[1] C. B. Vreugdenhil, Numerical methods for shallow-water flow, volume 13, Springer Science & Business Media, 1994.

[2] M. A. Abdelrahman, On the shallow water equations, Z NATURFORSCH A 72 (2017) 873–879.

[3] M. Brocchini, N. Dodd, Nonlinear shallow water equation modelling for coastal engineering, J Water Port Coast 134 (2008) 104–120.

[4] P. L.-F. Liu, Model equations for wave propagations from deep to shallow water, Adv Coastal Ocean En: (Volume 1) (1995) 125–157.

[5] C. Altomare et al., Applicability of smoothed particle hydrodynamics for estimation of sea wave impact on coastal structures, Coast Eng. 96 (2015) 1–12.

[6] E. Onate et al., The particle finite element method—an overview, Int. J. Comp. Meth. 1 (2004) 267–307.

[7] S. R. Idelsohn, E. Onate, F. D. Pin, The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves, IJNME 61 (2004) 964–989.

[8] M. Cremonesi et al., A state of the art review of the particle finite element method (pfem), Arch. Comp. Meth. Eng. 27 (2020) 1709–1735.

[9] M. Zhu, M. H. Scott, A pfem background mesh for simulating fluid and frame structure interaction, J. Str. Eng. 148 (2022) 04022051.

[10] O. Mavrouli et al., Damage analysis of masonry structures subjected to rockfalls, Landslides 14 (2017) 891–904.

[11] A. Montanino et al. Modelling with a meshfree approach the cornea-aqueous humor interaction during the air puff test. JMBBM, 77(2018), 205-216 .

[12] A. Montanino et al. Finite element formulation for compressible multiphase flows and its application to pyroclastic gravity currents. JCP, 451(2022), 110825.

[13] P.J. Ziółkowski et al. Adaptation of the arbitrary Lagrange–Euler approach to fluid–solid interaction on an example of high velocity flow over thin platelet. Continuum Mech. Therm., 33(6), pp. 2301-2314 2021

[14] P.J. Ziółkowski et al.. Fluid–solid interaction on a thin platelet with high-velocity flow: vibration modelling and experiment. Continuum Mech. Therm., pp. 1-27 (2022).

[15] E. Lorentz, S. Andrieux, Analysis of non-local models through energetic formulations, IJSS 40 (2003) 2905–2936.

[16] L. Ambrosio, V. M. Tortorelli, Approximation of functional depending on jumps by elliptic functional via t-convergence, Commun Pure Appl. Math 43 (1990) 999–1036.

[17] P. Sicsic, J.-J. Marigo, From gradient damage laws to Griffith’s theory of crack propagation, JElas 113 (2013) 55–74.

[18] J.-J. Marigo, Constitutive relations in plasticity, damage and fracture mechanics based on a work property, Nucl. Eng. Des. 114 (1989) 249–272.

[19] C. Comi, Computational modelling of gradient-enhanced damage in quasi-brittle materials, Mech. Cohes-fric Mat. 4 (1999) 17–36.

[20] K. Pham et al., Gradient damage models and their use to approximate brittle fracture, Int. Jour. Dam.Mech. 20 (2011) 618–652.

[21] C. Miehe et al., Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory, Int. J.Plasticity 84 (2016) 1–32.

[22] F. dell’Isola et al., At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola, Math. Mech. Solids 20 (2015) 887–928.

[23] Z. P. Bazant, G. Pijaudier-Cabot, Nonlocal continuum damage, localization instability and convergence (1988).

[24] Z. P. Bazant, M. Jirasek, Nonlocal integral formulations of plasticity and damage: survey of progress, J. Eng. Mech. 128 (2002) 1119–1149.

[25] S. Forest, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech. 135 (2009) 117–131.

[26] R. Peerlings et al., A critical comparison of nonlocal and gradient-enhanced softening continua, IJSS 38 (2001) 7723–7746.

[27] U. Mühlich et al., A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids, Arch. Appl. Mech. 83 (2013) 955–967.

[28] L. Zybell et al., Constitutive equations for porous plane-strain gradient elasticity obtained by homogenization, Arch. Appl. Mech. 79 (2009) 359–375.

[29] C. Oliver-Leblond et al., A micro-mechanics based strain gradient damage model: formulation and solution for the torsion of a cylindrical bar, Eur. J. Mech.-A/Solids 56 (2016) 19–30.

[30] L. Placidi et al., Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients, ZAMP 66 (2015) 3699–3725.

[31] L. Placidi et al., Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model, J.Eng.Math. 103 (2017) 1–21.

[32] Y. Rahali et al., Numerical identification of classical and nonclassical moduli of 3D woven textiles and analysis of scale effects, Comp. Str. 135 (2016) 122–139.

Published Results

The main findings have been published in the article Diffusion of a damaging fluid through a dam-shaped bidimensional body for the estimation of its lifetime by Scrofani, A., Barchiesi, E., Chiaia, B., Misra, A., & Placidi, L. (2025).

The research successfully demonstrated the effectiveness of a hemivariational principle for simulating mechanical degradation induced by the diffusion of a damaging fluid. Two models—a rectangular and a trapezoidal one—were compared. The figure below shows the geometry and boundary conditions for the two dam shapes.

The analyses were carried out using COMSOL Multiphysics, and the domains of the two geometries, together with their respective meshes, are shown in the figure.

A key aspect of the approach lies in the introduction of a direct coupling between the fluid concentration and the damage variable $\omega$. Unlike classical models, in this framework damage increases not only due to external loads but also due to the chemical presence of the fluid itself. This allows damage to be modeled as a monotonically increasing function, ranging from 0 to 1, which progressively reduces the material stiffness.

Regarding the influence of the diffusion coefficient $K_{DIF}$ on the service life, the parametric analysis revealed a nonlinear and highly realistic behavior. Under low diffusivity, the fluid remains confined near the contact surface, leading to extremely high concentration values in a small region and causing a slow evolution of total damage. Conversely, under high diffusivity, the fluid penetrates more easily and spreads over a larger area; this reduces the rate of local concentration growth and, paradoxically, may slow down the evolution of global damage after passing a minimum lifetime threshold. The plot of longevity versus $K_{DIF}$ therefore shows a minimum point, beyond which permeability helps distribute the degrading agent, slowing structural collapse.

Moreover, the concentration–damage coupling parameter $K_{c\omega }$ proved to be critical: even small variations in its value—for example, in the range from −4.5 to −5.5 m²/s²—produce significant changes in service life. A larger magnitude of this factor drastically accelerates the onset of damage under the same load, highlighting how the chemical nature of the fluid–solid interaction is decisive for durability.

Finally, the comparison between the rectangular and trapezoidal shapes scientifically confirmed traditional engineering intuition. The trapezoidal shape ensures that almost the entire structure remains in compression, whereas the rectangular shape exhibits large tensile regions at the base, where the trace of the strain tensor satisfies $G>0$. Since the damage activation threshold is ten times lower in tension than in compression, the rectangular model degrades much more rapidly. As a result, the trapezoidal dam exhibits a service life approximately 15% longer than the rectangular one, with damage evolution beginning only after more than 50 years, compared to the almost immediate onset observed in the rectangular case.

Summary of Key Findings

1. Effectiveness of the Hemivariational Model and Coupling Mechanism The study confirmed the effectiveness of a hemivariational principle for simulating mechanical decay induced by a degrading fluid. The central innovation is the direct coupling between fluid concentration and the damage variable ($\omega$), allowing damage to evolve monotonically from 0 to 1 and progressively reduce stiffness.
2. Influence of the Diffusion Coefficient ($K_{DIF}$) on Service Life The parametric analysis revealed a nonlinear relationship between diffusivity and structural lifetime:
   • Low diffusivity: fluid remains confined, concentration peaks locally, and total damage evolves slowly.
   • High diffusivity: fluid spreads more widely, reducing local concentration growth and potentially slowing global damage after a minimum lifetime is reached. The lifetime curve shows a minimum beyond which increased permeability slows collapse.
3. Sensitivity to the Concentration–Damage Coupling Factor ($K_{c\omega }$) Even small variations in $K_{c\omega }$ (from −4.5 to −5.5 m²/s²) significantly affect service life. A larger magnitude accelerates damage onset, underscoring the importance of chemical–mechanical coupling.
4. Geometric Superiority of the Trapezoidal Dam The trapezoidal shape keeps the structure predominantly in compression, while the rectangular shape exhibits tensile zones at the base.
   • Damage threshold is 10× lower in tension.
   • The rectangular dam degrades much faster.
   • The trapezoidal dam shows a 15% longer service life, with damage initiating after more than 50 years.

The website is continuously updated: new results, developments, and materials will be published here as the research progresses.