whittaker’s Reduction Method for Poincare’s Dynamical Equations

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Bingham’s heritage

A hundred years after the founding paper by E.C. Bingham, we briefly review the impact of the yield stress concept and current interest in it on the scientific community. We show that yield stress fluids have only emerged as a relevant fluid type, in both mechanics and physics, over the past 20 years, opening the way to a broad range of new study areas.     

EXTENDED MAC-MILLAN’S EQUATION

The principle of classical dynamics and Appell-Chetayev’s assumption are extended tonon-inertial frame,from which extended Mac-Millan’s equation is derived for non-holonomic system in non-inertial system.     

SET-VALUED CARISTI’S FIXED POINT THEOREM AND EEELAND’S VARIATIONAL PRINCIPLE

This paper proposes a formally stronger set-valued Caristi’s fixed point theorem and by using a simple method we give a direct proof for the equivalence between Ekeland’s variational principle and this set-valued Caristi’s fixed point theorem.The results stated in this paper improve and strengthen the corresponding results in[4].     

NIELSEN’S AND EULER’S OPERATORS OF HIGHER ORDER IN ANALYTICAL MECHANICS

In this paper,the definitions of Nielsen＇s and Euler＇s operators of higher order arepresented These operators are concerned in analysis for systems with holonomicconstraints and non-holonomic constraints of higher order. Some theorems that indicatingrelation between the two operators are established Moreover.using the theorems,the newequations of mechanical systems with constraints of higher order are derived Finally,anexample is given.     

A modified Darcy’s law

An approach to describe the turbulent flow through a complex geometry (e.g., urban area) by means of an analogy to flows through porous media is presented. Therefore, a modification of the original Darcy’s law is proposed, and its application is tested in a prototype problem with an idealized complex geometry using large eddy simulations. The numerical results indicate the validity of the modified Darcy’s law for the chosen setup.     

THE COMPLEMENTARY PROPERTY OF LINDELF’S WORK AND CHAPLYGIN’S WORK

Lindelf’s equation is derived by using the Vakonomic model,which shows that Lindelf’s work coincides with Vakonomic model. Chaplygin’s equation is derived by using Chetaev’s model, which shows that Chaplygin’s work coincides with Chetaev’s model. On basis of these, by improving the expressions of Chaplygin’s equation and Lindelf’s equation, the reasonable transition from Chaplygin’s equation to Lindelf’s equation is realized, the reasonable transition from Lindelf’s equation to Chaplygin’s equation is realized too. Finally, a typical example is given. The work of this paper shows that, just as the Vakonomic model and Chetaev’s model are complementary to each other, Lindelf’s work and Chaplygin’s work are complementary to each other too.     

A generalization of Prandtl’s and Spencer’s solutions on axisymmetric viscous flow

New analytical solutions for axisymmetric deformation of a viscous hollow circular cylinder on a rigid fibre are given. One of the solutions generalizes the famous Prandtl’s solution for compression of a rigid perfectly plastic layer between two rough, parallel plates and the other is a modification of Spencer’s solution for compression of an axisymmetric rigid perfectly plastic layer on a rigid fibre. All equations are satisfied exactly whereas some boundary conditions are approximated in a standard manner. Special attention is devoted to frictional interface conditions since these conditions result in additional limitations of the applicability of the solution when compared to that based on a rigid perfectly plastic models. In particular, difficulties with the convergence of numerical solutions under certain conditions can be explained with the use of results obtained. Therefore, the solutions can serve as benchmark problems for verifying numerical codes. The solutions are also adopted to predict the brittle fracture of fibres by means of an approach used in previous studies and confirmed by experiment.     

Dihedral ‘star’ tensegrity structures

This paper presents conditions for self-equilibrium and super stability of dihedral ‘star’ tensegrity structures, based on their dihedral symmetry. It is demonstrated that the structures are super stable if and only if they have an odd number of struts, and the struts are as close as possible to each other. Numerical investigations show that their prestress stability is sensitive to the geometry realisation.     

Front Solutions of Richards’ Equation

The design of a Compressed Air Energy Storage (CAES) plant requires knowledge of the pressure and temperature variations within the reservoir, for expected sets of plant operation. In the current work, a closed form approximate analytical solution for the pressure variations, in porous media reservoirs, was derived for conditions of steady periodic isothermal radial gas flow. Two different expressions for the pressure variation were obtained, one as an infinite series and the other as an integral, where the latter is the computationally preferred solution. In order to evaluate the model accuracy, a finite difference numerical solution of the full non-linear problem was developed. The accuracy of the analytical solution was confirmed through, both, error analysis and comparison against the numerical calculations. The analytical solution can be used to calculate the well pressure variations and the radius of the active region around the well. Examples of calculations are provided, and a parametric study is presented to demonstrate the sensitivity of the well pressure to pertinent parameters. The model could eventually yield improved CAES plant designs.     

Mullins’ effect in semicrystalline polymers

Observations are reported on isotactic polypropylene in uniaxial cyclic tensile tests at room temperature. A model is derived for the viscoplastic response of semicrystalline polymers at three-dimensional deformations with small strains. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. It is shown that polypropylene reveals some characteristic features of the Mullins effect that can be quantitatively predicted by the constitutive equations.     

Archie’s Law in Microsystems

Since 1942 Archie??s law is used every day to estimate, from electrical measurements, the quantity of oil present in oil fields. In this article, we perform the first experimental analysis of electric conductivity in well controlled models of porous media. We used microfluidic networks (called micromodels in the oil industry jargon), incorporating thousands of pores, with controlled wettability. Different electrode and pore geometries are considered. In all cases the evolution of the conductivity with the conductive fluid fraction (??saturation??) clearly reveals the presence of percolation thresholds, signaling that as the fraction of the conductive fluid decreases below some critical value, there are no more pathways involving only channels entirely filled with the conductive fluid that connect the electrodes. This behavior is observed in all cases, for all the network/electrode geometries and wetting properties we investigated, and is consequently likely to reflect a genuine behavior for microfluidic ??2D?? networks. The existing models??based on percolation theory or on mean field approach??reproduce correctly the structure of this behavior, but generally at a semi-quantitative level. The most successful case is obtained with the effective medium theory (EMT) model, with drainage and perpendicular electrodes. This outcome suggests that, despite the complexity of these systems, very simple models can describe correctly the physics of the system. Nonetheless, more precise modeling requires case-by-case studies. Our results are consistent with the current body of knowledge accumulated for decades on three-dimensional samples. The key point is that in 3D systems, owing to topological reasons, the threshold is extremely low in terms of water saturations. Archie??s law completely neglects the threshold effect. Nonetheless the percolation threshold should not be overlooked, and modeling should take this aspect systematically into account, as it is already done by several investigators.     

Prandtl’s formulation for the Saint–Venant’s torsion of homogeneous piezoelectric beams

The Saint–Venant torsional problem for homogeneous, monoclinic piezoelectric beams is formulated in terms of Prandtl’s stress function and electric displacement potential function. The analytical approach presented in this paper generalizes the known formulation of Prandtl’s solution which refers to homogeneous elastic beams. The Prandtl’s stress function and electric displacement potential function satisfy the so called coupled Dirichlet problem (CDP) in the cross-sectional domain. A direct and a variational formulation are developed. Exact analytical solutions for solid elliptical cross-section and hollow circular cross-section and an approximate solution based on a variational formulation for thin-walled closed cross-section are presented.     

TOUPIN-BERDICHEVSKll THEOREM CAN’T BE CONSIDERED AS A MATHEMATICAL EXPRESSION OF SAINT-VENANT’S PRINCIPLE

In this paper the condition and the conclusion of Toupin-Berdichevskii Theorem isexamined,whereby it is explained and demonstrated with an example that the theoremcan’t be considered as a mathematical expression of Saint-Venant’s Principle in Elasticity.     

Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.     

Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

     

ON THE MODIFIED CASTIGLIANO’S THEOREM

This paper gives the modified Castigliano’s theorem, which is more convenient and more extensive for applications than the classical Castigliano’s theorem.The modifications to the classical Castigliano ’s theorem are in following two respects, the first respect is that the expression of the partial derivative with respect to the concentrated load p of the complementary energy density in the clasical Castigliano ’s theorem is replaced by the expressions with teh products of the external loads and influence functions, the modification brings us greatest simplicity and greatest convenience for calculations under various loads; the second respect is that the expressions with the products of the inhomogeneous boundary displacements and the influence functions are introduced into the classical Castigliano’s theorem, the modification provides the theoretical fundamental for solving the problems of various boundary conditions.We show also the method of how to apply the modified Castigliano’s theorem to solve the problems of the surface structural mechanics.Finally, as a calculated example of the applications of the modified Castigliano’s theorem we solved the equation of the deflection surface of the rectangular plate with two adjacent edges built-in and other two adjacent edges free.     

Substructures’ coupling with nonlinear connecting elements

Nonlinear Dynamics - The analysis of complex structures is often very challenging since reliable data can only be obtained if the underlying model represents properly the real case. Thus,...     

Improvement of FEM’s dynamic property

The discretization size is limited by the sampling theorem, and the limit is one half of the wavelength of the highest frequency of the problem. However, one half of the wavelength is an ideal value. In general, the discretization size that can ensure the accuracy of the simulation is much smaller than this value in the traditional finite element method. The possible reason of this phenomenon is analyzed in this paper, and an efficient method is given to improve the simulation accuracy.