Abstract
“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. Achieved Results Over its 24-month duration, the MMWS project produced 19 peer-reviewed publications in leading international journals (average of nine per year, nearly three per researcher), with nine further manuscripts submitted or in advanced preparation. Results span the three planned research lines — constitutive modeling, model equations, and computational methods — while consistently emphasizing theoretical rigor and mathematical well-posedness over incremental technological implementation. On constitutive modeling, the project delivered a rigorous, well-posed theory of material dispersion in soft tissues, addressing a phenomenon long observed experimentally (in tendons, haemoglobin, and the pregnant cervix) but rarely incorporated into elastographic models. The resulting nonlinear Love-type equation extends the classical wave equation through higher-order mixed space-time derivatives without requiring additional boundary conditions, avoiding the ill-posedness that plagues naive nonlocal approaches such as the Boussinesq equation. Parallel work advanced nonlinear viscoelasticity through compressible and differential formulations, fractional-derivative models of anomalous relaxation, and applications to clinically relevant scenarios such as hepatic fibrosis monitoring in pediatric Fontan patients. A rigorous derivation of two-dimensional model equations in elastodynamics, including a long-incomplete nearly incompressible case, was also completed and extended to pre-deformed tissues. The Bologna Research Unit, coordinated by Andrea Mentrelli and including Francesca Brini, played a central role in bridging analytical constitutive derivations (Task 1, 2024) with computational and semi-analytical validation (Task 3, 2025). The unit's contributions were particularly influential in the treatment of fractional and dissipative viscoelastic models. Work on variable-order fractional linear viscoelasticity (Giusti, Colombaro, Garra, Garrappa, Mentrelli, 2024) clarified the mathematical structure underlying anomalous relaxation phenomena in soft biological media, while a companion study (Giusti, Mentrelli, Ruggeri, 2024) established an energetically consistent formulation for a nonlinear viscoelastic model compatible with fractional relaxation — directly supporting the project's broader goal of grounding dissipative constitutive laws in sound thermodynamic principles. On the computational side, the Bologna unit led the development of high-precision methods for transient wave propagation in dispersive-dissipative media. The steepest-descent-path approach to transient waves in linear dispersive media with dissipation (Mainardi, Mentrelli, González-Santander, 2025) provided an efficient and numerically robust alternative to standard inverse Laplace transform techniques, later revisited and extended to pulse-wave propagation in the viscoelastic Kelvin–Voigt model (González-Santander, Mainardi, Mentrelli, 2026). These contributions directly addressed one of the project's core computational challenges: reliably inverting noisy, band-limited experimental signals into physically meaningful viscoelastic parameters, a prerequisite for any future "digital twin" of elastographic imaging. A further outcome of particular significance to the Bologna unit concerns the extension of Rational Extended Thermodynamics (RET) to dispersive solids. Following a challenging attempt to apply a Chapman–Enskog-type expansion to model dispersive and dissipative phenomena, the team consulted Tommaso Ruggeri (Bologna), a lecturer at the project's Spring School. Ruggeri's insight — that classical parabolic laws (Navier–Stokes, Fourier, Fick, Darcy) can be recovered as vanishing-relaxation-time limits of more fundamental hyperbolic balance laws — led to a new closure strategy requiring new flux variables and balance equations, closed consistently with Galilean invariance, the entropy principle, and convexity of the entropy density, yielding a symmetric hyperbolic (hence thermodynamically consistent and well-posed) dispersive extension of the classical equations. This line of work is now being consolidated in a manuscript in preparation, "Rational Extended Thermodynamics for Dispersive Solids" (Ruggeri, Saccomandi, 2026), which represents a natural and direct extension of RET methodology — the theoretical framework at the core of the Bologna unit's broader research program — into the domain of nonlinear wave propagation in soft tissues. On computational methods more broadly, the project clarified the mathematical structure of the governing equations relevant to shock-capturing and high-resolution numerical schemes, and provided theoretical insight into why stiffness extraction from noisy, inhomogeneous, large-deformation ultrasound data remains an ill-conditioned inverse problem. Dissemination was substantial: the project catalyzed a national coordination initiative among PRIN 2022/PRIN PNRR 2022 projects on soft and biological materials mechanics, culminating in three "Mechanics of Soft, Heterogeneous, and Biological Materials" meetings (Bari 2024, Modena 2025, Naples 2026), a PhD-level Spring School ("New Theoretical Elastic Waves on Solids," featuring Aisling Ní Annaidh and Tommaso Ruggeri), and presentations at Giornate Signorini 2026, WASCOM 2025, MIMS 2025, and the Mathematics and Mechanics of Biological Tissues conference in Padova (2026). Twelve of the nineteen publications are Open Access, two are available as arXiv preprints under short embargo, and three are accessible on request via the project website. Overall, the project met and in several respects exceeded its planned objectives, establishing a coherent, thermodynamically consistent theoretical framework — spanning constitutive modeling, dispersive/dissipative model equations, and computational inversion — that directly informs the design of next-generation, physics-constrained elastography protocols, while training a cohort of early-career researchers (4 women, 6 men against a planned 3/6 split) in an interdisciplinary, internationally networked research environment.
Project details
Unibo Team Leader: Andrea Mentrelli
Unibo involved Department/s:
Dipartimento di Matematica
Coordinator:
Università degli Studi di PERUGIA(Italy)
Total Unibo Contribution: Euro (EUR) 23.000,00
Project Duration in months: 24
Start Date:
28/09/2023
End Date:
28/02/2026
