Meso‐scale thermo‐magneto‐mechanical constitutive model for magneto‐active elastomers

Magneto‐active elastomers (MAE) are one of many emerging smart materials. They have applications in mechanical, civil and biomedical engineering as actuators, such as grippers, and dampers, such as tunable vibration absorbers. MAE consist of a soft polymeric matrix filled with micron‐sized magnetizable particles. Set in a magnetic field, the particles displace, stiffening the entire composite by up to three orders of magnitude.

For modeling absorbers and isolators, introducing temperature is essential, due to the often greatly visco-elastic nature of elastomers.Recently, researchers have been investigating thermal properties of MAEs.Jäger et al. [15] have examined and analytically modeled the dependence of thermal conductivity on inclusion restructuring.Towards thermo-magnetomechanical modeling, Mehnert et al. [16] demonstrate a macro-scale formulation on two academic examples.Chen et al. [17] have presented a model of heat coupling in a magnetic medium, and Wen et al. [18] have demonstrated macro-scale temperature dependent magnetic properties.Of interest in this contribution is providing a mesoscale model for MAE vibration absorbers and isolators.

Kinematics
We designate a solid body by  0 in the reference configuration at time  = 0 (seconds), comprising material points .The body occupies the current configuration  at any time  with spatial points .The deformation map   ∶  →  describes the motion between these configurations.The deformation gradient is with determinant and isochoric part For the thermal and viscous parts of the problem, three intermediate deformations are useful: the elastic deformation,   , the viscous deformation,   , and the thermal deformation,   , which form a multiplicative decomposition of the total deformation As will be discussed in Section 3, the magnetization of a material is decoupled from viscous and thermal effects and is only weakly coupled to total deformation.Consequently, it is not necessary to be accounted for in the deformation decomposition.
In the following model, the thermo-mechanical coupling is strictly volumetric, so that it reads in which Θ is the difference between the absolute temperature  and some reference temperature  0 , Θ =  −  0 .In the reference configuration, the right Cauchy-Green strain is We define the magnetic field strength as the negative gradient of a scalar potential with its current configuration counterpart, An alternative choice of magnetic potential is the vector potential, whose curl field constitutes the magnetic induction, Discussion of trade-offs between the formulations takes place in Section 4. For both formulations, the magnetic induction fields are related by the deformation cofactor, or, using Nanson's Formula, The heat flux in the current configuration is  and is calculated from the gradient of the temperature with Fourier's Law with thermal conductivity .

Balance equations
Cauchy's Law governs mechanical equilibrium, where  is the Cauchy stress tensor,  is mass density in the current configuration,  is the body force, and ü is the acceleration.Gauss's Law and Ampere's Law govern magnetism, For magnetostatics, the current density  is zero.
From the second law of thermodynamics, see literature [19], heat conduction reads in which  is the specific heat capacity, θ is the temperature velocity, and  is a heat source.The external and internal heat are respectively defined as where  is the set of internal variables and Ψ is the Helmholtz free energy function, described in the next section.

Multiphysical coupling
By definition, thermo-magneto-mechanical modeling implies four forms of coupling: thermo-mechanical, magnetomechanical, thermo-magnetic, and thermo-magneto-mechanical. From Section 2, the kinematic thermo-mechanical coupling is a volumetric relation, implying that thermal loading does not generate shear deformations.However, shear visco-elastic effects dissipate energy, which, formulated above in the internal heat, produce a temperature increase.
Magneto-mechanical coupling is the uni-directional, or passive, magnetostriction: an external magnetic field induces deformation in magnetizable material, but deformation does not generate a magnetic field (only scales or rotates).
At the meso-scale, thermo-magnetic, and consequently thermo-magneto-mechanical coupling does not occur in normal environmental conditions.
The total Helmholtz free energy function is whose constituents are explained in the subsequent sections.

Hyperelasticity
The Neo-Hooke model, with shear parameter , describes the purely elastic part of the mechanical energy, where   is the first invariant of the isochoric part of the right Cauchy-Green strain.

Visco-hyperelasticity
To formulate the visco-hyperelastic behavior of the rubber matrix, we refer the work of Platen et al. [20], which defines the kinematic quantities deformation rate,   , Mandel stress,   , as well as the evolution law, in the intermediate viscous configuration, where dev  is the deviatoric stress component,   is the viscous retardation time, and Σ is a factor to rectify units with the strain rate.The advantage of the Platen formulation is its mathematical consistency.Existing visco-hyperelastic models make various assumptions about the underlying eigen-space of the viscous strain field and require special treatment for the symmetric strain tensors, see for example Dal and Kaliske [21].Finally, the Neo-Hooke hyperelastic model describes the elastic part of the viscous branches For both the elastic and inelastic branches, other hyperelastic functions are available to suit a polymer MAE matrix, such as the Yeoh model.

Magnetic model
Magnetic energy density, formulated in the magnetic field and for linear magnetization with permeability , reads Other magnetization relations, such as Ising or Langevin, could also be used.The negated energy function, when coupled within energy minimization problems, yields a max-min or saddle-point problem.

Thermal model
As stated before, thermal energy is related only to volumetric, elastic deformations.We use one of multiple common hyperelastic volumetric functions, where  is the bulk modulus.

Finite element method
For the numerical solution of the state equations, we select the Finite Element Method (FEM).Instead of solving the equations exactly at each point, we solve them on average over the entire considered domain.For such a formulation, we use the Galerkin method and multiply each equation with a test function -which can take on the meaning of the energy density conjugate -and then utilize Gauss's Theorem to reduce the order of the differential terms.The weak forms of Cauchy's Law, Gauss's Law, and heat conduction are These terms constitute the system residual.For solution via Newton-Raphson iteration, the linearized terms are also necessary.For an overview on deriving stress and stress-like quantities, as well as material tangents, see for example [19].The simulation model is descretized with trilinear hexahedron elements.To account for (near)-incompressibility in the elastomer matrix, we assume the pressure to be constant across the element domain and utilitize the Q1P0 formulation.Time integration takes place with the implicit Euler scheme.
The resulting nonlinear system of equations is unsymmetric in the mechanical sub-problem, due to viscous contributions, and in the thermo-mechanical problem, due to coupling.The scalar magnetic problem is a maximization problem, which is difficult or impossible to solve monolithically when coupled with a minimization problem.Consequently, we apply a staggered solution approach between the thermo-mechanical sub-problem and magnetic sub-problem.In this approach, at each load step, a solution is calculated for the magnetic partition and then for the thermo-mechanical partition.The resulting residual and energy norms are stored.Repeating this solution constitutes the staggered approach, in which the stored norms are used for calculating relative norms.When a staggered loop converges with sufficiently few Newton iterations, for example two magnetic and five thermo-mechanical, the load step is considered converged.

Example: Viscous deformation
To start, we model two inclusions in an elastomer block under magnetic loading and observe the viscous response.Table 1 displays the material parameters, and Table 2 the structural parameters.Figure 1 shows the structure of the mesh.The loading is a longitudinal magnetic field of strength 0.01 A/µm, which is applied on a linear ramp and then held.Figure 2 shows the deformation at the top of the sample, which continues to increase in magnitude once the field is constant, showcasing the viscous relaxation.F I G U R E 3 Magnetic field and temperature at the structure center (matrix).

Example: Viscous heat generation
In this next example we look at heat generation.First, a magnetic field is again applied along the direction of the inclusions, then an oscillating transverse displacement at the top of the structure.Figure 3 shows the temperature increase as the shear oscillations proceed.The oscillations in the magnetic field result from the application of the magnetic potential on the structure surface.The bumps in the temperature plot correspond to peaks in the shear stress.

CONCLUSIONS AND OUTLOOK
In this contribution, we have introduced a thermo-magneto-mechanical model and demonstrated it with an MAE at the mesoscale.Features of the model include three-field coupling, mathematical consistency of the visco-elastic formulation, and satisfactorily treating the problem of locking due to incompressibility.The example shown in Section 4.3 captures the three-field coupling, showing how temperature increase tracks with deformation.Many possibilities exist for further investigations.First, material parameter values could be tuned to reflect properties of particular MAE constituents.Further, larger structures could be discretized and simulated, to accurately capture the macroscale behavior of MAE.Also, to facilitate thermal model of MAE at the macroscale, a novel three-field homogenization scheme could be developed.

F I G U R E 2
Magnetic field and displacement at the structure top (matrix).
Material parameters.Structure parameters.
TA B L E 1 F I G U R E 1 Structure of the mesh.