5.13 Prescribed Active Contraction

Prescribed active contraction models allow the user to directly specify the time history of the active contractile stress.

Uniaxial Active Contraction

For this model, the active stress is acting along a prescribed direction given by the unit vector \(\mathbf{a}_{r}\) in the reference configuration. The \(2^{\text{nd}}\) Piola-Kirchhoff stress is

\[ \begin{equation} \mathbf{S}^{a}=T^{a}\mathbf{a}_{r}\otimes\mathbf{a}_{r}\,,\label{eq519} \end{equation} \]

and the Cauchy stress is

\[ \begin{equation} \boldsymbol{\sigma}^{a}=J^{-1}T^{a}\mathbf{a}\otimes\mathbf{a}\,,\label{eq520} \end{equation} \]

where \(T^{a}\) is the prescribed contractile stress and \(\mathbf{a}=\mathbf{F}\cdot\mathbf{a}_{r}\). Since \(\mathbf{S}^{a}\) is not a function of deformation, the material and spatial tangents are both zero.

Transversely Isotropic Active Contraction

In this case, the active stress is isotropic in a plane transverse to the direction \(\mathbf{a}_{r}\),

\[ \begin{equation} \mathbf{S}^{a}=T^{a}\left(\mathbf{I}-\mathbf{a}_{r}\otimes\mathbf{a}_{r}\right)\,,\label{eq521} \end{equation} \]

and the corresponding Cauchy stress is

\[ \begin{equation} \boldsymbol{\sigma}^{a}=J^{-1}T^{a}\left(\mathbf{B}-\mathbf{a}\otimes\mathbf{a}\right)\,,\label{eq522} \end{equation} \]

where \(\mathbf{B}=\mathbf{F}\cdot\mathbf{F}^{T}\) is the left Cauchy-Green tensor. The material and spatial tangents are zero.

Isotropic Active Contraction

An isotropic active contractile stress is given by

\[ \begin{equation} \mathbf{S}^{a}=T^{a}\mathbf{I}\label{eq523} \end{equation} \]

and the corresponding Cauchy stress is

\[ \begin{equation} \boldsymbol{\sigma}^{a}=J^{-1}T^{a}\mathbf{B}\,.\label{eq524} \end{equation} \]

The material and spatial tangents are zero.