Chapter 8: Doublet Mechanics and Acoustics for Discrete Media

In earlier chapters, we have dealt with the acoustic wave propagation in a continuous medium based on the classical continuum mechanics (CCM), which considers materials to be continuous. However, some bio- and nano-materials are not genuinely continuous, there are scaling factors that can be used to relate to their discrete property. The relatively newly developed Doublet Mechanics (DM) (also called Nanomechanics) is a new elastic theory dealing with materials that possess discrete microstructures (Ferrari et al., 1997). In DM, the fundamental microscopic unit of a material can be represented by a bundle of spheres or nodes separated by a finite distance (called inter-nodal distance) (Ferrari et al., 1997). Nodes in DM could be cells in biological cases or molecules of a solid. Nodes are either in contact with each other or separated by adhesive layers. In DM, the mechanical strain developed in a bundle due to its distortion is a function of the inter-nodal distance. Depending on the order and the discreteness of the system, it can be expressed as a multi-scale Mth-order Taylor expansion of the inter-nodal distances at its equilibrium configuration. It should be noted that the order (M) of the Taylor expansion should not be related to the degree of accuracy of this presentation in a common mathematic sense. In fact, the order of M represents the degree of the discreteness of a system; the higher the Mth-order a Taylor expansion includes, the more discrete nature of the system DM represents (Ferrari et al. 1997). When M = 1, i.e. only the first-order of the Taylor expansion is used; DM mechanics reduces to the COM case, i.e. the system described by DM becomes a system that does not have a discrete nature at all. In this case, the results obtained from DM completely agree with those predicted by CCM. When M = ∞, on the other hand, it describes a solid of single crystal which has a regular and periodic lattice structure. When ∞ > M > 1, discreteness of systems represented by DM are between continuous (nongranular) and periodically discrete. A recent success of DM in an ultrasonic application to characterization of elastic properties of tissue at relatively low frequency (<10MHz) shows that DM is quite versatile and powerful in description of wave propagation in a partially discrete and granular materials such as biological soft tissue (Liu & Ferrari, 2002).