Like the Phoenix, the idea of identifying forms of elastic potential capable of effectively and efficiently describing experimental data for rubber-like materials and soft tissues continuously resurfaces in my research activity.

This is a bipolar story, marked by highs and lows. As you all know, this problem was already present in Antonio Signorini's writings in the 1930s and 1940s and in Clifford Truesdell's works from the 1950s. The famous 1977 paper by Sir John Ball on polyconvexity seemed to provide an answer to this fundamental issue. However, solving a physico-mathematical problem is never a matter of a single, definitive identification. In real-world applications, these models require much more detailed responses than those concerning polyconvexity alone.

Imagine trying to create a digital twin of a brain to calibrate a surgical robot. In this case, a clear representation of the brain’s geometry is needed, which is already a significant challenge. But beyond that, an accurate mechanical response of the material is required—this is a constitutive problem that demands both an explicit and well-defined form of the elastic potential and a clear procedure for identifying the constitutive parameters. Knowing that this specific elastic potential is polyconvex is important but not sufficient, as it does not necessarily make it effective for the intended application.

In my 2017 work with Destrade and Sgura, we identified a possible approach. However, the challenges we encountered discouraged me, leading me to think that the best path forward was to rely on weakly nonlinear theory. Recently, new discoveries with Alain Goriely have reignited some hope. I have returned to this topic, and I now believe that physics-informed machine learning could provide a decisive contribution to the problem. I do not claim it will fully solve it, but it will certainly make it easier to identify functional forms that efficiently describe experimental data given a set of experimental parameters.

Destrade, Michel, Giuseppe Saccomandi, and Ivonne Sgura. "Methodical fitting for mathematical models of rubber-like materials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473.2198 (2017): 20160811.

Anssari-Benam, Afshin, Alain Goriely, and Giuseppe Saccomandi. "Generalised invariants and pseudo-universal relationships for hyperelastic materials: A new approach to constitutive modelling." Journal of the Mechanics and Physics of Solids 193 (2024): 105883.

Another obsession of mine is wave propagation in solid mechanics.

I find this subject fascinating because it represents a sort of "triple point" where the communities of continuum mechanics (including mathematicians interested in hyperbolic problems), nonlinear acoustics, and solid-state physics converge. This intersection creates, on the one hand, a great variety of perspectives and, on the other, opportunities for new and interesting synergies.

One thing that immediately struck me when reading the historical literature on the topic is how fragmented and often incomplete it is. A notable example is antiplane shear waves: these waves cannot propagate in general but only in certain classes of materials. This fact seems to be well known, yet, before my work, no one had seriously investigated what happens to these waves in the general case.

In any case, we are dealing with a vast subject. On one side, there is the study of constitutive equations designed to describe dispersive and viscous phenomena, which are the physical effects accompanying elastodynamic behavior. It is crucial to derive model equations correctly and to connect their solutions with the mechanics of these problems. There are many open mathematical questions. Even when well-established methods exist for tackling some aspects, their application is far from standardized.

In this context, I consider the study of dispersive models, the rigorous and general derivation of model equations, and spectral stability analysis to be my main priorities.

Amendola, A., de Castro Motta, J., Saccomandi, G., & Vergori, L. (2024). A constitutive model for transversely isotropic dispersive materials. Proceedings of the Royal Society A, 480(2281), 20230374.

Saccomandi, Giuseppe, and Luigi Vergori. "Waves in isotropic dispersive elastic solids." Wave Motion 116 (2023): 103066.

Nobili, Andrea, and Giuseppe Saccomandi. "Revisiting the Love hypothesis for introducing dispersion of longitudinal waves in elastic rods." European Journal of Mechanics-A/Solids 105 (2024): 105257.