On Differences of Statically and Dynamically Determined Elastic Modules of Materials Like Trabecular Bone

The linear elastic material law which is usually applied in simulations of bone behavior reads σ_ij = C_ijkl ε_kl. It contains up to 21 independent constants. In most applications only nine constants (orthotropic behavior) are used. The determination of these constants is troublesome. The most applied experimental method is based on ultrasonic wave propagation. As it is often recognized the elastic modules measured by this method differ significantly from those found by static testing. Whereas Young’s modules differ slightly only, the determination of shear modules by ultrasonic methods is extremely doubtful, especially in trabecular bone. To find reasons for this effect, wave propagations are simulated by Finite-Element-techniques. This is done for artificial structures and also for realistic models of trabecular bone based one μCT-data. It can be recognized that in structured media always three types of waves propagate through the material with different speeds. Unfortunately the shear wave which is to be measured is the slowest one. Even if no longitudinal waves disturb the measurements, at least bending waves appear and pretend some kind of shear mode. The different orientations of the trabeculae can cause longitudinal waves when shear waves are applied. The stimulation of the ultrasound is at first simulated as a half cycle or as a step function only. The realistic waves are superimpositions of several of such motions. Such a relatively simple simulation makes possible to distinguish the three wave types mentioned above. The superimpositions complicate the motion extremely. Also reflection, damping and variable cross sections make it almost impossible to identify the modules, especially the shear modules, in a certain manner.