Stability of active muscle tissue

Abstract

The notion of material stability is examined in the context of active muscle tissue modeling, where the nonlinear constitutive law is dependent both on the physiologically driven muscle contraction and finite mechanical deformation. First, the governing equations and constitutive laws for a general active-elastic material with a single preferred direction are linearized about a homogeneous underlying configuration with respect to both the active contraction and deformation gradient. In order to obtain mathematical restrictions analogous to those found in elastic materials, stability conditions are derived based on the propagation of homogeneous plane waves with real wave speeds, and the generalized acoustic tensor is obtained. Focusing on 2D motions, and considering a simplified, decoupled transversely isotropic energy function, the restriction on the active acoustic tensor is recast in terms of a generally applicable constitutive law, with specific attention paid to the fiber contribution. The implication of the material stability conditions on material parameters, active contraction, and elastic stretch is investigated for prototype material models of muscle tissue.

Appendices

Appendix 1: Derivatives related to invariants

We provide the derivatives of the invariants introduced in this study related to the deformation gradient and the active strain tensor. The second derivatives with respect to the deformation tensor are, in component form, provided as

$$\begin{aligned} \frac{\partial ^2 I_1}{\partial F_{i \alpha } \partial F_{j \beta }}&= 2 \delta _{ij} \delta _{\alpha \beta }, \end{aligned}$$

(8.1)

$$\begin{aligned} \frac{\partial ^2 I_2}{\partial F_{i \alpha } \partial F_{j \beta }}&= 2 \big (2 F_{j \beta } F_{i \alpha } - F_{j \alpha } F_{i \beta } + I_1 \delta _{ij} \delta _{\alpha \beta } - B_{ij} \delta _{\alpha \beta } - C_{\alpha \beta } \delta _{ij}\big ), \end{aligned}$$

(8.2)

$$\begin{aligned} \frac{\partial ^2 I_4}{\partial F_{i \alpha } \partial F_{j \beta }}&= 2 \delta _{ij} A_{\alpha } A_{\beta }, \end{aligned}$$

(8.3)

$$\begin{aligned} \frac{\partial ^2 I_5}{\partial F_{i \alpha } \partial F_{j \beta }}&= 2\big (\delta _{ij} A_{\alpha } C_{\beta \eta } A_{\eta } + A_{\alpha } F_{i \beta } F_{j \eta } A_{\eta } + A_{\alpha } B_{ij} A_{\beta }+ \delta _{\alpha \beta } F_{i \eta } A_{\eta } F_{j \gamma } A_{\gamma } \nonumber \\&\quad +\, A_{\beta } F_{j \alpha } F_{i \eta } A_{\eta }+ \delta _{ij} A_{\beta } C_{\alpha \gamma } A_{\gamma }\big ), \end{aligned}$$

(8.4)

$$\begin{aligned} \frac{\partial ^2 I_{4_{\mathrm{e}}}}{\partial F_{i \alpha } \partial F_{j \beta }}&= 2 I_{4_{\mathrm{a}}}^{-1} \delta _{ij} A_{\alpha } A_{\beta }, \end{aligned}$$

(8.5)

$$\begin{aligned} \frac{\partial ^2 I_{5_{\mathrm{e}}}}{\partial F_{i \alpha } \partial F_{j \beta }}&= 2 I_{4_{\mathrm{a}}}^{-1} \big ( \delta _{ij} A_{\alpha } C_{{\mathrm{a}}_{\beta \eta }}^{-1} C_{\eta \phi } A_{\phi } + A_{\alpha } F_{i \gamma } C_{{\mathrm{a}}_{\gamma \beta }}^{-1} F_{j \phi } A_{\phi } + A_{\alpha } F_{i \gamma } C_{{\mathrm{a}}_{\gamma \eta }}^{-1} F_{j \eta } A_{\beta } + C_{{\mathrm{a}}_{\alpha \beta }}^{-1} F_{j \eta } A_{\eta } F_{i \phi } A_{\phi } \nonumber \\&\qquad \quad +\, C_{{\mathrm{a}}_{\alpha \gamma }}^{-1} F_{j \gamma } A_{\beta } F_{i \phi } A_{\phi } + \delta _{ij} C_{{\mathrm{a}}_{\alpha \gamma }}^{-1} C_{\gamma \eta } A_{\eta } A_{\beta }\big ). \end{aligned}$$

(8.6)

Additionally, we provide the derivatives of the invariants related to the active stretch:

$$\begin{aligned} \frac{\partial I_{4_{\mathrm{e}}}}{\partial F_{{\mathrm{a}}_{B \beta }}}&= -2 I_4 I_{4_{\mathrm{a}}}^{-2} A_{\beta } F_{{\mathrm{a}}_{B \eta }} A_{\eta }, \end{aligned}$$

(8.7)

$$\begin{aligned} \frac{\partial I_{5_{\mathrm{e}}}}{\partial F_{{\mathrm{a}}_{B \beta }}}&= - 2\big ( I_{4_{\mathrm{a}}}^{-2} A_{\beta } F_{{\mathrm{a}}_{B \gamma }} A_{\gamma } A_{\phi } C_{\phi \eta } C_{{\mathrm{a}}_{\eta \mu }}^{-1} C_{\mu \alpha } A_{\alpha } + I_{4_{\mathrm{a}}}^{-1} C_{{\mathrm{a}}_{\beta \pi }}^{-1} C_{\pi \eta } A_{\eta } A_{\gamma } C_{\gamma \psi } F_{{\mathrm{a}}_{\psi B}}^{-1}\big ), \end{aligned}$$

(8.8)

$$\begin{aligned} \frac{\partial I_{4_{\mathrm{a}}}}{\partial F_{{\mathrm{a}}_{B \beta }}}&= 2 A_{\beta } F_{{\mathrm{a}}_{B \eta }} A_{\eta }, \end{aligned}$$

(8.9)

$$\begin{aligned} \frac{\partial I_{5_{\mathrm{a}}}}{\partial F_{{\mathrm{a}}_{B \beta }}}&= 2\big ( A_{\beta } F_{{\mathrm{a}}_{B \phi }} C_{{\mathrm{a}}_{\phi \eta }} A_{\eta } + C_{{\mathrm{a}}_{\beta \eta }} A_{\eta } F_{{\mathrm{a}}_{B \gamma }} A_{\gamma }\big ). \end{aligned}$$

(8.10)

Next, the mixed derivatives with respect to $\mathbf{F}$ and $\mathbf{F}_{\mathrm{a}}$ are written in component form

$$\begin{aligned} \frac{\partial ^2 I_{4_{\mathrm{e}}}}{\partial F_{i \alpha } \partial F_{{\mathrm{a}}_{B \beta }}}&= -4 I_{4_{\mathrm{a}}}^{-2} A_{\alpha } F_{i \gamma } A_{\gamma } A_{\beta } F_{{\mathrm{a}}_{B \eta }} A_{\eta }, \end{aligned}$$

(8.11)

$$\begin{aligned} \frac{\partial ^2 I_{5_{\mathrm{e}}}}{\partial F_{i \alpha } \partial F_{{\mathrm{a}}_{B \beta }}}&= -2 \Big [ 2 I_{4_{\mathrm{a}}}^{-2} \big ( A_{\alpha } F_{i \gamma } C_{{\mathrm{a}}_{\gamma \eta }}^{-1} C_{\eta \phi } A_{\phi } + C_{{\mathrm{a}}_{\alpha \gamma }}^{-1} C_{\gamma \eta } A_{\eta } F_{i \phi } A_{\phi }\big )A_{\beta } F_{{\mathrm{a}}_{B \mu }} A_{\mu } + I_{4_{\mathrm{a}}}^{-1} \big ( A_{\alpha } F_{i \gamma } C_{{\mathrm{a}}_{\gamma \beta }}^{-1} A_{\phi } C_{\phi \eta } F_{{\mathrm{a}}_{\eta B}} ^{-1}\nonumber \\&\qquad +\, A_{\alpha } F_{i \gamma } F_{{\mathrm{a}}_{\gamma B}}^{-1} C_{{\mathrm{a}}_{\beta \eta }}^{-1} C_{\eta \phi } A_{\phi } + F_{{\mathrm{a}}_{\alpha B}}^{-1} F_{i \phi } A_{\phi } C_{{\mathrm{a}}_{\beta \gamma }}^{-1} C_{\gamma \eta } A_{\eta } + C_{{\mathrm{a}}_{\alpha \beta }}^{-1} F_{i \phi } A_{\phi } A_{\eta } C_{\eta \gamma } F_{{\mathrm{a}}_{\gamma B}}^{-1}\big ) \Big ]. \end{aligned}$$

(8.12)

The second derivatives with respect to $\mathbf{F}_{\mathrm{a}}$ of the relevant invariants are given:

$$\begin{aligned} \frac{\partial ^2 I_{4_{\mathrm{e}}}}{\partial F_{{\mathrm{a}}_{B \beta }} \partial F_{{\mathrm{a}}_{D \gamma }}}&= 2 I_4 I_{4_{\mathrm{a}}}^{-2} A_{\beta } A_{\gamma } (4 I_{4_{\mathrm{a}}}^{-1} F_{{\mathrm{a}}_{B \phi }} A_{\phi } F_{{\mathrm{a}}_{D \eta }} A_{\eta } - \delta _{BD} ), \end{aligned}$$

(8.13)

$$\begin{aligned} \frac{\partial ^2 I_{5_{\mathrm{e}}}}{\partial F_{{\mathrm{a}}_{B \beta }} \partial F_{{\mathrm{a}}_{D \gamma }}}&= 2 \Big [ I_{4_{\mathrm{a}}}^{-2} \Big ( A_{\phi } C_{\phi \eta } C_{{\mathrm{a}}_{\eta \mu }}^{-1} C_{\mu \alpha } A_{\alpha }(4 I_{4_{\mathrm{a}}}^{-1}A_{\beta } F_{{\mathrm{a}}_{B \phi }} A_{\phi } A_{\gamma } F_{{\mathrm{a}}_{D \mu }} A_{\mu } - \delta _{BD} A_{\beta } A_{\gamma }) \nonumber \\&\qquad \qquad +\, 2C_{{\mathrm{a}}_{\beta \lambda }}^{-1} C_{\lambda \mu } A_{\mu } A_{\phi } C_{\phi \psi } F_{{\mathrm{a}}_{\psi B}}^{-1} A_{\gamma } F_{{\mathrm{a}}_{D \rho }} A_{\rho }+ 2 A_{\beta } F_{{\mathrm{a}}_{B \theta }} A_{\theta } C_{{\mathrm{a}}_{\gamma \mu }}^{-1} C_{\mu \alpha } A_{\alpha } A_{\phi } C_{\phi \eta } F_{{\mathrm{a}}_{\eta D}}^{-1} \Big ) \nonumber \\&\quad +\, I_{4_{\mathrm{a}}}^{-1} \Big ( C_{\lambda \phi } A_{\phi } A_{\mu } C_{\mu \psi } F_{{\mathrm{a}}_{\psi D}}^{-1} (C_{{\mathrm{a}}_{\beta \gamma }}^{-1} F_{{\mathrm{a}}_{\lambda B}}^{-1} + C_{{\mathrm{a}}_{\beta \lambda }}^{-1} F_{{\mathrm{a}}_{\gamma B}}^{-1}) + F_{{\mathrm{a}}_{\beta D}}^{-1} A_{\phi } C_{\phi \lambda } F_{{\mathrm{a}}_{\lambda B}}^{-1} C_{{\mathrm{a}}_{\gamma \psi }}^{-1} C_{\psi \mu } A_{\mu } \Big ) \Big ], \end{aligned}$$

(8.14)

$$\begin{aligned} \frac{\partial ^2 I_{4_{\mathrm{a}}}}{\partial F_{{\mathrm{a}}_{B \beta }} \partial F_{{\mathrm{a}}_{D \gamma }}}&= 2 \delta _{BD} A_{\beta } A_{\gamma }, \end{aligned}$$

(8.15)

$$\begin{aligned} \frac{\partial ^2 I_{5_{\mathrm{a}}}}{\partial F_{{\mathrm{a}}_{B \beta }} \partial F_{{\mathrm{a}}_{D \gamma }}}&= 2 \bigg [ A_{\beta } \delta _{BD} C_{{\mathrm{a}}_{\gamma \eta }} A_{\eta } + A_{\beta } F_{{\mathrm{a}}_{B \gamma }} F_{{\mathrm{a}}_{D \eta }} A_{\eta } + A_{\beta } F_{{\mathrm{a}}_{B \phi }} A_{\gamma } F_{{\mathrm{a}}_{D \phi }}+ \delta _{\beta \gamma } F_{{\mathrm{a}}_{B \phi }} A_{\phi } F_{{\mathrm{a}}_{D \eta }} A_{\eta } \nonumber \\&\quad +\, F_{{\mathrm{a}}_{D \beta }} F_{{\mathrm{a}}_{B \phi }} A_{\phi } A_{\gamma }+ C_{{\mathrm{a}}_{\beta \eta }} A_{\eta } \delta _{BD} A_{\gamma } \bigg ]. \end{aligned}$$

(8.16)

Appendix 2: Coefficients related to stability

The coefficients appearing in (6.7) given (6.12) are

$$\begin{aligned} H_1&= \mu \lambda _{\mathrm{a}}^{-4} \bigg [ \frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}^2} (\lambda ^2 \lambda _{\mathrm{a}}^{-2}-1)^2 + 4\frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}}(\lambda ^2 \lambda _{\mathrm{a}}^{-2}-1) + 2 \mu _{\mathrm{fib}} \bigg ], \end{aligned}$$

(9.1)

$$\begin{aligned} G&= \mu \lambda _{\mathrm{a}}^{-2}(\lambda ^2 \lambda _{\mathrm{a}}^{-2} - 1) \bigg [ \frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}} \partial I_{4_{\mathrm{a}}}} (\lambda ^2 \lambda _{\mathrm{a}}^{-2}-1) - \frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}^2} \lambda ^2 \lambda _{\mathrm{a}}^{-4} (\lambda ^2 \lambda _{\mathrm{a}}^{-2} -1) \nonumber \\&\qquad \qquad \qquad \qquad \qquad +\, 2 \frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}} - \frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}} \lambda _{\mathrm{a}}^{-2}(5\lambda ^2 \lambda _{\mathrm{a}}^{-2} -1 )- 2\mu _{\mathrm{fib}} \lambda _{\mathrm{a}}^{-2} \bigg ]- 2 \mu \mu _{\mathrm{fib}} \lambda ^2 \lambda _{\mathrm{a}}^{-6}, \end{aligned}$$

(9.2)

$$\begin{aligned} {\mathcal {L}}&= 2 \mu \lambda ^2 \lambda _{\mathrm{a}}^{-2} \bigg \{ \lambda _{\mathrm{a}}^{-2} (\lambda ^2 \lambda _{\mathrm{a}}^{-2} -1) \bigg [ \frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}^2} \lambda ^2 \lambda _{\mathrm{a}}^{-2} (\lambda ^2 \lambda _{\mathrm{a}}^{-2}-1) + \frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}}\bigg ( \frac{11}{2}\lambda ^2 \lambda _{\mathrm{a}}^{-2} - \frac{3}{2} \bigg ) + 3\mu _{\mathrm{fib}} \nonumber \\&-\, 2 \lambda _{\mathrm{a}}^2 \bigg ( \frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}} \partial I_{4_{\mathrm{a}}}} (\lambda ^2 \lambda _{\mathrm{a}}^{-2} -1) + 2 \frac{\mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}} \bigg ) \bigg ] + 2 \mu _{\mathrm{fib}} \lambda ^2 \lambda _{\mathrm{a}}^{-4} \bigg \} \nonumber \\&+\, \mu \bigg [ (\lambda ^2 \lambda _{\mathrm{a}}^{-2} -1)^2 \bigg ( 2 \frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}^2} \lambda _{\mathrm{a}}^2 + \frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}} \bigg ) + 2 \alpha (3 \lambda _{\mathrm{a}}^2 - 1) \bigg ], \end{aligned}$$

(9.3)

where $H_2$ is given by (6.18). Next, we provide the derivatives related to $\mu _{\mathrm{fib}}$ for each case of $\mu _{\mathrm{a}}$.

1.1 Derivatives for Case 1

For $\mu _{\mathrm{a}} = c_1(1 - I_{4_{\mathrm{a}}})$, we have

$$\begin{aligned} \frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}}&= -c_1, \end{aligned}$$

(9.4)

where all other derivatives are zero.

1.2 Derivatives for Case 2

For $\mu _{\mathrm{a}} = c_1(1 - I_{4_{\mathrm{a}}})\text{ exp }( (I_{4_{\mathrm{a}}}-1)/c_2)$ we have

$$\begin{aligned}&\frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}} = c_1 \bigg [ \frac{1}{c_2}(1 - I_{4_{\mathrm{a}}}) -1 \bigg ]\text{ e }^{(I_{4_{\mathrm{a}}}-1)/c_2}, \end{aligned}$$

(9.5)

$$\begin{aligned}&\frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}^2} = \frac{c_1}{c_2} \bigg [ \frac{1}{c_2}(1 - I_{4_{\mathrm{a}}}) - 2 \bigg ]\text{ e }^{(I_{4_{\mathrm{a}}}-1)/c_2}, \end{aligned}$$

(9.6)

with omitted derivatives equal to zero.

1.3 Derivatives for Case 3

For $\mu _{\mathrm{a}} = c_1(1 - I_{4_{\mathrm{a}}})\text{ exp }( -(I_{4_{\mathrm{e}}}-1)/c_2)$, we have

$$\begin{aligned}&\frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}} = -\frac{\mu _{\mathrm{a}}}{c_2}, \end{aligned}$$

(9.7)

$$\begin{aligned}&\frac{\partial \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{a}}}} = - c_1 \text{ e }^{-(I_{4_{\mathrm{e}}}-1)/c_2}, \end{aligned}$$

(9.8)

$$\begin{aligned}&\frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}}^2} = \frac{\mu _{\mathrm{a}}}{c_2^2}, \end{aligned}$$

(9.9)

$$\begin{aligned}&\frac{\partial ^2 \mu _{\mathrm{fib}}}{\partial I_{4_{\mathrm{e}}} \partial I_{4_{\mathrm{a}}}} = \frac{c_1}{c_2} \text{ e }^{-(I_{4_{\mathrm{e}}}-1)/c_2}, \end{aligned}$$

(9.10)

along with ${\partial ^2 \mu _{\mathrm{fib}}} / {\partial I_{4_{\mathrm{a}}}^2} = 0$.