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