in Fig. 2a. Later, we investigate the effect of inclusions with 217 three different geometries on the bandgap characteristics while keeping other material parameters 218 constant. 219 5.1. Quasi-static finite deformation and magnetic tunability of the phononic band gaps 220 Here, we analyze the quasi-static nonlinear large magnetodeformation and tunability of the 221 superimposed incremental anti-plane wave band gap characteristics of the HMSC at different levels 222 of normalized applied magnetic flux density. The nonlinear algebraic equation Eq. (14) is solved 223 to analyze the quasi-static nonlinear finite deformation. Figure 3(a) demonstrates the variation of 224 the stretch parameter λas a function of the normalized magnetic flux density ˜B2for a HMSC with 225 α= 10 ,β= 10 , and νa= 0.5. When the magnetic flux density is applied in the same direction as 226 the residual magnetic flux density, denoted by ( ˜B2> 0), the HMSC undergoes expansion along the 227 Table 1: Variation of the normalized bandgap width ( ∆˜ω1) and the frequency range ( ˜ω1max−˜ω2min) of the first anti-plane wave bandgap with normalized applied magnetic flux density ( ˜B2). ˜B2→ -10 -5 0 5 10 ∆˜ω1 0.0848 0.0809 0.0775 0.0907 0.1119 ˜ω1max−˜ω2min 0.4156 - 0.5004 0.3962 - 0.4771 0.3795 - 0.4570 0.4443 - 0.5350 0.5482 - 0.6601 13 (a) (b) Figure 4: Variation of the first and second band gap (a) widths, and (b) mean of the frequency limits as a function of normalized applied magnetic flux density ˜B2. x2direction ( λ< 1). Conversely, when the magnetic flux density is applied in the opposite direction 228 of the residual magnetic flux density, indicated by ( ˜B2< 0), the composite contracts in the x2229 direction ( λ> 1) due to the compressive magnetic stress. Furthermore, to explore the dependence 230 of the band gap characteristics on the applied magnetic flux density, we consider five different 231 cases: (I) ˜B2= 0, (II) ˜B2=−5,−10, and (III) ˜B2= 5,10. Figures 3(b-f) display the band 232 gaps along the boundaries of the irreducible first Brillouin zone for the composite when subjected 233 to the aforementioned five levels of magnetic flux density (˜B2), where ˜ω=ωL 2πrρ νaGa+νbGbis 234 the normalized frequency of the anti-plane wave. The band gap width in the band structure plots 235 is depicted by blue regions. The values of the stretch parameter λcorresponding to the above- 236 mentioned values of ˜B2are 1, 1.3227, 1.4934, 0.6759, and 0.6047, respectively. The width and 237 corresponding normalized frequency ranges of the first bandgap for the aforementioned applied 238 levels of normalized magnetic flux density ˜B2are listed in Table 1. It is observed from Fig. 3(b-f) 239 and Table 1 that the band gap size increases with an increase in the magnitude of applied magnetic 240 flux density in both cases, i.e., Case II: when the direction of applied magnetic flux density is along 241 the direction of residual magnetic flux density, and Case III: when the direction of applied magnetic 242 flux density is opposite to the direction of residual magnetic flux density. It is also observed that 243 the location of the band gap shifts towards a higher frequency range in both cases. Figure 4 (a) 244 14 illustrates the variation of the width of the first and second bandgaps as a function of the normalized 245 magnetic flux density ˜B2, while Fig. 4 (b) illustrates the variation of the mean of the normalized 246 band gap frequency limits ˜ωm. From Fig. 4, it is clearly seen that the rate of increase in the 247 width and mean of the frequency limits of band gaps is higher in the case of applied magnetic flux 248 density along the direction of residual magnetic flux density (˜B2>0). By magnetically actuating 249 the HMSC, the width of the first bandgap is increased by 44.38% when the applied magnetic flux 250 density is ˜B2= 10 and by 9.42% when the applied magnetic flux density is ˜B2=−10. This 251 observation shows that applying magnetic loading in the direction of the residual magnetic flux 252 density has a positive influence on the tunability of band gaps. 253 5.2. Parametric study of material parameter contrasts and volume fraction of phases 254 Firstly, we will investigate how the variation in the shear contrast parameter αaffects the 255 bandgap characteristics of a two-phase HMSC. The shear contrast significantly affects the width 256 and location of the bandgaps. Figure 5 (a-c) displays the normalized frequency ωas a function of 257 the wave vector kfor a magnetically actuated two-phase HMSC with three distinct values of shear 258 contrast α= 10, 20, and 30, respectively. In the numerical simulations for these three cases, we keep 259 other parameters constant, such as β= 30 ,˜B2= 10 , and νa= 0.5. The variation of the bandgap 260 widths of the first and second bandgaps with the shear contrast αis depicted in Fig. 6 (a). Figure 261 6 (b) illustrates the variation of the mean of the normalized frequency limits, ˜ωm, as a function of 262 the shear contrast α. From Fig. 6 (a), we observe that the first bandgap opens at α= 4.5, and the 263 width of the bandgap increases with shear contrast α, reaching a maximum at α= 15 , and then 264 decreasing. On the other hand, it is worth noting that the second bandgap opens at higher values of 265 shear contrast α= 16 , and the width of the second bandgap increases with an increase in α. From 266 Fig. 6 (b), we can see that the location of both bandgaps shifts towards the lower frequency range. 267 These inferences can be utilized in the design of wave-manipulating devices with applications in 268 the lower frequency range. 269 Next, we examine the impact of the magnetic contrast parameter βon the bandgap characteristics 270 of the two-phase HMSC with square inclusions. Figure 7 illustrates the band structures of the 271 composite for three different values of the magnetic contrast parameter: β= 10, 20, and 30. In the 272 numerical simulations, we take volume fraction νa= 0.5, shear contrast α= 30 , and normalized 273 magnetic flux density ˜B2= 10 . Figure 8(a) shows the variation of the widths of the first and second 274 15 (a) (b) (c) Figure 5: Band structures of the two-phase HMSC at different values of shear contrast parameter α, while keeping the magnetic contrast parameter β, and the normalized magnetic flux density ˜B2as constant. bandgaps with the magnetic contrast parameter β, while Fig. 8 (b) illustrates the variation of mean 275 of the normalized frequency limits ˜ω. Figures 7 and 8 reveal that the widths and the location of the 276 band gaps increases with an increase in the magnetic parameter contrast. However, the effect of β 277 on band gap tunability is much more modest than that of αfor the contrast values considered in the 278 present study. 279 The tunability of bandgaps in HMSC is strongly influenced by the volume fraction of the 280 inclusion νa. In order to extract the influence of volume fraction on the band gap characteristics, We 281 varies the volume fraction of inclusion from from 0 to 1, while other material parameters are kept 282 16 (a) (b) Figure 6: Variation of the first and second band gap (a) widths, and (b) mean of the frequency limits as a function of shear contrast parameter α. constants as: α= 30 ,β= 30 , and ˜B2= 10 . The volume fraction νa= 0represents the isotropic 283 homogeneous body made up of phase bonly while that of 1 corresponds to body made up of phase 284 aonly. Figures 9(a) and 9(b) illustrate the variation of the widths and mean of the frequency limits 285 of first and second bandgaps as a function of volume fraction of the inclusion phase νa, respectively. 286 The first bandgap opens at about νa= 0.25, and attain a maximum value at νa= 0.81, and closes at 287 νa= 0.93. The variation of second band gap width also follows the similar trend: band gap opens 288 atνa= 0.36and reaches maximum at νa= 0.56, and closes at νa= 0.64. It is evident from 289 Fig. 9(a) that the width of the both the band gaps attain maximum value when the concentration of 290 the harder phase ais a bit more than the softer phase b. The maximum value of the first bandgap 291 width is ∆˜ωmax = 0 .4252 , and the corresponding mean of the frequency limits is ˜ωm= 0.7656 . 292 The maximum width of the second bandgap is ∆˜ωmax = 0 .0492 , and the corresponding mean of 293 the frequency limits is ˜ωm= 0.7739 . From Fig. 9(b), it can be observed that the location of the 294 existing band gaps shifts towards higher frequency with an increase in the volume fraction of the 295 harder phases. These trends are similar to the observations reported in the literature (Shmuel, 2013) 296 regarding soft phononic composites. 297 17 (a) (b) (c) Figure 7: Band structures of the two-phase HMSC at different values of magnetic contrast parameter β, while keeping the shear contrast parameter α, and the normalized magnetic flux density ˜B2as constant. 5.3. Effect of geometry of inclusion phase 298 Finally, we investigate the dependence of the anti-plane wave band gap characteristics on 299 geometry of the inclusion phase a. To do so, we consider three configurations of the unit cells 300 as: (I) single central circular inclusion, (II) single central square inclusion, and (III) inclusion 301 at multiple locations) as displayed in top panel of Figs. 10(a-c), respectively. For a one-to-one 302 comparison of the different geometry of the inclusions, we extract the the band gaps for all three 303 cases at same volume fraction νa= 0.5and material parameters: α= 30 , and β= 10 . The unit 304 cell with circular inclusion is discretized with 1329 four noded quadrilateral elements while the 305 18 (a) (b) Figure 8: Variation of the first and second band gap (a) widths, and (b) mean of the frequency limits as a function of magnetic contrast parameter β. (a) (b) Figure 9: Variation of the first and second band gap (a) widths, and (b) mean of the frequency limits as a function of the volume fraction of inclusion phase νa. unit cells with square and multiple inclusions are discretized with 1156 four noded quadrilateral 306 elements. Middle panel of Fig. 10 displays the band structures for aforementioned three unit cells, 307 respectively, when the applied normalized magnetic flux density ˜B2= 0, while that at ˜B2= 10 , are 308 displayed in the bottom panel, respectively. As evident from Fig. 10, for both the values of ˜B2, the 309 unit cell with inclusion at multiple location exhibit the widest first band gap, while the unit cell with 310 19 circular inclusion exhibit highest second band gap. Further, as expected, the width and the location 311 of the band gaps for all the three cases is bound to be enhanced with the application of applied 312 magnetic flux density. The inferences from this section demonstrate the significant dependence of 313 the band gap widths and their location on the layout or micro-structure of the unit cell. Here, the 314 third layout of the unit cell is obtained by randomly putting inclusions at multiple locations, while 315 other two layouts are the with standard cross-sections of the inclusion. However, an optimized unit 316 cell layout with optimal distribution of phases for wide and tunable band gaps can be obtained using 317 (a) (b) (c) Figure 10: The layout of the unit cells with (a) single central circular inclusion, (b) single central square inclusion, and (c) inclusion at multiple locations, and corresponding band structures at two different levels of applied magnetic flux density ˜B2= 0(middle) and ˜B2= 10 (bottom). . 20 the topology optimization schemes (Bendsoe & Sigmund, 2003; Sigmund & Søndergaard Jensen, 318 2003) such as genetic algorithm (GA) (Manktelow et al., 2013; Hedayatrasa et al., 2016; Bortot 319 et al., 2018), gradient based optimization schemes (Bacigalupo et al., 2017; Sharma et al., 2022b,a; 320 Dalklint et al., 2022), etc. The finite element framework reported here may be used as a starting 321 point for the topology optimization of HMSCs in future study. 322 6. Conclusions 323 In this work, we have presented a theoretical model for extracting the anti-plane wave bandgaps 324 in finitely deformed two-phase 2D periodic HMSCs. Specifically, we implemented the finite 325 element method and the Bloch-Floquet theorem to solve the governing incremental anti-plane wave 326 equations. Through numerical computations of the band structures at various levels of applied 327 magnetic flux density, we characterized the magnetic tunability of anti-plane band gaps in the 328 representative hard-magnetic soft composite materials. At last, we explored the effect of material 329 parameter contrasts and geometry of the inclusion phase on the band gap characteristics. The 330 inferences drawn from the current study are summarized below: 331 1. We found that the two-phase HMSC under consideration contracts (i.e., experiences lateral 332 stretch with λ > 1) in the direction of applied normalized magnetic flux density when 333 the magnetic flux is applied in the opposite direction of the residual magnetic flux density. 334 Conversely, we observed expansion (i.e., lateral stretch with λ <1) of the composite when 335 residual magnetic flux density aligns with applied magnetic flux density. 336 2. It has been observed that magnetic loading has a positive impact on the characteristics of the 337 bandgap. When the magnitude of the applied magnetic flux density increases, the bandgap 338 widens and shifts towards higher frequencies. However, the rate at which the bandgap width 339 increases is greater when the applied magnetic flux density aligns with the direction of the 340 residual magnetic flux density. 341 3. In the context of material parameter contrasts, the shear contrast parameter αhas the most 342 significant effect on the band gap size. The influence of the magnetic parameter contrast β 343 was found to be more moderate for the range of contrast values examined in this study. An 344 increase in the shear contrast parameter αshifts the band gaps towards the lower frequency 345 21 range, while in contrast, an increase in the magnetic contrast parameter βshifts the band gap 346 location towards a higher frequency range. 347 4. Opening, closing, size and the location of the anti-plane wave band gaps exhibit a significant 348 dependence on the volume fraction of the constituent phases in the HMSC. Specifically, the 349 first bandgap width is maximized when the volume fraction, νa, is 0.81, while the second 350 bandgap width is maximized when νais 0.56. 351 5. We have demonstrated a notable dependence of the band gap widths and their location on 352 the layout or micro-structure of the unit cell. Among the three considered layouts of the unit 353 cells, the unit cell with inclusions at multiple locations exhibits the highest band gap width, 354 given the specified material parameters. 355 The results obtained in this study could serve as robust theoretical guidance for designing and 356 manufacturing of magnetically tunable hard-magnetic soft wave devices, as well as for wave-based 357 characterization techniques of HMSMs. The finite element framework presented in this work 358 establishes an initial foundation for future research on designing and developing wide and tunable 359 band gaps in HMSCs through topology optimization. Although our current focus has been on anti- 360 plane wave band gaps, exploring the tunability of in-plane wave band gaps in two-phase HMSCs 361 could be an immediate area of interest for further investigation. 362 Acknowledgment 363 The authors would like to express their gratitude to IIT Jodhpur for their financial support 364 through institute initial research Grant No. I/SEED/ATS/20220087. 365