The content of this paper is not a historiographical reconstruction. Instead, it intends to give a documented opinion, based also on historical considerations, about the postulation scheme which must be preferred in formulating novel mechanical models. Truesdell has claimed that his own preferred postulation to continuum mechanics, which builds the theory based on local integral balance laws, is historically grounded in the work of Leonhard Euler. Is this position, so strongly advocated by Truesdell, tenable? In the present paper, based on circumstantial evidence and by using the technique of logical reconstruction of "lacking" mathematical details in a "corrupted" flow of mathematical reasonings, as proposed by dell'Isola and Spagnuolo, we conjecture that, most likely, Euler's preferred postulation of mechanics was based on variational principles. This conjecture will need to be further substantiated, as, unfortunately, the greatest part of the corpus of Euler's works (mainly published in Latin only or even not completely printed yet) is not easily accessible; therefore, the careful philological analysis needed to fully confirm the proposed conjecture is, at the moment, impossible. In the literature, one finds two argumenta ab actoritate: the first, used by Truesdell, claims that balance laws postulation must be accepted because Euler is assumed to have chosen it; the second, used by Truesdellians, claims that such a postulation must be accepted because Truesdell has shown us the light. However, it is well-known that argumenta ab auctoritate sunt infirmissima (The arguments based on the authority of a "bigger" thinker are the weakest).

The relationship between balance laws and the Principle of Virtual Work as well as the structure of contact interactions in continua remain foundational issues in Mechanics. In this work, we revisit these issues within the distributional framework emphasized by Paul Germain. We show that while the Principle of Virtual Work implies balance of forces and moments for nth-gradient continua, balance laws alone do not suffice to characterize equilibrium for n ≥ 2. We then reexamine Noll's classical theorem asserting that surface contact forces depend solely on the unit normal to the surface and identify the precise role of his additional assumptions, namely the absence of edge and wedge contact interactions and the boundedness of the surface contact density on the space of oriented surfaces. We demonstrate that these hypotheses fail for general higher-gradient continua. Consequently, the presence of curvature-dependent surface contact forces in such materials does not conflict with Noll's theorem and refutes his claim of their nonexistence.

In this paper, we show one solution of the following synthesis problem: to find a planar, periodic, structure made up of straight bars linked by (perfect) hinges, which, once homogenized, can be modelled as a planar third-gradient one-dimensional (1D) continuum. One possible-solution structure is obtained by considering a suitably Modified Hart's Antiparallelograms Mechanism (MHAM). Such a mechanism has been conceived in order to get, once suitable kinematical constraints are added, what we call an MHAS, i.e., a Modified Hart's Antiparallelograms Structure. Each of these structures has a stress-free configuration, i.e., a configuration having vanishing deformation energy, which coincides with a circumference. By limiting to the case when all the mechanism bars are rigid and by replacing specific MHAM elements with two bars interconnected by elastic rotational joints, we get a truss structure which, once homogenized into an elastic 1 D continuum, can assume deformed shapes whose deformation energy is not vanishing only when its curvature is not constant. Choosing for the homogenized 1 D continuum, as a deformation measure, the derivative of curvature, and imposing a suitable rescaling of the rotational elastic moduli, we establish the asymptotic micro-macro relationship for its macro-deformation energy.

This paper investigates the nonlinear dynamics of the High-Temperature Superconducting (HTS) pinning magnetic levitation (MAGLEV) transit system under development at the University of L'Aquila. Due to its inherently weak damping characteristics, the MAGLEV system is particularly susceptible to external disturbances, such as mechanical or magnetic irregularities along the guideway. To analytically characterize its complex nonlinear dynamics, a simplified nonlinear single-degree-of-freedom model is developed, and the Multiple Scales Method (MSM) is employed as a solution technique. This approach enables the evaluation of how key design parameters influence the system's dynamic response. The analysis highlights the emergence of both primary and secondary resonances, which arise depending on system parameters and the nonlinear nature of the levitation force, potentially impacting not only performance but also stability. Finally, the analytical findings are validated against benchmark solutions obtained through direct numerical integration of the system's nonlinear equation of motion.

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.

This research extends a previous investigation initiated by the authors focusing on the impact of geometric nonlinearity on the propagation of seismic waves in a two-dimensional model of the Aterno Valley, which includes the urban region of L'Aquila. Starting from a Cauchy continuum framework, this study undertakes a comparative analysis between a traditional linearized formulation, based on infinitesimal strain energy, and a nonlinear model that incorporates the Green-Lagrange strain tensor. The computational domain describes a detailed geological cross-section of the valley, comprising three primary stratigraphic layers. In addition, the upper layer incorporates mechanical segmentation. Furthermore, the excitation is represented as a spherical wavefront applied at the basal boundary, emulating a deep, high-energy source equivalent to an earthquake with a magnitude of Mw >= 6. Finite element simulations were performed in COMSOL using a generalized-alpha time integration scheme. An examination was conducted on two distinct configurations of Young's modulus distribution within the upper stratum. The first configuration comprised a homogeneous rigid top layer characterized by a unicum body, while the second configuration incorporated soft inclusions within the upper layer to emulate its fragmentation. A comparative analysis was conducted on nonlinear and linear simulations using normalized energy deviation parameters. This approach facilitated the quantification of the relative discrepancies between the deformation energy densities exhibited by the two models. The results indicate that, while the average deviations between the linear and nonlinear formulations are generally moderate, substantial local discrepancies are observed in regions near interfaces with pronounced stiffness contrasts. Specifically, energy deviations in these areas can exceed 100% in certain regions. The observed localized amplifications and redistributions of energy indicate that geometric nonlinearities might significantly affect the spatial distribution of wave energy propagation. This effect is particularly pronounced in scenarios where the shallow layers are mechanically segmented or incorporate compliant inclusions. The temporal evolution of energy at specific reference points within the urban area exhibits scenario-dependent variations, suggesting that linear assumptions may inaccurately characterize site response under conditions of high-intensity excitation. The results confirm the assumption that geometric nonlinearities are significant factors in strongly heterogeneous basins. This emphasizes the need for precise geometric representation and realistic modeling of stiffness distributions to improve predictive accuracy and reliability.

A general third-gradient one-dimensional beam formulation is introduced through energy postulation. This novel formulation incorporates the effects of nonrigid (or deformable) boundary constraints by using the spring analogy. The Euler-Lagrange equations for equilibria and expressions for boundary conditions are derived. For certain specific cases, analytic solutions are found by reducing the problem to that of finding a single descriptor that accounts for the rotation of the cross section. Based upon the obtained analytical solutions, parametric studies are conducted by varying the stiffness coefficients and boundary spring coefficients. It is found that the deformable boundaries, represented by the boundary springs, modulate the highly nonlinear response of these curvature gradient beams. Both numerical and analytic solutions are compared, revealing exotic shapes when various external actions, such as couples, double couples, and boundary terms, are activated. Finally, an example of a microstructure, whose homogenized limit is representative of curvature-gradient beams, is presented. It is shown that the conceived continuum model describes the behavior of such a microstructure subjected to a concentrated boundary couple with high accuracy. This study, therefore, represents a significant advancement in the general formulation for higher-gradient energy.

We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.

In this article, a central result in the theory of linearized elasticity, namely, the reciprocity theorem, is outlined for a continuous solid body, performing the exegesis of the original formulation by Enrico Betti (1823-1892), comparing the proofs he provided with those permitted by modern approaches in continuum mechanics. For linear elastic structures under small displacements, rotations, and strains, legitimating the superposition of the structural responses under multiple loading systems, in the absence of volume forces Betti's theorem allows one to deduce relations involving only loading and displacements over the boundary surface, without requiring the knowledge of the body's elastic stiffness nor making reference explicitly to the distribution of strains and stresses (as it occurs for the conventional formulation of the virtual work). We discuss the auxiliary use of this theorem made by Betti in his works on continuum mechanics, inspired by the potential theory, in particular involving Green's identities. The outlined theoretical strategies, resting on the rigorous deduction of analytical expressions, can nowadays represent a reference for scholars prevalently devoted to repetitive numerical simulations, and an indispensable tool to investigate truly innovative material models.

Recently, it has been conjectured that a specific modular articulated bi-parallelogram microstructure is a solution for the problem of synthesis for a 1D continuum whose deformation elastic energy depends on the curvature gradient (dell'Isola et al. [Math. Mech. Complex Syst. 12 (2024)] and Terranova et al. [Comptes Rendus. M & eacute;canique 353 (2025)]). In this paper, we present an asymptotic procedure for getting the homogenized description of that microstructured system under large in-plane deformation. It is shown that such a periodic structure behaves macroscopically as a third gradient 1D continuum. The elastic energy stored in a module of such a microstructured system deformed by a gradient of curvature combined with extension is first established. This leads to the energy density of the effective 1D continuum. The latter involves three elastic stiffness coefficients related to the curvature gradient, the elongation, and the coupling of both, whose expressions are explicitly related to the morphology of the module, the stiffnesses of the micro bars, and the curvature. The strong formulation of the equilibrium condition of the microstructured system is then deduced following the Euler-Lagrange method of minimization of energy. The calculation of the first variation of the deformation energy allows for the determination of the generalized external forces which can be applied to the equivalent 1D continuum whose deformation energy depends on the gradient of curvature and extension: that is, normal and transverse force together with couple and double couple. Consequently, we determine the corresponding balance equations and the constitutive equations for forces and both couple and double couples. Some numerical examples are given in the case of newly introduced 1D continua loaded at their extremity by couples and double couples.

In this paper, we show, via an asymptotic expansion, that a 1D continuum model, with an internal variable, is suitable to describe the mechanical behavior of the Inverted Hart's micro-architectures, generalizing the treatment presented in Rizzi et al. (2026). An internal variable is needed to describe the kinematics of the 1D continuum: we find an expression for deformation energy, which depends on the gradient of curvature of a 1D continuum and on the introduced internal variable. We find, as a particular case, the results already presented in Rizzi et al. (2026): when the condensation of kinematical parameters is possible, the energy density of the 1D model depends only on the curvature gradient.

A two-dimensional (2D) reduced-order generalised continuum model within the framework of the three-dimensional (3D) deformations is deduced from a 3D Cauchy continuum model by imposing a micro-macro kinematical map, which is linear in the direction normal to the corneal surface. This kinematical assumption is plausible as the cornea thickness is much smaller than its diameter. We use the obtained 2D generalised continuum that incorporates a kinematically independent thickness to model the changes of shape induced in corneas: (1) by the changes of cornea mechanical properties whose aetiology can be found in the complex (and not completely understood yet) pathogenic process causing keratoconus, (2) by penetrating keratoplasty, and (3) degeneration of both patient residual corneal tissue and transplanted corneal tissue after transplant. We postulate that growth and regeneration phenomena occurring in the cornea shape it following the “elastic” solutions, which we have calculated. The preliminary obtained predictions seem to promise significant applicative developments and are in good qualitative agreement with experimental results: future investigations will need to improve the presented model by considering explicitly the remodelling phenomena and a more detailed analysis of the evolution of metabolically driven mechanical damage of corneal tissue and its visco-plasticity.

Our aim is to try to trace, in the history of mechanics, the first formulation of the principle of virtual work (PVW). This important question is, of course, connected with the origin of the concept of kinematics and its relation with the concept of dynamics. Now it is widely accepted that the Principle of Virtual Velocities (later called Virtual Work) was known in a geometrical form by the author of the Greek text Mechanica Problemata (The Mechanical Problems). Indeed, this text does not appear to be a theoretical treatise but rather a collection of solved exercises, mainly concerning statics, the functioning of machines and some dynamics. In this paper we present our exegesis of the first three problems of the Mechanica Problemata, because we believe that deeply understanding its content may allow us to clarify the Greek origin of the Principle of Virtual Work, to locate in space and time the birth of mathematized mechanics and to prove that Renaissance mechanics derives from Greek sources.

It is usually accepted in geophysics (and in civil engineering) that linear models can be used for describing an earthquake and the consequent seismic waves’ propagation. However, the large deformation experienced by the soil in these situations suggests that this paradigm requires more critical consideration. In fact, we claim that, in the vicinity of some discontinuities (that are common in all the geophysical applications of continuum models), the corresponding strain concentrations make the hypothesis of small deformation to be inadequate. In this paper, we verify the inappropriateness of the linear paradigm in a simple but reasonable case, with a view to a future application of this study to the effects of the 2009 L’Aquila earthquake. To this aim, we start with an analysis which is restricted to a two-dimensional body (i) with an inhomogeneity that resembles the Aterno River Valley, central Italy and (ii) with a non-linearity that is the most simple one, choosing the strain energy to be quadratic in the non-linear measures of deformation. More precisely, we consider a 2D piecewise homogeneous domain and a material that is viscoelastic isotropic and geometrically non-linear. We apply, to the bottom of such a domain, a seismic excitation and calculate the differences in the response between the linear and the geometrically non-linear cases. Using a suitably designed numerical model, we prove that, as conjectured, these differences not only originate near the pre-defined geometrical discontinuities but also propagate throughout the rest of the domain. Moreover, we find numerical predictions of the frequency ratios and ground acceleration time dependence and amplitude that produce, in the case of non-linear models, predictions which are closer to experimental evidence than those obtained using the corresponding linear model.

An inextensible 1D continuum whose deformation energy purely depends on the gradient of the associated curvature is introduced to describe the behavior of Zigzagged Articulated Parallelograms with Articulated Braces truss structures (ZAPAB structures) after homogenization. We choose a particular ZAPAB structure in which all but one of the constituting bars of the basic module do not change their length under applied loads. This judicious choice allows us to verify, through numerical simulations, that the corresponding 1D continuum indeed has a deformation energy that depends solely on the derivative of curvature. Thus, by employing a best-fitting approach based on the least squares method, we numerically identify the best stiffness coefficient (in the least squared sense) associated with the energy contribution due to the gradient of curvature, termed as double-bending stiffness. The presented simulations consider the case of uniformly distributed applied dead loads, and reveal a strong match between the current configurations of the proposed 1D continuum model, obtained numerically through the Finite Element Method, and the current configurations of the ZAPAB structure (for a selected number of basic modules), obtained through a discrete numerical approach, with the curves coinciding up to certain intrinsic error. These results require the development of an analytical micro–macro identification procedure. ZAPAB structures facilitate advances in the synthesis of tailored materials and the n-th gradient theory. We adopt a theory-driven approach with the expectation of devising materials with exotic behaviors. Specifically, we anticipate that material lines capable of not storing deformation energy under uniform bending (constant curvature) will be obtained after homogenization, thereby paving the way for future work that introduces complex materials built upon them. Our discussion is inspired by well-known pantographic structures, which serve as archetypes of second gradient materials designed in such a way that no deformation energy is stored under uniform extension.

Mechanical metamaterials consist of specially engineered features designed to tailor and enhance the mechanical properties of their constituent materials. In this context, 2D pantographic fabrics have gained attention for their unique deformation behavior, providing remarkable resilience and damage tolerance. This study explores micro-metric metamaterials with 3D pantographic motifs, aiming to transfer these properties to small scales. 3D micro-metric structures were designed using 2D pantographic fabrics arranged in multiple layers, each featuring unit cells with quasi-perfect pivots. Relatively large specimens of 3D micro-metric pantographs, measuring 158 m x 250 m x 450 m, were fabricated in various configurations using two-photon polymerization. These specimens were mechanically characterized through in-situ scanning electron microscopy microindentation under conditions of cyclic deformation. Structural failures were subsequently assessed via helium-ion microscopy. The 3D micro-metric pantographs exhibited complex mechanical properties, some aligning with those of 2D pantographic fabrics, while new properties, such as a dissipative response and softening, were identified. Nonetheless, the 3D micro-metric pantographs demonstrated great resilience against deformation and enhanced resistance to undesired out-of-plane motions, indicating their potential for novel applications in advanced engineering fields. Additionally, the findings can potentially lead to optimizing and enriching theoretical models describing the mechanical behavior of pantographic metamaterials.

We consider deformations of an elastic body having initially a spherical shape. Assumed deformation energy depends on the first and second gradient of displacements. We apply an equatorial line density of dead loads, that are forces per unit line length directed in radial direction and applied along the equator of the sphere. We restrict ourselves our analysis to the case of linearized second strain gradient isotropic elasticity (for which the more general energy was determined by Mindlin) with only one characteristic length. Differently to what happens in first gradient continua, i.e. in classic linear elasticity, we show that for the particular class second gradient continua considered here these forces do not determine infinite displacements in the direction of applied dead line forces. Instead, using a series method for the solution of the considered boundary-value problem, we demonstrate that the displacements are finite. So in the deformed configuration there is not the formation of an edge at the material points where the forces are applied. Further investigations are therefore needed for establishing if this elastic-regime edge formation is made possible: (I) either in the case of more general linear elastic constitutive equations or (II) only when large deformations are considered or (III) if non-elastic phenomena are involved.