Physics

In an effort that puts together a paper by Plebanski[1] with a matrix approach to the solution of Maxwell's equations in flat space by Moses[2], Maxwell's equations in 2 + 1 dimensional curved space are solved in two separate cases of the metric given by: a.Banados,Teitelboim and Zanelli[3] and b.Deser,Jackiw and 'tHooft[4] and Clement[5] to obtain the respective time - independent solutions;an extension to time - dependent solutions with the same point of view is also briefly indicated.

A geometrically nonlinear sandwich beam model founded on the modified couple stress Timoshenko beam theory with Kármán kinematics is derived and employed in the analysis of periodic sandwich structures. The constitutive model is based on the mechanical behavior of sandwich beams, with the bending response split into membrane-induced and local bending modes. A micromechanical approach based on the structural analysis of a unit cell is derived and utilized to obtain the stiffness properties of selected prismatic cores. The model is shown to be equivalent to the classical thick-face sandwich theory for the same basic assumptions. A two-node finite element interpolated with linear and cubic shape functions is proposed and its stiffness and geometric stiffness matrices are derived. Three examples illustrate the model capabilities in predicting deflections, stresses and critical buckling loads of elastic sandwich beams including elastic size effects. Good agreement is obtained throughout in comparisons with more involved finite element models.

The two full body problem concerns the dynamics of two spatially extended rigid bodies (e.g. rocky asteroids) subject to mutual gravitational interaction. In this note we deduce the Euler-Poincare and Hamiltonian equations of motion using the geometric mechanics formalism.

A new method to calculate the electric field inside a spherical shell with surface charge in terms of solid angle is presented. The integral can be readily carried out without invoking special functions typically used for this classical problem. For a flat surface of uniform charge density, the electric field normal to the surface is shown to be proportional to the solid angle subtended by the surface only.

Natural rubber (NR) is the most commonly used elastomer in the automotive industry thanks to its outstanding fatigue resistance. Strain-induced crystallization (SIC) is found to play a role of paramount importance in the great crack growth resistance of NR [1]. Typically, NR exhibits a lifetime reinforcement for non-relaxing loadings [2-3]. At the microscopic scale, fatigue striations were observed on the fracture surface of Diabolo samples tested in fatigue. They are the signature of SIC [2,4,5]. In order to provide additional information on the role of SIC in the fatigue crack growth resistance of NR, striations are investigated through post-mortem analysis after fatigue experiments using loading ranging from-0.25 to 0.25. No striation was observed in the case of tests performed at 90{\textdegree}C. This confirms that the formation of striation requires a certain crystallinity level in the material. At 23{\textdegree}C, two striation regimes were identified: small striation patches with different orientations (Regime 1) and zones with large and well-formed striations (Regime 2). Since fatigue striations are observed for all the loading ratios applied, they are therefore not the signature of the reinforcement. Nevertheless, increasing the minimum value of the strain amplified the striation phenomenon and the occurrence of Regime 2.

The concept of an observer and their associated rest space is defined in a pre-metric (i.e., projective-geometric) context that relates to time+space decompositions of the tangent bundle to space-time. The transformation from one observer to another when the two are in a state of relative motion is then defined, and its relationship to the Lorentz transformation is discussed. The group of all linear transformations that preserve the observer quadric, which generalizes the proper-time hyperboloid in Minkowski space, is defined and the reductions to some of its subgroups are described, as well as its extension to the group that preserves the fundamental quadric, which generalizes the light cone.

Direct current (DC) circuits are usually taught in upper-level physics curricula, and Kirchhoff's laws are stated and used to solve the steady-state currents. However, students are not often introduced to alternative techniques, such as variational principles, for solving circuits. Many authors have tried to derive the steady-state distribution of currents in circuits from variational principles, and the initial attempts were carried out by great physicists, such as Kirchhoff, Maxwell and Feynman. In this article, we shall review such variational principles in physics and illustrate how they can be used to solve DC circuits. We will also explore how they are related, in a fundamental way, to the entropy production principles of irreversible thermodynamics. We believe that the introduction of variational alternatives for solving circuits can be a good opportunity for the instructor to present the techniques of the calculus of variations and irreversible thermodynamics to students.

The fact that repulsive Rutherford scattering casts a paraboloidal shadow is rarely exploited in introductory mechanics textbooks. Another rarely used construction in such textbooks is the Hamilton vector, a cousin of the more famous Laplace-Runge-Lenz vector. We will show how the latter (Hamilton's vector) can be used to explain and clarify the former (paraboloidal shadow).

The paper explores the shadow of the repulsive Rutherford scattering - the portion of space entirely shielded from admitting any particle trajectory. The geometric properties of the projectile shadow are analyzed in detail in the fixed-target frame as well as in the center-of-mass frame, where both the charged projectile and the charged target cast their own respective shadows. In both frames the shadow is found to take an extremely simple, paraboloidal shape. In the fixed-target frame the target is precisely at the focus of this paraboloidal shape, while the focal points of the projectile and target shadows in the center-of-mass frame coincide. In the fixed-target frame the shadow takes on a universal form, independent of the underlying physical parameters, when expressed in properly scaled coordinates, thus revealing a natural length scale to the Rutherford scattering.

A well-posed stress-driven mixture is proposed for Timoshenko nano-beams. The model is a convex combination of local and nonlocal phases and circumvents some problems of ill-posedness emerged in strain-driven Eringen-like formulations for structures of nanotechnological interest. The nonlocal part of the mixture is the integral convolution between stress field and a bi-exponential averaging kernel function characterized by a scale parameter. The stress-driven mixture is equivalent to a differential problem equipped with constitutive boundary conditions involving bending and shear fields. Closed-form solutions of Timoshenko nano-beams for selected boundary and loading conditions are established by an effective analytical strategy. The numerical results exhibit a stiffening behavior in terms of scale parameter.