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Many works in elasticity have exploited the concept of gradient index (GRIN) lenses, borrowed from optics, for wave focusing and control. These effects are particularly attractive for cloaking, absorption or energy harvesting applications. Despite their potential, current lens designs suffer from limitations, mainly related to the difficulty in imaging point-like sources. Here, we exploit an alternative GRIN lens design, which enables a one-to-one correspondence between input and output phase, and allows to determine the focal length using the well-known thin lens equation, effectively establishing the elastic equivalent of the convex lens in optics. This is demonstrated analytically, obtaining a bijective relation between the location of a point-like source and its image, and the results are confirmed numerically and experimentally in an aluminium plate, where the lens is realized by introducing rows of circular cavities of variable diameters. Moreover, a proof-of-concept experiment demonstrates the possibility to image sources of flexural waves at the centimetre scale with subwavelength resolution. This research can extend applications of elastic GRIN lenses to new fields such as imaging and non-destructive testing, where the location of defects can be identified by focusing the scattered field. Multiple sources can be imaged simultaneously, and the combined effect of multiple lenses can also be used to design more complex systems, opening new possibilities in the technological exploitation of elastic wave manipulation.

Bioinspiration has widely been demonstrated to be a powerful approach for the design of innovative structures and devices. Recently, this concept has been extended to the field of elasticity, dynamics, and metamaterials. In this paper, we propose a seashell-inspired metasensor that can simultaneously perform spatial frequency mapping and act as a polarizer. The structure emerges from a universal parametric design that encompasses diverse spiral geometries with varying circular cross sections and curvature radii, all leading to tonotopic behavior. Adoption of an optimization process leads to a planar geometry that enables us to simultaneously achieve tonotopy for orthogonally polarized modes, leading to the possibility to control polarization as well as the spatial distribution of frequency maxima along the spiral axis. We demonstrate the versatility of the device and discuss the possible applications in the field of acoustics and sensing., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in APL Mater. 12, 041104 (2024) and may be found at https://doi.org/10.1063/5.0201722

The extended plane wave expansion (EPWE) formulation is derived to obtain the complex band structure of flexural waves in viscoelastic thin phononic crystal plates considering the Kirchhoff-Love plate theory. The presented formulation yields the evanescent behavior of flexural waves in periodic thin plates considering viscoelastic effects. The viscosity is modeled by the standard linear solid model (SLSM), typically used to closely model the behavior of polymers. It is observed that the viscoelasticity influences significantly both the propagating and evanescent Bloch modes. The highest wave attenuation of the viscoelastic phononic thin plate is found around a unit cell filling fraction of 0.37 for higher frequencies considering the least attenuated wave mode. This EPWE formulation broadens the suitable methods to handle evanescent flexural waves in 2-D thin periodic plate systems considering the effects of viscoelasticity on wave attenuation.

In this paper, we discuss the possibility of achieving tunable topologically protected edge modes through the application of uniaxial deformation in an auxetic metamaterial. The proposed structure consists of a thin slab with oriented cuts in a hexagonal lattice, where topologically protected band gaps are opened by introducing a controlled variation in the cut lengths. Numerical simulations demonstrate the existence of topologically protected and scatter-free wave propagation in the structure at the interface between two sub-domains with modified cells, in distinct frequency ranges. This only occurs in the presence of auxeticity. In addition, exploiting geometrical nonlinearity, the application of a uniaxial strain can be used to close the topological band gaps or to modify their frequency range, i.e., to weaken the localization effects or to shift the frequency at which they occur. The spatial and temporal variation of the applied strain field can thus be used for the dynamic tuning of metamaterial topological waveguiding properties, with applications in mechanical devices for logic operations and computations., Comment: 37 pages, 12 figures

We propose a new conceptual approach to reach unattained dissipative properties based on the friction of slender concentric sliding columns. We begin by searching for the optimal topology in the simplest telescopic system of two concentric columns. Interestingly, we obtain that the optimal shape parameters are material independent and scale invariant. Based on a multiscale self-similar reconstruction, we end-up with a theoretical optimal fractal limit system whose cross section resembles the classical Sierpi\'nski triangle. Our optimal construction is finally completed by considering the possibility of a complete plane tessellation. The direct comparison of the dissipation per unit volume with the material dissipation up to the elastic limit shows a great advantage: . Such result is already attained for a realistic case of three only scales of refinement leading almost (96%) the same dissipation of the fractal limit. We also show the possibility of easy recovering of the original configuration after dissipation and we believe that our schematic system can have interesting reliable applications in different technological fields. Interestingly, our multiscale dissipative mechanism is reminiscent of similar strategies observed in nature as a result of bioadaptation such as in the archetypical cases of bone, nacre and spider silk. Even though other phenomena such as inelastic behavior and full tridimensional optimization are surely important in such biological systems, we believe that the suggested dissipation mechanism and scale invariance properties can give insight also in the hierarchical structures observed in important biological examples.

Marine shells are designed by nature to ensure mechanical protection from predators and shelter for mollusks living inside them. A large amount of work has been done to study the multiscale mechanical properties of their complex microstructure and to draw inspiration for the design of impact-resistant biomimetic materials. Less is known regarding the dynamic behavior related to their structure at multiple scales. Here, we present a combined experimental and numerical study of the shells of two different species of gastropod sea snail belonging to the Turritellidae family, featuring a peculiar helicoconic shape with hierarchical spiral elements. The proposed procedure involves the use of micro-Computed Tomography scans for the accurate determination of geometry, Atomic Force Microscopy and Nanoindentation to evaluate local mechanical properties, surface morphology and heterogeneity, as well as Resonant Ultrasound Spectroscopy coupled with Finite Element Analysis simulations to determine global modal behavior. Results indicate that the specific features of the considered shells, in particular their helicoconic and hierarchical structure, can also be linked to their vibration attenuation behavior. Moreover, the proposed investigation method can be extended to the study of other natural systems, to determine their structure-related dynamic properties, ultimately aiding the design of bioinspired metamaterials and of structures with advanced vibration control., Comment: 28 pages, 20 figures

Bone is a stiff and though, hierarchical and continuously evolving material that optimizes its structure to respond to mechanical stimuli, which also govern growth and remodeling processes. However, a full understanding of the underlying mechanisms responsible for the cooperation of bone toughness and biological functions, with important implications in bone ageing, osteoporosis and tissue repair, has yet to be achieved. In particular, how micro-damage nucleation, needed for tissue remodeling, does not evolve into catastrophic failure in such a stiff material, still remains a partial enigma, cement lines, interfaces and sacrificial elements alone not providing a definitive answer to the question. Here, we bring to light a novel stress-based bone toughening mechanism, calling into play the nearly-symmetrical, chiral and hierarchical architecture of the osteons, demonstrating that their arrangement simultaneously gives rise to stress states alternating in sign along the osteon radius and to localized stress amplifications, both in the tensile and compressive regimes. This unveils a previously unforeseen synergistic mechanism allowing micro-damage accumulation without propagating cracks, kindled by the contrast between crack-opening due to tensile hoop stresses and crack-stopping due to compressive stresses at the crack tips in adjacent lamellae, which seal the crack ends conferring toughness to bone well beyond that predicted by current models. Furthermore, shear stresses occur at the lamellar interfaces, contributing with fluid flow to mechanically stimulate the osteocytes and to amplify the bone signalling. These results, obtained through original exact elastic solutions and confirmed by both FE fracture analyses and experimental tests on 3D-printed osteon prototypes, contribute with another piece in the puzzle to making the rational biophysical picture of bone mechanobiology complete.

Materials that are lightweight yet exhibit superior mechanical properties are of compelling importance for several technological applications that range from aircrafts to household appliances. Lightweight materials allow energy saving and reduce the amount of resources required for manufacturing. Researchers have expended significant efforts in the quest for such materials, which require new concepts in both tailoring material microstructure as well as structural design. Architectured materials, which take advantage of new engineering paradigms, have recently emerged as an exciting avenue to create bespoke combinations of desired macroscopic material responses. In some instances, rather unique structures have emerged from advanced geometrical concepts (e.g. gyroids, menger cubes, or origami/kirigami-based structures), while in others innovation has emerged from mimicking nature in bio-inspired materials (e.g. honeycomb structures, nacre, fish scales etc.). Beyond design, additive manufacturing has enabled the facile fabrication of complex geometrical and bio-inspired architectures, using computer aided design models. The combination of simulations and experiments on these structures has led to an enhancement of mechanical properties, including strength, stiffness and toughness. In this review, we provide a perspective on topologically engineered architectured materials that exhibit optimal mechanical behaviour and can be readily printed using additive manufacturing.

Nature has engineered complex designs to achieve advanced properties and functionalities through evolution, over millions of years. Many organisms have adapted to their living environment producing extremely efficient materials and structures exhibiting optimized mechanical, thermal, optical properties, which current technology is often unable to reproduce. These properties are often achieved using hierarchical structures spanning macro, meso, micro and nanoscales, widely observed in many natural materials like wood, bone, spider silk and sponges. Thus far, bioinspired approaches have been successful in identifying optimized structures in terms of quasi-static mechanical properties, such as strength, toughness, adhesion, but comparatively little work has been done as far as dynamic ones are concerned (e.g. vibration damping, noise insulation, sound amplification, etc.). In particular, relatively limited knowledge currently exists on how hierarchical structure can play a role in the optimization of natural structures, although concurrent length scales no doubt allow to address multiple frequency ranges. Here, we review the main work that has been done in the field of structural optimization for dynamic mechanical properties, highlighting some common traits and strategies in different biological systems. We also discuss the relevance to bioinspired materials, in particular in the field of phononic crystals and metamaterials, and the potential of exploiting natural designs for technological applications.

In this work, we demonstrate in a proof of concept experiment the efficient noise absorption of a 3-D printed panel designed with appropriately arranged space-coiling labyrinthine acoustic elementary cells of various sizes. The labyrinthine unit cells are analytically and numerically analysed to determine their absorption characteristics and then fabricated and experimentally tested in an impedance tube to verify the dependence of absorption characteristics on cell thickness and lateral size. The resonance frequency of the unit cell is seen to scale approximately linearly with respect to both thickness and lateral size in the considered range, enabling easy tunability of the working frequency. Using these data, a flat panel is designed and fabricated by arranging cells of different dimensions in a quasi-periodic lattice, exploiting the acoustic "rainbow" effect, i.e. superimposing the frequency response of the different cells to generate a wider absorption spectrum, covering the target frequency range, chosen between 800 and 1400 Hz. The panel is thinner and more lightweight compared to traditional sound absorbing solutions and designed in modular form, so as to be applicable to different geometries. The performance of the panel is experimentally validated in a small-scale reverberation room, and an absorption close to ideal values is demonstrated at the desired frequencies of operation. Thus, this work suggests a design procedure for noise-mitigation panel solutions and provides experimental proof of the versatility and effectiveness of labyrinthine metamaterials for tunable mid- to low-frequency sound attenuation.

The mechanical performance of reinforced composites is strongly influenced at different scales by the stress transferred at the matrix-fibre interfaces and at any surface where material discontinuity occurs. In particular, the mechanical response of elastomeric composites where the reinforcement is composed by cords with helically wound fibres is heavily compromised by fatigue and delamination phenomena occurring at cord-rubber as well as at the ply interfaces, since rubber and polymeric matrices are mainly vulnerable to the accumulation of deviatoric energy due to the shear stresses transferred across the surfaces. Despite the large diffusion of composites in a vast field of applications and the mature knowledge of their behaviour, some key mechanical aspects underlying failure mechanisms are still partially unclear. For example, stress amplification and strain localization are often difficult to predict by means of analytical solutions and averaging techniques that usually conceal stress gradients. In this work, we analyse coupling between torsional and tensile loads in twisted cords, which are adopted in many cases to reinforce composites and rubbers in tire applications. We provide a model characterized by an enriched cord-matrix mechanical interplay able to theoretically explain and predict actual stress distributions responsible for the onset of delamination and fatigue-guided phenomena that are experimentally observed in these composites. In particular, we demonstrate that the assumption of a monoclinic/trigonal behaviour for the mechanical response of the hierarchical strands allows to estimate, by means of analytical formulas and a homogenization approach, hitherto neglected shear stresses at the matrix-reinforcement interface. These stresses are transferred to the neighbouring regions, leading to post-elastic behaviour and failure events.

Clustering of ligand-binding receptors of different types on thickened isles of the cell membrane, namely lipid rafts, is an experimentally observed phenomenon. Although its influence on cell response is deeply investigated, the role of the coupling between mechanical processes and multiphysics involving the active receptors and the surrounding lipid membrane during ligand-binding has not yet been understood. Specifically, the focus of this work is on G-protein-coupled receptors (GPCRs), which regulate specific cell processes through chemical signalling pathways involving a synergistic balance between the cyclic Adenosine Monophosphate (cAMP) produced by active GPCRs in the intracellular environment and its efflux, mediated by Multidrug Resistance Proteins (MRPs) transporters. This paper develops a multiphysics approach based on the interplay among energetics, multiscale geometrical changes and mass balance of species, i.e. active GPCRs and MRPs, including diffusion and kinetics of binding and unbinding. Because the obtained energy depends upon both the kinematics and the changes of species densities, balance of mass and of linear momentum are coupled and govern the space-time evolution of the cell membrane. The mechanobiology involving remodelling and change of lipid ordering of the cell membrane allows to predict dynamics of transporters and active receptors -in agreement with experimentally observed cAMP levels- and how the latter trigger rafts formation and cluster on such sites. Within the current scientific debate on Severe Acute Respiratory Syndrome CoronaVirus 2 (SARS-CoV-2) and on the basis of the ascertained fact that lipid rafts often serve as an entry port for viruses, it is felt that approaches accounting for strong coupling among mechanobiological aspects could even turn helpful in better understanding membrane-mediated phenomena such as COVID-19 virus-cell interaction.

Graphyne nanotubes (GNTs) are nanostructures obtained from rolled up graphyne sheets, in the same way carbon nanotubes (CNTs) are obtained from graphene ones. Graphynes are 2D carbon-allotropes composed of atoms in sp and sp2 hybridized states. Similarly to conventional CNTs, GNTs can present different chiralities and electronic properties. Because of the acetylenic groups (triple bonds), GNTs exhibit large sidewall pores that influence their mechanical properties. In this work, we studied the mechanical response of GNTs under tensile stress using fully atomistic molecular dynamics simulations and density functional theory (DFT) calculations. Our results show that GNTs mechanical failure (fracture) occurs at larger strain values in comparison to corresponding CNTs, but paradoxically with smaller ultimate strength and Young's modulus values. This is a consequence of the combined effects of the existence of triple bonds and increased porosity/flexibility due to the presence of acetylenic groups.

The double peeling of detachment of non-linear adhesive tapes from a flat Poly(methylmethacrylate) (PMMA) surface has been investigated from both experimental and theoretical point of view. Double peeling tests show that, as the detachment process advances, the peeling angle stabilizes on a limiting value {\theta}lim corresponding to a critical pull-off force Fc above which the tape is completely detached from the substrate. This observed behavior is in good agreement with results obtained following the new theory of multiple peeling and taking into account the hardening-softening non-linear behavior of the experimentally tested adhesive tapes and clarifies some aspects of the experimental data. In particular, the theoretical model shows that the value of the limiting peeling angle depends on the geometry of the adhesive tape as well as on the stiffness properties and on the interfacial energy {\Delta}{\gamma}. Finally, theoretical predictions confirm that solutions with a peeling angle lower than {\theta}lim are unstable., Comment: 14 pages, 8 figures

Heterostructures formed by stacking layered materials require atomically clean interfaces. However, contaminants are usually trapped between the layers, aggregating into blisters. We report a process to remove such blisters, resulting in clean interfaces. We fabricate blister-free regions of graphene encapsulated in hexagonal boron nitride of$\sim$5000$\mu $m$^{2}$, limited only by the size of the exfoliated flakes. These have mobilities up to$\sim$180000cm$^2$V$^{-1}$s$^{-1}$ at room temperature, and$\sim$1.8$\times$10$^6$cm$^2$V$^{-1}$s$^{-1}$ at 9K. We further demonstrate the effectiveness of our approach by cleaning heterostructures assembled using graphene intentionally exposed to polymers and solvents. After cleaning, these samples reach similar high mobilities. We also showcase the general applicability of our approach to layered materials by cleaning blisters in other heterostructures based on MoS$_{2}$. This demonstrates that exposure of graphene to processing-related contaminants is compatible with the realization of high mobility samples, paving the way to the development of fab-based processes for the integration of layered materials in (opto)-electronic devices.

Continuum modeling of free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specification of the components of the shift vector that acts as an auxiliary variable. If only in-plane motions are considered the energy depends on an in-plane strain measure and the shift vector. The assumption of geometrical and material linearity leads to quadratic energy terms with respect to the shift vector, the strain tensor, and their combinations. Graphene's hexagonal symmetry reduces the number of independent moduli then to four. We evaluate these four material parameters using molecular calculations and the AIREBO potential and compare them with standard linear elastic constitutive modeling. The results of our calculations show that the predicted values are in reasonable agreement with those obtained solely from our molecular calculations as well as those from literature. To the best of our knowledge, this is the first attempt to measure mechanical properties when graphene is modeled as a hexagonal 2-lattice. This work targets at the continuum scale when the insight measurements comes from finer scales using atomistic simulations., Comment: 7 figures

The quest for large and low frequency band gaps is one of the principal objectives pursued in a number of engineering applications, ranging from noise absorption to vibration control, to seismic wave abatement. For this purpose, a plethora of complex architectures (including multi-phase materials) and multi-physics approaches have been proposed in the past, often involving difficulties in their practical realization. To address this issue, in this work we propose an easy-to-manufacture design able to open large, low frequency complete Lamb band gaps exploiting a suitable arrangement of masses and stiffnesses produced by cavities in a monolithic material. The performance of the designed structure is evaluated by numerical simulations and confirmed by Scanning Laser Doppler Vibrometer (SLDV) measurements on an isotropic polyvinyl chloride plate in which a square ring region of cross-like cavities is fabricated. The full wave field reconstruction clearly confirms the ability of even a limited number of unit cell rows of the proposed design to efficiently attenuate Lamb waves. In addition, numerical simulations show that the structure allows to shift of the central frequency of the BG through geometrical modifications. The design may be of interest for applications in which large BGs at low frequencies are required.

We propose a model for the mechanical behavior of protein materials. Based on a limited number of experimental macromolecular parameters (persistence and contour lengths, rate of unfolding dissipation) we obtain the macroscopic behavior of keratin fibers (human, cow, and rabbit hair), taking into account the damage and residual stretches effects which are fundamental in many functions of life. We support our theoretical results by showing that our model is robust and able to reproduce with high quantitive accuracy the cyclic experimental behavior of different keratinous protein materials we tested. We also show the capability of describing, even if with lower precision, the dissipation and permanent strain effects in spider silks.

Attenuating low-frequency sound remains a challenge, despite many advances in this direction. Recently developed acoustic metamaterials enable efficient subwavelength wave manipulation and attenuation due to exotic effects such as unusually high reflectivity, negative refraction or cloaking. In particular, labyrinthine acoustic metamaterials can provide broadband sound reduction and exhibit extremely high effective refractive index values due to their characteristic topological architecture. In this paper, we design a novel labyrinthine metamaterial with hybrid characteristics compared to previously proposed structures, by exploiting a spider web-inspired configuration. The developed metamaterial structure is characterized by additional tunability of the frequencies at which band gaps or negative group velocity modes occur, thus enabling versatility in the functionalities of the resulting structures. Time transient simulations demonstrate the effectiveness of the proposed metamaterials in manipulating wave fields in terms of transmission/reflection coefficients, amplitude attenuation and time delay properties in broadband frequency ranges. Results could find applications in the development of practical lightweight acoustic shielding structures with enhanced broadband wave-reflecting performance., Comment: 25 pages, 7 figures, submitted

We report the exfoliation of graphite in aqueous solutions under high shear rate [$\sim10^8s^{-1}$] turbulent flow conditions, with a 100\% exfoliation yield. The material is stabilized without centrifugation at concentrations up to 100 g/L using carboxymethylcellulose sodium salt to formulate conductive printable inks. The sheet resistance of blade coated films is below$\sim2\Omega/\square$. This is a simple and scalable production route for graphene-based conductive inks for large area printing in flexible electronics.

A combined experimental and numerical study on the variation of the elastic properties of defective single-crystal diamond is presented for the first time, by comparing nano-indentation measurements on MeV-ion-implanted samples with multi-scale modeling consisting of both ab initio atomistic calculations and meso-scale Finite Element Method (FEM) simulations. It is found that by locally introducing defects in the 2x10^18 - 5x10^21 cm-3 density range, a significant reduction of, as well as of density, can be induced in the diamond crystal structure without incurring in the graphitization of the material. Ab initio atomistic simulations confirm the experimental findings with a good degree of confidence. FEM simulations are further employed to verify the consistency of measured deformations with a stiffness reduction, and to derive strain and stress levels in the implanted region. Combining these experimental and numerical results, we also provide insight into the mechanism responsible for the depth dependence of the graphitization threshold in diamond. This work prospects the possibility of achieving accurate tunability of the mechanical properties of single-crystal diamond through defect engineering, with significant technological applications, i.e. the fabrication and control of the resonant frequency of diamond-based micromechanical resonators., Comment: 32 pages, 7 figures

This paper presents experimental data and analysis of the structural damage caused by swift-heavy ion irradiation of single-crystal diamond. The patterned buried structural damage is shown to generate, via swelling, a mirror-pattern on the sample surface, which remains largely damage-free. While extensive results are available for light ion implantations, this effect is reported here for the first time in the heavy ion regime, where a completely different range of input parameters (in terms of ion species, energy, stopping power, etc.) is available for customized irradiation. The chosen ion species are Au and Br, in the energy range 10-40 MeV. The observed patterns, as characterized by profilometry and atomic force microscopy, are reported in a series of model experiments, which show swelling patterns ranging from a few nm to above 200 nm. Moreover, a systematic phenomenological modelling is presented, in which surface swelling measurements are correlated to buried crystal damage. A comparison is made with data for light ion implantations, showing good compatibility with the proposed models. The modelling presented in this work can be useful for the design and realization of micropatterned surfaces in single crystal diamond, allowing to generate highly customized structures by combining appropriately chosen irradiation parameters and masks., Comment: 20 pages, 8 figures

In this paper we adopt a processing technology to develop elastomer plus nano-graphite hybrid composites with multifunctional properties. Beyond the improvements of the mechanical properties, the research findings demonstrate the synergistic effect of carbon black and graphite nanoplatelets to prepare rubber composite thermally conductive and to design a new class of shock absorbers. It was found that a critical GNPs/CB ratio was apt to reduce the strong interlayer forces among the GNPs sheets, which led to the efficiency on reinforcement in mechanical properties and improvements of the performance of the rubber composites.

Recently, it was proposed based on classical elasticity theory and experiments at macroscale, that the conformations of sheets inside cylindrical tubes present a universal behavior. A natural question is whether this behavior still holds at nanoscale. Based on molecular dynamics simulations and analytical modeling for graphene and boron nitride membranes confined inside carbon nanotubes, we show that the class of universality observed at macroscale is violated at nanoscale. The precise origins of these discrepancies is addressed and proven to be related to both surface and atomistic effects.

In this paper a modification of the classical Weibull Statistics is developed for nanoscale applications. It is called Nanoscale Weibull Statistics. A comparison between Nanoscale and classical Weibull Statistics applied to experimental results on fracture strength of carbon nanotubes clearly shows the effectiveness of the proposed modification. A Weibull's modulus around 3 is, for the first time, deduced for nanotubes. The approach can treat (also) a small number of structural defects, as required for nearly defect free structures (e.g., nanotubes) as well as a quantized crack propagation (e.g., as a consequence of the discrete nature of matter), allowing to remove the paradoxes caused by the presence of stress-intensifications.

A new quantum action-based theory, Dynamic Quantized Fracture Mechanics (DQFM), is presented that modifies continuum-based dynamic fracture mechanics. The crack propagation is assumed as quantized in both space and time. The static limit case corresponds to Quantized Fracture Mechanics (QFM), that we have recently developed to predict the strength of nanostructures.

In this paper we derive a statistical law of Life. It governs the probability of death, or complementary of survival, of the living organisms. We have deduced such a law coupling the widely used Weibull statistics, developed for describing the distribution of the strength of solids, with the universal model for ontogenetic growth only recently proposed by West and co-authors. The main idea presented in this paper is that cracks can propagate in solids and cause their failure as sick cells in living organisms can cause their death. Making a rough analogy, living organisms are found to behave as growing mechanical components under cyclic, i.e., fatigue, loadings and composed by a dynamic evolutionary material that, as an ineluctable fate, deteriorates. The implications on biological scaling laws are discussed. As an example of application, we apply such a statistical law to large data collections on human deaths due to cancer of various types recorded in Italy: a relevant agreement is observed.