【摘要】 本文以弹性结构理论与三维弹性力学的力学联系为背景,即以两者之间的变形联系、应变能联系、动能联系为依据,运用泛函分析的简单性质,对弹性结构理论中两类算子的正定性和紧致性作了统一证明。特别地,这些结论可用于具有相当广泛的边条件的弹性薄壳、组合弹性结构等结构理论。更多还原

【Abstract】 In the theory of elastic structures such as beams, plates and shells, we may define three kinds of spaces on the function space of all possible displacements: a Hilbert space of square integrable Uss = L2(F) (see eq. (1)) which has Uss = D(As), the domain of definition of the ’’ Structure Theory Operator’’ As, as its subspace; an inner product space Ups whose norm equals the square root of the strain energy Πs of the structure (see eq. (2)), together with its complete space Ups; and a Hilbert space Uks, whose norm equals the square root of the kinitic energy Ks of the structure (see eq. (3)). The operators TPs,ss:Ups→Ussand Tps,ks:Ups→Uksare then defined as mappings between the same elements in two different spaces.The purpose of this paper is to prove that A, is positive definite (which is equivalent to Tps,ss being bounded) and that Tps,ks is compact.On the other hand, the elastic structure may be dealt with as an elastic body with three dimensions. In 3-dimensional elasticity, three corresponding kinds of spaces are defined on the function space of displacements: Use and Use; Upe and Upe; and lastly Uke. The operators Tpe,se and Tpe,ke are similarly defined (see eqs. (5) to (8)). It was known in literatures that the "Elasticity Operator" A, is positive definite (i.e. Tpe,se is bounded)and Tep,ke is compact for a quite wide range of boundary conditions.Generally, there exist three connections in mechanics between the structural theory and elasticity. (i) The displacements in structural theory may be taken as the displacements in elasticity imposed with certain constraints described by equality (9), where the operator Tss,se maps Uss onto a subspace User of Use (ii) In structural theory, usually there are some assumptions about the stress state, so that there exists a connection (13) between IT, and strain energy Πer in elasticity theory with the same displacements. This shows that the operator Tps,pe: Ups-Uper is bounded, the correspondence between the elements of Tps,pe being the same as that for Tss,se (iii) In structural theory, the contribution of some displacement components to the kinetic energy is often omitted, so we have a third connection (14) between Ks and the kinetic energy Kre in elasticity, which shows that the operator Tse,ss the inverse operator of Tss,se, is bounded.The following conclusions are obtained by two auxiliary theorems about functional (see Figs. 1 and 2).For a structure with given boundary conditions, if the corresponding boundary conditions of the elastic body ensure the positive definiteness of the operator Ae, then the operater A, with the given boundary conditions is also positive definite.If such boundary conditions ensure the compactness of the operator Tpe,ke, then the operator Tps,ks is also compact.Especially, these conclusions are applicable to composite structures as well as to shells with boundary conditions of various types, such as fixed edge, hinged edge, free edge, movable edge in the direction norma to the middle surface and their conbinations.更多还原