Physics

We demonstrate how to derive Maxwell's equations, including Faraday's law and Maxwell's correction to Ampère's law, by generalizing the description of static electromagnetism to dynamical situations. Thereby, Faraday's law is introduced as a consequence of the relativity principle rather than an experimental fact, in contrast to the historical course and common textbook presentations. As a by-product, this procedure yields explicit expressions for the infinitesimal Lorentz and, upon integration, the finite Lorentz transformation. The proposed approach helps to elucidate the relation between Galilei and Lorentz transformations and provides an alternative derivation of the Lorentz transformation without explicitly referring to the speed of light.

A pedagogical derivation of the Huygens cycloidal pendulum, suitable for high-school students, is here presented. Our derivation rests only on simple algebraic and geometrical tricks, without the need of any Calculus concept.

An improved Iterative Physical Optics (IPO) image approximation method has been presented to dramatically increase the accuracy of the approximation and extend its applicability to PEC surfaces with smaller radii or larger curvatures. Starting from the first-order conventional PO image approximation, the IPO image approximation method iteratively correct the surface current to compensate the deviation of the electric field boundary condition on the PEC surfaces, making use of the local plane wave approximation. Numerical validations with two popular PEC surfaces, i.e., the parabolic dish antennas and the PEC spheres, are carried out and the results show that the IPO approximation method increases the surface current accuracy by more than two orders of magnitude, compared to the conventional PO image approximation method.

In this study, the Virtual Fields Method (VFM) is applied to identify constitutive parameters of hyperelastic models from a heterogeneous test. Digital image correlation (DIC) was used to estimate the displacement and strain fields required by the identification procedure. Two different hyperelastic models were considered: the Mooney model and the Ogden model. Applying the VFM to the Mooney model leads to a linear system that involves the hyperelastic parameters thanks to the linearity of the stress with respect to these parameters. In the case of the Ogden model, the stress is a nonlinear function of the hyperelastic parameters and a suitable procedure should be used to determine virtual fields leading to the best identification. This complicates the identification and affects its robustness. This is the reason why the sensitivity-based virtual field approach recently proposed in case of anisotropic plasticity by Marek et al. (2017) [1] has been successfully implemented to be applied in case of hyperelasticity. Results obtained clearly highlight the benefits of such an inverse identification approach in case of non-linear systems.

In this paper, a new inverse identification method is developed from full kinematic and thermal field measurements. It consists in reconstructing the heat source from two approaches, a first one that requires the measurement of the temperature field and the value of the thermophysical parameters, and a second one based on the measurement of the kinematics field and a thermo-hyperelastic model that contains the parameters to be identified. The identification does not require any boundary conditions since it is carried out at the local scale. In the present work, the method is applied to the identification of hyperelastic parameters from a heterogeneous heat source field. Due to large deformation undergone by the rubber specimen tested, a motion compensation technique is developed to plot the kinematic and thermal fields at the same points before reconstructing the heat source.

A second-gradient elastic (SGE) material is identified as the homogeneous solid equivalent to a periodic planar lattice characterized by a hexagonal unit cell, which is made up of three different linear elastic bars ordered in a way that the hexagonal symmetry is preserved and hinged at each node, so that the lattice bars are subject to pure axial strain while bending is excluded. Closed form-expressions for the identified non-local constitutive parameters are obtained by imposing the elastic energy equivalence between the lattice and the continuum solid, under remote displacement conditions having a dominant quadratic component. In order to generate equilibrated stresses, in the absence of body forces, the applied remote displacement has to be constrained, thus leading to the identification in a \lq condensed' form of a higher-order solid, so that imposition of further constraints becomes necessary to fully quantify the equivalent continuum. The identified SGE material reduces to an equivalent Cauchy material only in the limit of vanishing side length of hexagonal unit cell. The analysis of positive definiteness and symmetry of the equivalent constitutive tensors, the derivation of the second-gradient elastic properties from those of the higher-order solid in the \lq condensed' definition, and a numerical validation of the identification scheme are deferred to Part II of this study.

Positive definiteness and symmetry of the constitutive tensors describing a second-gradient elastic (SGE) material, which is energetically equivalent to a hexagonal planar lattice made up of axially deformable bars, are analyzed by exploiting the closed form-expressions obtained in part I of the present study in the \lq condensed' form. It is shown that, while the first-order approximation leads to an isotropic Cauchy material, a second-order identification procedure provides an equivalent model exhibiting non-locality, non-centrosymmetry, and anisotropy. The derivation of the constitutive properties for the SGE from those of the \lq condensed' one (obtained by considering a quadratic remote displacement which generates stress states satisfying equilibrium) is presented. Comparisons between the mechanical responses of the periodic lattice and of the equivalent SGE material under simple shear and uniaxial strain show the efficacy of the proposed identification procedure and therefore validate the proposed constitutive model. This model reveals that, at higher-order, a lattice material can be made equivalent to a second-gradient elastic material exhibiting an internal length, a finding which is now open for applications in micromechanics.

We describe an improvement on the magnetic scalar potential approach to the design of an electromagnet, which incorporates the need to wind the coil as a helix. Any magnetic field that can be described by a magnetic scalar potential is produced with high fidelity within a Target region; all fields are confined within a larger Return. The helical winding only affects the field in the Return.

Elastic waves guided along bars of rectangular cross section exhibit complex dispersion. This paper studies in-plane modes propagating at low frequencies in thin isotropic rectangular waveguides through experiments and numerical simulations. These modes result from the coupling at the edge between the first order shear horizontal mode S H 0 of phase velocity equal to the shear velocity V T and the first order symmetrical Lamb mode S 0 of phase velocity equal to the plate velocity V P . In the low frequency domain, the dispersion curves of these modes are close to those of Lamb modes propagating in plates of bulk wave velocities V P and V T . The dispersion curves of backward modes and the associated ZGV resonances are measured in a metal tape using non-contact laser ultrasonic techniques. Numerical calculations of in-plane modes in a soft ribbon of Poisson's ratio ν≈0.5 confirm that, due to very low shear velocity, backward waves and zero group velocity modes exist at frequencies that are hundreds of times lower than ZGV resonances in metal tapes of the same geometry. The results are compared to theoretical dispersion curves calculated using the method provided in Krushynska and Meleshko (J. Acoust. Soc. Am 129 , 2011).

A lattice of elastic rods organized in a parallelepiped geometry can be axially loaded up to an arbitrary amount without distortion and then be subject to incremental displacements. Using quasi-static homogenization theory, this lattice can be made equivalent to a prestressed elastic solid subject to incremental deformation, in such a way to obtain extremely localized mechanical responses. These responses can be analyzed with reference to a mechanical model which can, in principle, be realized, so that features such as for instance shear bands inclination, or emergence of a single shear band, or competition between micro (occurring in the lattice but not in the equivalent solid) and macro (present in both the lattice and the equivalent continuum) instabilities become all designable features. The analysis of localizations is performed using a Green's function-based perturbative approach to highlight the correspondence between micromechanics of the composite and homogenized response of the equivalent solid. The presented results, limited to quasi-static behaviour, provide a new understanding of strain localization in a continuum and open new possibilities for the realization and experimentation of materials exhibiting these extreme mechanical behaviours. Dynamic homogenization and vibrational localization are deferred to Part II of this study.