Accurate Determination of Stress Intensity Factor for Interface Crack by Finite Element Method

This paper presents the simple method to determine the complex stress intensity factor of interface crack problem by the finite element method. The proportional method is extended to the interface crack problem. In the present method, the stress values at the crack tip calculated by FEM are used and the stress intensity factors of interface crack are evaluated from the ratio of stress values between a given and a reference problems. A single interface crack in an infinite bi-material plate subjected to tension and shear is selected as the reference problem in this study. The accuracy of the present analysis is discussed through the results obtained by other methods. As the result, it is confirmed that the present method is useful for analyzing the interface crack problem. Introduction In recent years, use of composite materials has been increasing in wide engineering field and accurate evaluation of interface strength in dissimilar materials has become important. Many methods have been developed to calculate the stress intensity factors of an interface crack in dissimilar materials by using the finite element method (FEM) [1]. However, it is still not necessarily easy to analyze the stress intensity factors of interface crack by FEM because of the oscillatory stress singularity. In the crack problem in homogeneous material, Kisu et al [2] have proposed the simple method using the one point stress value near the crack tip. This method is called the proportional method and is based on the fact that the stress distribution near the crack tip is proportional to r / 1 . Recently, Nisitani et al [3, 4] have developed the crack tip stress method for calculating the highly accurate stress intensity factors by FEM The method is used only the stress values at the nodal point of crack tip analyzed by FEM Although the stress value at the crack tip calculated by FEM contain numerical error, the value is effective as a measure of the magnitude of singular stress field. In the interface crack problem, however, the highly accurate values of the stress intensity factors cannot be evaluated by using the crack tip stress values. This paper presents the simple method to calculate the stress intensity factors for interface crack by FEM. The proportional method is extended to the interface crack problem. The accuracy of the present analysis is verified through the comparing the present results with the results obtained by other researches. The calculation shows that the present method has the sufficient accuracy in the case of interface crack problems. Method of Analysis Principle of proportional method. In this section, the principle of the proportional method is explained by taking a two dimensional mode I crack problem as an example. In the fundamental concept of linear fracture mechanics, stress distribution near the crack tip is expressed by following equation when 0 = θ . r K I y π σ 2 / = (1) Key Engineering Materials Vols. 353-358 (2007) pp 3124-3127 Online available since 2007/Sep/10 at www.scientific.net © (2007) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.353-358.3124 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 169.234.136.213-21/01/13,03:23:10) When the distance from crack tip r= r0, y I K σ / is constant and the next relation between two different problems can be obtained theoretically. y I y I K K σ σ / * / * = (2) If the stress intensity factor K1* of the reference problem is known, we can easily obtained the K1 of given problem from the relation (2) because the stresses * y σ and y σ of reference and given problems can be calculated by FEM. In the interface crack problem, however, the highly accurate values of the stress intensity factors cannot be obtained by using the stress values near the interface crack tip because the elastic solution of the interface crack has an oscillatory stress singularity. Application of proportional method to interface crack problem. In this study, the proportional method is extended to interface crack problem. In the ordinary extrapolation method of interface crack problem, the stress intensity factors are evaluated by the following equations.         + = → Q Q r K y xy y r sin cos 2 lim 0 1 σ τ σ π ,         − = → Q Q r K xy y xy xy r sin cos 2 lim 0 2 τ σ τ τ π (3)       = a r Q 2 ln ε ,       + + = ) 1 /( ) 1 ( ln 2 1