Physics

In our analysis, we derive a model for a shell that is frictionally coupled to an elastic foundation. We use Kikuchi and Oden's model for Coulomb's law of static friction to derive a displacement-based static-friction condition for our shell model, and we prove the existence and the uniqueness of solutions with the aid of the works of Kinderlehrer and Stampacchia. As far as we are aware, this is the first derivation of a displacement-based friction condition, as only force or stress based friction conditions currently exist in the literature. For numerical analysis, we extend Kikuchi and Oden's model for Coulomb's law of static-friction to model a full two-body contact problem in curvilinear coordinates. Our numerical results indicate that if the shell has a relatively high Young's modulus or has a relatively high Poisson's ratio, and the contact region has a very high coefficient of friction or less curved, then the displacement field of the foundation predicted by both models are in better agreement.

A thermodynamic framework has been developed for a class of amorphous polymers used in fused deposition modeling (FDM), in order to predict the residual stresses and the accompanying distortion of the geometry of the printed part (warping). When a polymeric melt is cooled, the inhomogeneous distribution of temperature causes spatially varying volumetric shrinkage resulting in the generation of residual stresses. Shrinkage is incorporated into the framework by introducing an isotropic volumetric expansion/contraction in the kinematics of the body. We show that the parameter for shrinkage also appears in the systematically derived rate-type constitutive relation for the stress. The solidification of the melt around the glass transition temperature is emulated by drastically increasing the viscosity of the melt. In order to illustrate the usefulness and efficacy of the derived constitutive relation, we consider four ribbons of polymeric melt stacked on each other such as those extruded using a flat nozzle: each layer laid instantaneously and allowed to cool for one second before another layer is laid on it. Each layer cools, shrinks and warps until a new layer is laid, at which time the heat from the newly laid layer flows and heats up the bottom layers. The residual stresses of the existing and newly laid layers readjust to satisfy equilibrium. Such mechanical and thermal interactions amongst layers result in a complex distribution of residual stresses. The plane strain approximation predicts nearly equibiaxial tensile stress conditions in the core region of the solidified part, implying that a pre-existing crack in that region is likely to propagate and cause failure of the part during service. The free-end of the interface between the first and the second layer is subjected to the largest magnitude of combined shear and tension in the plane with a propensity for delamination.

Patch antennas are among the most popular radiating elements, yet their quasi-2-D structure reduces the degrees of freedom available to tailor their radiation pattern. To overcome this limitation, a possible solution consists in etching on a grounded substrate two concentric radiating elements and combining two modes (one for each element) with proper amplitude/phase relations. Although this technique leads, in principle, to an infinite number of possible configurations (i.e., each patch element can support an infinite number of modes), the theoretical and experimental verifications available in the literature are limited to the first two radiating modes (TM 11 and TM 21 ) of a circular patch antenna. Recently, we have shown that the design of this circular patch can be effectively performed by exploiting the topological properties of vortex fields and, in particular, by controlling the phase singularity exhibited by the higher order right-handed circularly polarized (RHCP) TM 21 mode of the circular patch. Since the number of RHCP higher order modes of a circular patch is infinite, we can in principle deal with an arbitrary number of phase singularity points, whose combined control leads to unprecedented possibilities to shape the radiation pattern of a circular patch. In this article, we present a complete design tool to determine the number and position of phase singularity points arising when combining the RHCP modes of a circular patch antenna and, eventually, manipulate them to synthesize the required radiation pattern. As a realistic application example, we show how the proposed tool can be used to effectively design a single antenna whose radiation pattern can be properly tailored to switch between two different states, i.e., a sector and a saddle shape, widely used in base stations for mobile and satellite communications, respectively.

This work presents a self-contained continuum formulation for coupled chemical, mechanical and thermal contact interactions. The formulation is very general and hence admits arbitrary geometry, deformation and material behavior. All model equations are derived rigorously from the balance laws of mass, momentum, energy and entropy in the framework of irreversible thermodynamics, thus exposing all the coupling present in the field equations and constitutive relations. In the process, the conjugated kinematic and kinetic variables for mechanical, thermal and chemical contact are identified, and the analogies between mechanical, thermal and chemical contact are highlighted. Particular focus is placed on the thermodynamics of chemical bonding distinguishing between exothermic and endothermic contact reactions. Distinction is also made between long-range, non-touching surface interactions and short-range, touching contact. For all constitutive relations examples are proposed and discussed comprehensively with particular focus on their coupling. Finally, three analytical test cases are presented that illustrate the thermo-chemo-mechanical contact coupling and are useful for verifying computational models. While the main novelty is the extension of existing contact formulations to chemical contact, the presented formulation also sheds new light on thermo-mechanical contact, since it is consistently derived from basic principles using only few assumptions.

A gauge theory of solids with conformal symmetry is formulated to model various electromechanical and magnetomechanical coupling phenomena. If the pulled back metric of the current configuration (the right Cauchy-Green tensor) is scaled with a constant, the volumetric part of the Lagrange density changes while the isochoric part remains invariant. However, upon a position dependent scaling, the isochoric part also loses invariance. In order to restore the invariance of the isochoric part, a 1-form compensating field is introduced and the notion of a gauge covariant derivative is utilized to minimally replace the Lagrangian. In view of obvious similarities with the Weyl geometry, the Weyl condition is imposed through the Lagrangian and a minimal coupling is employed so the 1-form could evolve. On deriving the Euler-Lagrange equations based on the action functional, we observe a close similarity with the governing equations for flexoelectricity under isochoric deformation if the exact part of 1-form is interpreted as the electric field and the anti-exact part as the polarization vector. Next, we model piezoelectricity and electrostriction phenomena by contracting the Weyl condition in various ways. Applying the Hodge decomposition theorem on the 1-form which leads to the curl of a pseudo-vector field and a vector field, we also model magnetomechanical phenomena. Identifying the pseudo-vector field with magnetic potential and the vector part with magnetization, flexomagnetism, piezomagnetism and magnetostriction phenomena under isochoric deformation are also modeled. Finally, we consider an analytical solution of the equations for piezoelectricity to provide an illustration on the insightful information that the present approach potentially provides.

A new continuous contact force model for contacting problems with regular or irregular contacting surfaces and energy dissipations in multibody systems is presented and discussed in this work. The model is developed according to Hertz law and a hysteresis damping force is introduced for modeling the energy dissipation during the contact process. As it is almost impossible to obtain an analytical solution based on the system dynamic equation, an approximate dynamic equation for the collision system is proposed, achieving a good approximation of the system dynamic equation. An approximate function between deformation velocity and deformation is founded on the approximate dynamic equation, then it is utilized to calculate the energy loss due to the damping force. The model is established through modifying the original formula of the hysteresis damping parameter derived by combining the energy balance and the law of conservation of linear momentum. Numerical results of five different continuous contact models reveal the capability of our new model as well as the effect of the geometry of the contacting surfaces on the dynamic system response.

Future developments of lighter, more compact and powerful motors-driven by environmental and sustainability considerations in the transportation industry-involve higher stresses, currents and electromagnetic fields. Strong couplings between mechanical, thermal and electromagnetic effects will consequently arise and a consistent multiphysics modeling approach is required for the motors' design. Typical simulations-the bulk of which are presented in the electrical engineering literature-involve a stepwise process, where the resolution of Maxwell's equations provides the Lorentz and magnetic forces which are subsequently used as the external body forces for the resolution of Newton's equations of motion. The work presented here proposes a multiphysics setting for the boundary value problem of electric motors. Using the direct approach of continuum mechanics, a general framework that couples the electromagnetic , thermal and mechanical fields is derived using the basic principles of thermodynamics. Particular attention is paid to the derivation of the coupled constitutive equations for isotropic materials under small strain but arbitrary magnetization. As a first application, the theory is employed for the analytical mod-eling of an idealized asynchronous motor for which we calculate the electric current, magnetic, stress and temperature fields as a function of the applied current and slip parameter. The different components of the stress tensor and body force vector are compared to their purely mechanical counterparts due to inertia, quantifying the significant influence of electromagnetic phenomena.

The electric field of a uniformly accelerated charge shows a plane of discontinuity, where the field extending only on one side of the plane, terminates abruptly on the plane with a finite value. This indicates a non-zero divergence of the electric field in a source-free region, implying a violation of Gauss law. In order to make the field compliant with Maxwell's equations everywhere, an additional field component, proportional to a δ -function at the plane of discontinuity, is required. Such a " δ -field" might be the electromagnetic field of the charge, moving with a uniform velocity approaching c , the speed of light, prior to the imposition of acceleration at infinity. However, some attempts to derive this δ -field for such a case, have not been entirely successful. Some of the claims of the derivation involve elaborate calculations with some not-so-obvious mathematical approximations. Since the result to be derived is already known from the constraint of its compliance with Maxwell's equations, and the derivation involves the familiar text-book expressions for the field of a uniformly moving charge, one would expect an easy, simple approach, to lead to the correct result. Here, starting from the electromagnetic field of a uniformly accelerated charge in the instantaneous rest frame, in terms of the position and motion of the charge at the retarded time, we derive this δ -field, consistent with Maxwell's equations, in a fairly simple manner. This is followed by a calculation of the energy in the δ -field, in an analytical manner without making any approximation, where we show that this energy is exactly the one that would be lost by the charge because of the radiation reaction on the charge, proportional to its rate of change of acceleration, that was imposed on it at a distant past.

In the framework of displacement-based equivalent single layer (ESL) plate theories for laminates, this paper presents a generic and automatic method to extend a basis higher-order shear deformation theory (polynomial, trigonometric, hyperbolic, ...) to a multilayer C 0 z higher-order shear deformation theory. The key idea is to enhance the description of the cross-sectional warping: the odd high-order C 1 z function of the basis model is replaced by one odd and one even high-order function and including the characteristic zig-zag behaviour by means of piecewise linear functions. In order to account for arbitrary lamination schemes, four such piecewise continuous functions are considered. The coefficients of these four warping functions are determined in such a manner that the interlaminar continuity as well as the homogeneity conditions at the plate's top and bottom surfaces are {\em a priori} exactly verified by the transverse shear stress field. These C 0 z ESL models all have the same number of DOF as the original basis HSDT. Numerical assessments are presented by referring to a strong-form Navier-type solution for laminates with arbitrary stacking sequences as well for a sandwich plate. In all practically relevant configurations for which laminated plate models are usually applied, the results obtained in terms of deflection, fundamental frequency and local stress response show that the proposed zig-zag models give better results than the basis models they are issued from.

We develop a direct geometric method to determine the orbital parameters and mass of a planet, and we then apply the method to Neptune using high-precision data for the other planets in the solar system. The method is direct in the sense that it does not involve curve fitting. This paper, thereby, offers a new pedagogical approach to orbital mechanics that could be valuable in a physics classroom.