1 Introduction

The wide application of bi-material structure to engineering,such as the composite material and the functional gradient materials,has greatly promoted the deepening of the research into interface fracture mechanics. The mechanical properties of the interface are closely related to the intensity,damage,and destruction of the structural material. Practical application shows that interface cracks constitute a major reason for failures of structural connection of different materials. Thus,the study of the stress field of the interface sustained such faults as bi- material cracks is of particularly theoretical and practical significance. Williams^[1] and Zak and Williams^[2] used the airy stress function and developed the eigen-function expansion method in order to study the single-material wedge for several combinations of homogeneous boundary conditions. England^[3],Erdogan^[4],Hein and Erdogan^[5],Erdogan and Wu^[6],and Elborgi et al.^[7] pointed out that the cause leading to overlapping near the ends of the crack is a physically unreasonable aspect of the oscillatory singularities. To correct this unsatisfactory feature, Common^[8] and Comininou and Schmueser^[9] introduced a closed crack tip model. Bogy^{[10, 11]} found that the stress singularity is purely determined by the material property mismatch and two joint angles of the bi-material corner. Generally speaking,the material property mismatch can be expressed in terms of the Dundurs parameters α and β^{[12, 13]},which are two non- dimensional parameters computed from the elastic constants of two bonded materials. The stress singularity order λ may be real or complex. Quite a good variety of researches on the problem of interface cracks are carried out^{[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]}. Most of researchers revealed that there is oscillatory singularity near the interface crack tip. The interface crack tip field stress and the displacement of joined materials are characterized by oscillatory singularity and overlapping of the crack planes,which defy physical laws.

The paper expands such methods as the complex variable function and the undetermined coefficients,frequently used to study the plane crack problems of composite materials,to study the bi-material interface crack problems. Under mixed loads in the crack plane,the mechanical problem of bi-material interface crack tip field can be reduced to a boundary value problem of partial differential equations. To describe the bi-material mechanical properties,the bi- material engineering parameters are introduced,and the relationship between the bi-material engineering parameters and the elastic constant of material is analyzed. Expression for the bi-material stress intensity factor,which reflects the strength of the bi-material crack tip elastic stress field,is obtained. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young’s modulus are given in the numerical analysis. 2 Mechanical model

The model for the orthotropic mixed type interface crack is shown in Fig. 1. The length of interface crack pierced in the center of the x-axis is 2a. The axes x and y are parallel to the principal directions of material elasticity,and at the same time under even and symmetrical stress load σ and shear load τ. Using the right tip of the crack as the origin of coordinates,a polar coordinate system is established.

Without considering the volume force and using elastic mechanics,the mechanical model for the orthotropic bi-material interface crack is^{[25, 26, 27, 28]}

in which (σ_y)_j and (τ_xy)_j are the stresses,and (u)_j and (v)_j are the displacements along the axles x and y,respectively. (1) is the control equation of orthotropic bi-material plate,(2) and (3) are the boundary conditions,and (4) is the continuous condition on the interface. The problem of bi-material interface crack is transformed into a boundary value problem of partial differential equations (1)−(4). Using the complex variable function,the boundary value of partial differential equations is obtained through the introduction of a new stress function containing real parameters.

To describe the mechanical properties of the bi-material interface,the bi-material engineering parameters e₁₂,f₁₂,g₁₂,and h₁₂ are introduced as follows:

of which E_j1 and E_j2 are Young’s moduli in the directions of x and y of material j (j = 1,2), respectively,v_j1 is Poisson’s radio of material j (j = 1,2),and Gj is the shear modulus of material j (j = 1,2). 3 Stress function

According to the complex variable function,the control equation of orthotropic bi-material plane can be transformed into a generalized re-harmonic equation. According to the partial differential equation and the complex variable function,there surely exists an analytical solution to the control equation,and the solution can be expressed as the real part and the image part of analytical function of complex variable z_jk (j,k = 1,2). Under the loading conditions,the stress function U_j is selected as the series involving two real stress singularity exponents λ₁ and λ₂ as follows:

of which A_jk,λ_t ,B_jk,λ_t,C_jk,λ_t,and D_jk,λ_t are undetermined coefficients.

Substituting (6) into the stress and stress function equation,the stress expressions can be rewritten as

Similarly,with the help of the stress-strain relationship and the strain-displacement relationship, the displacement expression with the stress function can be derived.

With the help of the boundary conditions and the continuous conditions (1)−(4),we can obtain two sets of non-homogeneous linear equations with 16 unknowns of coefficients A_jk,λ_t , B_jk,λ_t ,C_jk,λ_t ,and D_jk,λ_t (j,k,t = 1,2). In accordance with the nature of determinants,the coefficient matrix of solved sets of equations can be written as

It has been proved that the rank of the coefficient matrix of non-homogeneous linear equations with 16 unknowns is the same as that of the augmented matrix,which is 14. According to the linear algebra,there are infinite solutions to non-homogeneous linear equations with 16 unknowns. The augmented matrix for the non-homogeneous linear equations with 16 unknowns, by the elementary row transformation,can be reduced to the simplest form. Since the rank of the coefficient matrix is 14,16 undetermined coefficients A_jk,λ_t ,B_jk,λ_t ,C_jk,λ_t ,and D_jk,λ_t (j,k,t = 1,2) containing 2 free unknown values can be derived by substituting rλt into the stress function as a factor.

Since Rank (A_λt) = 14,the determinant of the coefficient matrix for the non-homogeneous linear equations is zero,that is,

in which the stress singularity exponents are Here,ε_t is a real bi-material elastic constant,while the value of n is determined by the boundary conditions. Let us plug the stress singularity exponents λ_t (10) into the stress function U_j (6). Consider the values of k,t,and n. Then,the stress function can be confirmed as the following series: Near the crack tip (z_jk → a ),let n = 0. We have Substituting (12) into (9),expanding function 1/ 1−2ε_t and tan (ε_tπ) as power series,and omitting third and higher orders of infinitesimal of ε_t,the quadratic equation of one variable of real bi- material elastic constant ε_t can be derived as When the discriminant is greater than zero,namely, the real root of bi-material elastic constant ε_t can be derived as As for the orthotropic bi-material mixed type interface crack,only when the bi-material engi- neering parameters satisfy (14),can two real bi-material elastic constants ε₁ and ε₂ be obtained by (15). Substituting ε₁ and ε₂ into (12),two real stress singularities λ₁ and λ₂ are obtained. 4 Stress intensity factor The mixed type stress intensity factor can be expressed as of which

Considering the stress singularity of the crack tip and the condition of mixed loads,and in (16) choosing

as z_jk → a,we have Obviously,(18) and (7) are different in constant multiple. Since the rank of the coefficient matrix for the non-homogeneous linear equations is 14 containing 2 unknown values,the influence of the constant multiple can be scrapped through the choice of free unknown values. Thus,(18) can be taken for the stress function of the crack tip.

Because the coefficients A_jk,λ_t ,B_jk,λ_t ,C_jk,λ_t ,and D_jk,λ_t (j,k,t = 1,2) contain free unknown values,the stress intensity factor (16) also contains free unknown values which cannot be determined. However,these can be determined through the uniqueness theorem of limit,and thus the expression for the stress intensity factor of the crack tip is obtained.

According to the uniqueness theorem of limit,when z_jk → a⁻ and z_jk → a⁺,

Through (19),the free unknown values can be derived when the discriminant of the charac- teristic equation is less than zero. Substituting the solutions to the non-homogeneous linear equations,the coefficient equation can be derived as of which α_j and β_j (j,k = 1,2) are the real part and the imaginary part of the characteristic roots of material j^[28],respectively. Substituting (20) and (11) into (7),the stress expressions can be obtained.

Substituting (20) into (16),the equations for the stress intensity factor for material j = 1 are

The stress intensity factor of material j = 2 can be expressed as follows:

Similarly,the expressions for the stress intensity factor of the orthotropic bi-material mixed type interface crack tip,with both the characteristic discriminants greater than zero or one being positive and the other being negative,can be derived in a similar way as (21) and (22).

Considering the intervention of the bi-material,the concept of bi-material stress intensity factor is introduced as follows:

Using (21) and (22),the expressions for the stress intensity factor of the orthotropic bi-material mixed type interface crack tip are

The stress intensity factor of the orthotropic bi-material mixed type interface crack tip characterizes mixed type fracture,with the type I bi-material stress intensity factor K_I related to the stress load σ and the type /bi-material stress intensity factor K_II related to both the shear load τ and the stress load σ. Meanwhile,the type I stress intensity factor K_I and the type II stress intensity factor KII are also related to the shape of the cracks and the bi-material engineering parameters,which means that they are also related to elastic constants of the two materials.

If the orthotropic bi-materials j = 1,j = 2 are the same,the bi-material stress intensity factor (24) of mixed type interface crack tip can be reduced to

This agrees with the result of the stress intensity factor of orthotropic composite plate I+II mixed type crack tip^[28]. 5 Stress field and displacement field near crack tip

Let

of which

Substituting (18),(20),(21),(22),and (26) into (7),the stress expressions of material j = 1, j = 2 can be obtained as follows.

For material j = 1,

For material j = 2,

By the relative formula,the displacement expression of materials j = 1 and j = 2 can be obtained as follows. For material j = 1,

For material j = 2, 6 Numerical analysis 6.1 Case 1

Assume the ratio of bi-material shear modulus μ = G₁/G₂. Assume the parameters for the two materials to be

Then,the relationship between the bi-material engineering parameters and the ratio of bi- material shear modulus is shown as Fig. 2.

The change of the ratio of bi-material shear modulus significantly influences the value of the bi-material engineering e₁₂,and e₁₂ increases with the increase in μ in a linear way. The curve slopes of the bi-material engineering parameters g₁₂ and h₁₂ are relatively small,showing that the change of the ratio of bi-material shear modulus μ has little influence on the bi-material engineering parameters g₁₂ and h₁₂. The curve of bi-material engineering parameters f₁₂ is a straight line,showing that the bi-material engineering parameter f₁₂ has nothing to do with the change of the ratio of bi-material shear modulus μ. In summary,it can be considered that the bi-material engineering parameter e₁₂ is a value showing the elastic characteristic of the bi-material,that is,showing the elastic characteristic by the change of the angle between the axle x and the axle y.

For simplicity,let σ = τ = 1,and a = 0.005. Using the bi-material elastic constant given in (31),when the ratio of bi-material shear modulus μ > 1.04,discriminant Δ > 0 is true,as shown in Fig. 3.

Figure 4 shows the relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus μ. The bi-material stress intensity factors K_I,K_II,and the modulus |K| increase with the increase in the ratio of bi-material shear modulus μ. When 1.04 < μ < 1.23,the increase rate of K_I is smaller than that of K_II (K_II increases significantly). When μ > 1.23,the increase rate of K_I is greater than that of K_II (K_I increases significantly), and K_I/K_II tends to approach 1,shown in Fig. 5.

6.2 Case 2

Assume the ratio of bi-material Young’s modulus γ = E₁₁/E₂₁. Assume the parameters for the two materials to be

Then,the relationship between the bi-material engineering parameters and the ratio of bi- material Young’s modulus is shown in Fig. 6.

The change of the ratio of bi-material Young’s modulus significantly influences the value of bi-material engineering e₁₂,and e₁₂ reduces with the increase in in a linear way. The curve slopes of the bi-material engineering parameters h₁₂ and f₁₂ are relatively small,showing that the change of the ratio of bi-material Young’s modulus has little influence on the bi-material engineering parameters h₁₂ and f₁₂. The bi-material engineering parameters g12 decreases with the increase in the ratio of bi-material Young’s modulus . In summary,it can be considered that the bi-material engineering parameter e₁₂ is a value showing the elastic characteristic of the bi-material,that is,showing the elastic characteristic by the change of the axle x.

For simplicity,let σ = τ = 1,and a = 0.005. Using the bi-material elastic constant given in (32) and the ratio of bi-material shear modulus 0 < γ < 0.260 1 or γ > 1.12,we have discriminant Δ > 0,as shown in Fig. 7.

Figure 8 shows the relationship between the bi-material stress intensity factor and the ration of bi-material Young’s modulus . The bi-material stress intensity factors K_I,K_II,and modulus |K| increase with the increase in the ratio of bi-material Young’s modulus . When 1.12 < γ < 2.15,the increase rate of KII is smaller than that of K_I (K_I increases significantly). When γ> 2.15,the increase rate of K_II is greater than that of KI (K_II increases significantly),and KI/K_II tends to approach 1,as shown in Fig. 9.

6.3 Case 3

After measuring three double dissimilar orthotropic materials,as shown in Table 1,the properties of materials are listed as follows.

After the elastic constants of each material are substituted into (6),(7),(11),(14),(23), and (19),as shown in Table 2,the stress singularity exponents can be obtained as shown in Table 3.

In Fig. 10,the changes of the stress (σ_x)_j ,(σ_y)_j ,and (τ_xy)_j with the polar angle θ are shown, where r/a = 0.1 and σ = τ = 1.

If the stress expressions (27) are rewritten as (σ_x)₁ = σF₁₁ +τF₁₂,(σ_y)₁=σF₂₁ +τF₂₂,and (τxy)₁=σF₃₁+τF₃₂,using the first group data of double materials,we can obtain Table 4.

Table 4 shows that the influence of the shear load τ on the stress field is larger than that of the stress load σ. That is to say that the shear load τ is an important factor which causes crack extension.

7 Conclusions

(i) The bi-material engineering parameters characteristic of bi-material mechanical proper- ties are introduced. With the elastic constants of the two materials,the relationship between the bi-material engineering parameter and the ratio of bi-material shear modulus μ and the re- lationship between the bi-material engineering parameter and the ratio of bi-material Young’s modulus are analyzed,considering the bi-material engineering parameter e12 as a parameter, which estimates the elastic mismatch across the bi-materials interface,similar to the Dundurs parameters.

(ii) As the bi-material engineering parameters satisfy certain conditions,the real bi-material elastic constant "t and the expression for real stress singularity exponents λt are obtained. With the two materials given,when the ratio of bi-material shear modulus μ > 1.04 and the ratio of bi-material shear modulus 0 < γ < 0.2601 or γ > 1.12,the discriminant Δ > 0 is true. The values for the real bi-material elastic constant "t and the real stress singularity exponents λ_t can be derived,which verifies the theoretical analysis of the present paper.

(iii) Considering the intervention of the bi-material,the concept of bi-material stress intensity factor characteristic of bi-material mechanical properties is introduced. The bi-material stress intensity factors K_I,K_II,and modulus |K| increase with the increase in the ratio of bi-material shear modulus μ and the ratio of bi-material Young’s modulus . When 1.04 < μ < 1.23,K_II increases significantly. When μ > 1.23,K_I increases significantly. When 1.12 < γ < 2.15,K_I increases significantly. When γ > 2.15,KII increases significantly.