# Tuning the Cell and Biological Tissue Environment through Magneto-Active Materials
## 5 Theoretical and Computational Mechanical Frameworks for Magneto-Active Polymers
### Constitutive Modelling of Magnetorheological Elastomers
#### 5.2.2 Micro-Mechanical-Based Models  
Compared to continuum approaches, the micro-mechanical-based models overcome an important limitation: the consideration of detailed microstructural information, such as the magnetic particles distribution within the polymeric matrix. These micro-mechanical models allow to account for the specific arrangement of the magnetic particles and, then, the magnetic interactions between particles, and with the surrounding matrix. This is an important consideration as the relative position between particles determines the nature of the internal magnetic forces, i.e., attraction or repulsion. The most common assumptions for these models are a purely isotropic distribution of the particles and an anisotropic distribution, forming perfectly aligned chain-like structures [154]. However, experimental evidence show that more complex arrangements are likely to be formed within the matrix [168].  
Regarding the modelling fundamentals of the micro-mechanical approaches, these explicitly consider the magnetic interactions between the particles that are embedded into the polymeric matrix. These internal magnetic interactions are the primary driver of geometrical changes in the MAP and mechanical properties variations under magnetic stimuli. A pioneering work is due to [PERSON] et al. [169], where the authors developed a one-dimensional lattice model. More recently, [PERSON] and co-authors [170] extended the latter model to predict the mechanical response of MAPs to a magnetic stimulus by means of variations in stiffness, contraction and expansion mechanisms. More recently, [PERSON] and [PERSON] [154] extended these approaches to finite deformations adding viscoelastic responses and considering a wide range of magnetic particle distributions. The common fundamental behind these models is the consideration of a magnetic contribution to the energy potential describing dipole-dipole interactions as:  
\[\psi_{magnetic}(\textbf{F},\textbf{M})=-\frac{1}{\text{V}_{\text{o}}}\frac{ \text{\mu}_{\text{o}}}{4\pi}\sum_{ij}\left[\frac{3\left(\textbf{m}_{\text{di} }\cdot\textbf{r}_{ij}\right)\left(\textbf{m}_{\text{di}}\cdot\textbf{r}_{ij} \right)}{\textbf{r}_{ij}{}^{5}}-\frac{\left  
between i-th and j-th particles. Note that the dependence with \(\mathbf{F}\) is implicitly accounted in \(\mathbf{r}_{ij}\), as these distances changes with the material deformation.  
In Figure 3 (adapted from [154]), we present a scheme with the main features of a micromechanical-based model and how these approaches can be scaled up to describe the macroscopic response of MAPs, while keeping relevant information from the microstructure.  
Alternative to these models, other micro-mechanical-based approaches propose variationally consistent phase-field theories [154], microstructurally guided explicit continuum models [171; 172], or FE\({}^{2}\) computational frameworks [173]. These approaches allow for considering further important features, such as heterogeneous magnetic fields within the MAP, or the explicit interaction between the particles and the matrix. However, they usually introduce much higher computational cost.