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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 716768, 9 pages
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.
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.
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 . The complex function method is developed by Rice and Sih . Erdogan  and England , 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 [5–16]. 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 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.
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.
4. Singularity Exponent
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 , 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
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).
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
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.
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.
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.
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.
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.
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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