Material forces in open system mechanics

The basic concern of the present work is the exploitation of the notion of material forces in the theoretical and computational analysis of open systems with particular application to biomechanical problems. Based on a completely dual framework for the spatial and the material motion problem, we introduce the balance equations for open system thermodynamics. In combination with the appropriate constitutive equations, they constitute the basis of the finite element formulation derived thereafter. For the spatial motion problem, the solution of the governing equations, basically the balance of mass and momentum, renders the discrete nodal values of the density and the deformation as primary unknowns. For the material motion problem, the computational analysis of the balance of momentum yields the discrete material node point forces which can be interpreted as driving forces for the local rearrangement of material inhomogeneities. Once the spatial motion problem has been solved, the material force method is nothing but a mere post-processing step from an algorithmic point of view. As a convincing benefit of the proposed strategy, the computation of the material forces is extremely cheap and requires no additional finite element data structure. In the context of biomechanics, material forces give further insight into complex biomechanically induced processes, such as functional adaption, morphogenesis, healing or growth. For example, material forces on the boundary can be considered as a measure of shape sensitivity and thus indicate morphological changes while material forces in the interior indicate the tendency to create new mass locally or to equilibrate local density inhomogeneities.