This tutorial depends on deal.II's step-44. Unless otherwise specified, all phyisical quantities have SI units.

Mathematical formulation

Consider a muscular structure (e.g. a block of muscle tissue) which, at time $t \geq 0$, occupies a region $\mathcal{B}_t \subset \mathbb{R}^3$. Given an activation profile $a = a(t)$ and a boundary strain $\varepsilon = \varepsilon(t)$, we wish to compute a displacement field $\mathbf{U}(\mathbf{X},t)$, a pressure field $p(\mathbf{X},t)$, and a dilation field $D(\mathbf{X},t)$ such that

$$\begin{gathered} \rho_0 \mathbf{U}_{tt} = \mathbf{Div} \ \mathbf{P(a,\mathbf{U},\mathbf{U}_t)} + \mathbf{f}_0 \quad \text{in } \mathcal{B}_0 \times (0,\infty), \\ J(U) - D = 0 \quad \text{in } \mathcal{B}_0 \times (0,\infty), \\ p - \Psi_{vol}'(D) = 0 \quad \text{in } \mathcal{B}_0 \times (0,\infty), \\ \mathbf{U} = \varepsilon(\cdot) L_0 \hat{\mathbf{i}} \quad \text{on } \Gamma_{0,D} \times (0,\infty), \\ \mathbf{P(a,\mathbf{U},\mathbf{U}_t)} \mathbf{N} = \mathbf{0} \quad \text{on } \Gamma_{0,N} \times (0,\infty), \\ \mathbf{U} = \mathbf{0}, \quad p = 0, \quad D = 1 \quad \text{at } t=0. \end{gathered}$$

Here, $\rho_0$ is the initial tissue density, $\mathbf{f}_0$ is a volumetric force per unit volume, $\Gamma_{0,D}$ represents boundary faces on which we impose strains, and $\Gamma_{0,N}$ is the collection of traction-free boundary faces. In addition, $\Psi_{vol}$ is the volumetric energy of the system given by

$$\Psi_{vol}(D) = \dfrac{\kappa}{4}\left( D^2 - 2 \ln D - 1 \right),$$

where $\kappa$ is the bulk modulus of muscle tissue.

Description of the stress tensor

The formulation described above is based on a volumetric-isochoric decomposition of the problem. In particular, the strain energy of the system can be thought as $\Psi = \Psi_{vol} + \Psi_{iso}$, although no explicit expression exists for $\Psi_{iso}$. The first Piola-Kirchhoff tensor is best described using the second