Andrea Mirandola, Ph.D. Candidate 37th cycle, University of Trento, DICAM
Roughly speaking, in mechanics we talk about instability, or bifurcation of equilibrium, when a body, subjected to external forces, can deform in more than one way, keeping constant the boundary conditions. Usually, the configuration assumed by the body corresponds to the one that allows it to store the minimum amount of energy. Among the different types of instability, during my Ph.D., I’m focusing on the one known as wrinkling. This particular kind of bifurcation is characterized by the formation of smooth surface waves and occurs typically in biological tissues such as the skin and arterial walls.
We are studying at which lateral deformation, called critical, a flat bilayer subject to compression stops to deform uniformly and starts to wrinkle. Such a model is called a bilayer because it is composed by a very thin and stiff upper film perfectly adhering on a much softer and semi-infinite substrate. Our model aims to describe the behavior of an arterial wall, therefore the substrate is a fiber-reinforced material described by the constitutive law due to Ogden-Gasser-Holzapfel, whereas a Neo-Hookean film is assumed. In such a model two family fibers (two groups of fibers having the same angle) are embedded within an elastic matrix. Up to now, our study leads to the formulation of a relatively simple model that fits results previously obtained. In particular, this allows us to obtain the so-called scaling laws of the problem, which are very simple relations that describe, in an approximated way, a phenomenon ruled by much more complex equations. Now, we are working, in collaboration with University Federico II (Naples), on the production of 3D-printed fibers embedded in silicone matrices in order to validate the analytical results obtained so far.
An extension of the model on which we are working is to take into account the plasticity of the film, which is the characteristic of a material to manifest residual deformations when the stresses exceed a certain threshold. The peculiarity of this type of material is that, unlike as mentioned at the beginning, the configuration assumed by the model does not correspond to the lower energy, since the one dissipated through plastic deformations is no longer recoverable, and the critical strain depends on the actual load path, thus making the treatment more complex.
In parallel, in the near future, our aim is to extend the fiber-reinforced bilayer model to more realistic geometries, from a biological point of view, such as the cylindrical one, typical of arteries.