Diversifying temporal responses of magnetoactive elastomers

Magnetoactive elastomers(MAEs) are able to deform significantly in response to the application of magnetic fields. Usually, a magnetic field which is harmonic in time usually results in a harmonic mechanical response of the MAEs. To render MAEs with the ability of responding or deforming diversely or anharmonically in time, in this work, we propose a hybrid MAE which is based on a rubber matrix embedded with both soft iron particles and hard NdFeB-alloy particles. Firstly, based on the principle of minimum free energy, we establish a theoretical model to study magnetomechanical behaviors of the proposed hybrid MAEs. Then, through both theoretical and experimental studies, we show that the response of a hybrid MAE sample to the applied magnetic field is usually complex, i.e. the deformation induced by a harmonic magnetic field in time is anharmonic. At last, the effect of two main factors, the state of magnetization and the amplitude of the applied magnetic field, is studied both experimentally and theoretically. This work provides a new idea of diversifying the temporal response of MAEs to the application of harmonic magnetic fields (harmonic in time). The hybrid MAEs may serve as a complement to the recently proposed 3D-printed hard MAEs which are able to deform inhomogeneously in space in response to a uniform magnetic field.


Introduction
In recent years, using the magnetic field to actuate or control the motion of materials is a new focus in soft active materials or soft robotics , Yoonho et al 2019, Liu et al 2019, Ren et al 2019. An interesting and well known example of magnetic active materials is MAE, an elastomer embedded with magnetic particles, owing to its large deformation in response to the applied magnetic field , Zhao et al 2011. Due to such strong magneto-elastic coupling, MAEs have found a wide range of applications in controllable actuators, rapid response regulators for mechanical systems, adaptively tuned vibration absorbers, and stiffness tunable mounts (Ivaneyko et al 2012, Filipcsei et al 2007, Yang and Sun 2014, Opie and Yim 2009.
Early efforts in MAEs are mainly focused on producing larger deformation of materials using a relatively small magnetic field h e . For example, under a magnetic field of 120 kA m −1 , 1.5% magnetostriction strain was obtained in a MAE bar embedded with iron particles at a volume ratio of 10% (Coquelle and Bossis 2005). Theoretical studies indicate that the magnetic force that deforms the material stems from the Maxwell stress s MW which is usually on the order of kPa (Liping and Pradeep 2013). Thus, the material should be soft enough to exhibit significant deformation. Also, since the magnetic field induced Maxwell stress increases with the permeability of the material, particles with high permeability, such as iron and iron oxides, were usually used to fabricate MAEs (Guan et al 2008, Rigbi and Jilken 1983, Zrinyi et al 1996. However, for a free standing MAE sample in a uniform magnetic field, the deformation is always uniform. To deform the sample inhomogeneously, a common way is to apply inhomogeneous mechanical constraints (mostly through boundary conditions), which is both inefficient and inconvenient (Feng et al 2017, Qi et al 2018.

Energy formulation and Euler-Largrange equations of a continuum body in magnetic fields
In this section, we establish the Euler-Lagrange equations and boundary conditions for a hybrid MAE through the principle of free energy minimization. Consider a deformable continuum body as shown in figure 2. Let the three-dimensional region occupied by the undeformed body be denoted V, with the boundary ∂V. We refer to the undeformed body as the reference configuration and the deformed body as the current configuration. The position of a material point A in V can be described by the coordinate system X K (K=1, 2, 3). After deformation χ, the material point A moves to a new position which is described by x k (k=1, 2, 3), a new coordinate system associated with the deformed body that occupies the region v. In the traditional continuum mechanics, we call X K the material or Lagrangian coordinates and x k the spatial or Eulerian coordinates. The base vectors for the material coordinate system and the spatial coordinate system are respectively denoted by I K and i k . The deformation c may be interpreted as the mapping from the material coordinates to the spatial coordinates, c = x X ( ). Based on the above setup, the deformation gradient is defined as . Throughout the whole work, we use, ' K' and, ' k' to represent the partial derivatives with respect to the coordinates X K and x k , respectively. The determinant of the deformation gradients is the so-called Jacobian which is denoted by º J det F ( ). The mass density field r X 0 ( ) of the undeformed body is related to its counterpart r x ( ) in the deformed body by r r = J X x 0 ( ) ( ) due to the assumption of local mass conservation. In the current configuration, the Maxwell equations are where e kln is the alternating symbol, b k is the magnetic flux density, μ 0 denotes the permeability of the vacuum, h k is the magnetic field, m k is the magnetization, and b r k is the residual magnetic flux density. For convenience, let x =h k k , , in which ξ is the magnetic potential. One may define the following Lagrangian counterparts of h k , m k , b k and b k r as where B L denotes the nominal magnetic flux density, H K denotes the nominal magnetic field, M K is the nominal magnetization, α kK is the shifter which may be viewed as the rotation from the coordinate frame I K to i k , and B L r is the nominal residual magnetic flux density. Then, the Lagrange form of Maxwell equations can be written as 1. Free energy of the system The total free energy of a magnetomechanical system can be expressed as (Liu 2014) Here, x M , k K U[ ]is the internal energy given by the following form: )is the internal energy function. Since a magnetized solid changes the magnetic field of the free space around it, the magnetic energy ( mag E in equation (5)) can be written as where  3 is the total space which is the sum of the domain of the deformed body plus the free space around it, and h k e is the externally applied magnetic field. Note that the first term on the right hand side of equation (7) corresponds to the energy associated with the change of the magnetic field by the magnetized body, and the second term corresponds to the energy of the external magnetic field. For the convenience of calculation, it is helpful to decompose the magnetic field h k into the externally applied field h e and the perturbation field h self . Finally, mech P in equation (5) is defined by denotes the velocity, f k is the body force and t kl is the surface traction.

Euler-Lagrange equations and boundary conditions
By the principle of minimum free energy, the equilibrium state of the system is determined by with the constraint of Maxwell equations (1) or (4), where  is the admissible space for the state variables.
To obtain the Euler-Lagrange equations associated with a minimizer of equation (9), we consider the variations of (1)magnetization where δ is a small number which controls the magnitude of the variations, M K and u k are the admissible variations of the field variables M K and x k , respectively. By applying variations (10) and (11) to (5) respectively, the Euler-Lagrange equations and associated boundary conditions are given by where Σ MW is the first Piola-Maxwell stress, Σ RE is the first Piola-remanence stress, º - is the jump of the quantity l evaluated at either side of the discontinuity surface, T rR is the reference traction on the boundary(force per unit reference area), N R is the outward unit vector normal to the surface in the reference configuration, and - V 3 is the free space around the deformed body. The detailed derivation can be found in appendices A and B. Σ MW and Σ RE can be written as m m m a S =- · is the Kronecker delta which is non-zero only if k=l. The same form of the Maxwell stresses can be found in Ball (1976)ʼs works. Using the relationship between the Cauchy stress and the first Piola-Kirchhoff stress, the Euler-Lagrange equations (12) can be further expressed in its Euler form as where n l is the outward unit vector normal to the surface in the current configuration.

Transverse isotropic nonlinear materials
Now we study a transverse isotropic hybrid MAE plate subjected to a uniform magnetic field(h e ) parallel to X 1 direction, in which the residual flux density B r is along the thickness direction(or X 1 direction), as presented in figure 3. The domain occupied by the hybrid MAE plate, in the reference configuration, is given by where L is the thickness of the hybrid MAE plate and R is the radius of the hybrid MAE plate. Under an applied uniform magnetic field, L and R become l and r. Here, we presume that the magnetic field changes slowly with time in order to separate the nonlinear dynamics from the phenomenon. So, the solution can be regarded as quasi-static. In addition, since the transverse isotropic film is thin (L is much smaller than R), we assume its deformation is uniform and the deformation gradient is given by is the stretch in the X i direction and λ 2 =λ 3 . We assume that the material is magnetostatically linear. The constitutive relation of the films is given by where m and h are the magnetization and the magnetic field in the X 1 direction, respectively. The stored or internal energy density function of the material is given by the following form ( where M is the counterpart of m defined in the reference configuration. We use the following Neo-Hookean hyperelasticity model to describe the mechanical property of the material (Treloar 1949, Deng et al 2014): Figure 3. The schematic of a hybrid MAE plate whose residual flux density B r is along the X 1 -direction. The film radius and thickness are prescribed as R=20 mm, L=5 mm.
where c 1 is the shear modulus, and k is the bulk modulus. For the 1D magneto-mechanical system, the total free energy equation (5) can be expressed as where the last term is a constraint to the incompressible materials , q is the Lagrangian multiplier which can be interpreted as the hydrostatic pressure. We assume that the relative permeability of the free space around the deformed body is equal to 1 and the perturbation field outside V vanishes because of the thin-film geometry (Coey 2011, Alameh et al 2015. The magnetic field and the perturbation field in the current configuration are given by Minimizing equation (22), carrying out the apropriate variational calculations with respect to λ 1 , λ 2 , and M yields the governing equations (the detailed derivations can be found in appendix C): To explore the response of a hybrid MAE to a harmonic magnetic field, as shown in figure 4(a), we assume that the frequency of the magnetic field is 1 Hz. As an example, the material properties used as in the calculation are listed below:c 1 =93 kPa,B r =1.2 mT, μ r =1.2. The amplitude of the applied magnetic field is 400Oe. By numerically solving equation (26), the strain is presented in figure 4(b). The result indicate that the deformation is non-sinusoidal under the sinusoidal magnetic field. Compared with the pure soft MAEs or H-MAEs (shown by figure 1(b)), we conclude that the hybrid MAEs may exhibit more diverse responses to a harmonic magnetic field, which provides us more freedom in designing magnetic actuating devices.

Experimental observations of the behavior of hybrid MAEs
To verify our theoretical predictions for the performance of hybrid MAEs, in this section, we design the following experiments: (1)the sample is loaded by the magnetic field with different amplitudes (section 3.1), 2 samples with different magnetization states are loaded by the same magnetic field (section 3.2).
In the experiment, the Ecoflex-0010 from Smooth-on Inc. was used as a matrix material. Iron powders with the mean diameter of m0.5 m were bought from NANGONG XINDUN ALLOY WELDING MATERIAL SPRAYING CO., China and used as the magnetically soft component. Neodymium-iron-boron (NdFeB) alloy powders with the mean diameter of m38 m were bought from GUANZHOU XINNUODE TRANSMISSION PART CO., China and used as the magnetically hard component. The overall concentration of the filler in all samples was 50wt% with the ratio between the magnetically soft and hard components being 1:1. Hybrid MAEs were prepared by mixing neodymium-iron-boron(NdFeB) and iron micro-particles with silicone elastomer (Ecoflex 00-10) at the aforementioned weight fraction. The mixture was poured into a mold to obtain desired geometry and then cured at 80°C for 10 min, in which the thickness(L) of the sample is 5 mm and the radius(R) is 20 mm. The cured material was uniformly magnetized along the thickness direction by applying an impulse magnetic field to produce B r in materials. The magnitude of B r can be tuned by changing the peak value of the pulsed magnetic field. The relationship between B r and the peak value of the pulsed magnetic field is presented in table 1. As shown in figure 5, the sample was loaded by using a customized solenoid that generated the AC magnetic field(h e ) and deformations of these materials were detected by using a Laser Scanning Vibrometer (Polytec OFV-5000, Germany). The experiments were carried out on the vibration isolation workstation(M-VIS3048-IG2-125A, newport, American). Throughout the whole work, the frequency of the magnetic field is fixed to 1 Hz.

The responses of the hybrid MAE thin film under different magnetic fields
Here, the sample is loaded by the magnetic field with different amplitudes (89Oe, 300Oe, and 600Oe). induced by σ MW is 2 Hz, double of that of the magnetic field, and the frequency of the deformation induced by σ RE is 1 Hz. To identify the contribution from σ MW and σ RE , in figures 6(a)-(i), we decompose the total displacement ΔL measured in the experiment into two parts: ΔL RE (1 Hz) and ΔL MW (2 Hz). As described in figures 6(a)-(ii), the ratio of the peak-peak value(PPV) of ΔL MW to the PPV of ΔL RE is equal to 7%, when the amplitude(h e A ) of h e being 89Oe. When h e A increases to 300Oe, the trough of the deformation flattened as present in figures 6(b)-(i) and the ratio of the PPV of ΔL MW to the PPV of ΔL RE value becomes 44% as shown in figures 6(b)-(ii). With h e A further increased to 600Oe, as shown in figures 6(c)-(i), the trough, which was originally negative (figures 6(a)-(i)), became positive. The ratio of the PPV of ΔL MW to the PPV of ΔL RE is equal to 88% as presented in figures 6(c)-(ii).
As we know, the consequence of the magnetic Maxwell stress is always stretching (the value of deformation is always positive no matter which direction the magnetic field is pointing at (upward or downward)). Meanwhile the body will be either stretched or compressed under a remanence stress depending on the direction of the applied magnetic field. As presented in figure 7, when the applied magnetic field is small, the Maxwell stress is smaller than the remanence stress in magnitude. However, with the increase of the magnitude of the magnetic field, the Maxwell stress would eventually exceed the remanence stress. Therefore, with the increase of the external magnetic field, the 2 Hz deformation increase faster than the 1 Hz deformation. It is possible to manipulate the waveform of ΔL by adjusting h e A . Note that the deformation lags behind the external magnetic field due to the viscoelastic effect.

Hybrid MAE's responses to different residual magnetic flux densities
From equation (26), the remanence stress σ RE is closely correlated to the residual magnetic flux density(B r ).
Here, samples with different B r are prepared and results are presented in figures 8(a)-(d). As presented in figure 8(a), when B r equals to 0 mT, since there is only σ MW in the sample, the ratio of the PPV of ΔL RE to the PPV of ΔL MW equals to 0 and the frequency of the deformation is 2 Hz, which is twice the frequency of the external magnetic field. As shown in figure 8(b), when the absolute value of B r ( B r ) is equal to 0.362 mT, σ RE appears. The ratio of the PPV of ΔL RE to the PPV of ΔL MW is equal to 21% and the waveforms begin to appear two positive peaks with different amplitudes within a period. With the increase of B r , the effect of σ RE is enhanced. When B r is equal to 2.194 mT, the ratio of the PPV of ΔL RE to the PPV of ΔL MW is equal to 113% and the positive peaks with different amplitudes becomes more obvious as shown in figure 8(c). Finally, when B r increases to 11.486 mT, the ratio of the PPV of ΔL RE to the PPV of ΔL MW is equal to 755% and the waveform becomes almost sinusoidal with a frequency of 1 Hz, as presented in figure 8(d). This conclusion also indicates that a good way, to manipulate the waveform of ΔL, is to adjust B r .

Conclusions
In this work, we aim to add more degrees of freedom in designing magnetic active materials by rendering them with the ability of responding or deforming diversely in time. Particularly, a hybrid MAE consisting of rubber matrix embedded with both iron particles and neodymium-iron-boron(NdFeB) particles is proposed. In order to better predict its response in external magnetic fields, we have developed an energy formulation for the magnetomechanical behaviors of general magnetoactive bodies based on the principle of minimum free energy. The theoretical and experimental studies both indicate that the waveforms for the responses of hybrid MAE samples largely depend on the magnetization of the sample and the amplitude of the applied magnetic field. We find that the total deformation measured in the experiment can be decomposed into two parts: the mechanical responses with the same frequency as the external magnetic field(ΔL 1 ) and the mechanical responses with a frequency twice of that for the external magnetic field(ΔL 2 ). With the increase of the external magnetic field, ΔL 2 increase faster than ΔL 1 . In addition, ΔL 1 is tuned by variation of the magnetization of the sample. These conclusions may provide us effective ways of further diversifying the responses of magnetoactive materials. It is undoubtedly useful to have a clear exploration of the magneto-elastic coupling. Besides, since motions in the real world are always complex in time and space, this work provides a new idea of diversifying the temporal response of MAEs to the application of harmonic magnetic fields (harmonic in time). The hybrid MAEs may serve as a complement to the recently proposed 3D-printed hard MAEs and is expected to be used in the soft robots to imitate more complex real motions, without manipulating the magnetic stimulation too much.

A.1. Variations of magnetization in the magnetic energy
We first consider the variation of the magnetization. Physically, the variation of the magnetization M K should result in the change of h H , k K self and B K because of the Maxwell equation (4). We assume that the variation M K results in It is noted that the perturbation field (H self ) far away from the solid must vanish, which can be described as (10)  A.10

By equation
The value M L is equal to 0 outside the deformed body. According to the Maxwell equations, the third term is equal to 0.

U[ ]ˆ( )
Since M K is arbitrary, the first Euler-Lagrange equation can be obtained as

Appendix B. Variations of deformation
In association with the variation (11), there will also be changes of F kK , -F Kk We noted that the reference quantities M H , K K e , and B K do not change with the variation u r . However, the current quantities m k ,h k and b k are affected by the variation of u k because of equation (2).

B.1. Variations of deformation in internal energy
Using the divergence theorem, we first rewrite the variation of deformation in the internal energy into the following form Eliminating q in (C.3) 2,3 and substituting (24) and (20)