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A plane problem for a crack between two anisotropic semi-infinite spaces under remote tensile-shear loading is considered. In the framework of the assumption that the crack faces are free of stresses an exact analytical solution of the problem is given on basis of the complex potentials approach. This solution possesses oscillating square root singularities in stresses and in the derivatives of the displacement jumps at the crack tips. To remove these singularities a new model founded on the introduction of the shear yield zones at the crack tips is suggested. This model is appropriate for the cases where interface adhesive layer is softer than the surrounding matrixes. Under this assumption the problem is reduced to the nonhomogeneous combined Dirichlet-Riemann boundary value problem with the conditions at infinity. An exact analytical solution of this problem is presented for the case of a single yield zone. The length of this zone is found from the finiteness of the shear stress at the end point of the zone. Due to such simulation the shear stress becomes finite at any point and the normal stress possesses only square root singularity at the crack tip. Therefore, the conventional stress intensity factor of the normal stress at the crack tip is used. The numerical illustration of the obtained solution is given.

Interface cracks in many cases are the main reason of composite constructions failure. Therefore, much attention has been paid to the investigation of interface fracture in the framework of the open crack model. Such model is connected with oscillating singularity, which was the most clearly investigated in the paper [

The problem of an interface crack between two different anisotropic materials is much more complicated and therefore less studied compared to an isotropic case. An analytical analysis of this problem in the framework of an open interface crack model was carried out in [

In many cases the interface is much softer than the adhered matrixes; therefore, thin yield zones develop at the crack tips along the interface. Accounting of such zones was performed in [

In this paper an interface crack with a yield zone based upon maximum shear stress criterion is studied. An exact analytical solution for the associated mathematical model is derived. This solution is free from an oscillation and, therefore, the conventional form of stress fracture criterion can be used.

The constitutive relations of elasticity for a linear anisotropic material in the absence of body forces in a fixed rectangular coordinate system

Substituting (

Assuming that all fields are independent of the coordinate _{1},_{2},_{3} are eigenvalue and eigenvector components of the following system:

The elements of the 3×3 matrices

Since (

Further a bimaterial composed of two different anisotropic semi-infinite spaces

Satisfying (

It should be noted that the vector function

The main attention in the following will be paid to the consideration of orthotropic materials with the axis of the material symmetry parallel to

It is worthy to note that the function

Let us assume further that a crack takes place at the section

A crack between two anisotropic materials with a yield zone.

The boundary conditions for the formulated problem are as follows:

Satisfying conditions (

The conditions at infinity follow from (

The solution of (

The stresses at the interface are found from (

The derivative of the displacement jumps is obtained by use of the formula (

After integrating the last relation, we obtain

It is clearly seen from (

Thus the boundary conditions for the considered model can be written as follows:

Satisfying the interface conditions (

Satisfaction of the boundary conditions (

Equations (

A particular solution of the nonhomogeneous combined Dirichlet-Riemann boundary value problem for certain right sides of (

A solution of the Dirichlet problem (

Using

The general solution of the problem (

Considering that for the validity of last equation the coefficient before

Using the obtained solutions the general solution of the nonhomogeneous combined Dirichlet-Riemann boundary value problem (

According to (

For an arbitrary position of the point

This equation should be solved with respect to

The normal stress at the interval

Substituting the formula (

Consider further the stress intensity factor (SIF) of the normal stress at the point

Considering that

This formula after some transformations takes the form

The derivative of the crack faces displacement jump at the interval

Substituting the expression (

The crack opening can be found by using the following formula:

The calculations were performed for a bimaterial composed of boron-epoxy orthotropic material (upper one) with

In Table

Yield zone length and SIF

| 1 | 2 | 3 | 4 | 5 |

| |||||

| 0.1318 | 0.3700 | 0.6581 | 0.9750 | 1.313 |

| |||||

| 1.644 | 3.396 | 5.161 | 6.931 | 8.704 |

The dependence of the same variables as in Table

Yield zone length and SIF

| 0 | 1 | 2 | 3 | 4 | 5 | 6 |

| |||||||

| 0.6581 | 1.902 | 4.039 | 7.100 | 11.04 | 15.77 | 21.19 |

| |||||||

| 5.161 | 5.049 | 5.107 | 5.290 | 5.576 | 5.954 | 6.425 |

The crack opening

Crack opening

The variation of the shear stress

Variation of the shear stress

An interface crack in a bimaterial anisotropic space under tensile-shear loading at infinity is considered. Plane strain conditions are assumed and the crack faces are free of stresses. Due to complex potentials approach an exact analytical solution (

As a result of the performed modeling the shear stress becomes restricted in the whole region and the normal stress has only square root singularity at the crack tip. It means that the stress intensity factor of the normal stress can be introduced and calculated in the conventional way. The dependencies of the mentioned stress intensity factor, yield zone length, crack opening, and the shear stress on the applied loading are illustrated in the tables and graphical forms. It is particularly shown that the stress intensity factor of the normal stress essentially depends on the normal applied stress and moderately on the external shear stress. Finally, it is worthy to mention that the results of the paper are obtained in a simple analytical form which is convenient for the engineering applications.

The data obtained in the published article can be used for future investigation on the fracture mechanics of composite materials related to an interface crack in anisotropic and isotropic materials. The data is available upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.