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

In this talk I will present two real-world 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. 
Well-posedness and finite element implementation of the problem are discussed. 

In the second part of the talk a one-dimensional, non-linear mathematical model of a bar under tension is examinated.
The mechanical energy encompasses the presence of a strain softening (Lennard-Jones) 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 non-linear regime.
It has been found that non-homogeneous branches bifurcates subcritically from the trivial one.
Non-homogeneous solutions are characterized by highly localized strain which, from the physical point of view, could be viewed as precursor of cracks.
The non-homogeneous branches also surprisingly feature re-entrant and frozen pattern behaviour.