A better understanding of the influence of material nonlinearities on the propagation of mechanical stress waves is necessary to generate insights into damage mechanisms of soft tissues subjected to rapid and strong external excitations. In this effort, the authors study the propagation of longitudinal stress waves through soft tissue. Emphasis is placed on the influence of nonlinear material behavior and nonuniform cross–section on the characteristics of the stress–wave propagation. The mechanical behavior of the soft tissue is represented by a nonlinear viscoelastic model that is obtained through a maximum dissipation, thermodynamically consistent construction. The effect of the tissue nonlinear mechanical behavior is studied through asymptotic analysis. Examining the obtained analytical approximation, it is possible to discern nonlinear wave front steepening and the effect of the material dissipation. The effects of a nonuniform cross–sectional area are investigated through numerical simulations. These studies can be applied to understand the effect of geometric features of axons on the propagation of longitudinal stress waves. For example, the diameter of an axon gradually increases near its ends, and varicosities/boutons along the axons represent concentrated cross–sectional area variations. Simulations are carried out to examine various aspects of the nonlinear wave propagation such as wave front steepening. This work can serve as a basis for better understanding the mechanical causes underlying mild traumatic brain injury caused by a head impact or explosive blast waves.

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