An Analysis of Mode III Doubly Periodic Crack-Tip Field of Orthotropic Composite Materials

Abstract

The mechanical behavior near the doubly periodic crack tips for orthotropic composite materials plate subjected to antiplane shear loading is studied. This is done by complex function theory and conformal mapping of the Jacobi elliptic function with the help of boundary conditions. The analytical solution of the crack-tips stress intensity factor and the expression of stress fields are obtained. Numerical examples are given to analyze the impact of the different transverse spacing, longitudinal spacing, and the ratio of cracks periods on stress intensity factors. The results show that the crack-tip field increases with reducing either the transverse spacing or the longitudinal spacing. At the same time, the crack-tip field increases with the decrease of the ratio of cracks periods. This shows that the distribution form makes an important effect on the crack-tip field, but the crack density parameter is not the only cause.

1. Introduction

Fiber reinforced composite materials are used widely and are characterized by the proportion of small, specific strength and large specific modulus, but there are usually many microcracks, microholes in the process of fatigue damage. Because of these microcracks, microholes are arranged in periodic characteristic more or less, and studying the interference between the doubly periodic cracks helps to deepen the understanding of the materials failure mechanism and plays an important role in fracture behavior of materials.

Doubly periodic crack problem is an important mechanical model for the study of the interaction between multiple cracks. The more important value of doubly periodic crack model is that it provides the limit properties of interference between multiple cracks in the process from disorder to order. This problem is solved by the superposition method, the single-cell method, and numerical methods. By using distribution dislocation method [1], the doubly periodic distribution of crack problem is turned into the integral equation. Pseudotraction method [2, 3] is also a good method to deal with this problem, but one needs to pay attention to the summation order and convergence issues. Chen and Lee [4], Chen and Lin [5], Dong and Lee [6], Yan and Jiang [7], Yan et al. [8], and Tong et al. [9] used the eigen function expansion variational method to solve the doubly periodic cracks and analyze the interaction of stress intensity factor in the different cracks arrangement of the corresponding rule. In 2011, Xiao et al. [10] obtained the exact solution to the antiplane problem of doubly periodic conducting rigid line inclusions of unequal size in piezoelectric materials. Chang et al. [11] discussed the anti-plane problem of doubly periodic cracks in the infinite homogeneous piezoelectric materials using conformal mapping and elliptic function and calculated the stress intensity factor and stress field in a closed form solution. Using two-dimensional boundary element method, Pasternak [12] analysed the intensity of mechanical, electric, and magnetic fields near the tips of magnetoelectroelastic medium containing doubly periodic sets of cracks or thin inclusions. Wang [13] and Wang and Feng [14] introduced a famous summation equation of complex variable function series with infinite number of terms and solved the interaction of multiple rows of periodic cracks with extremely high accuracy.

In this paper, the cracks-tip field of Mode??III double periodic cracks on plate subjected to anti-plane shear stress is studied through introducing the Westergaard stress function and conformal mapping of the Jacobi elliptic function. Stress intensity factor and stress fields on the cracks tip are obtained with the help of the boundary conditions. Numerical examples analyzed the impact of the different transverse spacing, longitudinal spacing, and the ratio of cracks periods on stress intensity factors. This research contributes to the understanding of the interaction between multiple cracks contained in the composite materials.

2. Crack Model

Figure 1 shows the infinite orthotropic composite plate with doubly periodic cracks. The crack length is 2𝑎, transverse spacing is 𝑏, longitudinal spacing is ℎ, and the plate surface is subjected to the shear force 𝜏.

Figure 1 

The crack model of doubly periodic cracks in the orthotropic composite plate.

From the theory of elasticity, strain-stress relationship [15–18] is 𝜏_𝑦𝑧=𝑄₄₄𝛾_𝑦𝑧,𝜏_𝑧𝑥=𝑄₅₅𝛾_𝑧𝑥,𝛾_𝑦𝑧=𝜕𝑤𝜕𝑦,𝛾_𝑧𝑥=𝜕𝑤,𝜕𝑥 (1) where 𝑄₄₄, 𝑄₅₅ are the stiffness coefficients in the elastic principal direction of the orthotropic materials.

The balancing equation of the orthotropic materials can be expressed as follows: 𝜕𝜏_𝑧𝑥+𝜕𝑥𝜕𝜏_𝑦𝑧𝜕𝑦=0. (2)

By substituting (1) into (2), the governing equation can be written as 𝑄₅₅𝜕²𝑤𝜕𝑥²+𝑄₄₄𝜕²𝑤𝜕𝑦²=0. (3)

Assuming 𝑤=𝑤(𝑥+𝑠𝑦), the characteristic equation of the governing equation (3) can be obtained as follows: 𝑄₄₄𝑠²+𝑄₅₅=0. (4)

The roots of characteristic equation (4) are 𝑠₁=𝑖𝑄₅₅𝑄₄₄=𝑖𝛽,𝑠₂=−𝑖𝛽. (5) Labeling 𝑧₁=𝑥+𝑠₁𝑦=𝑥₁+𝑖𝑦₁, (6) where 𝑥₁=𝑥, 𝑦₁=𝛽𝑦.

By (5) and (6), the governing equation (3) can be turned into generalized harmonic equation: ∇²₁𝜕𝑤=²𝜕𝑥²₁+𝜕²𝜕𝑦²₁𝑤=0. (7)

By the complex function theory, the governing equation (3) has a real analytic solution. We can choose it as follows: 𝑤=Re𝐹𝑧₁. (8)

Combining (1) with (8), the stress expression can be written as 𝜏_𝑥𝑧−𝑖𝜏_𝑦𝑧=𝑄₅₅Re𝐹^′𝑧₁+𝑖𝑄₄₄𝛽Im𝐹^′𝑧₁. (9)

Now, formulate the boundary conditions of the rectangle 𝑂𝐵𝐴𝐶𝐷𝐷^′𝐶^′𝐴^′𝐵^′. Here, 𝐵𝐴=𝐶𝐷=𝐷^′𝐶^′=𝐴^′𝐵^′=𝑎, 𝐴^′𝐴=𝑏,??and 𝐴𝐷=ℎ. The crack faces are free of traction, and it follows that 𝜏_𝑦𝑧=0onthe𝐵𝐴,𝐴^′𝐵^′,𝜏_𝑥𝑧=0onthe𝐶𝐷,𝐷^′𝐶^′. (10)

3. The Conformal Mapping of Jacobi Elliptical Function

Making use of the Jacobi elliptic function [19] 𝑧𝜁=sn¹𝐴,𝑘, (11)𝑧₁=𝐴∫^𝜁₀(𝑑𝑡/(1−𝑡²)(1−𝑘²𝑡²)) is an inverse function, where 𝐴 is a constant, and 𝑘 is the modulus of the function sn. The transform characteristic points in Figure 2 are indicated.

Figure 2 

Conformal mapping.

With the relation of the elliptic function 𝐴^′𝐴=𝑏𝐴𝐷ℎ=2𝐴𝐾(𝑘)𝐴𝐾𝑘^′, (12) where 𝑘^′2=1−𝑘² is called as complementary modulus, and 𝐾(𝑘) is the complete elliptic integral of the first kind: 𝐾(𝑘)=¹₀𝑑𝑡1−𝑡²1−𝑘²𝑡². (13)

From the mapping function, we can get that 𝑏𝐴=(𝑘)=ℎ2𝐾𝐾𝑘^′. (14) So, 𝑏ℎ=2𝐾(𝑘)𝐾𝑘^′=2∫¹₀𝑑𝑡/1−𝑡²1−𝑘²𝑡²∫¹₀𝑑𝑡/1−𝑡²1−𝑘^′2𝑡². (15)

The 𝑘,𝑘^′ can be obtained by numerical calculation. Using formula (11), the parameters can be obtained as follows: 𝑏₁()=sn𝑏/2−𝑎𝐴=(),𝑘cn𝑎/𝐴,𝑘dn(𝑎/𝐴,𝑘)=−𝑏^′₁,𝑐₁))()=sn((𝑏/2−𝑎+𝑖ℎ−𝑎𝐴=,𝑘dn𝑎/𝐴,𝑘^′𝑘=−𝑐^′₁. (16) Here cn, dn are Jacobi elliptic functions also.

4. The Solution of Crack-Tip Stress Field

Mapping the 𝑧-plane into 𝜁-plane, the boundary conditions are Φ(𝜁)=𝑖𝐹^′𝑧₁. (17) On (−1/𝑘,𝑐^′₁), (−1,𝑏^′₁), (𝑏₁,1), and (𝑐₁,1/𝑘), 𝑄₄₄𝛽ReΦ(𝜁)=0. (18) On the remainder of the real axis, 𝑄₅₅ImΦ(𝜁)=0. (19)

Using the Keldych-Sedov formula [16], the solution of Φ(𝜁) is Φ(𝜁)=𝑐₀𝜁²−1𝜁²−1/𝑘²𝜁−𝑐^′₁𝜁−𝑏^′₁𝜁−𝑐₁𝜁−𝑏₁, (20) where 𝑐₀ is an undetermined coefficient.

From (16), it can be simplified to Φ(𝜁)=𝑐₀𝜁²−1𝜁²−1/𝑘²𝜁−𝑐²₁𝜁−𝑏²₁. (21)

Considering the periodic cracks on the 𝑧-plane, the balancing condition of shear force is as follows: ^𝐵_𝐵′𝜏_𝑦𝑧𝑑𝑥=𝑏𝜏. (22)

Substituting (11), (12), and (21) into (22), 𝑐₀=𝑘𝐾(𝑘)𝜏𝑄₄₄𝛽𝐾𝑘_𝑟𝑐₁+𝑏₁, (23) where 𝑘²_𝑟=4𝑐₁𝑏₁/(𝑐₁+𝑏₁)².

Substituting 𝑐₀ into (21) gets Φ(𝜁)=𝑘𝐾(𝑘)𝜏𝑄₄₄𝛽𝐾𝑘_𝑟𝑐₁+𝑏₁𝜁²−1𝜁²−1/𝑘²𝜁−𝑐²₁𝜁−𝑏²₁. (24)

The stress field of double periodic cracks is 𝜏_𝑦𝑧=𝑄₄₄Φ(𝜁=(𝑘)𝜏𝛽Re)𝑘𝐾𝐾𝑘_𝑟𝑐₁+𝑏₁Re𝜁²−1𝜁²−1/𝑘²𝜁−𝑐²₁𝜁−𝑏²₁,𝜏_𝑥𝑧=𝑄₅₅Φ(𝜁=𝑄Im)₅₅𝑘𝐾(𝑘)𝜏𝑄₄₄𝛽𝐾𝑘_𝑟𝑐₁+𝑏₁Im𝜁²−1𝜁²−1/𝑘²𝜁−𝑐²₁𝜁−𝑏²₁. (25)

5. Stress Intensity Factor and the Interaction between Cracks

Considering periodicity, the stress intensity factor 𝐾_ΙΙΙ𝐵 near the tip 𝐵((𝑏/2)−𝑎,0) is defined by 𝐾_ΙΙΙ𝐵=lim_{()𝑧→𝑏/2−𝑎}𝑏2𝜋2−𝑎−𝑧𝜏_𝑦𝑧=𝑘^′𝐾𝑘_𝑟𝑏𝐾(𝑘)𝑐₁+𝑏₁2𝑎𝑏₁𝑐₁−𝑏₁Resn𝑎/𝐴,𝑘^′dn²𝑎/𝐴,𝑘^′√𝜋𝑎𝜏. (26) Labeling 𝐾^∗_ΙΙΙ𝐵=𝑘^′𝐾𝑘_𝑟𝑏𝐾(𝑘)𝑐₁+𝑏₁2𝑎𝑏₁𝑐₁−𝑏₁Resn𝑎/𝐴,𝑘^′dn²𝑎/𝐴,𝑘^′. (27) The stress intensity factor 𝐾_ΙΙΙ𝐶 near the tip 𝐶(𝑏/2,ℎ−𝑎) is defined by 𝐾_ΙΙΙ𝐶=lim_{()𝑧→𝑏/2+𝑖(ℎ−𝑎)}𝑏2𝜋𝑧−2+𝑖(ℎ−𝑎)𝜏_𝑦𝑧=𝐾(𝑘)𝐾𝑘_𝑟×Reℎ𝐾𝑘^′𝑐₁+𝑏₁𝑎𝑐₁𝑐₁−𝑏₁×1−dn²𝑎/𝐴,𝑘^′𝑘²1−dn²𝑎𝐴,𝑘^′×√𝜋𝑎𝜏. (28) Labeling 𝐾^∗_ΙΙΙ𝐶=𝐾(𝑘)𝐾𝑘_𝑟×Reℎ𝐾𝑘^′𝑐₁+𝑏₁𝑎𝑐₁𝑐₁−𝑏₁×1−dn²𝑎/𝐴,𝑘^′𝑘²1−dn²𝑎𝐴,𝑘^′. (29)

If the number of collinear cracks (density) increases, the strength of the crack tip field increases; similarly, if the number of parallel cracks (density) increases, the strength of the crack tip field increases. On the contrary, the crack tip field decreases with the increase of the period ratio 𝑏/ℎ. It shows that the distribution form of cracks has an important effect on crack-tip field. For the cracks of rectangular distribution, under the condition of the different ratios 𝑏/ℎ, the variation of the stress intensity factor of crack-tip 𝐵 is showed in Figures 3 and 4. In Figure 3, 𝐾^∗_ΙΙΙ𝐵 increases with increasing the ratio 2𝑎/𝑏. When 2𝑎/𝑏→1, 𝐾^∗_ΙΙΙ𝐵 grows quickly and tends to infinity. The reason is that the interaction is much smaller when the transverse spacing 𝑏 is longer in the initial phase of curve. In Figure 4, 𝐾^∗_ΙΙΙ𝐵 turns large rapidly and tends to infinity with increasing the ratio 𝑎/ℎ under the condition of the different ratio 𝑏/ℎ. At the end stage of curve, the longitudinal spacing is smaller, and the interaction is obvious. The cause is that the longitudinal spacing ℎ is reduced at the end stage of curve.

Figure 3 

Various SIF of crack-tip 𝐵 with 2𝑎/𝑏.

Figure 4 

Various SIF of crack-tip 𝐵 with 𝑎/ℎ for 𝑏/ℎ = 0.5, 1.0, 1.5, and 1.8.

The stress intensity factor of crack-tip 𝐶 is showed in Figure 5. The variation of the 𝐾^∗_ΙΙΙ𝐶??with the ratio 𝑎/ℎ increase is plotted in Figure 5 for several values of the dimensionless ratio 𝑏/ℎ. With the ratio 𝑏/ℎ reduction, the 𝐾^∗_ΙΙΙ𝐶??is increased; that is, the crack-tip 𝐵 makes an amplification effect on crack-tip 𝐶. It can be seen from Figure 6 that the stress intensity factor in crack-tip 𝐵 has a strong interaction, which is caused by the distribution of crack-tip 𝐶 when the longitudinal spacing ℎ is smaller. However, when the longitudinal spacing ℎ is larger, the crack-tip field remains unchanged; that is, the stress intensity factor is unchanged within a certain range of the longitudinal spacing ℎ. But, when ℎ grows bigger, the transverse spacing 𝑏 also increases, and the crack-tip field deduces; that is, the crack-tip 𝐶 field has a shielding effect on crack-tip 𝐵. Then, when ℎ grows relatively large, the changes in the crack-tip field are stabilized, which means that the crack tip 𝐶 has no interference effect on crack-tip 𝐵. From Figures 5 and 6, we can deduce that the important influence of the crack distribution form on crack-tip field is not just caused by the crack density parameter.

Figure 5 

Various SIF of crack-tip 𝐶 with 𝑎/ℎ.

Figure 6 

Various SIF of crack-tip 𝐵 with ℎ/𝑎 for 𝑎/𝑏 = 0.1, 0.2, 0.3, and 0.4.

6. Conclusions

(1)The analytical solution of the stress intensity factor of doubly periodic cracks in orthotropic composite materials is obtained under anti-plane mechanical load, by the introduction of the Jacobi elliptic function and the conformal mapping method.(2)It is easy to calculate the analytic form of stress intensity factor through numerical calculation to obtain the modulus 𝑘, which contributes to a better understanding of the nature of the fault of the interaction between multiple cracks in composite materials.(3)Analyses of the influence on stress intensity factor at different transverse spacing, longitudinal spacing, and the ratio 𝑏/ℎ are given. The results show that the important influence of the crack distribution form on crack-tip field is not just caused by the crack density parameter.

Acknowledgments

This work was financially supported by the Science and Technology Major Project of the Ministry of Education of China (208022) and the Doctoral Initiating Project of the Taiyuan University of Science and Technology (20122005).