Two theorems on electromagnetic bistatic scattering

For an axially incident plane wave, two theorems are stated in which sufficient conditions are imposed on the constitutive parameters epsilon and mu of a three-dimensional scatterer to ensure the identity of scattering patterns in orthogonal planes, both near and far zone, such as E and H planes. The theorems represent an extension of existing scattering theorems that apply to bodies of revolution. The theorems are proven analytically, and the results are validated through detailed finite-difference-time-domain and method-of-moments computer simulations on a few noncanonical complex shapes characterized by quite general causal permittivity epsilon and permeability mu functions. The selected materials have Lorentzian functional forms and encompass ordinary materials as well as left-handed materials as special cases.     

Two shift theorems leading to lower bounds to eigenvalues

Summary Two theorems are proved by which the other bound can be obtained from the Rayleigh quotient, i.e., the bound which cannot be obtained by the Rayleigh-Ritz method.The method presented here utilizes a shift operation which is the redistribution of weight functions in the Rayleigh quotient.     

Reciprocal theorems and variational theorems in nonlocal elastodynamics

The paper is concerned with the linear theory of nonlocal elastodynamics for inhomogeneous and anisotropic solids. Reciprocal theorems and variational theorems are established.     

Two bracketing theorems characterizing the eigensolution for the h-version of the finite element method

This paper demonstrates that the classical inclusion principle is in general not valid for the h-version of the finite element method. Whereas the inclusion principle is valid for second-order systems discretized by the h-version of the finite element method, provided linear interpolation functions are used as admissible functions, the principle is not valid for fourth-order systems. To characterize the computed eigenvalues for fourth-order systems discretized by the h-version of the finite element method, this paper formulates two bracketing theorems.     

Energy theorems in structural mechanics

Conclusion Corresponding relationships between certain incremental equilibrium equations and compatibility equations can be derived for structures undergoing gross deformation. Investigation of these relationships leads to the development of a new principle of virtual work and a form of energy called stress energy. The concept of stress energy has been developed by considering the potential energy of the forces applied to the structure or mechanism and its relationship to Libove's from of complementary energy and Charlton's virtual work method has been discussed.It is hoped that this paper helps to clarify the situation with regard to the use of virtual work methods and energy theorems as applied to structures undergoing gross deformation, whilst at the same time indicating a possible application to cable and suspension structures.     

On some classical theorems in micropolar elasticity

By means of suitable a priori estimates, some classical theorems for the system of micropolar elasticity are proved in arbitrary domains. No assumptions are made upon the behaviour of solutions at infinity and the coefficients of the equations may suitably grow at large spatial distance.     

Quantum Bochkov-Kuzovlev work fluctuation theorems

The quantum version of the Bochkov-Kuzovlev identity is derived on the basis of the appropriate definition of work as the difference of the measured internal energies of a quantum system at the beginning and the end of an external action on the system given by a prescribed protocol. According to the spirit of the original Bochkov-Kuzovlev approach, we adopt the 'exclusive' viewpoint, meaning that the coupling to the external work source is not counted as part of the internal energy. The corresponding canonical and microcanonical quantum fluctuation theorems are derived as well, and are compared with the respective theorems obtained within the 'inclusive' approach. The relations between the quantum inclusive work w, the exclusive work w(0) and the dissipated work w(dis), are discussed and clarified. We show by an explicit example that w(0) and w(dis) are distinct stochastic quantities obeying different statistics.     

Structural theorems for symbolic summation

Starting with Karr’s structural theorem for summation—the discrete version of Liouville’s structural theorem for integration—we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can be applied for harmonic sums which arise frequently in particle physics.     

Some theorems in the theory of microstretch thermopiezoelectricity

The electromagnetic theory of microstretch thermoelasticity is an adequate tool to describe the behaviour of porous bodies, animal bones and solids with deformable microstructures. In this paper we study the linear theory of microstretch thermopiezoelectricity. First, we establish a reciprocity relation which involves two processes at different instants. This relation forms the basis of a uniqueness result and a reciprocal theorem. Then, we study the continuous dependence of solutions upon initial data and body loads. A variational characterization of solutions is also presented. Finally, we investigate the effect of a concentrated heat supply and the effect of a concentrated volume charge density in an unbounded homogeneous and isotropic body.     

Bounding theorems for stress intensity factors

Bounding theorems for the changes in elastic energy due to a crack in a finite body and crack tip Stress Intensity Factors are established by use of the variational principles of the theory of elasticity. It is shown that the change in elastic energy due to a crack in a finite body is larger than the change in energy due to the same crack in an infinite body with the same boundary conditions when stresses are prescribed on the external boundary, and smaller when displacements are prescribed. The energy changes can be expressed as functions of the crack tip Stress Intensity Factors and for special loadings, bounds for single mode Stress Intensity Factors are obtained. The obtained inequalities are in agreement with known numerical solutions of finite cracked bodies.     

Some theorems on generalized thermoelastic diffusion

The objective of the present work is to establish a convolutional type (Gurtin in Arch. Rat. Mech. Anal. 16:34?C50, 1964) variational theorem and a reciprocity theorem for the linear theory of generalized thermoelastic diffusion for homogeneous and isotropic elastic solids.     

Matrix theorems of multiconductor transmission lines

Abstract

The purpose of this article is to present three matrix theoremes pertaining to the analysis of electrical multiconductor transmission lines. In principle, the formulations are also applicable to coupled mechanical systems.     

Uniqueness and reciprocity theorems in generalized linear micropolar thermoviscoelasticity

A model of the equations of generalized linear micropolar thermoviscoelasticity is given. The formulation is applied to the coupled theory as well as to five generalizations, the Lord-Shulman theory with one relaxation time, the Green-Lindsay theory with two relaxation times, the Green-Naghdi theories of type II (without energy dissipation) and of type III, and the Chandrasekharaiah-Tzou theory with dual-phase-lag. Using Laplace transforms, a uniqueness theorem for this model is proved, restrictions on relaxation functions are deduced and the dynamic reciprocity theorem is derived. The cases of generalized linear micropolar thermoviscoelasticity of Kelvin-Voigt model, generalized linear micropolar thermoelasticity, generalized thermoviscoelasticity and generalized thermoelasticity can be obtained from the given general model.     

Circle theorems for inviscid steady flows

General circle theorems which localize the complex eigenfrequencies arising in the linear stability analysis of conservative steady flows are given. Howard's circle theorem for incompressible plane parallel flow is contained as a special case. Two applications are considered: swirling flow of an inviscid incompressible fluid, and rotating flow of an inviscid, incompressible, perfectly conducting magnetofluid with an axial magnetic field. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.     

Continuous dependence theorems in linear elasticity on exterior domains

By using the weight function approach, we prove some continuous dependence theorems for solutions to the equations of linear elasticity in an exterior domain in $\text{R}$³. Among other things, these theorems allow the elasticities to be unbounded as well as the density to be infinitesimal at large spatial distance from the boundary.     

Limit theorems for sequences of random trees

We consider a random tree and introduce a metric in the space of trees to define the “mean tree” as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law.     

Minimum theorems in 3D incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142, 1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144, 2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.     

Two neighbouring quasi-trapezoidal holes in discs with minimum s.c.f.

Using a two dimensional photoelastic technique, hole shapes have been optimised in diametrically loaded circular discs with two neighbouring holes leading to minimum stress concentration factor (s.c.f.). Two neighbouring holes symmetrically located side by side with the load axis perpendicular to the line joining the centres of the holes are considered. Results are given for a range of disc diameter/hole diameter ratios (11.44≥D/d≥4.16). Optimised quasi-trapezoidal hole geometries, stress distributions around these holes and the effect of tilting the load direction are presented. In comparison with circular holes, the s.c.fs. have been reduced up to about 14% with quasi-trapezoidal holes at regions of peak stresses and up to 23% at peak tensile stress regions.     

The area theorems for two-core nonlinear directional couplers

The area theorems, describing propagation of ultrashort pulses of resonant laser radiation in a two-core nonlinear directional coupler, are formulated for the first time. It is shown that such a system is characterized by a finite, bounded above set of areas of the stationary propagating pulses. We also predict quantization of the limit cycles and possible existence of the regime of periodic complete transfer of the radiation energy from one fiber to another.