In 1809, almost a century before the first human flight, the polymath Thomas Young opened his lecture “On the functions of the heart and arteries” with the following statement:
“The mechanical motions, which take place in an animal body, are regulated by the same general laws as the motions of inanimate bodies… As far, therefore, as the functions of animal life depend on the locomotions of the solids or fluids, those functions must be capable of being illustrated by the consideration of the mechanical laws of moving bodies: and it is obvious, that the inquiry, in what manner, and in what degree, the circulation of the blood depends on the muscular and elastic powers of the heart and of the arteries, supposing the nature of those powers to be known, must become simply a question belonging to the most refined departments of the theory of hydraulics.”
This prophetic statement is materialized, now, more than ever. An increasing number of engineers, mathematicians, and physicists apply analytical, numerical, and experimental methods that have matured in those disciplines, to answer questions in biology. Mechanical stability analysis, a method well known to aeronautical, mechanical, and civic engineers, provides valuable insights into developmental processes (cortical folding, pattern formation in gastrointestinal tissues), neurological disorders (schizophrenia, polymicrogyria, lissencephaly), arterial pathologies (atherosclerosis, aneurysms, thrombosis), to name a few. The intrinsic complexity of these tissues poses quite a few modeling challenges. Many soft tissues have a layered structure, each layer has different, highly nonlinear and anisotropic mechanical behavior. Due to differential growth taking place in the layers, residual stresses are developed. Moreover, the boundary and interface conditions in soft biological tissues are not known to the same level of certainty as in engineering structures.
In this talk, I will be discussing the application of linear bifurcation analysis to soft tissues within the framework of nonlinear elasticity. In particular, I will discuss twist buckling in arteries (as found in femoropopliteal and testicular arteries), how sensitive are stability margins to load configuration at the onset of instability (i.e. dead-load, follower force, fluid-pressure), and how the joint effect of residual stresses and interfacial conditions can induce spontaneous buckling (as found in esophagus and airways). Given time, I will also discuss the problem of sensitivity of vascular constitutive models to uncertainties in residual stresses data.