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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 716768, 9 pages
http://dx.doi.org/10.1155/2013/716768
Research Article

Oscillatory Singularity Behaviors Near Interface Crack Tip for Mode II of Orthotropic Bimaterial

School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China

Received 20 December 2012; Revised 18 May 2013; Accepted 22 May 2013

Academic Editor: Ray K. L. Su

Copyright © 2013 Xiaomei Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The fracture behaviors near the interface crack tip for mode II of orthotropic bimaterial are discussed. The oscillatory singularity fields are researched. The stress functions are chosen which contain twelve undetermined coefficients and an unknown singularity exponent. Based on the boundary conditions and linear independence, the system of twelve nonhomogeneous linear equations is derived. According to the condition for the system of nonhomogeneous linear equations which has a solution, the singularity exponent is determined. Total coefficients are found by means of successive elimination of the unknowns. The theoretical formulae of stress intensity factors and analytic solutions of stress field near the interface crack tip are obtained. The crack tip field is shown by figures.

1. Introduction

Many researchers have studied the singularity behavior near interface crack tip for isotropic, orthotropic, or anisotropic bimaterial. The method of eigenfunction expansions is used by Williams [1]. The complex function method is developed by Rice and Sih [2]. Erdogan [3] and England [4], present and research various interface crack problems. These academic authorities play a leading role in fracture mechanics for interface crack, and they have made the tremendous contribution. The subsequent papers are published one after another over several decades [516]. In this paper, the solution method is proposed to research the singularity behavior near interface crack tip for mode II of orthotropic bimaterial.

It can be seen that the following differences exist between this method and previous methods by comparison. (1)The stress function contains twelve undetermined coefficients, rather than eight.(2)The system of twelve nonhomogeneous linear equations is deduced based on the boundary conditions, rather than the system of eight homogeneous linear equations. (3)The characteristic equation is found by using the condition for the system of nonhomogeneous linear equations that possess a solution, rather than being based on the condition for the system of homogeneous linear equations which has a nontrivial solution.(4)In order to determine total coefficients, we only need to solve the system of nonhomogeneous linear equations, rather than to solve the system of homogeneous linear equations first, and then use the load conditions at infinity.(5)The stress intensity factors are defined by right-hand limit and left-hand limit rather than by limit. (6)The oscillatory singularity fields near interface crack tip of three orthotropic bimaterial are illustrated by the help of two tables and seven figures.

2. Mechanical Model

The plane is the upper orthotropic material , and its elastic constants are and . The plane is the lower orthotropic material , and its elastic constants are , and . , is the crack surfaces, and , is the bonded interface of two dissimilar materials.

The stress functions , satisfy the governing equations [1620]:

The boundary conditions of the interface crack for mode II are as follows:

The geometric and load conditions for the investigated problem are given as shown in Figure 1.

fig1
Figure 1: Interface crack for mode II of orthotropic bimaterial.

3. Stress Function

The relationships between the stresses and the stress functions can be obtained asin which the stress functions of right side of (5a)–(5c) contain twelve undetermined coefficients , , and an unknown singularity exponent.

Considering the boundary conditions (2), (3), and (4), the functions of right side of (5a)–(5c) can be chosen as in which

From (5a)–(5c) and (6), an unknown number is called singularity exponent, and sometimes it is also called eigenvalue.

4. Singularity Exponent

If the singularity exponent is the complex number, the stress near the interface crack tip shows oscillatory singularity. We let [116]

Substituting (9) into (6), it can be found as

Equations (7) and (8) are substituted into (10), and we have at the crack surfaces.

Substituting (11), (5b), and (5c) into the boundary condition (2), by the help of the functions and which are linearly independent, the four homogeneous linear equations can be deduced as

Similarly, we can find other eight linear equations to which are omitted here. By in condition (4), two nonhomogeneous linear equations are contained in the eight linear equations.

The system of nonhomogeneous linear equations (12a) to is solved by eliminating the unknowns. To let the rank of the coefficient matrix for the system (12a)– and the rank of the augmented matrix be equal, the rank must be 8; that is, . For this reason [21], the numbermust satisfy the following characteristic equation: in which bimaterial parameters are given as

The singularity exponent of (9) is a complex number, so in (13), and in (14a) and (14b) for determinate bimaterial. To the equation (13) has a solution, the discrimination rule can be deduced as : if and in (14a) satisfy and , then the negative root of (13) is the bielastic constant .Substitutingsolved by (13) into (9), the complex singularity exponent can be obtained.

The system of eight equations which remained through sequence elimination is solved by means of the inverse sequence backsubstitution. We found all coefficients as follows: in which , ; , . By (15a) and (15b), the stress functionsof the right side of (5a)–(5c) contain practically eight coefficients.

5. Stress Intensity Factors

Considering the stress expressions (5b) and (5c) and also the load condition in (4), the stress intensity factors are defined as

In order to express the change process ofcompletely, it is necessary that the factor of (11): , is substituted into the expressions (16b) and (17b). At the same time, the minus is used in (16c) because the load condition is given in (4).

Substituting (10), (7), (8), (15a), and (15b) into (16a)–(16c), (17a)–(17c), and (18), the stress intensity factors of interface crack for mode II can be deduced as in which

6. Stress

From (10), (8), and (7), it can be found as follows that in the vicinity of the crack tip.

By (21), it can be known that

Substituting (22), (15a), (15b), (19), and (8) into (5a)–(5c), the stresses in the vicinity of the interface crack tipfor mode II of orthotropic bimaterial can be expressed as follows:in whichandare given by (20a) and (20b).

Note that (23a)–(23c) of the stress field contain the following terms: The right side of (24) tends to change the sign rapidly at the crack tip , and therefore the stresses possess an oscillatory character.

7. Oscillatory Field

The test and calculus results [16, 2224] for the mechanicals properties of three orthotropic bimaterial are shown in Tables 1 and 2.

tab1
Table 1: Mechanical properties of each orthotropic material.
tab2
Table 2: Mechanical properties of three orthotropic bimaterials.

From (19), the normalized stress intensity factorsanddepend on the length of crackand the bielastic constant. The variations of the stress intensity factors versus crack length are plotted in Figure 2. The variations of the stress intensity factors with bielastic constant are illustrated in Figure 3. The minute variations of factorcan be observed by the inner small figure of Figure 3. The factorsandincrease almost linearly when the lengthor the constantincreases as shown in Figures 2 and 3.

716768.fig.002
Figure 2: Normalized stress intensity factors as a function of .
716768.fig.003
Figure 3: Normalized stress intensity factors as a function of when .

By (23a)–(23c), the normalized stresses, , anddepend on the polar angle and the polar radius. The variations of the normalized stresses for the above three bimaterials with respect to polar angleare plotted in Figure 4. Figure 4 shows that three stresses have alternately the increase and decrease with increase, their maximum values can be always reached at both sides of the crack.

716768.fig.004
Figure 4: Normalized stresses versus for .

Figures 5 and 6 show the variations of the normalized stresses for the above three bimaterials with respect to polar radiusforand. Two figures mean that the stresses, , andare the monotonic decreasing functions ofon the two half-planes.

716768.fig.005
Figure 5: Normalized stresses versus for .
716768.fig.006
Figure 6: Normalized stresses versus for .

Figures 7 and 8 show the variations of the normalized logarithmic stresses with respect to the polar radiusforand. It can be found that the stress distribution in Figures 7 and 8 is not straight line and does not parallel to each other. Such distributions mean oscillatory singularity state.

716768.fig.007
Figure 7: Normalized stresses versus for .
716768.fig.008
Figure 8: Normalized stresses versus for .

8. Conclusion

From the above derivation, the following results are very significant.(1)New stress functions are chosen. (2)The system of twelve nonhomogeneous linear equations is derived. (3)The characteristic equation can be given.(4)By the help of the distinction rule, the complex singularity exponentcan be found. (5)Total coefficients are determined.(6)For the oscillatory field, the theoretical formulae of stress intensity factors and the analytic solutions for stresses near interface crack tip of mode II are yielded, respectively. The stress intensity factors and the stresses show mixed crack characteristics.(7)The crack tip fields for oscillatory singularities are illustrated in Figures 2 to 8.

Acknowledgment

This work was supported by the Natural Science Foundation of Shanxi province (no. 2011011021-3), and the doctoral fund of Taiyuan University of Science and Technology (no. 20102028).

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