Stress integration procedures for a biaxial isotropic material model of biological membranes and for hysteretic models of muscle fibres and surfactant

Stress calculation for a biaxial isotropic material model of a biological membrane and for hysteretic models of muscle fibres and surfactant is presented in the paper. The non-linear elastic membrane model is defined by uniaxial and biaxial stress-stretch relations, while the hysteretic models of tissue fibres and surfactant are described by the stress-stretch and surface tension-surfactant area ratio constitutive relationships, respectively. The conditions when tissue is or is not covered by surfactant are considered. It is assumed that the material is subjected to cyclic loading. Quasi-static and steady conditions are considered. The models are implemented in large strain finite element incremented-iterative analysis of shell deformations. Numerical examples demonstrate characteristics of the computational procedures and structural response of biological membranes when subjected to cyclic loading. Hysteretic response of biological membranes subjected to cyclic loading is caused by hysteresis of fibres and hysteresis of surfactant. The hysteretic effects may play an important role in the physiology of human body.