关于小变形假设的几何与数学分析

GEOMETRIC AND MATHEMATICAL ANALYSIS OF THE ASSUMPTION OF SMALL DEFORMATION

小变形是材料力学最基本的概念之一。材料力学中众多的推理和演绎都是在小变形假设条件下完成的,若对概念的认识不清晰,会得到错误的结果。本文通过某超静定结构建立变形协调方程的过程中,将同一小变形假设应用于不同的几何关系却得到不同的变形协调方程的例子,指出了小变形假设使用中的误区所在,并进一步通过几何和数学分析辨析了小变形假设的本质、含义及应用条件,澄清误解,为今后类似小变形概念的应用提供了理论保证。

Small deformation, one of the basic assumptions in mechanics, plays a critical role in the derivations and the applications of many theories in mechanic, for example, the beam and plate theories, and the mechanics of materials. The interpretation and the use of the small deformation assumption are important since wrong results may come from the misunderstanding of the concept of "small deformation". This paper studies a simple example of statically indeterminate problems. In the statically indeterminate truss, the equilibrium equations derived from the free-body diagram of the member must be complemented by relations involving deformations and obtained from the geometry of the problem. The importance of the concept of small deformation in geometric relationships is shown. It is indicated that the same definition of small deformation but different geometric relationships can result in different complementary equations in which, of course, only one of them is correct. The wrong use of small deformation assumption in the geometric relationships is pointed out to further highlight the correct concept of the small deformation.