Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

Abstract

Abstract:The collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone. PACS

FCAA Related News, Events and Books (FCAA–Volume 20–2–2017)

“…The main achievements were reported in several state-of-the-art papers [10,16,24,25,29]. Utilizing the generalized Hertzian contact law and algebra of dimensionless fractional Rabotnov operators, which was developed recently by Prof. Rossikhin [29,31,32], the theory of impact of viscoelastic bodies has been constructed during last five years by Prof Rossikhin and his research team [27,29,30,34,36,37].…”

Modelling of the collision of two viscoelastic spherical shells

“…It has been shown in Rossikhin and Shitikova (2013a) that on each of these waves the following relationships are fulfilled: the geometric condition of compatibility (Thomas 1961) …”

“…2 Scheme of the propagating wave-strip along the spherical shell surface (Rossikhin and Shitikova 2013a) derivative with respect to the normal to the wave surface, ξ is the coordinate directed along the normal to the spherical shell, λ i and ξ i are the components of the unit vectors λ and ξ directed along the normal to the wave and spherical surfaces, respectively, [Z] = Z + − Z − is the discontinuity in the desired field Z, where "+" and "-" denote that the given value is calculated "ahead of" and "behind" the surface of strong discontinuity, and Latin indices take on the values 1, 2, 3.…”

“…The only paper considering the analysis of two colliding fractionally damped spherical shells in modeling blunt human head impacts has been recently appeared (Rossikhin and Shitikova 2013a), wherein the contact force is represented using linear approach via fractional derivative standard linear solid model. It was assumed that during impact process the microstructure of the shells' materials changed only in the contact domain, i.e., both shells remained elastic except the parts involved in the contact interaction which exhibited locally viscoelastic properties.…”

Fractional Calculus in Biomechanics

“…Fractional viscoelastic models were adopted to describe the mechanical behavior of the brain (Kobayashi et al 2012), arteries (Craiem et al 2008;Smyth and Green 2015;Palomarez-Ruiz et al 2015), human fat pad (Freed and Diethelm 2006), and periodontal ligament (Bosiakov et al 2015). The utilization of the fractional calculus for modeling the dynamics of biomechanical objects is presented in Rossikhin and Shitikova (2013a), wherein the collision of two elastic or viscoelastic spherical shells has been analytically simulated to investigate the dynamic response of the human head under the action of another head or some spherical object; in doing so the viscoelastic characteristics of the shells have been described by the standard linear solid model with a fractional derivative. In addition to whole biological systems, the relaxation processes are also carried out in cells and cellular structures.…”