弹塑性柱体自由扭转问题的渐近解法

AN ASYMPTOTIC METHOD OF SOLUTION TO THE ELASTOPLASTICITY PRISM FREE-TORSION PROBLEM

具有幂级数型强化律的弹塑性平面问题已有较为一般的渐近解法。本文首先在空间问题的范畴将该方法推广于应变硬化具有Ramberge-Osgood方程型的材料,并归为一系列G.Lamé方程的求解。由此引入拄体自由扭转问题的基本假设后,除了对应于摄动参数一次幂项的方程以及边界条件和通常的弹性柱体一样外,前述一系列G.Lamé方程将化为一系列Poisson方程的Neumann问题。最后计算了椭圆截面柱体的扭转问题,当将得到的渐近解简化为圆截面柱的情形时,和解析解的展开式完全一致。

There has been a general method to obtain the asymptotic solution of the plane elasto-plasticity problem with strain-hardening of a power series model. In this paper, the method is firstly extended to the material with strain-hardening of Ramberge-Osgood model in the category of the three-dimensional problem, with the problem reduced to the solution of a series of G. Lame's equations. Then, by using of the assumption of torsion problem, except that the equation and the boundary condition corresponding to the 1st power of perturbation parameter are the same as those for the elastic prism, the above mentioned boundary value problems arc reduced to Neumann's problem of Poission's equations naturally. Finally, the resulting system of linear boundary value problems are solved for the case of the prism with a ellipse cross-section. When the asymptotic solution is reduced to the case of circle cross-section, it coincides with the expansion of the known analytical solution.