Continuum-mechanical modelling across scales : homogenisation methods and their application to microstructurally-based skeletal muscle modelling

Abstract

A variety of materials, such as biological soft tissues, exhibit large inter- and intra-subject microstructural variations that cannot be captured with individual material tests on the macroscopic observation scale. In such scenarios, multiscale modelling approaches are used instead, which explicitly incorporate the microstructure and provide macroscopic quantities through suitable homogenisation methods. This enables the description of biological soft tissue behaviour arising from microstructural changes, for example, those caused by disease. This thesis, therefore, deals with the multiscale continuum-mechanical modelling of materials and the particular application of such methods to the description of skeletal muscle tissue. Besides a general introduction to the subject, novel analytical estimates for the effective macroscopic potential of two-phase, hyperelastic, incompressible solids are presented. These are based on the so-called tangent second-order homogenisation method and are applicable for highly nonlinear, anisotropic material behaviour at large strains. Subsequently, a novel multiscale model for skeletal muscle is presented, which describes the macroscopic behaviour of soft tissue as a direct consequence of properties at smaller scales, such as the stiffness and arrangement of individual collagen fibres. The methods and models presented in this thesis are discussed by means of representative examples, thus demonstrating their merits in comparison to alternative approaches.