# 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
#### 5.2.1 Continuum-Based Models  
Some of the seminal works are due to [PERSON] and [PERSON] [156; 163]. In these works, the authors established the fundamentals to develop magneto-mechanical formulations for finite strains, solving analytically some classical boundary value problems. In this regard, they proposed a general free energy, depending on both mechanical (i.e., deformation gradient tensor \(\mathbf{F}\)) and magnetic variables. Regarding the magnetic variables, there are three main quantities that, expressed in the spatial configuration, read as the magnetic field \(\mathbf{h}\), magnetisation \(\mathbf{m}\), and magnetic induction \(\mathbf{b}\). These variables can be alternatively expressed in the material configuration, respectively, as:  
\[\begin{split}\mathbf{H}&=\mathbf{h}\mathbf{F}\\ \mathbf{M}&=\mathbf{m}\mathbf{F}\\ \mathbf{B}&=\text{det}(\mathbf{F})\mathbf{b}\mathbf{F }^{-\mathbf{T}}\end{split} \tag{4}\]  
Moreover, these variables are related in the bulk as:  
\[\mathbf{B}=\text{det}(\mathbf{F})\mathbf{\mu}_{\text{o}}\mathbf{C}^{-1}( \mathbf{H}+\mathbf{M}) \tag{5}\]  
where \(\mathbf{\mu}_{\text{o}}\) is the vacuum magnetic permeability.  
Similar to the calculation of the stress tensors from thermodynamic principles in hyperelasticity, the mechanical and magnetic constitutive equations can be derived from the total energy potential as, for example:  
\[\begin{split}\mathbf{P}&=\frac{\partial\mathbf{ \psi}(\mathbf{F},\mathbf{M})}{\partial\mathbf{F}}\\ \mathbf{B}&=\frac{\partial\mathbf{\psi}(\mathbf{F},\mathbf{M})}{\partial\mathbf{M}}\end{split} \tag{6}\]where **P** is the first Piola-Kirchhoff stress tensor. It is worth mentioning that the magnetic problem allows for taking any magnetic variable as the independent one, which choice will be determined by the specific application.  
Therefore, these constitutive equations, along with the corresponding governing equations, provide a general framework for magneto-active elastomers. The main governing equations are the mechanical balance and magnetostatics [PERSON]'s equations:  
\[\
abla\cdot\textbf{P}+\text{f}_{\text{o}}=0 \tag{7}\] \[\
abla\cdot\textbf{B}=0\] \[\
abla\textbf{xH}=0\]  
where \(\
abla\) is the material gradient and f\({}_{\text{o}}\) is the external mechanical body forces.  
From these bases, other authors have proposed significant advances including other features. In this regard, [PERSON] and co-workers [164] extended this framework to account for the underlying transverse isotropy of certain MAPs. Moreover, one limitation of these previous models was that the polymeric matrix is, by nature, a viscoelastic material. Therefore, the MAP composite will show time-dependent responses, such as strain rate dependence or relaxation mechanisms. These issues were addressed by [PERSON] et al. [165] and [PERSON] [166], who included viscoelastic behaviours. Very recently, [PERSON] et al. [124] proposed a constitutive model to describe the response of soft MAPs reinforced with hard magnetic particles, and [PERSON] [167] extended this framework to incorporate time-dependent responses.