The trabecular bone can adapt its form to mechanical loads and form structures that are both lightweight and very stiff. In this sense, it is a problem similar to structural optimization, especially topology optimization. The natural phenomenon leading to mechanical optimization of the bone structures is called trabecular bone remodeling. The main assumption and the benchmark for the numerical models of the phenomenon is the observation that the strain energy density on the structural surface is constant. This constant value corresponds to the homeostatic strain energy density, the state of bone tissue with a perfect balance of the loss, and gain of the bone mass. We assumed that the trabecular bone can form an optimal structure. The idea behind the investigation is to carry out studies on the ground of mechanics and to interpret clinical observations in the context of the results obtained from the optimization studies. In this way, clinical observations have been confirmed by strict arguments based on mechanics, leading to the unequivocal conclusion that equalization of the strain energy density on the trabecular bone surface allows minimizing the strain energy in the whole structure of the bone. This proves the veracity of the assumption that the remodeling process leads to the formation of the structure with the highest stiffness. In addition, this article elaborates on two new aspects of the remodeling phenomenon resulting directly from the considerations in the field of shape optimization important for numerical simulation. The first one concerns the influence of surface curvature on the remodeling process. The second one concerns the role of the bone surface where different loads are analyzed. Both aspects show the need of actual trabecular bone geometry model for the simulation of the trabecular bone remodeling phenomenon.
Volume free strain energy density method for applications to blunt V-notches
