不同加载条件下平面应力圆盘应力和位移的全场解

摘要

不同加载条件下平面应力圆盘(简称巴西圆盘)应力场和位移场的研究,对于弹性力学、岩石力学和断裂力学而言具有重要的理论价值和工程实用价值.为得到不同加载条件下巴西圆盘应力场和位移场的数学表达式,依据弹性理论和巴西圆盘集中载荷作用下应力场的幂级数展开式,得到巴西圆盘在集中载荷作用下位移场的数学表达式;通过数理分析得到均布载荷作用下巴西圆盘试件应力场的数学表达式,并获得位移场的数学表达式.计算结果表明:径向应力σr、周向应力σθ和径向位移u均关于θ=0轴对称,切应力τrθ和切向位移v均关于θ=0轴反对称;在集中力作用点或分布载荷的边界点,应力场发生剧烈变化,位移场只有外围边界处的位移有较大变化,因此相同条件下,载荷类型对应力场的影响要大于对位移场的影响;另一方面,载荷类型的不同只对其作用点或作用区间附近(即ρ较大时)的应力场或位移场有重要影响,对离载荷作用点或作用区间较远的地方(即ρ较小时)的应力场或位移场影响极小,这一结论与圣维南原理完全一致.此外,径向位移u随p的增大而增大,切向位移v在ρ =0.7附近较大,在圆盘的中心和四周均较小.进一步的分析结果表明,国外有关学者所得均布载荷作用下巴西圆盘应力场的幂级数公式可进一步化简、合并,本文结果是均布载荷作用下巴西圆盘应力场的最简洁形式.%Investigation on the stress displacement field for the plane stress disk,which is also called the Brazilian disk,has an important theoretical and practical value for elastic mechanics,rock mechanics and fracture mechanics.In order to obtain the closed-form solution to stresses and displacements in the Brazilian disk under different loading conditions,based on the theory of elasticity and a series solution of stress for the Brazilian disk subjected to a pair of compressive forces,the explicit expressions of displacements were obtained for the Brazilian disk under diametral-compression loading.The series solutions to stresses were then derived by using mathematical analysis method and the explicit expressions of displacements were obtained subjected to uniformly distributed pressure.The calculated results showed that the stressesσθ,σrand radial displacement u were symmetric to the loading line θ =0,and the stress τrθ and tangential displacement v were anti-symmetricto θ =0.At the loading point of a concentrated force and the starting point or the end point of the distributed pressure,the stress field and displacement field had a abrupt change.For the same condition,the effect of the loading types on the displacements was less than that on the stresses.On the other hand,the loading type only had an important effect on the stress or displacement distributions near the loading point (i.e.,when p was larger),and the effect on the stress or displacement distributions was very small at points away from the loading point or loading range (i.e.,when p is smaller).This conclusion agreed well with the Saint-Venant principle.In addition,the radial displacement increased with the increase ofp,and it had a maximum value nearp =0.7.The further analysis showed that the series formula obtained by other researcher can be simplified and combined,and the formula derived from the present paper is the simplest form for the Brazilian disk loaded by pressure.