“**THE MATHEMATICS AND MECHANICS OF NONLINEAR WAVE PROPAGATION IN SOLIDS**” (MMWS) is a PRIN 2022 granted project focused on the understanding of the wave motion in soft tissues in finite deformation regime, by employing nonlinear solid mechanics, continuum mechanics and rigorous mathematics.

We first focus on the framework of the actual technology of transient elastography with the goal to improve elastograms in precision and spatial resolution, but we also consider applications to shock waves in brain and other soft tissues. The project is organized in three work packages: the first is about the constitutive modeling, finite deformations and extended thermodynamics; the second is about wave propagation issues described by dissipative/dispersive equations or hyperbolic systems of nonlinear elasticity; the third is about computational and algorithmic aspects of continuum mechanics and hyperbolic systems.

For linear incompressible materials, the only elastic parameter is the shear modulus. The stiffness of a given material is therefore measured directly using this parameter. A nonlinear model for an elastic material contains more than one constitutive parameter and different in nonlinear elasticity models are characterized by a complete different set of elastic moduli. This means that if we use full nonlinear elasticity to assess the stiffness of soft tissues it is first necessary to define an universal measure of such material property. A possibility to solve this problem is the weakly nonlinear theory where the elastic stored energy is a polynomial of degree three or four in the strains of the finite deformations. This is a very robust theory of continuum mechanics where the constitutive parameters can be determined uniquely and precisely from the curve fitting of experimental data by means of a standard linear regression technique. Moreover, new ideas to model the diffusive and dispersive phenomena of soft tissues based on the theory of extended thermodynamics will be considered. In so doing, is possible to derive a rigorous multiple scale approach to deduce the general expression of the classical model equations for wave propagation in soft tissues taking into account finite deformations. This aspect will be analyzed in a general and rigorous theoretical framework.

The project will focus on some innovative computational methods based on numerical algorithms for the Riemann problem obtained via Godunov iterations and FEM/FD taking into account stochastic filtering algorithms. This because to consider the possibility of real world applications it is necessary to take into account the difficulty to extract information from experimental data since real tissues have a large distortion and the ultrasound imaging-derived data are usually corrupted by noises of various nature. In so doing we will produce a sort of “digital twin” of elastography based on a robust mechanical modeling, rigorous continuum mechanics and a new generation of computational algorithms able to work in finte deformation and with dispersive equations or hyperbolic systems.