对带有微结构的弹性固体理论的再研究

对现有的带有微结构的弹性固体理论进行了再研究,并指出由于引进许多记号而使诸基本方程的推导过程复杂化的原因。针对该理论中存在的问题,提出更为普遍的功率能率原理,并借助此原理和广义Piola定理即可自然地推导出局部和非局部理论的运动方程、能率和能量的均衡方程、本构方程和边界条件。这些基本方程和边界条件连同初始条件一起即可用来解决带有微结构的弹性固体动力学理论的混合问题。     

Finite element analysis of arches undergoing large rotations - I: Theoretical comparison

In this two part paper, the first part deals with five different nonlinear theories applicable to the analysis of arches in the context of solving the large displacement and the large rotation problem. These theories include, classical theory, first-order shear deformation theory, third-order shear deformation theory, modified classical theory and the Donnell-type theory. All the theories are developed using the Total Lagrangian approach. Simplifications and assumptions used in each of the theory are discussed. Explicit strain displacement gradient relations and element independent equilibrium equations in terms of displacement gradients are given for all the theories. Limitations of each of theory are discussed. In the second part of this paper, application of these theories for the classification of arch geometries is considered.     

Definability for equational theories of commutative groupoids

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.     

Sulla struttura formale delle teorie fisiche

Starting from the observation that among physical quantities of every physical theory there are many ones that are naturally referred to the basic geometrical and chronometrical objects, it is shown how one may obtain a classification scheme for the physical quantities of every physical theory. This permits to assign a definite behaviour of physical quantities under time reversal and space inversion. The association of physical quantities with chrono-geometrical objects gives a motivation for the introduction in physical theories of the notions of multivectors, fiber bundles, exterior differential forms and connection theory. Typical equations of physics are shown to be described by a single mathematical process, the coboundary process (exterior differential). The basic equations of physics are then divided into two classes: metric independent equations, i.e. purely topological equations, and metric dependent equations, the phenomenological equations. Classification schemes for the main physical theories close the paper.     

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

We derive an asymptotic solution of the vacuum Einstein equations that describe the propagation and diffraction of a localized, large‐amplitude, rapidly varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations. © 2007 Wiley Periodicals, Inc.     

On using the more-accurate equations of thin coatings in the theory of axisymmetric contact problems for composite foundations

More-accurate equations describing the axisymmetric deformations of elastic, thin-walled elements (coatings) are derived using the asymptotic analysis of the solution to the first fundamental problem of the theory of elasticity for a layer. The notable difference distinguishing these relations from the classical, Kirchhoff-Love and Reissner-Timoshenko equations of flexure of plates, and their modifications /1/, is, that there are no concentrated forces at the edges of the stamp when the corresponding contact problems are solved. Moreover, the formulas obtained contain the equations of classical theory as a special case. The solutions obtained using various applied theories are compared with the corresponding solution obtained using the equations of the theory of elasticity, using the example of the axisymmetric contact problem of impressing a plane circular stamp into a layer lying on a Fuss-Winkler foundation. The characteristic parameters of the problem in question are computed by numerical methods.     

Theory of plasticity and creep taking account of microstrains

A relationship between the theories of plasticity and creep of the type /1, 2/ and theories based on the concept of slip is set up. A most logical structure is proposed for the constitutive equations of the theory which is convenient for engineering calculations.

It has been shown /3/ that the theory of slip /4/ results from the theories /1, 2/. However, it remains unclear whether a deeper connection exists between these theories. Moreover, the connection between creep theories constructed using the approach in /1, 2/ and creep theories based on the slip concept was not generally examined. A survey of the development of polycrystalline strain theory /5/ yields a complete representation of the state of matters in plasticity and creep theories.     

On global meromorphic solutions of second-order linear differential equations with meromorphic coefficients

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.     

高阶代数微分方程解的增长级(英文)

本文研究了高阶代数微分方程解的增长级的问题.利用亚纯函数的Nevanlinna值分布理论和微分方程的一些技巧,得到了一个更精确和更一般的结论,推广了何育赞和Laine的一些理论.     

On complicated models of continuous media in the general theory of relativity

In the context of the general theory of relativity, the system of Euler's equations is obtained from the variational equation under the assumption that the Lagrangian of the material depends on supplementary (as compared with classical theories) thermodynamic parameters, and when possible irreversible processes are taken into account. It is shown that, for a thermodynamically closed system, the equations of momenta for a continuous medium are a consequence of the field equations. The form of the energymomentum tensor of the material is considered when the arguments include the Lagrangian of the derivatives of the supplementary thermodynamic parameters.     

Multiexponential models of (1+1)-dimensional dilaton gravity and Toda-Liouville integrable models

We study general properties of a class of two-dimensional dilaton gravity (DG) theories with potentials containing several exponential terms. We isolate and thoroughly study a subclass of such theories in which the equations of motion reduce to Toda and Liouville equations. We show that the equation parameters must satisfy a certain constraint, which we find and solve for the most general multiexponential model. It follows from the constraint that integrable Toda equations in DG theories generally cannot appear without accompanying Liouville equations. The most difficult problem in the two-dimensional Toda-Liouville (TL) DG is to solve the energy and momentum constraints. We discuss this problem using the simplest examples and identify the main obstacles to solving it analytically. We then consider a subclass of integrable two-dimensional theories where scalar matter fields satisfy the Toda equations and the two-dimensional metric is trivial. We consider the simplest case in some detail. In this example, we show how to obtain the general solution. We also show how to simply derive wavelike solutions of general TL systems. In the DG theory, these solutions describe nonlinear waves coupled to gravity and also static states and cosmologies. For static states and cosmologies, we propose and study a more general one-dimensional TL model typically emerging in one-dimensional reductions of higher-dimensional gravity and supergravity theories. We especially attend to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible.     

Approximate variational-iterative solutions for a class of nonlinear operator problems

A general theory for the approximate solution of nonlinear operator equations where the solution lies in some real or complex Hilbert space is developed. It is shown that the techniques can be applied to a variety of problems taken from the theories of plasma physics, colloid theory, and the theory of current flow in nerve membranes.     

重建极性连续统理论的基本定律和原理(Ⅱ)——微态连续统理论和偶应力理论

重建微态连续统理论和偶应力理论的动量和动量矩均衡定律以及能量守恒定律,并由这些定律自然地推导出相应的局部和非局部均衡方程。这些结果可由耦合型微极连续统理论过渡和归结而得到。把推导出的结果和传统的质量和微惯性守恒定律以及熵不等式结合在一起就构成微态连续统理论和偶应力理论的基本均衡定律和方程体系。还弄清了以前的各种连续统理论的不完整性层次。最后,给出了几种特殊情形。     

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.     

The contact of a solid with an elastic half-space through a thin coating

More accurate equations of the deformation of thin plates, which are more convenient for solving contact problems for bodies with coatings and containing, as a special case, the equations of all known applied theories, are derived by an asymptotic analysis of the first fundamental problem of the theory of elasticity. The equations of the deformation of thin-walled elastic bodies are classified, their qualitative correspondence to the equations of the theory of elasticity is clarified, and the forms of the features that arise along the shift lines of the boundary conditions in the corresponding contact problems are established. A criterion for selecting approximate models to describe the properties of the coatings depending on the geometrical and mechanical characteristics of the coating and the substrate and also on their degree of adhesion is given.     

Universally coupled massive gravity

We derive Einstein’s equations from a linear theory in flat space-time using free-field gauge invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. We adapt these results to yield universally coupled massive variants of Einstein’s equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The Freund-Maheshwari-Schonberg theory is therefore not the unique universally coupled massive generalization of Einstein’s theory, although it is privileged in some respects. The theories we derive are a subset of those found by Ogievetsky and Polubarinov by other means. The question of positive energy, which continues to be discussed, might be addressed numerically in spherical symmetry. We briefly comment on the issue of causality with two observable metrics and the need for gauge freedom and address some criticisms by Padmanabhan of field derivations of Einstein-like equations along the way. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 311–336, May, 2007.     

Extended thermodynamics—a theory of symmetric hyperbolic field equations

Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models

The free vibration response of single-walled carbon nanotubes (SWCNTs) is investigated in this work using various nonlocal beam theories. To this end, the nonlocal elasticity equations of Eringen are incorporated into the various classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Reddy beam theory (RBT) to consider the size-effects on the vibration analysis of SWCNTs. The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations of each nonlocal beam theory corresponding to four commonly used boundary conditions. Then molecular dynamics (MD) simulation is implemented to obtain fundamental frequencies of nanotubes with different chiralities and values of aspect ratio to compare them with the results obtained by the nonlocal beam models. Through the fitting of the two series of numerical results, appropriate values of nonlocal parameter are derived relevant to each type of chirality, nonlocal beam model, and boundary conditions. It is found that in contrast to the chirality, the type of nonlocal beam model and boundary conditions make difference between the calibrated values of nonlocal parameter corresponding to each one.     

Fundamental solution of generalized thermo-viscoelasticity using the finite element method

The present paper is aimed at an investigation of the temperature, displacement, and stress in a viscoelastic half space of Kelven–Voigt type. The formulation is applied according to three theories of generalized thermoelasticity: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, as well as the coupled theory. The nondimensional governing equations are solved by the finite element method. Numerical results for the temperature distribution, displacement, and thermal stress are represented graphically. Comparisons are made with the results predicted by the CD, L-S, and G-L theories in the presence and absence of the viscoelastic relaxation time.