On-demand contactless programming of nonlinear elastic moduli in hard magnetic soft beam based broadband active lattice materials

Engineered honeycomb lattice materials with high specific strength and stiffness along with the advantage of programmable direction-dependent mechanical tailorability are being increasingly adopted for various advanced multifunctional applications. To use these artificial microstructures with unprecedented mechanical properties in the design of different application-specific structures, it is essential to investigate the effective elastic moduli and their dependence on the microstructural geometry and the physics of deformation at the elementary level. While it is possible to have a wide range of effective mechanical properties based on their designed microstructural geometry, most of the recent advancements in this field lead to passive mechanical properties, meaning it is not possible to actively modulate the lattice-level properties after they are manufactured. Thus the on-demand control of mechanical properties is lacking, which is crucial for a range of multi-functional applications in advanced structural systems. To address this issue, we propose a new class of lattice materials wherein the beam-level multi-physical deformation behavior can be exploited as a function of external stimuli like magnetic field by considering hard magnetic soft beams. More interestingly, effective property modulation at the lattice level would be contactless without the necessity of having a complex network of electrical circuits embedded within the microstructure. We have developed a semi-analytical model for the nonlinear effective elastic properties of such programmable lattice materials under large deformation, wherein the mechanical properties can be modulated in an expanded design space of microstructural geometry and magnetic field. The numerical results show that the effective properties can be actively modulated as a function of the magnetic field covering a wide range (including programmable state transition with on-demand positive and negative values), leading to the behavior of soft polymer to stiff metals in a single lattice microstructure according to operational demands.


Introduction
Metamaterials are a class of materials with properties that are not found or are rare in naturally-occurring materials. By engineering them at the micro-level, we can achieve desired properties like high specific stiffness or strength at the macro level. In such materials, the effective properties are characterized by their structural configuration and not by their intrinsic material properties alone. We can modulate the global mechanical properties of mechanical metamaterials by controlling their microstructural geometric parameters, which can be tuned to present unprecedented novel characteristics like negative elastic moduli, auxetic characteristics, extreme multi-physical properties, meta-fluid properties, high crushing resistance, shock absorption characteristics, negative mass density, etc [1][2][3][4][5][6][7][8][9][10][11][12]. Lattice-based materials are a class of metamaterials that have a typical feature of unit cell periodicity [13][14][15][16][17]. The geometric characteristics of the periodic unit cell determine the overall mechanical properties of these latticebased metamaterials [15,18]. Such materials can meet various application-specific requirements owing to their multiphysical properties that range over varying length scales (nano to macro). These engineered materials have drawn the attention of the scientific community significantly in the recent past, especially owing to the advancements in additive manufacturing techniques. Metamaterials, when compared to conventional composites, provide a larger design input parameter space for property modulation, and while achieving more extreme and multifunctional properties [19][20][21][22][23].
Lattice-based materials, being light and able to exhibit high stiffness, are used in several lightweight systems like sandwich structures [24][25][26][27][28][29]. Further, the lattice-type structural configurations are found in plenty across different length scales (including nano and micro) of naturally occurring matters [30][31][32][33][34]. Recent trends in engineered materials try to propose intuitive microstructural configurations in a forward framework, or designs identified through computer simulations such as topology optimization. Anisotropy tailoring for lattices made up of multiple materials is presented in [35], where their differing properties are utilized to enhance the design space beyond the geometry of microstructure only. The voltage-dependent Young's moduli of piezo-electric latticebased microstructures are shown recently to have values that vary from positive to negative [36]. The randomness and disorder in geometric parameters considering the irregularities in manufacturing are studied by considering voronoi honeycombs and through quasi-random configurations [33,[37][38][39][40][41].
The effect of intrinsic stresses in honeycomb lattices is characterized in a recent study [42]. Besides a wide range of investigations concerning 2D lattices, multiple analyses of 3D microstructures and cellular foams have recently been reported [43][44][45][46]. On careful examination of the above presented brief literature review, we find that for periodic lattice structures, in general, a unit cell-based approach is considered to analyze the effective mechanical properties [15].
While it is possible to have a wide range of effective mechanical properties based on their designed microstructural geometry, most of the recent advancements in this field lead to passive mechanical properties, meaning it is not possible to actively modulate the lattice-level properties after they are manufactured. Thus the on-demand control of mechanical properties is lacking, which is crucial for a range of multifunctional applications in advanced structural systems. To address this issue, we propose a new class of lattice materials wherein the beam-level multi-physical deformation behavior can be exploited as a function of external stimuli like magnetic field by considering hard magnetic soft (HMS) beams [47,48]. Since the effective properties of lattices (such as deformation under different loading conditions, vibration, wave propagation, buckling, etc) are essentially derived from the beam-level deformation mechanics, it is possible to control such effective lattice-level properties actively by the intensity and direction of magnetic field. More interestingly, effective property modulation at the lattice level would be contactless without the necessity of having a complex network of electrical circuits embedded within the microstructure [49,50]. The focus of this work is to investigate the effective nonlinear elastic properties of HMS beam lattice materials, which is essential to analyze these active lattices for static, dynamic and instability characteristics.
HMS materials are a novel class of smart materials that are obtained by embedding particles of hard magnetic materials into soft materials [47,48]. These are named so because they are soft mechanically, yet hard magnetically. Here the residual magnetic flux is reoriented using an external magnetic field. By hard magnetic, it is implied that high-coercivity ferromagnetic materials are embedded in soft materials [51][52][53]. This high coercivity helps them retain high magnetic flux density. HMS beams can undergo complex large deformations under various magnetic actuations. These HMS structures can be printed using additive manufacturing techniques by using the magnetic actuation field. Different motion shapes of HMS structures can be realized by varying the magnetic field, opening the design space in soft robotics, biomedical devices and flexible electronics [53]. In this article, we conceptualize a lattice network of such HMS beams under the combined influence of magnetic field and mechanical loads, where broadband stiffness modulation can be realized at the global level. It would permit active modulation of elastic properties of the honeycomb lattice (leading to a seamless on-demand conversion of hard and soft material under normal and shear modes of deformation) by intelligently using the desired residual magnetic flux density, keeping the material properties, geometry of microstructure and lattice density unaltered. For doing so, it is imperative that we have an efficient model which could determine the effective elastic constants of lattice materials in terms of the applied magnetic field and far-field mechanical forces along with geometry the of microstructure and intrinsic material properties under normal and shear modes of lattice-level deformations. Further, nonlinear large deformation analyses concerning effective elastic properties of lattice metamaterials are scarce to find in the literature.
In this article, we aim to develop a semi-analytical model for the nonlinear effective elastic properties of programmable lattice materials under large deformation, wherein the mechanical properties can be modulated in an expanded design space of microstructural geometry and magnetic field. We will first develop a numerical model to obtain the nonlinear large deflection profile of HMS beams under a combined magnetic field and mechanical load. The compound mechanics of elementary beam-level transverse and bending deflections would in turn be exploited in a semi-analytical unit cell based framework to evaluate the effective elastic moduli of honeycomb lattice materials (refer to figure 1). Thus we integrate the beam-level multi-physical deformation mechanics and the lattice geometry to have an expanded design space of active and passive parameters. In this work, our focus is essentially on hexagonal lattice materials (refer to figure 1(C)) for demonstrating the concept of contactless active modulation. While such hexagonal lattices are directly relevant to obtaining results for auxetic and non-auxetic configurations along with other lattices like rhombic and rectangular forms, the proposed approach of contactless active modulation through HMS beam networks is generic and it can be extended to a wide range of two and three-dimensional lattices by considering appropriate unit cells. Hereafter, this paper is structured as, section 2: the numerical model for the analysis of HMS beams under combined magnetic field and mechanical load, followed by the development of a lattice-level semi-analytical framework for the nonlinear elastic moduli of metamaterials, section 3: numerical results concerning contactless active modulation of the elastic moduli along with adequate validation at the beam and lattice levels, section 4: conclusions and perspective of the current research.

Nonlinear mechanics of HMS beam lattices
In this section, we would first present the beam-level mechanics of HMS beams, followed by a unit cell-based analysis for evaluating lattice-level effective properties. To understand the mechanics of HMS structures, it is necessary to theoretically model them. However, the modeling of HMS structures is a challenge due to the geometric and material nonlinearities. The geometric nonlinearity arises due to large deformation, while the magneto-elastic coupling gives rise to material nonlinearity. Numerical simulation seems to be the solution for predicting the mechanical behaviors of these types of structures rather than trying to find out closed-form analytical solutions. But even the numerical methods to the likes of FEM prove to be time-consuming in the cases of large deformation coupled multi-physical problems. This necessitates the development of an effective theoretical model for HMS structures. Various efforts have been made in the recent past to develop a theoretical model for HMS materials. A nonlinear field theory has been developed recently considering the coupling of magnetic field and finite deformation of HMS materials. The simulation was conducted using a FE software, the results of which agreed well with the experimental results [47]. Another study considered different volume fractions of the hard magnetic materials and studied the magnetization and shear moduli [54]. A magneto-visco-hyperelastic model was developed and calculations for simple deformations were performed using the FE method [55]. The dynamic features in HMS beams were also investigated [56]. Theoretical modeling of HMS beams considering extreme bending deformations was done and then solved analytically in a recent study [57].
Though recent studies have tried to understand the mechanics of the HMS structures, very little has been done to develop a theoretical model for the same without the use of any commercial software. Also as pointed out earlier these simulations are time-consuming due to the increased iterations owing to large deformation. The problem increases when dealing with dynamical parameters. In this section, we present a simple, effective and efficient theoretical model that accounts for the complex shape transition of the HMS structures by accommodating the exact expression for the curvature of the centerline. This will be used to model the unit cell of lattice materials (refer to the bottom-up framework presented in figures 1(A)-(E)), which will then be used for the evaluation of the five inplane elastic constants of HMS hexagonal lattices. First, we would analyze the local-level deformation mechanics of the beam-like members accounting for both axial as well as bending deformation (refer to figure 1(E)). Then, we would consider a unit cell that is made up of such beam elements and analyze them to obtain the formulations of the five in-plane elastic constants in semi-analytical form. It is to be noted that at this point the impact of geometry of lattice microstructure is incorporated into the metamaterial design (refer to figures 1(C) and (D)). The effective elastic constants thus derived considering a unit cell are representative of the effective elastic properties of the entire lattice (refer to figure 1(D)), which could then be utilized in the design of various industrial structures as demonstrated in figures 1(A) and (B).

Deriving the governing equations for the HMS beam
Based on the requirement of analyzing the lattices under combined magneto-mechanical loading, we first develop the deformation physics of HMS beams under the combined application of magnetic field and mechanical loading. Note that analytical developments of HMS beams have not been undertaken under such combined loading in the literature. The HMS beam is analyzed using the Euler-Bernoulli beam theory, considering planar deformations. In the proposed theoretical model, the centerline of the beam may either be extensible or inextensible. The theoretical model is developed by deriving the governing equations in terms of the rotation angle, for the beam with both external load and magnetic field. The principle of minimum potential energy is used to derive the governing equations. Here the total strain energy U comprises magnetic potential energy U M and elastic potential energy U E .
Let us consider a HMS beam of length l, having a crosssectional area A. The centerline strain is considered to be small, hence any change in A will be neglected. Let E be the intrinsic Young's modulus of the beam and I be the second moment of inertia. To account for the large deformation of the beam, its geometric exact nonlinearity needs to be considered. Thus, the stretch ratio λ and centerline rotationθ are included in the formulation (refer to figure 1(E)). The axial and transverse displacements of the beam centerline are denoted byū andv, respectively.
2.1.1. Magnetic potential energy of the HMS beam. Let the residual magnetic flux density of the beam in the reference configuration be denoted as Br 0 and as Br in the current configuration. The magnetic field applied externally be denoted by Bē and the angle that Bē makes with the horizontal line beᾱ. The residual magnetic flux density is considered to be constant in magnitude and to be acting in the longitudinal direction or along the beam length. The magnetic potential energy per unit volume of the current configuration [47] is given as: where µ 0 represents the vacuum (or, air) permeability. The magnetic interaction due to the high residual magnetization and coercivity of the embedded hard-magnetic particles is captured by the above equation even for the very large deformations case. The residual magnetic flux density of the beam in the current configuration, i.e. Br is given as: Here S takes care of the non-uniform magnetization. Since we have assumed Br 0 to be constant and in longitudinal direction of the beam, so S can be either +1 (alongx-direction) or −1 (opposite tox-direction) and may be considered as a coefficient. To capture complex deformation shapes of the beam, S can be designed as a function of the beam length, i.e. S(x). The local beam axes are represented byx andȳ denoting axial and lateral directions, respectively. The externally applied magnetic field Bē is given as: Now we substitute equations (2) and (3) into equation (1) to get: The magnetic potential energy of the beam can then be expressed as: Using the relation between strain of the beam centreline ε and stretch ratio λ, ε = λ − 1, the above equation can be expressed as: 2.1.2. Elastic potential energy of the HMS beam. The total elastic potential energy of the beam, U E consisting of membrane strain energy due to stretch of the beam's centerline, U mem and the strain energy due to bending of the beam, U bend is given as: where Φ denotes the strain energy density function defined in per unit of the undeformed volume [58].

Priniciple of minimum potential energy.
In this subsection we will derive the governing equations for the HMS beam considering both externally applied magnetic field and mechanical load using the principle of minimum potential energy. The variation of the magnetic potential energy, U M (refer to equation (6)) is given as: ∂θ. (8) Similarly the variation of the elastic potential energy, U E (refer to equation (7)) is given as: The HMS beam is considered to be acted upon by the following external loads namely, a distributed force withp(x) acting alongx-direction andq(x) acting alongȳ-direction, a distributed momentm(x) and a concentrated force ⃗ F = Fxî + Fȳĵ acting on the tip of the beam, i.e. atx = l. Note that we have taken a generalized loading condition for the purpose of beamlevel derivation here, among which some of the components would be considered as zero during the lattice-level derivation. The displacement vector at tip of the beam is ⃗ d =ūî +vĵ. The variation of the work done by the external forces is given as: Note that equations (8)-(10) account for virtual displacements of four types, i.e. ∂ū, ∂v, ∂θ and ∂ε. In order to derive the governing equations and the boundary condition of the beam system, the redundant virtual displacements can be replaced and thus a unified problem with only two virtual displacements, namely, ∂θ and ∂ε can be setup. According to the Euler-Bernoulli framework, we can relate the four virtual displacements using two equations (or two constraints) and this reduces the problem to two virtual displacements (two variables) which simplifies the problem under consideration. The geometric relationships betweenū,v,θ and ε, are given as: The value of displacements atx = 0 are fixed as zero, to eliminate the effect of rigid body displacement of the system. The centerline displacements are then given as: The variation of equations (12a) and (12b) gives: Substituting the equations (13a) and (13b) into equation (10) gives: Note that the above equation is expressed in terms of only two virtual displacements, ∂ε and ∂θ. Now using the following integral property [59]: the equation (14) can be rewritten as: The principle of minimum potential energy gives: here U = U M + U E . Therefore the above equation becomes: Inserting equations (8), (9) and (16) into equation (18) gives two governing equations, one involving ∂ε and the other involving ∂θ. ∂ε: ∂θ: with the following boundary conditions (adopted based on lattice-level periodicity): denotes the nominal stress. To simplify the governing equations and solve the beam model, we will introduce the following non-dimensional quantities that will make the results more general: Equations (19) and (20), after non-dimensionalization using parameters of equation (22), are thus expressed as: and

Inextensible HMS beam model.
For an inextensible beam, the centerline strain is zero. Thus the governing equation for an inextensible HMS beam model can be obtained by putting ε = 0 in equation (24) (while the equation (23) can be ignored since the variation of centerline strain is not meaningful for inextensible beams):

Solving the coupled set of equations.
The equations (23) and (24) are coupled equations where the first equation is a non-linear equation whereas the second equation is a second-order differential equation. This nonlinearity in the equations arises from the inherent geometric and material non-linearities due to the beam curvature and the material properties respectively. Previously, the Galerkin method has been used by effectively discretizing the infinite dimensional beam [60,61]. This simplifies the complex set of coupled non-linear equations into two sets of a linear system of equations by choosing a base function set that is complete. This method proves to be more complicated to solve when the current generic beam is considered involving both mechanical and magnetic loading. Hence, we deploy an intuitive method that can give us both ε(x) andθ(x) from the non-linear set of governing equations, as discussed in the following subsection.
2.1.6. Numerical model for solving the coupled differential equations. The two governing equations can be solved using the boundary conditions from equation (21) to getθ(x). To obtain ε(x) from the coupled equations we generally need two more boundary conditions resulting from the application of the external loads; but in our approach, we use an intuitive numerical method by which we can obtain the function ε(x) without the necessity of the two boundary conditions resulting from the external loads.
In the analyses of HMS beams, we generally have an idea of the range of the values that ε can lie within. This is what we mainly exploit in our approach. The 'inextensibility assumption' as discussed by [60] evaluates the maximum force that can be applied is 500 N, which stems from the fact that they consider a maximum ε value 0.05. Beyond this the assumption becomes invalid. Since all our analyses are based on the magnitude of forces well within the range of 100 N, we consider a broad range of ε = [0.00004, 0.01].
We follow the following steps in solving the governing equations. Firstly, we split the governing equations in (23) and (24)  For each such value ofξ, we then iterate over a range of values of ε (preferably 100 values) and obtain the corresponding data ofθ vsξ for all these iterations by solving the Primary equation using 'bvp4c' function in MATLAB for solving boundary value problems (with a rotationally restrained boundary condition at both ends; refer to equation (21). From theθ vsξ data we obtain the value ofθ for the current iteration'sξ and then retain the value. Then, during each of these iterations, we calculate the value of the left-hand side of the Primary equation and then find the ε value which minimizes the left-hand side of the Secondary equation. This gives us a value for bothθ and ε for each iteration overξs in [0, 1]. Thus we now obtain the functional mapping using numerical data fromξ to bothθ and ε.
For analyzing hexagonal lattices maintaining unit cell level periodic boundary conditions, we need to analyze beams with one end fixed and the other end rotationally restrained condition, while applying a transverse point load at the rotationally restrained end and a magnetic field in any arbitrary orientation. A computational framework of the beam with this boundary and loading condition is sufficient for the subsequent lattice-level analysis (in-plane) for evaluating the two Young's moduli, two Poisson's ratios and shear modulus. In the further analysis, since our focus is solely on the magnetic field and concentrated load, we setP =Q =M = 0 andσ N = ε.

Effective in-plane elastic properties of active hexagonal lattices
The effective mechanical properties of the proposed active lattice-based material are defined as a function of the unit cell geometry, intrinsic material properties of the beam-like members (including the multi-physical properties of the hard magnetic particles) and applied magnetic field. The lattice here is conceived as a periodic beam network. Since the deformation physics of the constituting beams is active in nature and depends on the applied magnetic field, the global deformation behavior of the lattice becomes a function of the applied magnetic field. Subsequently, the effective elastic properties of the lattice can be actively modulated as a function of the magnetic field and applied external mechanical stress. Note that since we consider large nonlinear deformation of the lattices, the effective elastic properties become dependent on the lattice-level externally applied remote stress. The active component in the proposed lattice metamaterial can be controlled through the intensity and direction of magnetic fields, while the influence of magnetic fields can be accentuated based on the distribution and orientation of the hard magnetic particles within the beam-like members. In this article, we consider an external magnetic field actuation acting in direction-2 of the lattice (refer to figure 2), where beams have uniformly distributed hard magnetic particles.
In this section, the deflection values at the beam-level obtained using the numerical model (as described in the preceding subsection) are used to further evaluate the homogenized effective elastic properties (E 1 , E 2 , ν 12 , ν 21 and G 12 ) at the lattice level, considering a unit cell-based approach. To take care of the effect of large deformation analysis, we consider incremental geometry updation at the unit cell level [62]. The initial configuration S 0 of hexagonal honeycomb is considered to be no-load configuration having load step dependent cell walls of length l 0 s and inclination angle θ 0 (refer to figures 2(A) and 3(A)). At ith load step, in the deformed configuration S i , the load step dependent length and inclination angle of the cell walls become l i s and θ i , respectively. Note that we consider the width b and thickness t of all cell walls to be independent of the load step (refer to figures 2(B) and (C)).

Longitudinal young's modulus E 1 .
To derive the nonlinear longitudinal Young's modulus, we consider a uniform incremental compressive stress field σ i 1 (= σ i−1 1 + ∆σ i 1 ) being applied to the unit cell along direction-1 (i.e. in the longitudinal direction). This stress produces a force P i that acts at the beam end points (refer to figure 2(D)). The equivalent force P i acting on the beam end can be expressed as: We substitute this force and the externally applied magnetic flux density B (non-dimensional quantity representing the magnetic actuation) in the governing equations (refer to equations (23) and (24)), which are then solved using the above described numerical model to get axial deformation δ i x and transverse deformation δ i y at the tip of the beam under the influence of both B and P i . The boundary condition of the beams is considered as one end fixed and the other end rotationally restrained. The moment M i that bends the cell wall (refer figure 2(D)) is given as: where l i x and l i s are related by: Total deflection in direction −1 is given as: Strain along the direction of σ i 1 is given as: Note that we calculate the strain with respect to the reference configuration S 0 . Now the longitudinal Young's modulus can be expressed as (corresponding to ith increment):

Poisson's ratio ν 12 .
To obtain Poisson's ratio ν 12 , we calculate the strain in direction 2 resulting from the stress σ i 1 acting along the direction-1 as shown in figure 2(D). The total deflection in direction-2 (δ 2 ) is obtained as: Now, the total strain along direction-2 is given as: From the equations (30) and (33), the Poisson's ratio ν 12 is given as: To obtain the nonlinear transverse Young's modulus E 2 , we consider a uniform incremental compressive stress field σ i 2 (= σ i−1 2 + ∆σ i 2 ) in direction-2 being applied on the unit cell. This results in a vertical force W i at the end points (refer to figure 2(E)). W i is expressed as: We substitute this force and the externally applied magnetic flux density B in the governing equations (refer to equations (23) and (24)), which are then solved using the above described numerical model to get axial deformation δ x i and transverse deformation δ y i at the tip of the beam under the influence of both Bē and W i . The boundary condition of the beams is considered as one end fixed and the other end rotationally restrained. The total lateral force (acting in direction-2) at point O is 2W i , accounting for the axial deformation of the vertical member OC. The moment M i obtained from the free body diagram in figure 2(E) is given as: Total deflection, therefore, in direction-2 is given as: The strain in direction-2 can be expressed as: Thus, transverse Young's modulus acting along direction-2 (i.e. the transverse direction) corresponding to the ith increment is given as:

Poisson's ratio ν 21 .
For quantifying the Poisson's ratio ν 21 , we need to obtain the strain in direction 1, resulting from the stress applied along the direction-2. The deformation along direction-1 (i.e. in the longitudinal direction) due to σ i 2 is given as (refer to figure 2(E)): Thus, strain due to the above calculated deflection in direction-1 can then be given as: From the equations (38) and (41), the Poisson's ratio ν 21 can be expressed as:

Shear modulus G 12 .
To derive of the shear modulus G 12 , we consider the shear strain arising from the bending deformations as well as that due to axial deformations. For the case of shear modulus, the vertical cell walls are also considered to be load step dependent. In the initial configuration S 0 of hexagonal honeycomb, the vertical cell wall is considered to have length h 0 s while the slant cell walls have length l 0 s and inclination angle is θ 0 (refer to figure 3(A)). At ith load step, in the deformed configuration S i (refer to figure 3(B)), the load step dependent length of the vertical cell wall becomes h i s and that of slant cell walls and their inclination angle become l i s and θ i , respectively. The deformation occurs under the effect of anti-clockwise mode of incremental shear stress τ i (= τ i−1 + ∆τ i ), as considered in the current analysis (note that the shear moduli under a given magnetic field would be same for clock-wise and anti-clockwise modes for the current scenario). This shear stress field produces forces F i 1 and F i 2 in the slant members, which are given as (refer to figure 3(D)): The force produced in the vertical cell member is given as (refer to figure 3(C)): The moment M i v generated in the vertical member at both the ends (refer to figure 3(C)) is given as: No axial force is produced in the vertical cell wall under the applied shear stress. This moment M v gets distributed in the two slant members. Note that the two slant members deform differently depending on the external magnetic force B. At certain portion of a particular slant member, B acts along the deformation caused due to mechanical stress and in other portion it acts in opposite direction to the applied stress resulting in different deformation of the two slant members. Thus the moment M v does not get distributed equally on the two slant cell walls. The moment generated on slant member 1 is denoted by M i s1 while that on the slant member 2 is M i s2 (refer to figure 3(D)). The sum of the two moments is equal to M v , i.e. M v = M i s1 + M i s2 . Thus we see that the externally applied magnetic flux density B causes different deformation of the slant members making it an asymmetric problem. The force F i 1 gets distributed on both the slant members as F i 1a and F i Note that these forces act in conjunction with the externally applied magnetic flux density B. We use the components of these forces (F i 1a , F i 1 b and F i 2 ) acting on the slant members along the axial and transverse directions. Using these external loads the governing equations (refer to equations (23) and (24) along with satisfying the other compatibility condition pertaining to the rotation of the slant members 1 and 2 and vertical member. The rotation of the vertical cell wall and that of the slant members will be same to maintain the integrity of the structure, i.e. ϕ i = ϕ i 1 = ϕ i 2 . Here ϕ i 1 and ϕ i 2 are the rotations of the slant members, which are given as: where l i x and l i s are related by: Thus the shear strain γ i v of the vertical cell wall in the diretion of applied incremental shear stress τ i is given as: The shear strain γ i s of the slant cell wall in the direction of applied incremental shear stress τ i is given as: The total shear strain γ i t is therefore given by: The shear modulus G 12 can, therefore, be expressed as: Note that in this analysis, we have considered anticlockwise shear stress. When the shear stress acts in the clockwise sense, the slant members 1 and 2 behave opposite to the present anticlockwise shear case, but the total shear strain under a specific value of shear stress remains the same. Thus the shear modulus is same in both cases. This is unlike the case for E 1 , E 2 , ν 12 and ν 21 which change under tensile and compressive stress fields.
The expressions of five in-plane effective elastic moduli are presented in equations (31), (34), (39), (42) and (51). A careful consideration of these equations would reveal that the elastic moduli depend on different deflection components and the applied far-field stress. The deflection components further depend on the applied far-field stress (mechanical loading) and magnetic field. Thus the effective elastic properties under a certain value of applied stress are functions of the magnetic field, leading to an on-demand active modulation capability.

Elastic moduli considering only the bending deformation
In the preceding subsection, we have presented the generic framework for evaluating the nonlinear elastic properties considering both axial and transverse deflections of the beam-like elements. The five in-plane elastic moduli, considering only the bending deformation (ignoring the axial deformation part), i.e., δ i x = 0, based on the previously derived equations (refer equations (31), (34), (39), (42), (51)) lead to: The above expressions are useful in the case of axially rigid beams, where the axial deformation can be neglected because of dominant bending deformation [15]. So the axial deformation in these structures can be neglected, which in our case will lead to low-density thin-walled honeycombs. It is interesting to note that the Poisson's ratios (ν 12 and ν 21 ) for such lightweight honeycomb lattices are independent of the external loads. When the beam-like cell walls are assumed to be axially rigid (i.e. the axial deformations are neglected), we find that the Poisson's ratios are dependent only on the geometry of the microstructure. These observations agree with those available in literature [36,63,64], in which the impact of other multiphysical behaviors is considered.

Results and discussion
In this section we will present numerical results to characterize the effect of magnetic field intensity comprehensively following the numerical model proposed in the section above. Before moving forward with the numerical investigation, the validity of the numerical model for a HMS beam under the effect of both mechanical load and magnetic flux density (refer to section 2.1.6), needs to be established. The validated numerical model is then used for the quantification of the elastic moduli of lattice-based structure. At the lattice level, we first investigate the nonlinear effective elastic properties without considering any magnetic field and compare the results with available literature. It provides a validation of the proposed nonlinear large deformation framework at the lattice level. After establishing adequate validation both at the beam level and lattice level, we demonstrate the on-demand modulation of effective elastic properties of the lattice metamaterial as a function of the magnetic field.
For beam level validation under magnetic field and mechanical load, we have compared the results obtained from the numerical model with those available in the literature individually for the two cases. Figure 4 shows that the results of the current analysis considering only mechanical load (and no external magnetic intensity) are close to those of reference literature, affirming the validity of the numerical model. Note that we have considered different magnitudes of loading and boundary conditions for this purpose, including the specific boundary condition necessary (one end fixed and the other end rotationally restrained, refer to figure 4(E)) for the lattice-level analysis. Figure 5 shows that the results of the current analysis considering only external magnetic intensity (and no mechanical load) are also in good agreement with those of reference literature, ascertaining the validity of the numerical model further (considering both deformation and rotational angles). Beam-level validations of the nonlinear deformation under both mechanical and magnetic loading ensure that the beam model is accurate enough for adopting it to the subsequent lattice level analysis (note that the unit cell level mechanics here is primarily dependent on the beam level deformation, which we have extensively validated considering nonlinearity, mechanical load, magnetic field and appropriate boundary condition).
After having beam-level validations considering both mechanical load and magnetic field, we focus on lattice-level validation of the nonlinear elastic properties. Figure 6 presents the comparison of results for a hexagonal lattice using the results obtained from the proposed numerical model with those available in the literature. For all the nonlinear elastic properties, we find that the results match quite well for different values of far-field mechanical stress under the normal and shear modes. The numerical results converge to the widely accepted closed-form solutions [15] in case of small deformation. Thus, figures 4-6 validate the proposed numerical framework extensively at two levels, one at the beam level and the other at the lattice level, considering nonlinear large deformation under mechanical load and magnetic field. In the following paragraphs, we first investigate the nonlinear elastic properties of lattices without any magnetic field, followed by exploring the effect of magnetic field for active on-demand modulation of these effective elastic properties. The results are presented both for compressive and tensile far-field stresses separately (note that the shear modulus remains unaltered for clockwise and anti-clockwise directions). Figures 7 and 8 investigate the nonlinear variation of five in-plane elastic moduli under large deformation due to only mechanical stresses (no external magnetic field). The results are presented considering both auxetic and non-auxetic microstructures considering different cell angles. The effect of beam-level axial deformation is studied by comparing the elastic moduli calculated based on only bending deformation and the combined effect of bending and axial deformation. Significant variation of the elastic moduli at higher values of applied far-field stresses affirms that it is necessary to incorporate the effect of nonlinearity due to the large deformation of beams and subsequent incremental change in the unit cell geometry. The Poisson's ratios are not affected by applied stress when only bending deformation is considered, while these  [15], G&M refers to the reference literature [62] and P&G&M refers to the reference literature [65]. The cell angles considered in this analysis are based on the availability of results from literature. become stress-dependent when the coupled effect of bending and axial deformations are accounted. In the following paragraphs, we would explore the influence of external magnetic fields in addition to the far-field mechanical stresses. The combined effect of transverse bending and axial deformation is considered (along with a comparative perspective of the scenario when only beam-level bending deformations are considered) at the beam level for evaluating the effective elastic properties of the lattices under the application of both far-field mechanical stresses and magnetic fields. Note that, for active effective elastic property modulation, we have applied the magnetic field only vertically as shown in figures 2 and 3. In this section, we will present the results considering two cases for better clarity in understanding the multi-physical behavior, one mechanical load dominated and the other magnetic field intensity B dominated. For the former case, we take comparatively lesser values of B (ranging from −0.1 to 0.1) and higher for the latter (ranging from −10 to 10). In all the ana- mm, 2 and 10 −2 respectively. Note that these microstructural parameters can also be varied (/designed) for expanding the range of active property modulation in future studies. Figure 9 presents the variation in elastic moduli E 1 , E 2 and G 12 of hexagonal HMS lattices in mechanical load-dominated case with externally applied stress, considering both the bending and axial deformations at beam level. The right vertical axis of these plots presents the only bending deformation case, while the left vertical axis presents the combined bending and axial deformation. Note that the three elastic moduli are shown considering non-dimensional forms as E 1 /E, E 2 /E and G 12 /E, where E represents the intrinsic equivalent Young's modulus of the constituting beam members making up the unit cell. From the figures, we find that initially the elastic moduli both for only bending and for bending and axial deformation case match with each other while they tend to differ with increasing applied stress values. This is because as the stress increases, the axial deformation component of the total deformation, which is initially very small also increases significantly. As the axial deformation increases, the elastic moduli decrease when compared to only the bending case (refer to equations (31), (39) and (51)). It may be noted that though the combined axial and bending deformation case is more complicated and computationally expensive, it provides more accurate results compared to the only bending deformation case. We notice from the figures (primarily based on the combined effect of axial and bending deformations) that, in general, for each nonauxetic orientation θ of the microstructure, the longitudinal elastic modulus decreases first and then begins to increase with increasing stress σ 1 values. Also as B decreases, E 1 /E decreases, for each θ. For auxetic cases, the longitudinal elastic modulus decreases with increasing stress σ 1 values. As B decreases, E 1 /E increases, for each θ. The transverse elastic modulus decreases for the non-auxetic cases while it increases for the auxetic cases with increasing stress σ 2 values. Here as B decreases, E 2 /E increases, for each θ. The in-plane shear modulus increases with increasing stress τ values. Also, in general, as B decreases, G 12 /E, increases, for each θ. Figure 10 presents the variation in Poisson's ratios ν 12 and ν 21 of hexagonal HMS lattices in the mechanical loaddominated case with externally applied stress, considering both the bending and axial deformations. Here we have not presented the only bending deformation case because for this case the Poisson's ratios are independent of the external stresses and magnetic field intensity. They depend only on the geometric configurations as evident from the expressions in equations (52b) and (52d). From the figures, we notice that for each microstructural non-auxetic orientation θ, the Poisson's ratio, ν 12 decreases with increasing stress σ 1 values. In auxetic case, for each θ, the Poisson's ratio, ν 12 increases with increasing stress σ 1 values. Also as B decreases, ν 12 decreases, for each θ, both in auxetic and non-auxetic cases. The transverse Poisson's ratio, ν 21 decreases with increasing stress σ 2 values. With decrease in B, ν 21 increases, for each θ, both in auxetic and non-auxetic cases. Figure 11 presents the variation in elastic moduli E 1 , E 2 and G 12 of hexagonal HMS lattices in the magnetic field intensity-dominated case with externally applied stress, considering both the bending and axial deformations. The right vertical axis of these plots presents the only bending deformation case, while the left vertical axis presents the combined bending and axial deformation. Note that the three elastic moduli are shown considering non-dimensional forms as E 1 /E, E 2 /E and G 12 /E, where E represents the intrinsic equivalent Young's modulus of the constituting beam members making up the unit cell. From the figures, we find that the elastic moduli both for only bending and for bending and axial deformation case follow a similar trend though their values differ with increasing applied stress values. This is because of the fact that as the stress increases, the axial deformation component of the total deformation becomes dominant. As the axial deformation increases, the elastic moduli decrease when compared to only bending case (refer to equations (31), (39) and (51)). The results show that a wide range of effective elastic moduli (including positive and negative values) can be achieved with their active programmable feature based on the coupled effect of applied stress and magnetic field. It may be noted that though the combined axial and bending deformation case is more complicated and computationally expensive, it provides more accurate results compared to the only bending deformation case. We notice from the figures (primarily based on the combined effect of axial and bending deformations) that for each microstructural orientation θ, the longitudinal and transverse elastic modulus either increase or decrease with increasing stress σ 1 values, depending on the values of B. The nature and values of the effective elastic moduli depend on the relative influence of mechanical stress and magnetic field on the beam-level deformations. If they are opposite in nature, the combined effect may lead to negative values of effective elastic moduli. Further, other extreme values (very high and very low) can also be realized actively by exploiting such coupled influences. The in-plane shear modulus shows an increasing trend with increasing stress τ values as a general trend. Figure 12 presents the variation in Poisson's ratios ν 12 and ν 21 of hexagonal HMS lattices in the magnetic field intensitydominated case with externally applied stress, considering both the bending and axial deformations. Here we have not presented the only bending deformation case because for this case the Poisson's ratios are independent of the external stresses and magnetic field intensity. They depend only on the geometric configurations as evident from the expressions in equations (52b) and (52d). From the figures, we notice that Note that the above discussed results (all figures concerning elastic moduli and Poisson's ratio's presented in the main paper) correspond to the external far-field stresses that are compressive in nature. The results with the hexagonal lattice being subjected to tensile external far-field stresses are presented in the supplementary document. In this paragraph and the subsequent ones, we will discuss the same. We have referred to the figures in supplementary document with a prefix S. Figure  S1 presents the comparison of results for a hexagonal lattice using the results obtained from the proposed numerical model with those available in the literature. Since the lattice is subjected to far-field tensile stress, the validation and subsequent results are presented for the four in-plane elastic moduli (E 1 , E 2 , ν 12 and ν 21 ) . For all these nonlinear elastic properties, we find that the results show good agreement with literature [62] for different values of far-field mechanical stress. The numerical results converge to the widely accepted closed-form solutions [15] in case of small deformation. Figures S2 and S3 investigate the nonlinear variation of the four in-plane elastic moduli, as mentioned above, under large deformation due to only mechanical tensile stresses with no external magnetic field, considering different cell angles. The effect of beam-level axial deformation is studied by comparing the elastic moduli calculated based on only bending deformation and the combined effect of bending and axial deformation. Here we observe significant variation of the elastic moduli at higher values of applied far-field tensile stresses which reaffirms the necessity of incorporating the effect of nonlinearity due to the large deformation of beams and subsequent incremental change in the unit cell geometry. The Poisson's ratios are not affected by applied stress when only bending deformation is considered, while these become stress-dependent when the coupled effect of bending and axial deformations are accounted. Note that the lattice properties for the tensile case are same as that of the compressive one. Figure S4 presents the variation in elastic moduli E 1 , E 2 and G 12 of hexagonal HMS lattices in mechanical load-dominated case with externally applied tensile stress, considering both the bending and axial deformations at beam level. The right vertical axis of these plots presents the only bending deformation case, while the left vertical axis presents the combined bending and axial deformation. Note that the two elastic moduli are shown considering non-dimensional forms as E 1 /E and E 2 /E, where E represents the intrinsic equivalent Young's modulus of the constituting beam members making up the unit cell. From the figures, we find that initially the elastic moduli both for only bending and for bending and axial deformation case match with each other while they tend to differ with increasing applied stress values. This is because as the stress increases, the axial deformation component of the total deformation, which is initially very small also increases significantly. Since the formulation of elastic moduli for the tensile case is similar to that of the compressive case (refer to equations (31), (39) and (51)), we observe that as the axial deformation increases, the elastic moduli decrease when compared to only the bending case. We notice from the figures (primarily based on the combined effect of axial and bending deformations) that for each non-auxetic orientation θ and broadly for all auxetic configuration of the microstructure, the longitudinal elastic modulus increases with increasing stress σ 1 values. Also, for the non-auxetic case, as B decreases, E 1 /E increases, for each θ. For the auxetic case, as B decreases, E 1 /E decreases, for each θ. The transverse elastic modulus increases for the non-auxetic cases while it decreases for the auxteic cases with increasing stress σ 2 values. Here as B decreases, E 2 /E decreases, for each θ. Figure S5 presents the variation in Poisson's ratios ν 12 and ν 21 of hexagonal HMS lattices in the mechanical loaddominated case with externally applied tensile stress, considering both the bending and axial deformations. Here we have not presented the only bending deformation case because for this case the Poisson's ratios are independent of the external stresses and magnetic field intensity and are dependent only on the geometric configurations. From the figures, we notice that for each microstructural non-auxetic orientation θ, the Poisson's ratio, ν 12 increases with increasing stress σ 1 values. In auxetic case, for each θ, the Poisson's ratio, ν 12 decreases with increasing stress σ 1 values. Also as B decreases, ν 12 decreases, for each θ, both in auxetic and non-auxetic cases. The transverse Poisson's ratio, ν 21 increases with increasing stress σ 2 values. With decrease in B, ν 21 increases, for each θ, both in auxetic and non-auxetic cases. Figure S6 presents the variation in elastic moduli E 1 , E 2 and G 12 of hexagonal HMS lattices in the magnetic field intensity-dominated case with externally applied tensile stress, considering both the bending and axial deformations. The right vertical axis of these plots presents the only bending deformation case, while the left vertical axis presents the combined bending and axial deformation. Note that the two elastic moduli are shown considering non-dimensional forms as E 1 /E and E 2 /E, where E represents the intrinsic equivalent Young's modulus of the constituting beam members making up the unit cell. From the figures, we find that the elastic moduli both for only bending and for bending and axial deformation case follow a similar trend though their values differ with increasing applied stress values. This is because of the fact that as the stress increases, the axial deformation component of the total deformation becomes dominant. Also, as the axial deformation increases, the elastic moduli decrease when compared to only bending case. The results show that a wide range of effective elastic moduli (including positive and negative values) can be achieved with their active programmable feature based on the coupled effect of applied stress and magnetic field. It may be noted that though the combined axial and bending deformation case is more complicated and computationally expensive, it provides more accurate results compared to the only bending deformation case. We notice from the figures (primarily based on the combined effect of axial and bending deformations) that for each microstructural orientation θ, the elastic moduli either increase or decrease with increasing stress σ 1 values, depending on the values of B. The nature and values of the effective elastic moduli depend on the relative influence of mechanical stress and magnetic field on the beam-level deformations. If they are opposite in nature, the combined effect may lead to negative values of effective elastic moduli. Further, other extreme values (very high and very low) can also be realized actively by exploiting such coupled influences. Figure S7 presents the variation in Poisson's ratios ν 12 and ν 21 of hexagonal HMS lattices in the magnetic field intensitydominated case with externally applied tensile stress, considering both the bending and axial deformations. Here we have not presented the only bending deformation case because for this case the Poisson's ratios are independent of the external stresses and magnetic field intensity and are dependent only on the geometric configurations. From the figures, we notice that for each microstructural orientation θ, the Poisson's ratio, ν 12 shows a varying trend with increasing stress σ 1 values based on the value of B. It either increases or decreases with stress based on the beam-level deflections which in turn are influenced by the combined effect of both mechanical load and B values. The transverse Poisson's ratio, ν 21 increases with increasing stress σ 2 values. With decrease in B, ν 21 increases, for each θ, both in auxetic and non-auxetic cases.
The numerical results presented in this section demonstrate that the nonlinear Young's moduli, shear modulus and Poisson's ratios can be actively modulated as a function of externally applied far-field mechanical stress (tensile and compressive) and magnetic field. It can be noticed that the trends of nonlinear variations of elastic moduli differ significantly in the mechanical stress dominated and magnetic intensity dominated cases, as presented in figures 9-12. The extent of such modulation can further be controlled by microstructural design parameters such as cell angles, length and thickness of the cell walls, and intrinsic material properties. We note that it is possible to achieve a wide range of variation in the elastic properties in an on-demand and contactless paradigm including active transition between negative and positive values of the elastic moduli and Poisson's ratios. Effective elastic moduli being one of the fundamental properties of a material, the capability of having active control through a contactless framework would essentially lead to on-demand programming of a range of static and dynamic structural behavior, including direction-dependent deformation, vibration, wave propagation and control, impact and penetration resistance, energy absorption, shape morphing, robotic motion and actuation.
In this paper, we have focused on the hexagonal lattices with auxetic and non-auxetic configurations. It is primarily because the hexagonal lattice-based forms are widely encountered in naturally occurring and artificial structures across the macro, micro and even nanoscales (like graphene and hBN nanostructures, woods and bones microstructures and sandwich structures' core) [15,33,66]. Due to their high specific strength and stiffness along with high energy absorption capability and crushing resistance, and at the same time being lightweight, these hexagonal lattices have a wide range of structural and industrial applications [67,68]. Though the present study is confined only to hexagonal lattices, the developed semi-analytical framework is directly applicable to various other 2D lattice forms such as rhombic and rectangular (by considering special cases of having the vertical cell wall length and cell angles as zero, respectively). A range of other active 2D and 3D periodic lattices can also be analyzed based on the proposed framework by considering appropriate unit cells. The current focus of this paper is effective elastic moduli and their active control that would essentially affect the stiffness. However, other mechanical properties of lattices such as failure strength [69] can also be controlled actively following a similar framework considering beam-level bending and axial deformations as proposed here.
It may be noted that the focus of this work is theoretical development of the computational framework and proposition of the concept of contactless active modulation of effective elastic properties. However, future research will follow the prospective manufacturing of the proposed lattices. With the recent advancements in 3D and 4D printing (/additive manufacturing) [70][71][72][73], it is possible to fabricate complex lattice structures along with the inclusion of magnetoactive particles.

Conclusions and perspective
With the quest of achieving contactless on-demand modulation of effective elastic properties, we propose a new class of lattice materials in this article wherein the beam-level multiphysical deformation behavior can be exploited as a function of external magnetic field by considering HMS beams. A semianalytical bottom-up multi-level approach is developed for evaluating the active nonlinear elastic properties of the lattices under large deformation. In the first stage, we have developed a computational framework for analyzing the multi-physical nonlinear deformation behavior of HMS beams under the combined application of mechanical load and magnetic field. Subsequently, the beam-level deformation behavior is integrated with unit cell-based mechanics of lattices for obtaining the effective elastic properties of lattice materials in an efficient semi-analytical framework.
Before demonstrating the active modulation of elastic properties, the developed computational framework is validated extensively both at the beam level and lattice level considering different boundary conditions, nonlinearity mechanical loading and magnetic fields. The numerical results are systematically presented first exploring the nonlinear effective elastic moduli under only far-field mechanical stresses, followed by investigating the coupled influence of external mechanical stresses and magnetic fields. For investigating the coupled influence with more clarity, we have adopted two different possible scenarios of mechanical load dominated case and magnetic field dominated case depending on the relative intensity of applied far-field stress and magnetic field. The results demonstrate that the nonlinear Young's moduli, shear modulus and Poisson's ratios can be actively modulated as a function of externally applied far-field mechanical stress and magnetic field within a broadband of respective values. The extent of such modulation can further be controlled by microstructural design parameters such as cell angles, length and thickness of the cell walls, and intrinsic material properties. It is possible to achieve an active transition between negative and positive values of the elastic moduli and Poisson's ratios. More interestingly, the active programming of effective elastic properties at the lattice level can be achieved in a contactless arrangement without the necessity of having a complex network of electrical circuits embedded within the microstructure as in the case of piezoelectric lattices.
In general, tremendous progress in the field of lattice-based metamaterials has led to the current capabilities where it is possible to have a wide range of effective mechanical properties based on their designed microstructural geometry in a passive framework, meaning it is not possible to actively modulate the lattice-level properties after they are manufactured. Thus the on-demand control of mechanical properties is lacking, which is crucial for a range of multi-functional applications in advanced structural and mechanical systems. The current development concerning active lattices addresses this issue with the additional advantage of contactless nonlinear modulation. By externally applying different values of the magnetic field intensity, we can get different elastic properties, and that too from a distance. Essentially, this will help in optimizing the material utilization to an extreme extent by controlling the stiffness of a structure based on operational demands. For example, the stiffness of a structure can be actively increased during an operational condition when higher magnitudes of loads are experienced to keep the deformations under control or the natural frequencies need to be increased to avoid resonance under dynamic loading. The stiffness can also be actively reduced to allow large deformation and shape control for (soft-)robotic motions or increased energy absorption and avert sudden failure. The numerical results in this paper show that the effective properties can be actively programmed as a function of the magnetic field covering a wide range (including programmable state transition with on-demand positive and negative values), leading to the behavior of soft polymer to stiff metals in a single lattice microstructure according to operational demands. Effective elastic moduli being one of the fundamental properties of a material, the capability of having active control would essentially lead to on-demand programming of a range of static and dynamic structural behavior, including direction-dependent deformation, vibration, wave propagation and control, impact and penetration resistance, energy absorption, shape morphing, robotic motion and actuation.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).