Analytical dynamics of continuous medium and its application

Finite difference methods in dynamics of continuous media , Finite difference methods in dynamics of continuous media , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

The general features of the dynamics of continuous media are investigated within the framework of the Einstein-Cartan gravitation theory using a formalism for the description of congruence geometry for the stream lines in the continuous medium. Raichaudkhur-type equations are derived for the space with twisting which are applicable to the investigation of the singularity problem in the gravitation theory. It is demonstrated that the spur of the space-time twisting tensor can directly affect the volumetric divergence of the autoparallel, while the twist pseudospur can affect the rotation of the congruence of the stream lines in the continuous medium. Using the investigated formalism, metrics are found and investigated for the uniform, rotating, nonstationary cosmologic model.

M. A. Lavrent'ev,"Cumulative charge and its operation," Usp. Mat. Nauk, 12, No. 4 (76) (1957). G. Birkhoff, D. McDougall, E. Pugh, and G. Taylor, "Explosives with lined cavities," J. Appl. Phys., 19, 563 (1948). M. V. Rubtsov, "Boundary layer in intersecting plane jets with small viscosity," in: Dynamics of Continuous Media [in Russian], No. 51, Hydrodynamics Institute, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (1981). M. I. Vishik and L. A. Lyusternik, "Regular singularity and boundary layer for linear differential equations with a small parameter," Usp. Mat. Nauk, 12, No. 5(77) (1957). J. D. Cole, Perturbation Methods in Applied Mathematics, Parabolic Press (1968). M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press (1964). M. A. Lavrent'ev and B. V. Shabat, Methods in Theory of Functions in Complex Variables [in Russian], Nauka, Moscow (1973).

Dynamics of continuous media:

Finite Difference Methods in the Dynamics of Continuous Media (Julian L. Davis)

1. Newton’s foundation of classical mechanics rests on the concepts of absolute time and absolute space. Thus, analytically, any event can be labelled in time and space provided a choice is made of an origin, a unit of time, and a frame of spatial coordinates. For example, the frame might originate at the center of mass of the solar system and its axes might point towards fixed stars. Of course, any other frame which is invariantly related to this one would be also admissible. As is well known, the laws of mechanics remain unaffected if the frame of spatial coordinates is made to undergo a rectilinear, uniform translation with respect to Newton’s absolute space, keeping the absolute time undisturbed. One is thus led to the notion of Galilean frames of reference. The principle of inertia may be stated as follows: in absence of interactions with other bodies, the velocity of a point mass remains constant in direction and magnitude in any Galilean frame. The fact that the validity of this principle in one Galilean frame implies its validity in any other Galilean frame follows immediately from the transformation laws governing these frames. Let us label a point in space by arbitrary Cartesian coordinates,1 not necessarily orthogonal. Then the transformation laws are as follows:

This issue of Journal of Physics: Conference Series contains ten selected technical papers that were presented at the 8th Symposium on the Mechanics of Slender Structures (MoSS2019) and show a wide research and application of slender structure. The meeting was held in Changsha, China, from 24th to 26th May 2019. This conference runs under the auspices of the Institute of Physics Applied Mechanics Group and forms a continuation of a successful meeting series on the Mechanics of Slender Structures first held in Northampton, UK, in 2006, followed by the events hosted in Baltimore, USA in 2008, in San Sebastian, Spain in 2010, in Harbin, China in 2013, in Northampton, U.K. in 2015, in Shanghai, China in 2016, and in Mérida, Spain 2017. This conference is also sponsored by the Chinse Society of Theoretical and Applied Mechanics and jointly organized by Hunan University, Harbin Institute of Technology and Xi’an University of Technology. The aim of MoSS 2019 is to bring together the international leading scientists of mechanics of cable, string and other slender structure and to present their original and latest research work. Slender structure refers to a component whose size in one dimension is much larger than that in other dimensions. Applications of slender structures include terrestrial, marine and space systems. With the development of science and technology, slender structures are developing in both large and micro directions, and the working environment is becoming more and more diverse and complex. Moving elastic elements such as ropes, cables, belts and tethers are pivotal components of many engineering systems. Their lengths often vary when the system is in operation. The applications include vertical transportation installations and, more recently, space tether propulsion systems. Traction drive elevator installations employ ropes and belts of variable length as a means of suspension, and also for the compensation of tensile forces over the traction sheave. In cranes and mine hoists, cables and ropes are subject to length variation in order to carry payloads. Tethers experiencing extension and retraction are important components of offshore and marine installations, as well as being proposed for a variety of different space vehicle propulsion systems based on different applications of momentum exchange and electrodynamic interactions with planetary magnetic fields. Furthermore, cables, beams, columns, towers and other slender poles and rods are used extensively in mechanical and civil engineering; they are common in machinery, automotive components, rails, tunnels, girder bridges, arch bridges, cable-stayed bridges, suspension bridges, high-rise buildings, masts and large-span roof systems of buildings and stadiums. Also, suspended cables are applied as electricity transmission lines. Chains are used in various power transmission systems that include such mechanical systems as chain drives and chain saws. Moving conveyor belts are essential components in various material handling installations and textile manufacturing systems involve slender continua such as yarns composed of staple fibers. This meeting brought together experts from various fields: structural mechanics, thermomechanics, dynamics, vibration and control, structure-media interaction, structural health monitoring, materials science and applied mathematics to discuss the current state of research as well as rising trends and direction for future research in the area of mechanics of slender structures. The event is aimed at improving the understanding of structural properties and behaviour of slender structures. The papers presented at the conference covered analytical, numerical and experimental research in various applications of slender structures. The conference programme was arranged around the following seven keynote lectures: Keynote Lecture 1: On the interesting behaviour of a beam with an inclined roller by Stefano Lenci, Polytechnic University of Marche (Italy). Keynote Lecture 2: The seismic design and safety control of high-speed railway track-bridge system based on train safety traveling performance by Lizhong Jiang, Central South University (China). Keynote Lecture 3: Historical development and challenges in aeroelastic flutter of long-span bridges by Zhengqing Chen, Hunan University (China). Keynote Lecture 4: Flutter control of long-span bridges by Walter Lacarbonara, Sapienza University (Italy). Keynote Lecture 5: Energy harvesting enhanced by double-jumping: an l-shaped beam case by Li-Qun Chen, Harbin Institute of Technology (China). Keynote Lecture 6: Time as an imperfection: what do we know about the effect of dynamics on buckling? by Gert van der Heijden, University College London (UK). Keynote Lecture 7: Dynamics of continuous media: from time-varying and nonlinear systems to flexible multibody systems by Weidong Zhu, University of Maryland Baltimore County (USA). Additionally, eight minial sessions involving i) cable, string and other soft-slender structure, ii) beam, bridge, railway engineering and structure-media interaction, iii) plate, membrane, panel and sandwich structures and iv) material, conductors, control and numerical method were addressed in this symposium. The articles presented in this volume are arranged alphabetically by the first author details and are extended peer-reviewed versions of the papers presented at the conference. The Editors and Organizing Committee gratefully acknowledges support received from the co-sponsoring institutions and would like to thank the authors for their hard work and high quality contributions.

Share Icon Share Twitter Facebook Reddit LinkedIn Reprints and Permissions Cite Icon Cite Search Site Citation Philippe Tourrenc, Clifford M. Will; Relativity and Gravitation. Physics Today 1 June 1998; 51 (6): 66–67. https://doi.org/10.1063/1.882263 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentPhysics Today Search Advanced Search

After introducing the Hamiltonian representation for continuous systems, we present a technique for calculating the evolutions of continuous systems submitted to any fields, starting from the solutions for simple motions.

Fractional derivative stress-strain relations are derived for compressible viscoelastic materials based on the continuum mechanics. Several types of stress tensor and strain tensors are specified to describe the dynamics of continuous media. Consequently there are many equivalent expressions of stress-strain relations. If memory effect is not taken into account, these relations are equivalently transformed from one to another by suitable tensor operations. However, if memory effect is included in the mechanics of the materials, different types of stress-strain relations can be derived depending on the choice of the type of stress tensor, or equivalently the choice of the strain energy function. In this paper, several types of fractional derivative stress-strain relations are proposed.

This chapter will systematically discuss the differential geometry, kinematics and dynamics of deformation in continuous media. To discuss deformation geometry, the deformation gradients will be introduced in the local curvilinear coordinate system, and the Green and Cauchy strain tensors will be presented. The length and angle changes will be discussed through Green and Cauchy strain tensors. The velocity gradient will be introduced for discussion of the kinematics, and the material derivatives of deformation gradient, infinitesimal line element, area and volume in the deformed configuration will be presented. The Cauchy stress and couple stress tensors will be defined to discuss the dynamics of continuous media, and the local balances for the Cauchy momentum and angular momentum will be discussed. Piola-Kirchhoff stress tensors will be presented and the Boussinesq and Kirchhoff local balance of momentum will be discussed. The local principles of the energy conservation will be discussed by the virtual work principle. This chapter will present an important foundation of continuum mechanics. From such a foundation, one can further understand other approximate existing theories in deformable body and fluids.

The method of correlations, which has been widely used in the theory of turbulence, is applied to the case of a field that satisfies linear equations. It is found that most of the difficulties encountered in the theory of turbulence disappear when the equations are linear and that a number of general results can be obtained.In particular, it is possible to use an analog of Gibbs' procedure of replacing a time average by an ensemble average. The mathematical foundations of this procedure are discussed and shown to be considerably different than for dynamical systems with a finite number of degrees of freedom, to which the method is usually applied. In particular, it is shown that it can be applied to dissipative fields excited by a randomly varying force.Three examples are treated in some detail: statistically uniform but nonisotropic fields; statistically nonuniform, dissipative fields excited by a random force; and the scattering of monochromatic radiation by inhomogeneities of the refractive index.The method of ensembles, as applied to these problems, appears to have a mathematical foundation that is rather simpler than in other cases that have been considered in the past, but several basic and unsolved problems are mentioned.

A method and algorithm for rebuilding a surface triangulation in three-dimensional space defined by an STL file is proposed. An initial surface in 3D space (STL file) is represented as a polyhedron composed of triangular faces. The method is based on the analytical representation of the surface as a piecewise polynomial function. This function is built on a polyhedral surface composed of triangles and satisfies the following requirements: 1) within one face, the function is an algebraic polynomial of the third degree; 2) the function is continuous on the entire surface and preserves the continuity of the first partial derivatives; 3) the surface determined by the function passes through the vertices of the initial triangulated surface. The restructuring of computational meshes is required in cases of distortion of the shape of cells when solving problems of mathematical physics using mesh methods (finite-difference, FEM, etc.). Cell distortion can be due to various reasons. These can be large distortions of moving Lagrangian meshes in the calculations in the current configuration, with instability of the hourglass type, with distortion of the faces of the interface between interacting gaseous, liquid and elastoplastic bodies. The rebuilding of the mesh reduces to solving the problem of constructing a smooth surface passing through the nodes of an existing triangulated surface or part of it. Later the nodes of the new mesh are placed on the constructed smooth surface with existing requirements for the size and shape of the cells. The construction of a smooth piecewise polynomial surface is based on the ideas of spline approximation and reduces to the building of a cubic polynomial on each triangular face, taking into account the smooth conjugation of polynomial pieces of the surface constructed on adjacent faces. The proposed method for rebuilding surface triangulation can be useful for calculating the motion of deformable bodies when solving problems of the dynamics of continuous media on immovable Euler grids.

LXXXVI. <i>On the transformation of the equation of motion of the dynamics of continuous media in the restricted principle of relativity</i>

We generalize the Hamilton equations for dynamical processes with relaxation. We introduce a dissipative Poisson bracket in terms of the dissipation function. We obtain the universal structure of the relaxation terms in the equations for the dynamics of condensed media and verify this result for structureless liquids, elastic solids, and quantum liquids. In the examples of the condensed media under consideration, we obtain expressions for the dissipative Poisson brackets for the complete set of dynamical parameters.