从受拉杆变形浅析弹性力学中的最小势能原理

ON PRINCIPLE OF MINIMUM POTENTIAL ENERGY IN ELASTICITY FROM DEFORMATION OF BARS SUBJECTED TO TENSION

最小势能原理是弹性力学中较难理解的知识点。本文通过对弹性杆轴向受拉时的弹性势能、外力势能和总势能的变分分析,得出总势能变分是位移变分或应变变分的二阶无穷小量,并在位移变分或应变变分为零时,总势能取得极小值。弹性杆轴向受力变形的分析应验了最小势能原理,有助于对一般情况下最小势能原理的深刻理解。

The principle of minimum potential energy in elasticity is comparatively difficult to understand.In this paper,by variational analyses on elastic potential energy,external force potential energy and total potential energy of elastic bars subjected to tension,it is obtained that the variation of total potential energy is the second order infinitesimal of the displacement variation or strain variation.Furthermore,when the displacement variation or strain variation equals to zero,the total potential energy reaches the minimum.The principle of minimum potential energy is thereby examined by the analysis of deformation for elastic bars under axial forces,which is helpful to profoundly understand the principle of minimum potential energy in general.