圆筒在内外压力作用下的有限位移问题

本文用逐次逼近法求得这个边值问题的一次解和二次解,从而获致位移场,应变场和应力场的二级近似公式,我们的结果还表明:在变形后,(i)圆筒任一截面必位移至另一仍与筒轴垂直的平面上;(ii)应变分量E_RR(2)与E_ΦΦ(2)之和以及应力分量∑_RR(2)与∑_ΦΦ(2)之和在整个圆筒内均不保持恒定。后一效应是经典弹性理论里所没有的,它对∑_ZZ(2)的产生承担责任,此外,∑_ZZ(2)与(∑_RR(2)+∑_ΦΦ(2))之间呈现线性关系,其比例系数仅与圆筒的材料有关。

On the Finite Displacement Problem of a Hollow Cylinder under Internal and External Pressures

Using the method of successive approximations we find of this boundary-value problem the first-and second-order solutions. And then we obtain the formulae in the second approximation for the displacement, strain, and stress fields. Also, our results show that after deformation (i) a cross-section of the cylinder must be displaced into a plane section perpendicular to the central axis of the cylinder; and (ii) neither the sum of the strain components E_RR(2) and E_φφ(2) nor the sum of the stress components ∑_RR(2) and ∑_φφ(2) maintains contant throughout the cylinder. The latter effect, which is absent from classical elasticity, bears responsibility for the presence of the ∑_zz(2) Moreover, there exhibits a linear relation between ∑_zz(2) and (∑_RR(2)+∑_φφ(2)), with the proportionality coefficients depending only on the material of the cylinder.