Modelling the passive and active response of skeletal muscles within the adapted Voigt representation framework

Abstract

We present a constitutive model for the passive and active response of skeletal muscles. At variance with more classical approaches, the model is developed exploiting adapted Voigt representations of strain and stress tensors within the context of nonlinear Cauchy elasticity. This framework allows us to identify non-trivial stress-strain relations in a rather direct way from experimental data, enhancing the mechanical interpretability of the material functions that describe the tissue response and obtaining additional insight on the distinct role of the contractile fibres and of the surrounding extracellular matrix. We propose a two-material model, with an additive splitting of the stress contributions, in which only one component depends on an activation parameter. The constitutive model for the passive behaviour satisfactorily predicts the nonlinear stress response to elongation at different relative orientations with respect to the fibre direction and highlights the dominant role of the extracellular matrix. The activation model, essentially determined by the mechanics of the contractile fibres, captures well the isometric stress response through the prescription of an elasto-plastic evolution of the along-fibre active strain.

Figures (7)

• Figure 1: Experimental data sets used for the construction of the model: uniaxial tensile tests performed at quasi-static rate on (a) passive takaza2013anisotropic and (b) tetanised hawkins1994comprehensive samples of skeletal muscle tissue. The reported data collect the component of the Cauchy stress tensor $\tau_{xx}^\alpha$ as a function of the Hencky stretch ratio $\lambda=\log\left(\tfrac{L}{L_0}\right)$, measured along a direction inclined at an angle $\alpha$ with respect to the fibre orientation.
• Figure 2: Schematic representation of key features of the passive response, emerging from the experimental data set takaza2013anisotropic, obtained in uniaxial tensile tests at different fibre alignments. The along-fibre stress increases nonlinearly with the strain (blue dots), displaying a transition between two linear regimes (blue and red lines). The measurements at inclination angles of $30^\circ,45^\circ,60^\circ$ show an analogous trend (purple, green, and orange dots) with however different stiffnesses at small strains. In the cross-fibre direction, the stress increases linearly with the strain (red dots). At large enough strains, the stiffness of the tissue appears to be the same regardless of the direction of stretching with respect to the fibres and coinciding with the cross-fibre one (red lines).
• Figure 3: Fit of the experimental data in the passive regime at small strains. The proposed stress--strain fitting function \ref{['eq:tau_xx_fit_tensile_1']}-\ref{['eq:tau_xx_fit_tensile_2']} is linear in the strain, for sufficiently small $\lambda$, with stiffness $m_l^\mathrm{m}(\alpha)$ dependent on the orientation of the fibre \ref{['eq:mu_f_alpha_tensile']}. The two material parameters $m^\mathrm{m}_0,m^\mathrm{m}_t$ are the longitudinal and transverse stiffness, respectively, and are obtained by interpolation of the corresponding data (blue and red lines, respectively). The remaining (purple, green, and orange) curves are, instead, predictions of the model.
• Figure 4: The coefficients $r^\mathrm{m}$ and $s^\mathrm{m}$, which account for the nonlinear transition between the two linear regimes in \ref{['eq:tau_xx_fit_tensile_1']}-\ref{['eq:tau_xx_fit_tensile_2']}, as functions of the fibre orientation. The values of $r^\mathrm{m}$ and $s^\mathrm{m}$ (black dots) are interpolating parameters from fitting \ref{['eq:tau_xx_fit_tensile_1']}-\ref{['eq:tau_xx_fit_tensile_2']} against the data. Solid curves represent interpolating functions of such parameters depending on $\alpha$.
• Figure 5: Fitting of the model \ref{['eq:model_tensile']} on the data by takaza2013anisotropic for the passive response of the muscle tissue. The measurements associated with stretching at fibre orientations of $0^\circ,45^\circ,90^\circ$ have been used to identify the material functions \ref{['eq:material_functions_c_passive']} and calibrate the material parameters in \ref{['eq:tau_xx_fit_tensile_2']}, while we test the predictive ability of the proposed model on the measurements at $30^\circ$ and $60^\circ$.