Method Of Asymptotic Splittings Research Articles

Validation of Bredt's formulas by the method of asymptotic splitting

AbstractThe equations describing torsion of prismatic bars with thin‐walled closed cross sections, known as Bredt's formulas, are verified using the method of asymptotic splitting. In particular, the strong formulation of the Saint Venant problem of a straight beam is expanded asymptotically. At first, well‐known technical assumptions of the shear stress distribution are validated. Further, the influence of a transverse force acting on the beam is considered. This shear force causes a deformation of the cross section and therefore an adaption of Bredt's formulas. Two distinct formulations of the shear center, called the kinematic and the energetic shear center, are obtained. The latter are verified in numerical experiments.

Enveloping branes and brane-world singularities.

The existence of envelopes is studied for systems of differential equations in connection with the method of asymptotic splittings which allows one to determine the singularity structure of the solutions. The result is applied to brane-worlds consisting of a 3-brane in a five-dimensional bulk, in the presence of an analog of a bulk perfect fluid parameterizing a generic class of bulk matter. We find that all flat brane solutions suffer from a finite-distance singularity contrary to previous claims. We then study the possibility of avoiding finite-distance singularities by cutting the bulk and gluing regular solutions at the position of the brane. Further imposing physical conditions such as finite Planck mass on the brane and positive energy conditions on the bulk fluid, excludes, however, this possibility as well.

Hybrid asymptotic-direct approach to the problem of finite vibrations of a curved layered strip

A multi-stage approach for the mathematical modeling in the field of nonlinear problems of mechanics of thin-walled structures is the subject of the present paper. A combination of the asymptotic, direct, and numerical methods for consistent and efficient analysis of problems of structural mechanics is presented on the example of plane problem of finite vibrations of a thin curved strip with material inhomogeneity. The method of asymptotic splitting allows for a consistent dimensional reduction of the original two-dimensional continuous problem as the thickness is small: the leading-order solution of the full system of equations of the theory of elasticity results in a one-dimensional formulation of the reduced theory and a problem in the cross-section. The direct approach to a material line extends the results to the geometrically nonlinear range. The appropriate finite element formulation allows for practical applications of the theory; with the numerical solution of the reduced problem, we restore the distributions of stresses, strains, and displacements over the thickness. Numerically and analytically investigated convergence of the solutions of various problems in the original (two-dimensional) and reduced (one-dimensional) models as the thickness tends to zero justifies the analytical conclusion that the curvature and variation of the material properties over the thickness do not require special treatment for classical Kirchhoff’s rods. Further terms of the asymptotic expansion lead to a model with shear and extension, in which curvature appears in a nontrivial way.

Brane singularities and their avoidance

The singularity structure and the corresponding asymptotic behavior of a 3-brane coupled to a scalar field or to a perfect fluid in a five-dimensional bulk is analyzed in full generality using the method of asymptotic splittings. In the case of the scalar field, it is shown that the collapse singularity at a finite distance from the brane can be avoided only at the expense of making the brane world-volume positively or negatively curved. In the case where the bulk field content is parametrized by an analog of perfect fluid with an arbitrary equation of state P = γρ between the ‘pressure’ P and the ‘density’ ρ, our results depend crucially on the constant fluid parameter γ. (i) For γ > −1/2, the flat brane solution suffers from a collapse singularity at a finite distance that disappears in the curved case. (ii) For γ < −1, the singularity cannot be avoided and it becomes of the big rip type for a flat brane. (iii) For −1 < γ ⩽ −1/2, the surprising result is found that while the curved brane solution is singular, the flat brane is not, opening the possibility for a revival of the self-tuning proposal.

Asymptotics of flat, radiation universes in quadratic gravity

We consider the asymptotics of flat, radiation-dominated isotropic universes in four-dimensional theories with quadratic curvature corrections which may arise when contributions related to the string parameter α′ are switched on. We show that all such universes are singular initially, and calculate all early time asymptotics using the method of asymptotic splittings. We also examine the late time asymptotic behaviour of these models and show that there are no solutions which diverge as et2, the known radiation solution of general relativity is essentially the only late time asymptotic possibility in these models.

Edge-localized effects in buckling and vibrations of a shell with free in circumferential direction ends

Edge vibrations of a cylindrical shell, as well as buckling localized in a neighboring edge for cylindrical and conical shells are analyzed. The method of asymptotic splitting has been applied and simple analytical solutions have been proposed.

Longitudinal–Transverse Bending of Layered Beams in a Three-Dimensional Formulation

Three-dimensional equations of the elasticity theory for layered beams are solved by the method of asymptotic splitting without additional hypotheses or restrictions.