This article examines the force deduced by Ampère (1775–1826). In 1820 Oersted conducted his revolutionary experiment: a current-carrying wire capable of deflecting a compass needle. To explain this phenomenon Ampère presented two crucial hypotheses: he postulated the existence of forces between current-carrying conductors and suggested that electric currents existed not only in conductive wires, but also inside magnets and even within the Earth. To deduce his force Ampère developed his famous cases of equilibrium in which a mobile circuit is kept at rest between two opposing forces or between two opposing torques. In 1822 Ampère obtained his final expression, a central force consistent with Newton’s third law of motion. According to Ampère’s force, the action between two current elements is proportional to the product of their intensities, to the product of their infinitesimal lengths, and varies inversely with the square of their distance. Furthermore, this force depends on three angles. According to Maxwell, Ampère’s force should always remain the most important formula of electrodynamics. However, despite this positive evaluation, Ampère’s force disappeared from modern textbooks, being replaced by Grassmann’s theoretical force from 1845, which does not comply with Newton’s third law. In 1822 Ampère predicted the existence of longitudinal repulsive forces between two collinear and parallel elements with currents in the same direction. This force should not exist according to Grassmann’s law. Ampère created the bridge experiment to test this prediction and it was successful. Maxwell, while aware of both expressions (Ampère’s and Grassmann’s forces), expressly preferred Ampère’s force.
Nonlocal constitutive models have been widely developed to account for size/gradient effects or to circumvent the difficulties associated with excessive spatial localization. Two main thermodynamic frameworks have been proposed to incorporate some information regarding the spatial distribution of internal variables in a consistent manner: formulations based on an extended energy flux and those relying on an extended entropy flux. Although these approaches are often regarded as equivalent, their respective implications on the thermodynamic structure of the governing equations remain insufficiently understood. In this work, a comparison between nonlocal models formulated with an extended energy flux and with an extended entropy flux is conducted within the general framework of continuum thermodynamics. Both gradient-based and integral-based nonlocal approaches are considered in a unified manner. The corresponding energy and entropy balance equations are derived, along with the associated dissipation inequalities, evolution equations for internal degrees of freedom, and heat diffusion equations. It is shown that the two formulations lead to identical evolution equations under uniform temperature conditions. However, some differences arise in the presence of temperature gradients, notably through the coupling between nonlocal interactions and the temperature field in the extended entropy flux formulation. The analysis also clarifies under which assumptions some formulations presented as extended entropy flux approaches should rather be interpreted as extended energy flux frameworks. The results provide a theoretical basis to assess the relevance and limitations of extended energy and entropy flux formulations of nonlocal constitutive models in thermomechanically coupled problems.
The aim of this work is to study theoretically and numerically the effect of the inhomogeneity of the yield stress on void coalescence for isotropic porous ductile materials. Such inhomogeneities may arise due to strain hardening prior to void coalescence. Sequential limit analysis is applied to a cylindrical unit cell containing a coaxial cylindrical void in a von Mises matrix material with a radially dependent yield stress. An estimate of the coalescence criterion is obtained for combined tensile and shear loading. The criterion relies on 1D integrals, but two approximations are also provided to obtain analytical expressions. Numerical limit analysis based on FFT simulations is performed to get exact, up to numerical errors, coalescence stresses, and to assess the theoretical expressions. A good agreement is found between the analytical coalescence criterion and the numerical results for a large set of void shape, porosities and yield stress distributions. The coalescence criterion is finally used to assess the effect of strain-hardening on the orientation of void coalescence plane, as well as to describe the full yield locus — accounting for both void growth and coalescence — of porous isotropic materials, for axisymmetric loading conditions.
This paper introduces a novel mathematical framework, termed the pseudo-continuous media approach, for treating strong discontinuities (such as cracks, interfaces, and contact surfaces) within continuum mechanics. Traditional methods typically require explicit geometric representation of discontinuities, leading to complex meshing and algorithmic challenges. In contrast, our method leverages distribution theory to embed the discontinuity conditions directly into the governing equations, allowing the problem to be posed on a simpler, uncracked domain. Our distributional approach is conceptually aligned with the rigorous framework of Special Functions of Bounded Variation (SBV) which provides a natural setting for functions with jump discontinuities. We present the theoretical foundation and the corresponding finite element formulation. The method demonstrates significant advantages in geometric simplification, mathematical elegance, and numerical robustness compared to traditional techniques like the eXtended Finite Element Method (XFEM), positioning itself as a novel numerical methodology that implements the SBV philosophy using a standard finite element framework.
The 2025 commemoration of the 250th anniversary of André-Marie Ampère’s birth in 1775 was inspired in large part by the legacy he left through his creation of the new branch of physics he called electrodynamics. Primarily accomplished between 1820 and 1826, Ampère’s achievement was precipitated by Hans Christian Oersted’s 1820 discovery of an unexpected interaction between a magnetic needle and an electric current. While many factors contributed to Ampère’s successful response to Oersted’s discovery, there was some serendipity to its timing that is best appreciated by locating it within the biographical context of Ampère’s academic career, his tumultuous personal life, his extensive philosophical investigations, and his tenuous religious faith. It is noteworthy that Oersted’s discovery came at a point in time when Ampère’s status on all these fronts was relatively stable. The fortuitous state of equipoise he enjoyed in 1820 facilitated his engagement with electrodynamics to an extent that would have been extremely difficult to achieve during the previous decade. From Ampère’s religious perspective, the perspicacious timing of Oersted’s discovery could readily be attributed to divine providence. The timely balance between the romantic and analytic components of Ampère’s mentality in 1820 is aptly symbolized by the first of the four equilibrium apparatuses he invented to ground his derivation of the force law at the heart of his electrodynamics.
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.
This study presents a finite element (FE) simulation framework for determining the shear wave velocity of the feline cornea and comparing it with human, canine, and keratoconic corneas. A hyper-viscoelastic material model was implemented in ABAQUS, combining a Neo-Hookean hyperelastic formulation with a generalized Maxwell viscoelastic model represented by a Prony series. The corneal geometry incorporated species-specific thickness, curvature, and diameter parameters under physiological intraocular pressure (15 mmHg). Shear wave propagation was simulated using excitation pressures of 15 000-30 000 Pa. The calculated shear wave velocity in the feline cornea ranged from 5.26 m/s to 5.43 m/s, showing an increasing trend with excitation pressure. Comparative results indicated the following interspecies relationship: c(s)(,keratoconus) < c(s)(,human) < c(s)(,feline) < c(s)(,canine). These findings demonstrate that the feline cornea exhibits biomechanical characteristics closer to the canine cornea, reflecting similar hyper-viscoelastic responses. The model provides a validated computational basis for evaluating corneal stiffness and supports future shear wave elastography studies in comparative and veterinary ophthalmology.
In the context of the growing industrial use of architected materials, it is crucial to predict the critical loads at which their mechanical response transitions from linear to nonlinear. The sources of nonlinearity can be multiple, including material nonlinearity and/or multiscale buckling. The set of critical loadings defines a surface in stress or strain space that delineates the region within which the behavior remains elastic, also referred to as the linearity domain. This article presents a numerical investigation of the connections between the symmetries of a periodic architected material and the corresponding symmetries of its linearity domain. This investigation yields novel insights: (i) the rotational symmetry order of the linearity domain is directly related to that of the underlying architected material; (ii) the material's chirality manifests itself in the geometry of its linearity domain, which in this case appears tilted; and (iii) the angle of this tilt is correlated with the angle of the parent mesostructure.
We derive various models of assemblies of slender anisotropic linearly elastic beams through an asymptotic analysis taking into account a triplet of small parameters associated with the slenderness of the beams but also the thinness and the stiffness of a very thin third body which connects them. Our models allow the description of the mechanical constraint of the linkage between two beams which strongly depends on the relative orders of magnitude of the previous parameters.
The aim of this work is to revisit the formulation of FFT-based methods for heterogeneous materials in the periodic setting. These numerical methods are based on the iterative resolution of an auxiliary problem involving a reference homogeneous material and a polarization tensor. Assuming a description of the fields by Fourier series, we show the equivalence between three discretization approaches based either on the strong and the weak formulation of the problem. A special emphasis is put on the representation of the local fields described by Fourier series including the possibility of using non-uniform grid. Numerical experiments are performed on a model problem of conductivity with a checkerboard microstructure for which an analytical solution allows to assess the effect of Fourier modes together with the grid discretization. The occurrence of oscillations is finally addressed by studying (generalized) discrete Green operators (still in the context of Fourier series) and interface spreading approaches based on smoothing techniques.
X-ray Computed Tomography (XCT) combined with Digital Volume Correlation (DVC) enables for internal displacement and strain measurements in deforming materials. The long acquisition time of tomographic scans restricts analyses to a few static loading steps and prevents from time-dependent or nonlinear mechanism quantification. Projection-based DVC (P-DVC) addresses this limitation by exploiting projections acquired during continuous loading, providing substantially higher temporal sampling. This study assesses a spacetime framework of projection-enhanced DVC for the in situ investigation of a 3D-printed lattice. Global DVC displacement fields are used to construct reduced spatial bases, namely, (i) a pure data-driven basis, (ii) a mechanics-only basis derived from an elastic compression solution, and (iii) a hybrid mechanics-data basis combining both. P-DVC then exploits the projections to identify the temporal amplitudes associated with these modes, thereby reconstructing the time-resolved kinematics with a resolution of 3.7 s, nearly two orders of magnitude faster than conventional 3D-to-3D DVC. Nonlinear mechanisms were detected ahead of their clear expression in tomographic reconstructions. The hybrid reduced-order approach provided interpretable kinematic fields while preserving the predictive accuracy of data-driven strategies. The projection-enhanced DVC framework thus enabled for temporally dense and spatially resolved in situ measurements, providing the type of rich datasets required for the identification and learning of complex constitutive models.
When two semi-infinite periodic media are joined together, a localized interface mode may exist, whose frequency belongs to their common band gap. Moreover, if certain spatial symmetries are satisfied, this mode is topologically protected and thus is robust to defects. A method has recently been proposed to identify the existence and the frequency of this mode, based on the computation of surface impedances at all the frequencies in the gap. In this work, we approximate the surface impedances thanks to high-frequency effective models, and therefore get a prediction of topologically protected interface states while only computing the solution of an eigenvalue problem at the edges of the bandgaps. We also show that the nearby eigenvalues high-frequency effective models give rise to a better approximation of the surface impedance.
This paper presents an innovative hybrid approach that combines Hidden Markov Models (HMM) with Radial Basis Function Neural Networks (RBFNN) for the automatic classification of mechanical faults in rolling element bearings using vibration signal analysis. The signals are sourced from the well-established database (https://engineering.case.edu/bearingdatacenter/welcome), widely used in fault diagnosis research. Raw signals are preprocessed to extract relevant features across time, frequency, and time-frequency domains, including wavelet packet decomposition. To enhance classification robustness and reduce computational complexity, dimensionality reduction is performed using Principal Component Analysis (PCA), complemented by Fisher score-based feature selection. HMMs are trained to capture the temporal dynamics of the signals, while RBFNNs leverage the reduced feature space for fine-grained classification. A comprehensive performance comparison is conducted between standalone HMM and RBFNN models, as well as their integration within the hybrid HMM-RBFNN system. Experimental results demonstrate that the proposed hybrid method significantly improves classification accuracy, highlighting its potential for industrial predictive maintenance applications.
This review traces the historical trajectory of electricity in agriculture, from the earliest observations of electrical phenomena to the emergence of cold plasmas. Looking back to Antiquity and then to the Enlightenment, it underlines Abb & eacute; Bertholon's 18th-century efforts to channel atmospheric electricity to stimulate crops, using devices such as the electro-v & eacute;g & eacute;tom & egrave;tre. Although these early electroculture experiments relied on neither quantitative dosimetry nor rigorous methodology, they foreshadowed the idea of a controlled transfer of electrical energy to plants. Then the review examines the historical development of galvanism, electrochemistry, and the physics of gaseous discharges throughout the 19th and 20th centuries, which collectively laid the foundations for contemporary cold-plasma technologies. In the 21st century, plasma agriculture has emerged as an interdisciplinary approach integrating electrical, chemical, radiative, thermal, and fluid-mechanical effects. Applications include seed treatment (preconditioning, seed priming), stimulation of plant growth, soil and water treatment, and decontamination of agri-food products. The review thus reassesses Abb & eacute; Bertholon's contributions as those of a methodological precursor and shows how his intuition of a "vivifying electricity" resonates with modern cold-plasma science. Finally, it argues that plasma agriculture can transform an Enlightenment intuition into a reproducible experimental framework for sustainable agriculture and food safety.
In this article, we develop an analytical approach to characterize the breathing mode vibration of a thermoelastic nanosphere submerged in an incompressible fluid. The inclusion of temperature is under the concept of heat wave and the energy equation is combined with elastic theory in the fluid-structure interaction method. The bi-harmonic function is derived from the coupling of the velocity and temperature fields by the coupled thermoelasticity theory. Whereas for an incompressible fluid, these two fields are decoupled. This leads to the convenience of separating thermal conduction and dynamic viscosity parts in the frequency equation. The validation of frequency equation is confirmed by comparing other literatures. The thermal damping and viscosity are represented by P & eacute;clet number and Reynolds number respectively. The effects of two parameters on the vibration of the system are analyzed with multiple plots. The analysis could be a useful interpretation of experimental observation and an applicable measurement for vibrational and rheological properties of solids and fluids.
Digital Rock Physics (DRP) analysis is a widely employed technique for predicting transport parameters from 3D images of core samples. However, the effects of image resolution and spatial discretization of the DRP mesh grid have rarely been systematically studied in detail. To address this issue, we examine a generic sand pack, representing a homogeneous porous medium. This sample was imaged using X-ray micro-tomography at three different spatial resolutions (6, 3, and 1.5 microns/voxel). Permeability is then numerically evaluated by solving the Stokes flow equations using a finite volume method. The processed meshes for converged macroscopic evaluations consist of 105 million to 58 billion cells, necessitating the alternative use of a two-step upscaling method. By employing both methods, this study analyzes the respective influences of image resolution and spatial discretization. Significant effects are observed from both image resolution and spatial discretization, the analysis of which can contribute to identifying optimal strategies for enhancing the accuracy of permeability evaluation.
In this paper, we compare four algorithms solving the quasi-static contact between one elastic body and one rigid obstacle in two dimensions. With the Coulomb's law of friction, this example exhibits multiple solutions if the friction coefficient is larger than 3. After describing the numerical methods tested in this paper, we study the influence of the parameters of the algorithms on the nature of the obtained solution at convergence. On this academic example, we also compute two existing criteria on the uniqueness of the solution. Finally, for friction coefficient larger than 3, we compute a new sliding solution and observe that not all approaches are able to find it.
Shear fracture studies of three-dimensional (3D) printing polymers with interfaces were rarely reported due to their complicated mechanics and material issues. In this study, a short-beam shear fracture approach was employed to characterize the mode-II shear fracture toughness of polyamide specimens of three printing surface angles made with selective laser sintering (SLS). Results show that a pure shear crack only existed if the initial crack propagated along the printing interface. In other cases, initial cracks kinked right after crack initiation, so no valid shear fracture toughness was measured. A simple model based on linear elastic fracture mechanics including anisotropic fracture toughnesses was proposed to predict the crack kinking angles. The prediction agreed with the measurements well and was more reasonable than the prediction based on the maximum tensile stress criterion.
The Double Generator Boundary Augmented bracket structure is a double generator bracket formulation, tailored to model continuum thermodynamics. Based on the idea of bracket generated formulations, this framework encompasses balance principles and thermodynamics laws within a unique expression. The present paper develops the methodology to derive this structure from classical equations of continuum thermodynamics for two examples. We consider first a unidimensional small strains generalized standard material, with a general quadratic dissipation potential. Then, we consider the example of large strain thermo-visco-elastodynamics, within the multisymplectic framework. We derive, for the first time, a multisymplectic Poisson bracket for thermo-(visco)-elastodynamics. Eventually, both formulations are shown to recover exactly balance principles and thermodynamics laws. This paper sets grounds necessary to develop variational integrators from the Double Generator Boundary Augmented bracket structure.
Pantographic unit cells are well-known and used both for deployable structures and metamaterials design. It is surely less known that with minor changes, moving the nodes in the reference configuration or using angulated elements, this unit cells can be used to build curved shapes in the strain-free configuration. Curved pantographs are employable in several technical applications. One interesting application is related with robot arms. The peculiar characteristic of pantographic structures, linear or curved, is the existence of a floppy mode, i.e. a zero energy mode, which ensures the existence of a branch of the equilibrium path without strain in all its parts. This characteristic provides the deployability of the pantographic structures and several exotic mechanical behaviours in metamaterials based on the same pattern. We discuss some results obtained by a mechanical digital twin capable of: (i) verifying the deployability, i.e. the existence of a floppy mode when it is not prevented by constraints, of the considered scheme checking for the whole equilibrium path the absence of strain on all the springs modelling the problem; (ii) providing some information, useful for a preliminary structural design, about the mechanical behaviour of a simple structural scheme in the hypothesis of large displacements when the floppy mode is prevented by a large enough number of constraints.
