# Tuning the Cell and Biological Tissue Environment through Magneto-Active Materials
## 5 Theoretical and Computational Mechanical Frameworks for Magneto-Active Polymers
### Constitutive Modelling of Magneto-Active Hydrogels  
When modelling MAPs using a hydrogel as polymeric matrix (magneto-active hydrogels, MAHs), to the already complex problem presented in the previous section, we need to incorporate a solvent diffusing within the system formed by the polymeric network and magnetic particles. This feature implies that, along with the mechanical and magnetic problems, we will have a solvent diffusion process occurring in parallel. This diffusion will directly interplay with the mechanical behaviour of the MAH by incorporating a volumetric response due to the incompressibility condition of the solvent. Therefore, the volumetric mechanical stress of the MAH will be determined by the amount of solvent within the polymeric network that, in turn, will temporarily diffuse influencing the mechanical balance. Overall, the MAH system combines multiple difficulties associated with nonlinearities and coupled responses at different time scales. The short-time response of these materials presents strain rate dependences that determine the nonlinear instantaneous mechanical response [174; 175]. After this initial response, the polymeric network experiences a viscoelastic relaxation process resulting in a continuous decrease in stiffness until complete relaxation [174; 176; 177]. Then, as a long-time response, the mechanical deformations within the MAH lead to heterogeneous distribution of solvent concentration that evolve via solvent diffusion process [178; 179]. In addition, note that these diffusion-mechanical processes are also constantly affected by the interaction between magnetic particles with the external magnetic field. An overall description of the mechanics, magnetics and diffusion process interplay is shown in Figure 4 (adapted from [157]). Here, an MAH is considered  
Figure 3: Scheme of microstructural-based approach, accounting for both magnetic particles and polymer matrix components, and how these models can be scaled up to the micromechanical level. Figure adapted from [PERSON] and [PERSON] [154]. with a given initial swelling ration. When this is immersed in a solvent, solvent diffusion occurs until reaching a stable swelling ratio. Then, a magnetic field is applied, leading to mechanical deformation due to magnetostriction response. Such a mechanical deformation results into a heterogeneous distribution of the solvent concentration which, in turn, leads to a continuous solvent diffusion until reaching a new stable state. Therefore, the complete modelling of MAHs needs to account, in a coupled manner, for magnetics, solvent diffusion, and mechanics. A seminal work by [PERSON] and co-authors [158] proposed a nonequilibrium theory for coupled diffusion and finite deformations. Further extensions of this theory are due to [PERSON] et al. [180], [PERSON] et al. [181] and [PERSON] et al. [182]. More recently, viscoelasticity has been included in these modelling frameworks [183]. In recent years, diffusion-mechanical theories have been coupled to the magnetostatics problem. In this regard, the governing equations presented in Section 5.2 complement the diffusion equation:  
\[\frac{\partial c_{\mathrm{s}}}{\partial t}+\
abla\cdot\mathbf{J^{s}}=r^{s} \tag{9}\]  
where \(c_{\mathrm{s}}\) is the solvent concentration, \(\mathbf{J^{s}}\) is the nominal flux of the solvent, and \(r^{s}\) is a source term for the number of solvent molecules injected into unit reference volume per unit time. Now defining the total energy potential, depending on the solvent concentration as well, \(\psi(\mathbf{F},\mathbf{M},c_{\mathrm{s}})\), from thermodynamic principles, this solvent concentration can be related to a chemical potential (\(\mu_{\mathrm{s}}\)) as:  
\[\mu_{\mathrm{s}}=\frac{\partial\psi(\mathbf{F},\mathbf{M},c_{\mathrm{s}})}{ \partial c_{\mathrm{s}}} \tag{10}\]  
Following these ideas, [PERSON] and co-authors [184] developed a constitutive model coupling mechanical, magnetic, and chemical fields.