# Activation of a muscle as a mapping of stress-strain curves

## TL;DR

This paper critically examines the active strain model for muscle contraction, demonstrating its limitations in replicating experimental stress-stretch data, and proposes a novel mixture active strain model involving two solid phases for improved accuracy.

## Contribution

The paper introduces a mixture active strain model that better captures muscle behavior by considering two phases, addressing limitations of the traditional active strain approach.

## Key findings

- Active strain model cannot reproduce experimental stress-stretch curves.
- Mixture active strain model successfully captures uniaxial muscle deformation.
- Proposed model offers a more accurate representation of muscle contractility.

## Abstract

The mathematical modeling of the contraction of a muscle is a crucial problem in biomechanics. Several different models of muscle activation exist in literature. A possible approach to contractility is the so-called active strain: it is based on a multiplicative decomposition of the deformation gradient into an active contribution, accounting for the muscle activation, and an elastic one, due to the passive deformation of the body. We show that the active strain approach does not allow to recover the experimental stress-stretch curve corresponding to a uniaxial deformation of a skeletal muscle, whatever the functional form of the strain energy. To overcome such difficulty, we introduce an alternative model, that we call mixture active strain approach, where the muscle is composed of two different solid phases and only one of them actively contributes to the active behavior of the muscle.