Mathematics and Mechanics of Solids

Francesco Brioschi (1824–1897), renowned for his contributions to the Theory of Algebraic Equations and the Theory of Invariants, was also an applied mathematician, theoretical mechanician, and hydraulic engineer. His scientific and technical expertise, coupled with his proven managerial skills, led him to hold significant political responsibilities. He was the founder and director of the Istituto Tecnico Superiore, later the Polytechnic University of Milan, which, thanks to him, became a driving force for the Italy’s technological development after its unification. This paper will discuss the development of his epistemological views starting from his contributions to Mathematical Physics and Analytical Mechanics, strongly inspired by his mentor Gabrio Piola, but not as thoroughly informed by the orthodox Lagrangian vision characteristic of Piola. Furthermore, we will examine his contributions to Hydraulics, which have a much more applied flavor and analyze his epistemological approach to the use of Mathematics in physical modeling. We will highlight the methodological differences that emerge in his approach to Mechanics and Hydraulics, which render his epistemology somewhat inconsistent and wavering. This weakness in his views on science, however, does not diminish the value of his contributions to Italian industrial, technological, and scientific growth: his political vision, evidenced by his institutional commitment, was remarkably precise and effective.

We consider a frame-indifferent variational model for magnetoelastic solids in the large-strain setting with the magnetization field defined on the unknown deformed configuration. Through a simultaneous linearization of the deformation and sharp-interface limit of the magnetization with respect to the easy axes, performed in terms of Γ-convergence, we identify a variational model describing the formation of domain structures and accounting for magnetostriction in the small-strain setting. Our analysis incorporates the effect of magnetic fields and, for uniaxial materials, ensures the regularity of minimizers of the effective energy.

Although nonlocal elasticity theories have been widely used to study various nonlocal problems, including problems of nanostructures and problems of harmonic plane waves with a huge number of published papers, they have not been proven to be well-posed at all for the general case. Therefore, it is extremely necessary and important to prove their well-posedness. In this paper, it has been proven that the stress-driven fully nonlocal elasticity theory is ill-posed for any harmonic plane wave problem in the sense of no solution, while the stress-driven two-phase nonlocal elasticity theory is well-posed for all problems of harmonic plane waves. It implies from this fact that we cannot use the stress-driven fully nonlocal elasticity theory to study problems of harmonic plane waves, while the stress-driven two-phase nonlocal elasticity theory is good tool for this task. It is remarkable that the ill-posedness of the stress-driven fully nonlocal elasticity theory for all problems of harmonic plane waves is contrary to the well-posedness of nanobeam problems of this theory. To illustrate the well-posedness of the stress-driven two-phase nonlocal elasticity theory, the reflection of (shear horizontal) waves from the traction-free surface of stress-driven two-phase nonlocal isotropic elastic half-spaces is investigated.

The effective properties of thermoelastic composite materials with ellipsoidal inclusions are determined. Inclusions are embedded in a homogeneous matrix, forming a three-dimensional periodic structure. Constituents are assumed to be isotropic materials. The effective properties are evaluated using a semi-analytical approach. In this method, the governing equations of the local problems over a unit cell are obtained via asymptotic homogenization and solved using the finite element method. In the proposed methodology, the local problems within the periodic cell are transformed into boundary value problems defined over one-eighth of the unit cell. This reduction is achieved by implementing the symmetry properties of the unit cell and the involved fields, assuming the periodicity of the microstructure and that the material properties are modeled as even functions with respect to the local coordinate system centered at the unit cell. Then, the effective elastic coefficients ( C ¯ ), thermal expansion coefficients ( α ¯ ), and thermal conductivity ( κ ¯ ) are obtained. Furthermore, a COMSOL Multiphysics program is implemented to compute these effective coefficients.

VAM (velocità ascensionale media) is a measurement that quantifies a cyclist’s climbing ability. We show that to minimize the time to attain a given height gain—which is tantamount to maximizing VAM—a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. This article is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that—for given start and end points, and for any fixed average-power constraint—the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.

This paper presents a study of thermoelasticity theory grounded in the recently developed two-temperature Moore–Gibson–Thompson (2TMGT) generalized thermoelasticity theory. We derive the fundamental governing equations for a homogeneous and isotropic medium within the context of 2TMGT. A primary objective is to establish key theoretical results, beginning with the uniqueness theorem for a mixed initial-boundary value problem in linear thermoelasticity under the current framework. Furthermore, we formulate a variational principle based on an appropriate functional corresponding to the governing equations of motion. This variational formulation offers deeper insights into the interactions between the mechanical and thermal aspects of the system. In addition, a reciprocity theorem is derived through the application of the Laplace transform method. The generalized results presented herein not only extend classical thermoelastic theorems but also enhance their applicability to a broader range of physical scenarios.

A widely adopted formulation of the incompressible elastic incremental deformation and the Steigmann–Ogden model of surface elasticity are used to study the effects of surface elasticity and an applied surface pressure on the gravity-driven instability of incompressible elastic layers. An explicit gravity parameter–wavenumber relation is derived to determine the critical condition for the gravity-driven instability, which reduces to the known formula in the absence of the surface elasticity and the applied surface pressure. Detailed results are shown to illustrate the effects of an applied surface pressure and various surface elasticity parameters on the critical value for instability and the associated wavenumber of unstable mode. Our results show that the applied surface pressure has a crucial effect on the critical value for the instability, and the effects of surface elasticity on the critical value of instability heavily depend on the magnitudes of various surface elasticity parameters.

This paper aims to determine the surface patterns of corneal diseases under the effects of swelling, such as in keratoconus (KC). Swelling is a mechanism that responds to changes in pressure within a medium. The cornea allows aqueous humor to flow from the stroma into the anterior chamber, following the laws of mechanics to reach mechanical equilibrium. The geometry of the cornea is altered to change its curvature, and it is reformed through the process of inflating and shearing. The theory of shells is divided into two theories: membrane theory and bending theory. These theories are used to determine stress and its corresponding shell deformation, providing solutions for bending, changes in shell thickness, and axially symmetrical load deformation, respectively. The change in corneal shape due to swelling is calculated and demonstrated under constant intraocular pressure (IOP) and the corneal load. This estimates the potential deformations of the corneal structure in response to the pressure and load imposed by the equilibrium equations. Moreover, the model can estimate the tensile properties of corneal tissue, such as its elastic modulus and stiffness. This model helps understand the mechanical stress–strain relations, permanent structural changes, and surface patterns of corneal deformations.

This study investigates the planar transient dynamics of a linearized string interacting with a moving mass. Unlike conventional models that idealize the moving system as a point mass or particle, the mass is modeled more realistically as a finite body that maintains dual contact with the string at closely spaced points. This modeling choice naturally incorporates the rotational inertia of the moving mass into the formulation. For a prescribed motion of the moving mass along the string, the coupled governing equations are derived in terms of the Dirac delta function. In the limiting case where the two contact points approach at the mass center, the formulation reduces to an equivalent monopole–dipole representation. The governing equations are discretized using the Galerkin method and integrated in time via a Runge–Kutta scheme. The formulation is first applied to a stretched string subjected to a uniformly moving mass. Particular attention is given to the trajectory paradox reported in earlier studies for moving point masses near a boundary. Using the present two-point contact model, this paradox is examined in terms of trajectory at the front contact, the separation between the actual and virtual centers of mass, and the variation in contact-point spacing induced by abrupt responses near the boundary. Subsequently, a new system, in which a string is vertically hung and carries a uniformly moving mass, is analyzed. Trajectory convergence is demonstrated for both ascending and descending motions. Similar to the trajectory paradox, a jump-like response appears near the upper fixed boundary when the mass ascends at higher velocities. At these velocities, the two-point contact model yields divergent results across the entire domain, indicating limitations in its applicability. This behavior highlights the need for future studies that incorporate nonlinear effects to more accurately capture the observed dynamics.

Cylindrical structures are widely employed in micro-resonators, sensors, and energy harvesting devices owing to their high space utilization and efficient electromechanical coupling, especially with the advancement of microelectromechanical systems and high-frequency communication technologies. Multilayer concentric cylindrical structures with cylindrical PN and Schottky junctions, which are the core carriers of micro high-power devices, can regulate the behavior of elastic waves through the surface and interface effects of the junction regions, providing new solutions for high-performance acoustic-electric devices. This study theoretically investigates the propagation of shear cylindrical waves in an infinitely long, multilayered piezoelectric semiconductor concentric cylinder structure containing multiple PN and Schottky junctions. The work aims to comprehensively examine the influence of mechano-electric-carrier coupling on wave energy efficiency and to quantify the critical steady-state carrier concentration along with the synergistic multi-junction effects. By considering the coupling among mechanical displacement, electric potential, and charge carrier perturbation, the fundamental governing equations for this structure are meticulously derived in the cylindrical coordinate system. Subsequently, integrating the physical attributes of cylindrical junctions, a mathematical model for the propagation of shear cylindrical waves is established. Leveraging the spectral method, the calculation problem of the dispersion and attenuation curves of shear cylindrical waves is transformed into the calculation problem of the eigenvalues of the square matrix, which shows the joint influence of the multiple cylindrical PN and Schottky junctions on the dispersion and attenuation characteristics of the shear cylindrical waves. Through numerical computations of the dispersion and attenuation curves, it is evident that cylindrical Schottky and PN junctions respectively manifest surface and interface effects. These effects serve to reduce wave attenuation and inhibit the conversion of mechanical energy into electrical energy. Specifically, an increase in the steady-state charge carrier concentration intensifies these effects, leading to the attainment of a peak value.

In nanocomposites, debonding originates at the interface between reinforcing particles and the matrix. Since experimental characterization and computational modeling of particle debonding are highly resource-intensive, the development of efficient predictive strategies is essential. In this study, a machine learning framework is proposed to predict debonding evolution using data obtained from finite element simulations. A multiscale modeling approach using representative volume elements (RVEs) was implemented to analyze debonding in the interface area of polyethylene–graphene nanocomposites. Debonding at the interface between graphene nanosheets and the polymer was modeled using a cohesive-zone model defined by traction–separation curves. Cohesive-zone parameters for polyethylene-graphene were sourced from molecular dynamics simulations. RVEs were generated with a Python script and analyzed using the Abaqus explicit solver. The study examined the effects of volume fraction, aspect ratio, and loading conditions on particle debonding. The results closely matched the published literature and effectively captured the trends in debonding behavior. A dataset comprising input parameters (e.g., volume fraction, aspect ratio) and the corresponding output (debonding percentage) was fed into a 10-layer artificial neural network. The network was trained using a gradient descent algorithm with various activation functions. The trained model demonstrated reasonable generalization capabilities with minimal overfitting. Subsequently, the model was used to predict debonding percentages for unseen input data. The results accurately captured the trends in debonding behavior for varying volume fractions and aspect ratios. The observed discrepancies were attributed to the limited dataset and the absence of debonding in specific samples.

This paper proposes a discrete kinematic model inspired by the principles of cellular automata and swarm-based models, aimed at simulating deformation phenomena in complex materials. Each point in the system represents a material unit that interacts locally with its neighbors according to rules inspired by solid mechanics, without resorting to global differential equations or force balances. This approach is part of a broader tradition of geometric and kinematic models, from ancient cosmologies to modern computational structures, and is particularly effective in the study of disordered or non-linear systems, where continuous methods are inapplicable or computationally burdensome. Kinematic modeling has historically favored representations of points without orientation, as evidenced in the models of Eudoxo and Descartes. However, such approaches are limited in describing phenomena in which directional orientation is crucial, as in the case of anisotropic deformations of materials. In this study, we introduce a kinematic model based on points with intrinsic direction, extending the traditional representation to a full vector description. Each point in the system possesses a position and a direction vector, allowing the analysis of local interactions that include both translations and rotations. The proposed model is distinguished by its ability to represent emergent phenomena arising from local interactions, offering a new perspective in the simulation of complex mechanical behavior. The intrinsic direction of the points makes it possible to capture anisotropic effects and to more accurately model the dynamics of real material systems. This extension of the kinematics-discrete paradigm opens up new possibilities in the computational modeling of materials, providing a versatile tool for the analysis of complex systems in engineering and materials physics. The potential of the proposed model is discussed in relation to its ability to describe, with reduced computational cost, both reversible elastic phenomena and irreversible processes such as plasticization and damage propagation. The presented framework thus emerges as a promising tool for the qualitative and quantitative analysis of heterogeneous materials, with possible applications in the physical, engineering, and computational fields.

In a previous paper, we proposed a symplectic version of the Brezis–Ekeland–Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. We applied it to the standard plasticity. The object of this work is to extend the previous formalism to non-associated plasticity. For this aim, we generalize the concept of bipotential to dynamical systems. The keystone idea is to define a symplectic bipotential. We present a method to build it from a bipotential. Next, we generalize the symplectic Brezis–Ekeland–Nayroles principle to the non-associated dissipative laws. We apply it to the non-associated plasticity and to the unilateral contact law with Coulomb’s dry friction.

In this paper, we consider Saint-Venant torsion of a functionally graded micropolar (Cosserat) beam, where the elastic moduli vary within the plane of the beam cross-section. We state a well-posedness result for the corresponding Neumann boundary-value problem in the weak Sobolev-space setting and show the existence and uniqueness of the solution up to the natural rigid micropolar mode. In view of symmetry, the circular and annular cases reduce to an ordinary differential equation for the radial amplitude. Numerical examples are presented for the stress and couple-stress fields and for the torsional rigidity. The characteristic micropolar length associated with the chosen material parameters is compared with the size of the cross-section, and the classical limit is recovered as a special case.

It has recently become understood that, in dealing with plane strain deformations in polar linearly isotropic elasticity, the couple-stress theory predicts that neither the trace of the couple-stress tensor nor the encountered couple-stress and non-symmetric stress fields are, or can generally be considered independent of the out-of-plane co-ordinate. The present communication proceeds beyond the bounds of polar linear elasticity and initiates an investigation regarding the extent to which these observations hold when polar material response is attributed to large plane strain elastic deformations. Clear distinction is made and separate attention is given to hyperelastic polar material behaviour that is due to inherent, probably unknown material properties, from possible relevant cases in which the source of such behaviour is known and well understood; such as the case that fibres embedded in a solid material are capable to resist deformation. The differences and similarities observed between these two categories of potential applications is also demonstrated with a brief discussion of a pair of fundamental boundary value problems, namely, the pure bending problem of an incompressible fibre-reinforced slab and the problem of pure azimuthal shear of an incompressible isotropic tube.

We develop a continuum variational theory describing the coupling between magnetic textures and electric polarization in systems where spatial gradients of the spin field generate an electrical response. The framework is formulated through a thermodynamically consistent Landau–Ginzburg free-energy functional that couples the spin field S, electric polarization P, and electric potential Φ. The model introduces a gradient magnetoelectric interaction of the form λ ijk P i ∂ j S k , linking polarization to spatial variations of the spin texture. Within this formulation, spatial inhomogeneities of the magnetization act as geometric sources of electric polarization, yielding the relation P = − λχ ∇ ⁡ × S for isotropic media. To elucidate the physical implications of this coupling, we analyze several illustrative boundary-value problems. In particular, a periodic spin spiral is shown to generate a corresponding polarization wave, predicting an electrical voltage signal that can serve as an experimental signature of spin–polarization coupling. A confined magnetic texture similarly produces localized polarization structures directly tied to spin curvature. The theory further admits nonlinear topological equilibrium solutions, including axisymmetric skyrmions whose spin textures generate annular polarization rings. Explicit analytical profiles for Néel- and Bloch-type skyrmions reveal distinct polarization vortices and characteristic chiral signatures associated with the topology of the spin field. Linearization around a uniform configuration yields closed-form dispersion relations for transverse and longitudinal spin waves. The analysis shows that the gradient magnetoelectric coupling selectively softens the transverse branch by renormalizing the exchange stiffness, thereby modifying the propagation of spin excitations. Phase diagrams constructed from the analytical solutions demonstrate how the effective stiffness A eff = A − λ 2 χ governs skyrmion size, polarization strength, and the transition between isolated skyrmions and skyrmion lattice phases. Together, these results establish a unified continuum framework linking magnetic topology, induced polarization, and spin-wave dynamics. The model provides a predictive tool for analyzing gradient magnetoelectric effects in multiferroic and chiral magnetic materials and suggests new strategies for electrical detection and control of complex spin textures in spintronic devices.

The analysis of the classical Lennard-Jones (LJ) oscillator under harmonic excitation reveals a fascinating phenomenon—a rich interplay between quasi-periodic motion and intermittent chaos. It is shown that chaotic oscillations can emerge even at low forcing amplitudes, while the system remains largely non-chaotic across a broad parameter range. The emergence of strange attractors composed of densely organized regular patterns represents another distinctive and persistent feature of the LJ oscillator. The analysis employs bifurcation diagrams, the largest Lyapunov exponents (LLEs), correlation dimension (CD), Shannon entropy (SE), and other tools. A comparison with the Duffing oscillator reveals a substantial discrepancy in oscillation regimes across the whole range of forcing amplitudes. To ensure numerical stability, the variable-step Adams–Bashforth–Moulton (ABM) method in an explicit–implicit formulation was used.

Acoustic pressure measurement in specific regions is essential for many engineering applications. However, due to factors such as inaccessibility, high temperature, or geometric constraints, sensors cannot always be placed directly near the target area. In such cases, measurements are taken from sensors located at accessible positions, and the acoustic pressure in the region of interest is subsequently estimated from the recorded data. Accurate reconstruction of this pressure field is therefore crucial. In this study, we address the estimation of acoustic pressure at arbitrary points within a rectangular enclosure by designing an optimized sensor weighting through evolutionary-based optimization algorithms. Three algorithms, namely genetic algorithm (GA), bees algorithm (BA), and particle swarm optimization (PSO), are compared in terms of their ability to determine the optimal sensor weights for accurate pressure estimation. A numerical investigation is presented to assess the overall performance of the three optimization algorithms.

This study develops an efficient computational framework for analyzing the acoustic behavior of structures coated with sound-absorbing materials in shallow water environments, explicitly accounting for seafloor reflection effects. Geometric modeling is carried out using isogeometric analysis with Catmull–Clark subdivision surfaces, ensuring high geometric fidelity. The isogeometric boundary element method (IGA-BEM) is then formulated with half-space fundamental solutions to incorporate both impedance boundary conditions of absorbing surfaces and reflection from the seabed. To overcome the computational challenges associated with fully populated system matrices, the fast multipole method (FMM) is integrated to achieve significant reductions in memory usage and computation time. Numerical experiments involving a spherical model and a submarine model demonstrate excellent agreement with analytical benchmarks and confirm the robustness of the proposed approach for complex geometries. The results underline the necessity of jointly considering seafloor reflection and absorbing coatings in high-accuracy ocean acoustic simulations, thereby offering a reliable and scalable tool for marine engineering applications.

In the present paper, the linearized theory of a transversely isotropic electroactive elastomer is used to analyze plane and surface wave characteristics. The transversely isotropic elastomer is considered with electroactive particles aligned in a preferred direction in the presence of an external applied field. The equations of motion are specialized to a two-dimensional setting and homogeneous plane and Rayleigh-type surface waves are considered. The two-dimensional governing equations are solved to obtain plane and surface wave solutions. Plane wave solutions of the governing equations lead to a velocity equation which indicates the propagation of two plane waves. The surface wave solutions satisfy the required boundary conditions which lead to a secular equation for Rayleigh-type wave speed. A numerical example is considered to illustrate graphically the wave speeds of plane and Rayleigh-type waves.