复合材料层合梁接触问题的线性化与能量解法

本文将非线性的接触问题线性化,即假设已知带可调参数的接触区载荷分布模态以及接触区宽度,再反求圆柱压头的半径。首先根据叠加原理将受力状态分解成对称和反对称状态,然后用正交完备的三角级数和勒让德级数构造这两种受力状态的位移场,并应用最小势能原理确定位移场中的待定系数,从而确定层合梁的位移场和应力场。载荷分布模态中的可调参数可根据接触区表面的位移协调条件确定,从而求得圆柱压头的半径。最后,由于在给定接触区的条件下压头曲率与压头合力成正比,故可以得到压头曲率在不同的接触区下随载荷变化的直线族。根据这些直线族,可以由已知的压头曲率和外载荷确定接触区尺寸。计算结果显示,这种解法的收敛性非常好。根据物理方程与根据平衡方程得到的层合梁剪应力分布的一致性非常好。而且,在远离接触区处的应力分布同经典层合梁理论结果的一致性也非常好。 

LINEARIZATION AND ENERGY METHOD ABOUT CONTACT PROBLEM OF COMPOSITE BEAM

The nonlinear contact problem of composite beams is linearized by an inverse method, that is to say, the loading distribution on the contact zone with the adjustable parameter and the contact width are assumed to be given and the curvature of the cylinder is to be solved. By means of the principle of superposition, the loading state is decomposed into symmetric and antisymmetric ones. The trigonometric series and Legendre series are applied to describe the displacement field of the above two loading states and the principle of minimum potential energy is used to determine the unknown coefficients of the above series. Then the displacement and stress fields of the composite beam are known. The adjustable parameter of loading distribution is used to satisfy the compatibility conditions of displacements along the contact surface. By the way the indentor curvature is determined. Then many families of straight lines passing through the origin of the indentor curvature varying as the loading with different contact zones can be figured out. Based on the above straight lines, the contact zone can be determined from the known indentor curvature and loading. From the computational results, it can be shown that the displacements and stresses converge very well and the distribution of shearing stress obtained from the constitutive equation and from the equilibrium equation agree with each other very well. The distribution of stress of the composite beam has the local effects, and at the place far from local loading the distribution of stress is the same as the results of classical beam theory.