Séminaire LJKGéométrieImages: Calcul des variations

Le Mercredi 12 Février 2014 à 13h30 en

Séminaire de Guido VITALE (Université Joseph Fourier)

Applications of Calculus of Variations in Mechanics: Traction Force Reconstruction and Tension Test

Résumé:

In this talk I will present two realworld applications of Calculus of Variations in Solid Mechanics.
Constrained minimization of regularized functionals is the main issue in both case, attacked with analytical and numerical tools.
The first part of the talk concerns with Force Traction Microscopy.
The latter is an inversion method that allows to obtain the stress field applied by a living cell on an elastic medium from the pointwise knowledge of the displacement produced by the cell itself.
This classical biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled in a variational framework. In such a case, a variation of the error functional under Tichonov regularization is operated in view of its minimization.
This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem.
Wellposedness and finite element implementation of the problem are discussed.
In the second part of the talk a onedimensional, nonlinear mathematical model of a bar under tension is examinated.
The mechanical energy encompasses the presence of a strain softening (LennardJones) potential coupled with two quadratic terms penalizing both strain gradients and displacements.
Linear stability of the trivial homogeneous solution is first investigated analitically and the results are supplied with numerical solutions in the full nonlinear regime.
It has been found that nonhomogeneous branches bifurcates subcritically from the trivial one.
Nonhomogeneous solutions are characterized by highly localized strain which, from the physical point of view, could be viewed as precursor of cracks.
The nonhomogeneous branches also surprisingly feature reentrant and frozen pattern behaviour.
