半無限軟鐵磁性區域表面V型缺口底部裂紋所引致之磁漏現象及其破壞力學分析

Abstract

本文探討半無限軟鐵磁性區域表面V型缺口底部裂紋所引致之磁漏現象及其破壞力學分析問題，由於材料構件本身結構會存在一些微小缺陷問題，一般可利用漏磁檢測技術進行現場即時檢測，找出缺陷的尺寸及位置。首先，基於1973年Pao與Yeh所發表之軟鐵磁材料之磁彈耦合線性化理論，本文採用再簡化分析模型(擾動態磁場忽略不計)。其次，透過Schwarz-Christoffel 變換，映射函數將表面V型缺口底部裂紋映射至單位圓內，並用有理函數近似之，並利用解析函數區域內部的值可由邊界上的值來表示的特性，將位移、應力及邊界條件以解析函數來表示之，其中，彈性力學及磁場的複變函數解法是以Muskhelishvili Method為基本理論。雖然所有問題皆可以用冪級數進行求解，但因其收斂太慢，需要取到非常高的項數才能達到所要求之精度，而本文所使用之有理函數近似及疊代法便可取代原本的映射函數，最後求得磁漏場及裂紋尖端應力強度因子之數值結果。

This research deals with the problem of a sharp crack emanating from the root of a V-notch in a semi-infinite soft ferromagnetic solid under the uniform magnetic field at infinity. First, based on the linear theory of magnetoelasticity for soft ferromagnetic materials of Pao and Yeh in 1973. The linear theory has made it possible to obtain analytical results for boundary value problems of magnetoelastic interactions. Moreover, we assume that the strain is so small that its back-coupling can be neglected. Second, a mapping function that maps a V-notch with a crack into a unit circle is obtained by the Schwarz-Christoffel transformation. Also, the complex variable method with the rational mapping function technique is used in the analysis for both magnetic field and mechanical field. Otherwise, Muskhelishvili shows that a closed solution can be obtained if a rational mapping function is used. Stress components and boundary conditions are expressed in terms of two complex stress functions and the magnetic field intensity, it can be obtained by taking Cauchy integrals. The method to form the rational mapping function is briefly discussed in the following. The magnetic flux leakage as well as the singularity of stress intensity in the vicinity of crack tip can also be determined.