Solution of edge crack problem by using a novel weight function formulation
✍ Scribed by Chen, Y. Z. ;Chen, R. S.
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 402 KB
- Volume
- 12
- Category
- Article
- ISSN
- 1069-8299
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✦ Synopsis
A fundamental field for the edge crack problem is suggested, and the field is composed of the singular displacement field and the complementary regular field. The boundary displacement of the fundamental field plays the role of the weight function in the edge crack problem. After multiplying the boundary traction in the physical problem with the weight function and performing integration along the boundary, the stress intensity factor at the crack tip is obtainable. Numerical examples are given to demonstrate the use of the suggested weight function approach.
KEY WORDS weight function approach; edge crack problem
1. Introduction
An earlier version of the weight function formulation for the crack problem can be seen from some references.lS2 Later, a more detailed description of the weight function formulation was carried out.3 The formulation depends on the properties of the eigenfunction expansion form in the crack problem. It is found that the terms corresponding to the negative eigenvalue play an important role in the analysis. Within the terms having the negative eigenvalue, the displacement expansion form with the order of O(r-1'2) becomes the singular displacement field, where r denotes the distance of a point to the crack tip. This field is called the /I1-field hereafter. In the analysis, we introduce the complementary regular field, called the B2-field. The b2-field is defined such that the tractions along the boundary caused by the B1-field are opposite to those cased by the B2-field. The superposition of the /I1-field and the B2-field will give the fundamental field, which is called the /I-field. In the analysis, the displacement of the /I-field along the boundary of the cracked body becomes the weight function. Clearly, the B1-field is given beforehand, and the B2-field should be evaluated by the numerical method. After multiplying the given boundary traction and the obtained weight function, the stress intensity factor at the crack tip is obtainable.
The merit of the weight function is that, once a particular boundary value problem is solved, all the boundary value problems with the same geometry can be solved immediately.