A Frame-Independent Solution to Saint-Venant’s Flexure Problem

The paper illustrates a solution approach for the Saint-Venant flexure problem which preserves a pure objective tensor form, thus yielding, for sections of arbitrary geometry, representations of stress and displacement fields that exploit exclusively frame-independent quantities. The implications of the availability of an objective solution to the shear warpage problem are discussed and supplemented by several analytical and numerical solutions. The derivation of tensor expressions for the shear center and the shear flexibility tensor is also illustrated. Furthermore, a Cesaro-like integration procedure is provided whereby the derivation of a frame-independent representation of the displacements field for the shear loading case is systematically carried out via the use of Gibbs’ algebra. The objective framework presented in this paper is further exploited in a companion article (Serpieri, in J. Elast. (2013)) to prove the coincidence of energetic and kinematic definitions of the shear flexibility tensor and of the shear principal axes.     

A Brief Account of G.D.Birkhoff’s Problem in the Problem of Three Bodies

The present article gives a historical survery of G.D.Birkhoff’s seventh problem which is an inquiry about the topological structure of the set of definition of the reduced differential equations of motion.Recent advances in the problem and their meaning have been briefly indicated.The classical 3-body problem concerns how the three particles should move under their mutual Newtonian attraction.By a particle we mean a goometrical point endorsed with a constant positive number m which is called mass.Expressed mathematically,the problem appears as to solving of the following system of differential equations:     

On Saint-Venant’s Problem with Concentrated Loads

The classical Saint-Venant problem is to find a solution of the traction problem of elastostatics in a finite cylinder ?? loaded over its bases. We prove that the problem has a unique solution for equilibrated surface forces $\hat{ \boldsymbol { s}}\in W^{-1,q}(\partial\Omega)$ , with q??(2?? ₀,+??) for some positive ? ₀ depending on ??. Hence $\hat{ \boldsymbol { s}}$ can model force acting on ???, concentrated on sets of zero Lebesgue surface measure of ???. Moreover, if $\hat{ \boldsymbol { s}}$ is equilibrated on each basis, we give a simple proof of the Toupin estimate expressing Saint-Venant??s principle.     

General Melnikov Approach to Implicit ODE’s

Journal of Dynamics and Differential Equations - Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed...     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

On a representation of an elastic potential in A. A. Il’yushin’s generalized strain space

For elastic isotropic materials under finite strains, we consider an elastic potential in the form of a function of invariants of the Hencky logarithmic strain measure. For such a potential, we propose a representation in A. A. Il’yushin’s generalized strain space. This representation is used to construct an approximation to the elastic potential for incompressiblematerials; this approximation permits exactly describing the stress-strain, compression, and pure shear diagrams.     

On an Eigenvalue Problem Related to the Buckling of Sandwich Beams

This paper gives various properties of eigenvalue problems related to the buckling of sandwichbeams.The applications of eigenfunctions are also indicated.     

Authors’ response to the rebuttal by A. Cushman

This is a response to the rebuttal by A. Cushman regarding our paper “Fault-tolerant sliding mode attitude control for flexible spacecraft under loss of actuator effectiveness” (Nonlinear Dyn. doi:, 2011).     

A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range

Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.     

On the Brézis–Nirenberg Problem

We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983): $-\Delta u=\lambda u+ |u|^{2^\ast-2}u, \quad u\in H_0^1(\Omega),$ where Ω is a bounded smooth domain of R ^N (N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ₁, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions.     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

On the Completeness of Hu Hai chang’s Solution

In this paper, the completeness of Hu Hai-chang’s solution is proved in the case of convex regions in z-direction under a supplementary condition. On the other hand, for those non-convex regions in z-direction, Hu Hai-chang’s solution is proved to be incomplete.     

On the Completeness of Hu Hai chang’s Solution

In this paper,the completeness of Hu Hai-chang’s solution is proved in the case of convex regions in z-direction under a supplementary con-dition.On the other hand,for those non-convex regions in z-direction,Hu Hai-chang’s solution is proved to be incomplete.