Peierls框架与裂纹尖端位错行为

PEIERLS' FRAME AND THE BEHAVIOR OF DISLOCATION AROUND THE TIP OF CRACK

本文在Peieris框架下对裂纹尖端位错成核与发射问题进行了严格的数学分析。在修正Rice设想的基础上,建立了一组新的控制方程。应力场与位错密度场分别表示成第一类与第二类切比雪夫多项式的级数。相应的张开位移与滑错位移可以用三角级数表示。通过离散的方法,控制方程转化为一组非线性代数方程。用Newton—Raphson方法求解这组方程,得到远场为纯剪、纯拉及两者复合情况下的解答。计算结果清楚地揭示了裂纹尖端位错成核与发射过程。

This paper presents a strict mathematical analysis for the problem of the nucleation and the radiation of the dislocation at crack tip, under the Peierls' frame. On the basis of the revised Rice's hypothesis, the author established a new system of governing equations, in which the stress field and the field of dislocation density are expressed respectively as the first and second Chebyschev polynomials series and the opening displacement and the slip displacement of dislocation may be expressed with trigonometrical series. And, by the discrete method, these governing equations may be reduced to a system of nonlinear algebraic equations solved by the Newton-Raphson's method. The solutions of pure shear, pure tension and shear and tension combined far fields are obtained. The computational results clearly show the processes of nucleation and the radiation of the dislocation around the tip of the crack.