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

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 [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 𝑦>0 is the upper orthotropic material (𝑗=1), and its elastic constants are 𝐸₁₁,𝐸₁₂,𝜈₁₁ and 𝜇₁. The plane 𝑦<0 is the lower orthotropic material (𝑗=2), and its elastic constants are 𝐸₂₁,𝐸₂₂,𝜈₂₁, and 𝜇₂. 𝑦=0, |𝑥|<𝑎 is the crack surfaces, and 𝑦=0, |𝑥|>𝑎 is the bonded interface of two dissimilar materials.

The stress functions 𝑈_𝑗(𝑥,𝑦), (𝑗=1,2) satisfy the governing equations [16–20]: 𝑏_22𝑗𝜕⁴𝑈_𝑗𝜕𝑥⁴+2𝑏_12𝑗+𝑏_66𝑗𝜕⁴𝑈_𝑗𝜕𝑥²𝜕𝑦²+𝑏_11𝑗𝜕⁴𝑈_𝑗𝜕𝑦⁴=0,𝑗=1,2. (1)

The boundary conditions of the interface crack for mode II are as follows: 𝑦=0,|𝑥|<𝑎∶𝜎_𝑦1=𝜎_𝑦2𝜏=0,_𝑥𝑦1=𝜏_𝑥𝑦2|𝑥|𝜎=0,𝑦=0,>𝑎∶_𝑦1=𝜎_𝑦2,𝜏_𝑥𝑦1=𝜏_𝑥𝑦2,(𝑢)₁=(𝑢)₂,(𝑣)₁=(𝑣)₂,𝑦⟶+∞∶𝜎_𝑦1=𝜎_𝑦2=0,𝜏_𝑥𝑦1=𝜏_𝑥𝑦2=𝜏. (2)(3)(4)

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

(a)  Geometric state

(b) Shear load

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 as𝜎_𝑥𝑗=𝜕²𝑈_𝑗𝜕𝑦²=²_𝑘=1Re𝐴_{𝑗𝑘,𝜆}+𝑖𝐶_{𝑗𝑘,𝜆}−𝑖𝐵_{𝑗𝑘,𝜆}+𝑖𝐶_{𝑗𝑘,𝜆}×−𝛽²_𝑗𝑘𝑈_{𝑗𝑘,𝜆}𝑧_𝑗𝑘,𝜎_𝑦𝑗=𝜕²𝑈_𝑗𝜕𝑥²=²_𝑘=1Re𝐴_{𝑗𝑘,𝜆}+𝑖𝐶_{𝑗𝑘,𝜆}−𝑖𝐵_{𝑗𝑘,𝜆}+𝑖𝐶_{𝑗𝑘,𝜆}𝑈_{𝑗𝑘,𝜆}𝑧_𝑗𝑘,𝜏_𝑥𝑦𝑗𝜕=−²𝑈_𝑗𝜕𝑥𝜕𝑦=−²_𝑘=1Re𝐴_{𝑗𝑘,𝜆}+𝑖𝐶_{𝑗𝑘,𝜆}−𝑖𝐵_{𝑗𝑘,𝜆}+𝑖𝐶_{𝑗𝑘,𝜆}×𝑖𝛽_𝑗𝑘𝑈_{𝑗𝑘,𝜆}𝑧_𝑗𝑘,𝑗=1,2 (5a)(5b)(5c)in which the stress functions 𝑈_𝑗 of right side of (5a)–(5c) contain twelve undetermined coefficients 𝐴_{𝑗𝑘,𝜆}, 𝐵_{𝑗𝑘,𝜆}, 𝐶_{𝑗𝑘,𝜆}(𝑗,𝑘=1,2) and an unknown singularity exponent𝜆.

Considering the boundary conditions (2), (3), and (4), the functions 𝑈_{𝑗𝑘,𝜆}(𝑧_𝑗𝑘) of right side of (5a)–(5c) can be chosen as 𝑈_{𝑗𝑘,𝜆}𝑧_𝑗𝑘𝑎=𝜏1−²𝑧²_𝑗𝑘^𝜆𝑧=𝜏^{−2𝜆𝑗𝑘}𝑧²_𝑗𝑘−𝑎^2−𝜆,𝑗,𝑘=1,2. (6) in which 𝑧_𝑗𝑘=𝑥+𝑠_𝑗𝑘𝑦=𝑥+𝑖𝛽_𝑗𝑘𝑧𝑦,𝑗,𝑘=1,2_𝑗𝑘−𝑎=𝑥−𝑎+𝑖𝛽_𝑗𝑘𝑦=𝑟cos𝜃+𝑖𝛽_𝑗𝑘,.sin𝜃𝑗,𝑘=1,2 (7)(8)

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

4. Singularity Exponent

If the singularity exponent 𝜆 is the complex number, the stress near the interface crack tip shows oscillatory singularity. We let [1–16] 1𝜆=−2+𝑖𝜀. (9)

Substituting (9) into (6), it can be found as 𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘𝑎=𝜏1−²𝑧²_𝑗𝑘^{−(1/2)+𝑖𝜀}𝑧=𝜏^{1−𝑖2𝜀𝑗𝑘}𝑧²_𝑗𝑘−𝑎^{2(1/2)−𝑖𝜀},𝑗,𝑘=1,2. (10)

Equations (7) and (8) are substituted into (10), and we have 𝑈_{𝑗𝑘,𝜀}𝑧_{𝑗𝑘𝑦=0,|𝑥|<𝑎}=𝑒^{(−1)𝑗𝜀𝜋}𝜏𝑥√𝑎²−𝑥²(−1)^𝑗−1×√sin2𝜀ln𝑎²−𝑥²𝑥√−𝑖cos2𝜀ln𝑎²−𝑥²𝑥 (11) at the crack surfaces.

Substituting (11), (5b), and (5c) into the boundary condition (2), by the help of the functions cos(2𝜀ln((√𝑎²−𝑥²)/𝑥)) and sin(2𝜀ln((√𝑎²−𝑥²)/𝑥)) which are linearly independent, the four homogeneous linear equations can be deduced as𝐴_11,𝜀+𝐴_12,𝜀−𝐵_11,𝜀−𝐵_12,𝜀+2𝐶_11,𝜀+2𝐶_12,𝜀=0,−𝐴_21,𝜀−𝐴_22,𝜀+𝐵_21,𝜀+𝐵_22,𝜀−2𝐶_21,𝜀−2𝐶_22,𝜀=0,−𝛽₁₁𝐴_11,𝜀−𝛽₁₂𝐴_12,𝜀−𝛽₁₁𝐵_11,𝜀−𝛽₁₂𝐵_12,𝜀𝛽=0,₂₁𝐴_21,𝜀+𝛽₂₂𝐴_22,𝜀+𝛽₂₁𝐵_21,𝜀+𝛽₂₂𝐵_22,𝜀=0. (12a)(12b)(12c)(12d)

Similarly, we can find other eight linear equations (12e) to (12l) which are omitted here. By (𝜏_𝑥𝑦)₁=(𝜏_𝑥𝑦)₂=𝜏 in condition (4), two nonhomogeneous linear equations are contained in the eight linear equations.

The system of nonhomogeneous linear equations (12a) to (12l) is solved by eliminating the unknowns. To let the rank of the coefficient matrix 𝐴_𝜆 for the system (12a)–(12l) and the rank of the augmented matrix 𝐴_𝜆=(𝐴_𝜆,𝑏_II) be equal, the rank must be 8; that is, rank(𝐴_𝜆)=rank(𝐴_𝜆)=8. For this reason [21], the number𝜀must satisfy the following characteristic equation: 𝑒₁₂−𝑓₁₂𝑓₁₂−4𝑔₁₂ℎ₁₂𝜀²=0 (13) in which bimaterial parameters are given as 𝑒₁₂=1𝜇₁−1𝜇₂,𝑓₁₂=𝛽₁₁𝛽₁₂−𝜈₁₁𝐸₁₁−𝛽₂₁𝛽₂₂−𝜈₂₁𝐸₂₁,𝑔₁₂=𝛽₁₁+𝛽₁₂𝐸₁₁+𝛽₂₁+𝛽₂₂𝐸₂₁,ℎ₁₂=𝛽₁₁+𝛽₁₂𝐸₁₁𝛽₁₁𝛽₁₂+𝛽₂₁+𝛽₂₂𝐸₂₁𝛽₂₁𝛽₂₂. (14a)(14b)

The singularity exponent 𝜆 of (9) is a complex number, so 𝜀≠0 in (13), and 𝑒₁₂≠0,𝑓₁₂≠0,𝑔₁₂>0,ℎ₁₂>0 in (14a) and (14b) for determinate bimaterial. To the equation (13) has a solution, the discrimination rule can be deduced as (∗) D₁₂: if 𝑒₁₂ and 𝑓₁₂ in (14a) satisfy 𝑒₁₂>0 and 𝑓₁₂>0, then the negative root of (13) is the bielastic constant 𝜀.Substituting𝜀solved by (13) into (9), the complex singularity exponent 𝜆 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: 𝐴_{𝑗𝑘,𝜀}+𝐶_{𝑗𝑘,𝜀}=(−1)^𝑘𝑓₁₂+2𝛽_𝑗𝑘^∗𝑔₁₂𝜀𝛽_𝑗2−𝛽_𝑗1𝑓₁₂,𝐵_{𝑗𝑘,𝜀}−𝐶_{𝑗𝑘,𝜀}=(−1)^𝑘−1𝑓₁₂−2𝛽_𝑗𝑘^∗𝑔₁₂𝜀𝛽_𝑗2−𝛽_𝑗1𝑓₁₂,𝑗,𝑘=1,2. (15a)(15b)in which 𝑘=1, 𝑘^∗=2; 𝑘=2, 𝑘^∗=1. By (15a) and (15b), the stress functions𝑈_𝑗of 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 as𝑘⁺_2𝑗=−²_𝑘=1lim_{𝑧𝑗𝑘→𝑎+}Re2𝜋𝑧_𝑗𝑘−𝑎^{(1/2)−𝑖𝜀}×𝐴_{𝑗𝑘,𝜀}+𝐶_{𝑗𝑘,𝜀}−𝑖𝐵_{𝑗𝑘,𝜀}−𝐶_{𝑗𝑘,𝜀}×𝑖𝛽_𝑗𝑘𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘,𝑘⁻_2𝑗=−²_𝑘=1lim_{𝑧𝑗𝑘→𝑎−}Re2𝜋𝑧_𝑗𝑘−𝑎^{(1/2)−𝑖𝜀}×𝐴_{𝑗𝑘,𝜀}+𝐶_{𝑗𝑘,𝜀}−𝑖𝐵_{𝑗𝑘,𝜀}−𝐶_{𝑗𝑘,𝜀}×(−1)^𝑗−1(2𝜀−𝑖)𝑖𝛽_𝑗𝑘×𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘,,𝑘𝑗=1,2₂=𝑘⁺₂−𝑘⁻₂2=12²_𝑗=1𝑘⁺_2𝑗2−12²_𝑗=1𝑘⁻_2𝑗2. (16a)(16b)(16c)

𝑘⁺_1𝑗=²_𝑘=1lim_{𝑧𝑗𝑘→𝑎+}Re2𝜋𝑧_𝑗𝑘−𝑎^{(1/2)−𝑖𝜀}×𝐴_{𝑗𝑘,𝜀}+𝐶_{𝑗𝑘,𝜀}−𝑖𝐵_{𝑗𝑘,𝜀}−𝐶_{𝑗𝑘,𝜀}×𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘,𝑘⁻_1𝑗=²_𝑘=1lim_{𝑧𝑗𝑘→𝑎−}Re2𝜋𝑧_𝑗𝑘−𝑎^{(1/2)−𝑖𝜀}×𝐴_{𝑗𝑘,𝜀}+𝐶_{𝑗𝑘,𝜀}−𝑖𝐵_{𝑗𝑘,𝜀}−𝐶_{𝑗𝑘,𝜀}×(−1)^𝑗−1(2𝜀−𝑖)𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘,,𝑘𝑗=1,2₁=𝑘⁺₁+𝑘⁻₁2=12²_𝑗=1𝑘⁺_1𝑗2+12²_𝑗=1𝑘⁻_1𝑗2, (17a)(17b)(17c)𝑘=𝑘₂+𝑖𝑘₁. (18) In order to express the change process of𝑧_𝑗𝑘→𝑎⁻completely, it is necessary that the factor of (11): 𝑒^{(−1)𝑗𝜀𝜋}(−1)^𝑗−1[sin(2𝜀ln((√𝑎²−𝑥²)/𝑥))−𝑖cos(2𝜀ln((√𝑎²−𝑥²)/𝑥))]=(−1)^𝑗−1(2𝜀−𝑖),(𝜀→0) 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 𝑘₂=𝜏(𝜋𝑎)^1/2𝑝(𝜀),𝑘₁=−𝜏(𝜋𝑎)^1/22𝑔₁₂𝜀𝑓₁₂𝑞(𝜀),𝑘=𝑘₂+𝑖𝑘₁ (19) in which𝑝(𝜀)=12()]()−[()](,𝑞(𝜀)=1{[1+1+2𝜀𝑐ℎ𝜀𝜋cos𝜀ln𝜋𝑎1+1−2𝜀𝑐ℎ𝜀𝜋sin𝜀ln𝜋𝑎)}2()]()−[()](.{[1−1+2𝜀𝑐ℎ𝜀𝜋cos𝜀ln𝜋𝑎1−1−2𝜀𝑐ℎ𝜀𝜋sin𝜀ln𝜋𝑎)} (20a)(20b)

6. Stress

From (10), (8), and (7), it can be found as follows that lim_{𝑧𝑗𝑘→𝑎}2𝜋𝑧_𝑗𝑘−𝑎^{(1/2)−𝑖𝜀}𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘=𝜏(𝜋𝑎)^{(1/2)−𝑖𝜀},𝑗,𝑘=1,2 (21) in the vicinity of the crack tip.

By (21), it can be known that 𝑈_{𝑗𝑘,𝜀}𝑧_𝑗𝑘=𝜏(𝜋𝑎)^{(1/2)−𝑖𝜀}(2𝜋)^{(1/2)−𝑖𝜀}1𝑧_𝑗𝑘−𝑎^{(1/2)−𝑖𝜀},𝑧_𝑗𝑘⟶𝑎;𝑗,𝑘=1,2. (22)

Substituting (22), (15a), (15b), (19), and (8) into (5a)–(5c), the stresses in the vicinity of the interface crack tip(𝑧_𝑗𝑘→𝑎,𝑟→0;𝑗,𝑘=1,2)for mode II of orthotropic bimaterial can be expressed as follows:𝜎_𝑥𝑗=𝑘₂(2𝜋𝑟)^1/2√2𝛽_𝑗2−𝛽_𝑗1𝑝(𝜀)×𝜋cos4()𝜋−𝜀ln𝜋𝑎⋅cos𝜀ln2𝜋𝑟−sin4(⋅𝛽−𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Re^2𝑗1cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}𝛽−Re^2𝑗2cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}−𝜋cos4()𝜋+𝜀ln𝜋𝑎cos𝜀ln2𝜋𝑟+sin4(⋅𝛽+𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Im^2𝑗1cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}𝛽−Im^2𝑗2cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}+𝑘₁(2𝜋𝑟)^1/2√2𝛽_𝑗1𝛽_𝑗2𝛽_𝑗2−𝛽_𝑗1𝑞(𝜀)×𝜋cos4()𝜋+𝜀ln𝜋𝑎⋅cos𝜀ln2𝜋𝑟+sin4(⋅𝛽+𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Re^𝑗2cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}𝛽−Re^𝑗1cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}+𝜋cos4()𝜋−𝜀ln𝜋𝑎cos𝜀ln2𝜋𝑟−sin4(⋅𝛽−𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Im^𝑗2cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}𝛽−Im^𝑗1cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀},𝜎_𝑦𝑗=𝑘₂(2𝜋𝑟)^1/2√2𝛽_𝑗2−𝛽_𝑗1𝑝(𝜀)×𝜋cos4()𝜋−𝜀ln𝜋𝑎⋅cos𝜀ln2𝜋𝑟−sin4(⋅1−𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Recos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}1−Recos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}−𝜋cos4()𝜋+𝜀ln𝜋𝑎cos𝜀ln2𝜋𝑟+sin4(⋅1+𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Imcos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}1−Imcos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}+𝑘₁(2𝜋𝑟)^1/2√2𝛽_𝑗2−𝛽_𝑗1𝑞(𝜀)×𝜋cos4()𝜋+𝜀ln𝜋𝑎⋅cos𝜀ln2𝜋𝑟+sin4(⋅𝛽+𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Re^𝑗2cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}𝛽−Re^𝑗1cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}+𝜋cos4()𝜋−𝜀ln𝜋𝑎cos𝜀ln2𝜋𝑟−sin4(⋅𝛽−𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Im^𝑗2cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}𝛽−Im^𝑗1cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀},𝜏_𝑥𝑦𝑗=𝑘₂(2𝜋𝑟)^1/2√2𝛽_𝑗2−𝛽_𝑗1𝑝(𝜀)×𝜋cos4()𝜋+𝜀ln𝜋𝑎⋅cos𝜀ln2𝜋𝑟+sin4(⋅𝛽+𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Re^𝑗2cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}𝛽−Re^𝑗1cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}+𝜋cos4()𝜋−𝜀ln𝜋𝑎cos𝜀ln2𝜋𝑟−sin4(⋅𝛽−𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Im^𝑗2cos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}𝛽−Im^𝑗1cos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}+𝑘₁(2𝜋𝑟)^1/2√2𝛽_𝑗1𝛽_𝑗2𝛽_𝑗2−𝛽_𝑗1𝑞(𝜀)×𝜋cos4()𝜋−𝜀ln𝜋𝑎⋅cos𝜀ln2𝜋𝑟−sin4(⋅1−𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Recos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}1−Recos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀}−𝜋cos4()𝜋+𝜀ln𝜋𝑎cos𝜀ln2𝜋𝑟+sin4(⋅1+𝜀ln𝜋𝑎sin𝜀ln2𝜋𝑟)Imcos𝜃+𝑖𝛽_𝑗2sin𝜃^{(1/2)−𝑖𝜀}1−Imcos𝜃+𝑖𝛽_𝑗1sin𝜃^{(1/2)−𝑖𝜀} (23a)(23b)(23c)in which𝑝(𝜀)and𝑞(𝜀)are given by (20a) and (20b).

Note that (23a)–(23c) of the stress field contain the following terms: 𝜎_𝑥𝑗𝜎_𝑦𝑗𝜏_𝑥𝑦𝑗⊃1(2𝜋𝑟)^1/2()()cos𝜀ln2𝜋𝑟sin𝜀ln2𝜋𝑟. (24) The right side of (24) tends to change the sign rapidly at the crack tip (𝑟→0), and therefore the stresses possess an oscillatory character.

7. Oscillatory Field

The test and calculus results [16, 22–24] for the mechanicals properties of three orthotropic bimaterial are shown in Tables 1 and 2.

Table 1 

Mechanical properties of each orthotropic material.

Table 2 

Mechanical properties of three orthotropic bimaterials.

From (19), the normalized stress intensity factors𝑘₂/𝜏and𝑘₁/𝜏depend on the length of crack𝑎and the bielastic constant𝜀. 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 factor𝑘₁/𝜏can be observed by the inner small figure of Figure 3. The factors𝑘₂/𝜏and𝑘₁/𝜏increase almost linearly when the length𝑎or the constant𝜀increases as shown in Figures 2 and 3.

Figure 2 

Normalized stress intensity factors as a function of 𝑎.

Figure 3 

Normalized stress intensity factors as a function of 𝜀 when 𝑎=1.

By (23a)–(23c), the normalized stresses𝜎_𝑥/𝜏, 𝜎_𝑦/𝜏, and𝜏_𝑥𝑦/𝜏depend on the polar angle 𝜃 and the polar radius𝑟. The variations of the normalized stresses for the above three bimaterials with respect to polar angle𝜃are plotted in Figure 4. Figure 4 shows that three stresses have alternately the increase and decrease with 𝜃 increase, their maximum values can be always reached at both sides of the crack.

Figure 4 

Normalized stresses versus 𝜃 for 𝑟/𝑎=0.0001.

Figures 5 and 6 show the variations of the normalized stresses for the above three bimaterials with respect to polar radius𝑟for𝜃=±45^∘and𝜃=±60^∘. Two figures mean that the stresses𝜎_𝑥/𝜏, 𝜎_𝑦/𝜏, and𝜏_𝑥𝑦/𝜏are the monotonic decreasing functions of𝑟/𝑎on the two half-planes.

Figure 5 

Normalized stresses versus 𝑟 for 𝜃=±45∘.

Figure 6 

Normalized stresses versus 𝑟 for 𝜃=±60∘.

Figures 7 and 8 show the variations of the normalized logarithmic stresses with respect to the polar radius𝑟for𝜃=±45^∘and𝜃=±60^∘. 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.

Figure 7 

Normalized stresses versus 𝑟 for 𝜃=±45∘.

Figure 8 

Normalized stresses versus 𝑟 for 𝜃=±60∘.

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 exponent𝜆can 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).