Model of the structure of the near tip area of interface crack in a piece-homogeneous elastic-plastic body

Abstract

The model of interface crack, which assumes the existence near its tip the area of the contact of faces interactive by the law of dry friction, and the plastic zone with the small scale destruction zone in the part adjoining the crack tip with the high level of tensile and shear deformations is extended in the article. Modelling a plastic zone under the plane strain conditions by the line of displacement rupture inclined to the interface and using the Wiener-Hopf method the equations for calculating the length and orientation of the plastic zone, the sizes of the contact faces area and of the destruction zone, the stress singularity indexes near the crack tip under the conditions of the prevailing shear or tensile loading were obtained. It is established that due to the formation of a lateral plastic zone near the crack tip at the prevailing tensile loading, the length of the contact area depends not only on the configuration but also on the module of loading. Herewith its value appears to be on a few more orders than if a plastic zone was absent. The change of the stress singularity index at approach to the crack tip is revealed.

Keywords

Interface crack contact friction plastic zone destruction zone Wiener-Hopf method

1. Introduction

A crack on the interface of different materials is a widespread defect in the structure of composites, glued and welded joints, covering, etc. The crack propagation between the parts of products and constructions with different physical-mechanical characteristics (piece-homogeneous bodies) in the process of exploitation is able to result in their partial or complete fracture. The necessity of the prognostication of strength, ensuring reliability and durability of the use of machine parts and elements of constructions, made of piece-homogeneous materials, cause the lasting and deep interest of research workers to the problem of interface cracks.

The research on interface cracks within the framework of a classic model of crack-cut (A.A. Griffith), founded in the 1960s and discussed in the works by Williams [45], Cherepanov [7], Erdogan [17], England [16], Rice and Sih [40], Malyshev and Salganik [34] and others, discovered physically uncorrected spatial oscillations of displacements of crack faces with the overlapping at approaching to the crack tip. The alternative model of an interface crack with the contact of part of the faces adjacent to the tip, was developed afterwards by Comninou [12,13], Comninou and Schmueser [11], Comninou and Dundurs [10], Gautesen and Dundurs [19], and later by Antipov [2], Loboda [32], Ostryk [36] and others, allowed to avoid the above-mentioned shortcoming of a classic model, however at predominance of the tensile loading predicted extremely small sizes of contact area, less distances between atoms on which the use of continual theory of elasticity becomes improper.

Among other mechanisms of spatial oscillations removal, Rice [39] assumed the formation of a small scale plastic zone near the tip of interface crack. The plasticity influence on the stress-strain state near the tip was confirmed in the works by Kaminskii et al. [21,22]. Their research on the plastic zone in adhesive material educed only the power feature of stresses near the tip. This result, as well as conclusions of a cohesive zone model research [42,43], points to the necessity of accounting joint nonideality not only due to the plasticity of adhesion layer but also due to its elastic properties. In particular, Antipov et al. [3], modelling a soft interface layer by certain effective contact conditions with the jump of displacements, showed the presence of stresses near the tip of the crack in the condition of anti-plain shear of logarithmic singularity.

However, taking into account only one of the factors listed above does not provide a physically correct, realistic enough and complete description of an interface if there is a presence of crack in it. Therefore, the development of an interface crack complex model, which takes into account the most substantial factors that influence the stress-strain state of piece-homogeneous body near the tip, is actual. According to the definition by Cherepanov [8], a complex of small structural elements of the area near the crack tip forms the thin structure of the tip.

The complexity of defining equations of plasticity theories does not allow us to get their exact analytical solutions. At the same time, it is desirable to know the approximate sizes of structural elements of near tip area for the numeral simulation of plastic zone taking into account the material destruction and the possibility of the crack faces contact near the tip. Such possibility is given by the approximate Leonov–Panasiuk–Dugdale model of plastic zone [14,31] which represents this zone as the line of rupture of the tangential displacement. In fact, the plastic zone area is as if cut out conditionally and her boundaries are sewn together by the corresponding boundary conditions in this model for the avoidance of the necessity of the plasticity theory nonlinear equations solution. As a result the initial problem comes to elastic theory boundary-value problem, which can be solved by the known analytical methods.

In this work the complex model of crack on the plane interface of two parts of piece-homogeneous elastic-plastic body with different mechanical properties for the plain strain conditions is developed for cases, when the sizes of contact area of the crack faces are much larger or much smaller than the sizes of a small scale plastic zone. Such size correspondence on the initial stage of plastic zone development occurs at the predominance in the external loading of shear or tension load. For the receipt of exact analytical solution a plastic zone is described within the framework of Leonov–Panasiuk–Dugdale model. In accordance with experimental data [23,24,29], it is foreseen that in the near tip part of plastic zone a very small area of material destruction with the high level of both shear and tension deformation is formed.

2. A model of interface crack in the prevailing shear loading conditions

Research by Comninou and Schmueser [11], Comninou and Dundurs [10], Gautesen and Dundurs [19] and others educed that at the predominance a shear external loading the faces of internal crack on the plane interface of division of two different homogeneous isotropic materials near one of the tips have a considerable area of faces contact and far less contact near the area of an opposite tip. In this part of the article we examine a thin structure of the tip of a crack (after terminology by G.P. Cherepanov) with the considerable contact of the faces in its vicinity.

In accordance with the complex model [20,25,26] we assume that from the tip of a crack in more plastic material a small scale lateral plastic stripe, which contains in part, adjoining to the tip, a zone of destruction of material of considerably less sizes spreads. Their determination in the plain strain conditions comes to the solution of two separate problems about a side initial plastic zone and about a destruction zone. This is examined below.

2.1. Determination of an initial plastic zone

We examine a problem about the initial stage of development at the plain strain conditions of plastic zone from the tip of a crack, located in a piece-homogeneous plane on the rectilinear interface of two different isotropic elastic-plastic materials with the shear moduli G ₁, G ₂ and Poisson’s ratios 𝜈₁, 𝜈₂, which are under the action of the external loading which provides the considerable contact of faces. This stage consists of a formation near the tip of a thin lateral plastic zone in material with elastic constants G ₁, 𝜈₁, which is assumed by a more plastic one. A plastic zone is modelled by the straight line of rupture of the tangential displacements [14,31,37] which propagate from the tip of a crack under an angle 𝛼 to the interface. On the line of rupture the tangential stress is equal to the shear yield strength of the first material 𝜏_1s.

Since the length s of a contact area at the prevailing shear loading near one of the tips is close to length L of the crack, that is why it, as well as the length of a crack, is much larger than the length l of an initial plastic zone. It allows to consider an interface crack with contacting faces which interact by the law of dry friction as a semi-infinite, and to formulate a condition on infinity as a requirement to the possibility of sewing together at l ≪ r ≪ s the solution of the problem (Fig. 1) with the solution of the corresponding problem of elasticity theory about the interface crack of finite length with a contact zone near the tip in the absence of lateral line of rupture, as found by Comninou [13].

As a result we come to the static problem of elastic theory with boundary conditions in a kind of: $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle 0,\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =\langle u_{r}\rangle =0;\nonumber\\ \displaystyle {\theta} & = & \displaystyle {\alpha},\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =0;\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle \pm {\pi},\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =0,\quad {\tau}_{r{\theta}}=-{\mu}{\sigma}_{{\theta}};\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle {\alpha},\quad r\lt l,\quad {\tau}_{r{\theta}}={\tau}_{1};\quad {\theta}={\alpha},\quad r>l,\quad \langle u_{r}\rangle =0;\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle {\alpha},\quad r\rightarrow \infty ,\quad {\tau}_{r{\theta}}=K_{\mathit{II}}F\left({\alpha}\right)r^{{\lambda}}+o\left(1/r\right);\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F\left({\alpha}\right)=\frac{\left(2{\pi}\right)^{{\lambda}}}{2}\left\{\left[2+{\lambda}\left(1+{\beta}\right)\right]\cos \left(2+{\lambda}\right){\alpha}-{\lambda}\left(1+{\beta}\right)\cos {\lambda}{\alpha}\right\};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\beta}=\frac{\left(1+e{\kappa}_{2}\right)-\left(e+{\kappa}_{1}\right)}{\left(1+e{\kappa}_{2}\right)+\left(e+{\kappa}_{1}\right)},\quad e=\frac{G_{1}}{G_{2}}\quad \left(G_{1}\lt G_{2}\right),\quad {\kappa}_{1(2)}=3-4{\nu}_{1(2)};\nonumber\end{eqnarray}$ where 〈f〉 is a jump of f; 𝜏₁ = 𝜏_1s sgn (K _II F (𝛼)); 𝜇 is a friction coefficient between the crack faces; 𝛽 is a Dundurs parameter [15], |𝛽| ≤ 0,5; 𝜆 is a root of equation ctg𝜆𝜋 + 𝜇𝛽 = 0 on an interval (−1, 0). At the shear of upper material in direction 𝜃 = 𝜋 a friction coefficient must be negative. We determined the stress intensity factor at the end of the interface shear crack K _II from the solution of the external problem and assume that it is a priori known. For providing a compressed stress on the crack faces in a contact area (𝜃 = 𝜋, 𝜎_𝜃 < 0) a K _II must meet the condition 𝛽K _II > 0 (see the analysis in [30]).

The solution of the formulated boundary-value problem (1)–(3) by the Wiener-Hopf method (Addition A) results in the next expression for length of plastic stripe: $\begin{eqnarray}\displaystyle l=\left(\frac{K_{\mathit{II}}F\left({\alpha}\right)Q^{+}\left(-1\right)J\left(0\right)}{{\tau}_{1}Q^{+}\left(-{\lambda}-1\right)J\left({\lambda}\right)}\right)^{-1/{\lambda}},\quad J\left(x\right)=\exp \left[\frac{1}{2{\pi}}\mathop{\int }_{-\infty }^{\infty }\frac{\ln G\left(\mathit{it}\right)}{it+\left(x+1\right)}dt\right]. & & \displaystyle\end{eqnarray}$ (4)

For the determination of the angle of an initial plastic zone slope we use the condition of maximum speed of the dissipation of energy in a zone [9], which is equal to $\begin{eqnarray}\displaystyle \frac{dW\left(t\right)}{dt}=\frac{1-{\nu}_{1}}{G_{1}}\frac{{\tau}_{1s}^{2}}{\left(1-{\lambda}\right)\left(2+{\lambda}\right)}\frac{1}{Q^{-}\left(1\right)Q^{+}\left(-1\right)}\left(\frac{|K_{\mathit{II}}|}{{\tau}_{1s}}\right)^{-2/{\lambda}}\frac{1}{K_{\mathit{II}}}\frac{dK_{\mathit{II}}}{dt}w\left({\alpha}\right), & & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle w\left({\alpha}\right)=\frac{J_{1}\left(0\right)}{J\left(0\right)}\left(\frac{|F\left({\alpha}\right)|Q^{+}\left(-1\right)J\left(0\right)}{Q^{+}\left(-{\lambda}-1\right)J\left({\lambda}\right)}\right)^{-2/{\lambda}},\quad J_{1}\left(x\right)=\exp \left[\frac{1}{2{\pi}}\int _{-\infty }^{\infty }\frac{\ln G\left(\mathit{it}\right)}{it-\left(x+1\right)}dt\right]. & & \displaystyle \nonumber\end{eqnarray}$ Accepting that K _II is the positive growing or negative decreasing function of time, we come to the condition w (𝛼) → max.

According to (4), the length of a plastic zone grows nonlinear with the increase of the external load which is included in the stress intensity factor K _II. In addition, the length of a plastic zone is larger than the shear yield strength of the first material in which it arises.

From this solution we found a normal stress near the crack tip on the line of rupture, which is necessary for the determination of a destruction zone: $\begin{eqnarray}\displaystyle {\sigma}_{{\theta}}\left(r,{\alpha}\right)={\tau}_{1}\left[f_{1}\left({\alpha},e,{\nu}_{1},{\nu}_{2}\right)+\mathop{\sum }_{i}f_{2i}\left({\alpha},e,{\nu}_{1},{\nu}_{2}\right)r^{{\lambda}_{i}}\right]\quad \left(r\ll l\right), & & \displaystyle\end{eqnarray}$ (6) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle f_{1}=\displaystyle \frac{N\left(-1\right)}{D_{1}\left(-1\right)},\quad f_{2i}=\frac{N\left(-1-{\lambda}_{i}\right)}{D_{1}^{\prime }\left(-1-{\lambda}_{i}\right)}\frac{Q^{+}\left(-1-{\lambda}_{i}\right)}{Q^{+}\left(-1\right)}\frac{J\left({\lambda}_{i}\right)}{J\left(0\right)}\frac{{\lambda}l^{-{\lambda}_{i}}}{{\lambda}_{i}\left({\lambda}_{i}-{\lambda}\right)},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle D_{1}^{\prime }\left(p\right)=dD_{1}\left(p\right)/dp,\quad N\left(p\right)=N_{1}\left(p\right)+{\mu}N_{2}\left(p\right),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle N_{1}\left(p\right) &=& \displaystyle \left(1+{\kappa}_{1}\right)\left[\left(1+{\kappa}_{1}\right)^{2}u_{11}+e^{2}\left(1+{\kappa}_{2}\right)^{2}u_{12}-\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)u_{13}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left(1-e\right)\big[\left(1+{\kappa}_{1}\right)^{2}v_{11}+2e\left(1+{\kappa}_{2}\right)\left(1+e{\kappa}_{2}\right)v_{12}+\left(1+{\kappa}_{1}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times\, \left(e\left(1+{\kappa}_{2}\right)v_{13}+\left(1-e\right)v_{14}\right)\big],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle u_{11}\left(p\right) & = & \displaystyle 0,5\left[\sin 2{\alpha}-\sin 2p\left({\pi}-{\alpha}\right)\right],\nonumber\\ \displaystyle u_{12}\left(p\right) & = & \displaystyle 2p\left(p-1\right)\sin ^{2}{\alpha}\sin 2p{\alpha}-p\sin 2{\alpha}+0,5\left[\sin 2{\alpha}+\sin 2p\left({\pi}-{\alpha}\right)\right],\nonumber\\ \displaystyle u_{13}\left(p\right) & = & \displaystyle \left(p-1\right)\sin 2{\alpha}\cos 2p{\alpha},\nonumber\\ \displaystyle v_{11}\left(p\right) & = & \displaystyle 2\left(p-1\right)\sin 2{\alpha}\sin ^{2}p{\alpha}-p\sin 2{\alpha}+\sin 2p\left({\pi}-{\alpha}\right),\nonumber\\ \displaystyle v_{12}\left(p\right) & = & \displaystyle 2p\left(p-1\right)\sin ^{2}{\alpha}\sin p\left({\pi}-2{\alpha}\right)\cos p{\pi},\nonumber\\ \displaystyle v_{13}\left(p\right) & = & \displaystyle 4\left(p-1\right)\left[p\sin ^{2}{\alpha}\sin p{\pi}\cos p\left({\pi}-2{\alpha}\right)-\sin 2{\alpha}\sin ^{2}p{\alpha}\right]+v_{11}\left(p\right),\nonumber\\ \displaystyle v_{14}\left(p\right) & = & \displaystyle 2\left(p-1\right)\left[p\sin ^{2}{\alpha}\sin 2p\left({\pi}-{\alpha}\right)-\sin 2{\alpha}\sin ^{2}p{\alpha}\right];\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle N_{2}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)\left[\left(1+{\kappa}_{1}\right)^{2}u_{21}+e^{2}\left(1+{\kappa}_{2}\right)^{2}u_{22}-2\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)u_{23}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -4\left(1-e\right)\left[\left(1+{\kappa}_{1}\right)^{2}v_{21}+\left(1+e{\kappa}_{2}\right)\left(e\left(1+{\kappa}_{2}\right)+2\left(1-e\right)\right)v_{22}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\left.\left(1+{\kappa}_{1}\right)\left(e\left(1+{\kappa}_{2}\right)v_{23}+\left(1-e\right)\right)v_{24}\right],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle u_{21}\left(p\right) & = & \displaystyle \sin ^{2}{\alpha}-\sin ^{2}p\left({\pi}-{\alpha}\right),\quad u_{22}\left(p\right)=4p\left(p-1\right)\sin ^{2}{\alpha}\cos ^{2}p{\alpha}+u_{21}\left(p\right),\nonumber\\ \displaystyle u_{23}\left(p\right) & = & \displaystyle 2\left(p-1\right)\sin ^{2}{\alpha}\cos ^{2}p{\alpha}+u_{21}\left(p\right),\nonumber\\ \displaystyle v_{21}\left(p\right) & = & \displaystyle \left(p\cos ^{2}p{\alpha}+\sin ^{2}p{\alpha}\right)\sin ^{2}{\alpha}-\sin ^{2}p\left({\pi}-{\alpha}\right),\nonumber\\ \displaystyle v_{22}\left(p\right) & = & \displaystyle p\left(p-1\right)\sin ^{2}{\alpha}\sin p\left({\pi}-2{\alpha}\right)\sin p{\pi},\nonumber\\ \displaystyle v_{23}\left(p\right) & = & \displaystyle v_{21}\left(p\right)+v_{22}\left(p\right)+2p\left(p-1\right)\sin ^{2}{\alpha}\cos ^{2}p{\alpha},\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle v_{24}\left(p\right) & = & \displaystyle p^{2}\sin ^{2}{\alpha}\left(\cos 2p{\alpha}+\sin ^{2}p\left({\pi}-{\alpha}\right)\right)+\sin ^{2}{\alpha}∼\sin ^{2}p{\alpha}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -p\sin ^{2}{\alpha}∼\sin p{\pi}\sin p\left({\pi}-2{\alpha}\right)-\sin ^{2}p\left({\pi}-{\alpha}\right);\nonumber\end{eqnarray}$ 𝜆_i are the roots of equation $\begin{eqnarray}\displaystyle D_{1}\left(-1-x\right)=0, & & \displaystyle\end{eqnarray}$ (7) which satisfies a condition Re𝜆_i > −1 (𝜆_i ≠ 0).

In the first stage of the plastic zone development, when the destruction zone has not yet appeared, the boundary conditions (accepted in (1)) are (〈u _𝜃(r, ±𝜋)〉 = 0, 〈u _𝜃(r, 𝛼)〉 = 0), a crack opening at its tip equals a zero, which is why in accordance with the deformation criterion of fracture the appearance of an initial plastic zone does not influence the conditions of a crack start directly.

2.2. Determination of destruction zone parameters

As shown in the calculations, Eq. (7) has roots in a stripe −1 < Rex < 0, which indicates the conservation of stress concentration at the crack tip and provides further development of plastic zone by a formation in a destruction zone of material with the high level of both shear and tensile deformations. In relation to this, we shall model the destruction zone by the line of length d, on which not only tangent but normal displacement as well undergo a rupture, and a normal stress in this line is equal to the tear resistance of the first material 𝜎₁.

According to the experimental data [23,24,29] it is foreseen that the length of destruction zone is much smaller than those of both a contact zone and all plastic zone. Therefore, the investigated body can be considered as a piece-homogeneous plane that contains a semi-infinity interface crack with contacting faces by the law of dry friction. From the tip O of crack under an angle 𝛼 to the interface the semi-infinity straight line of rupture propagates and it consists of two parts (Fig. 2).

In the part OO’, which joins to the crack tip, the tangential and normal displacements try ruptures, and tangential and normal stresses are equal to 𝜏₁ and 𝜎₁. In the second part the only tangential displacement tries a rupture, and tangential stress is equal to 𝜏₁. The static problem of elasticity theory with boundary conditions corresponds to the following model: $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle 0,\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =\langle u_{r}\rangle =0;\nonumber\\ \displaystyle {\theta} & = & \displaystyle \pm {\pi},\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =0,\quad {\tau}_{r{\theta}}=-{\mu}{\sigma}_{{\theta}};\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle {\alpha},\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0;\quad {\theta}={\alpha},\quad {\tau}_{r{\theta}}={\tau}_{1};\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle {\alpha},\quad r\lt d,\quad {\sigma}_{{\theta}}={\sigma}_{1};\quad {\theta}={\alpha},\quad r>d,\quad \langle u_{{\theta}}\rangle =0.\end{eqnarray}$ (9)

At d << r << l the main terms of expansions of stresses in asymptotic series coincide with the main terms of expansions of stresses in asymptotic series at r → 0 in a problem which is represented in Fig. 1, the solution of which is found in a previous section. Taking into account (6), we formulate a condition on infinity in a kind of $\begin{eqnarray}\displaystyle {\theta}={\alpha},\quad r\rightarrow \infty ,\quad {\sigma}_{{\theta}}={\tau}_{1}\left[f_{1}\left({\alpha},e,{\nu}_{1},{\nu}_{2}\right)+\mathop{\sum }_{i=1}^{2}f_{2i}\left({\alpha},e,{\nu}_{1},{\nu}_{2}\right)r^{{\lambda}_{i}}\right]+o\left(1/r\right), & & \displaystyle\end{eqnarray}$ (10) where only a constant and singular at r → 0 items, which correspond to two real roots 𝜆_i of the equation (7) in a stripe −1 < Rex < 0 is taken into account. It is foreseen that on that stage of a plastic zone development, when the destruction zone has not appeared yet, the normal stress near the crack tip at 𝜃 = 𝛼 is tensile (𝜎_𝜃 > 0). The implementation of this supposition is checked up directly at numerical calculations.

The solution of the formulated boundary problem is found by the method of Wiener-Hopf analogous to the solution of a previous problem. It brings to the next equation for determination of the length of the destruction zone $\begin{eqnarray}\displaystyle \mathop{\sum }_{i=1}^{2}\frac{f_{2i}d^{{\lambda}_{i}}K^{+}\left(-1-{\lambda}_{i}\right)J_{2}\left(0\right)}{\left(1+{\lambda}_{i}\right)K^{+}\left(-1\right)J_{2}\left({\lambda}_{i}\right)}=\frac{{\sigma}_{1}}{{\tau}_{1}}-f_{1}; & & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle K^{+}\left(p\right) & = & \displaystyle \frac{{\Gamma}\left(1-p\right)}{{\Gamma}\left(1/2-p\right)};\quad J_{2}\left(x\right)=\exp \left[\frac{1}{2{\pi}}\int _{-\infty }^{\infty }\frac{\ln H\left(\mathit{it}\right)}{it+\left(x+1\right)}dt\right],\nonumber\\ \displaystyle H\left(p\right) & = & \displaystyle -\frac{\cos p{\pi}D_{2}\left(p\right)}{2\sin ^{2}p{\pi}D_{1}\left(p\right)},\quad D_{2}\left(p\right)=D_{20}\left(p\right)+4{\mu}D_{21}\left(p\right),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle D_{20}\left(p\right) & = & \displaystyle 2s_{1}\left(p\right)s_{4}\left(p\right)-\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)d_{2}\left(p\right)s_{5}\left(p\right)+e\left(1+{\kappa}_{2}\right)\cos p{\pi}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times\, [\left(1+{\kappa}_{1}\right)^{2}d_{3}\left(p\right)d_{5}\left(p\right)-4e^{2}\left(1+{\kappa}_{2}\right)^{2}d_{1}\left(p\right)d_{4}\left(p\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\,2 \left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)\cos p{\pi}∼s_{3}\left(p\right)],\nonumber\\ \displaystyle D_{21}\left(p\right) & = & \displaystyle e^{2}\left(1+{\kappa}_{2}\right)^{2}\left\{\left[2\left(1-e\right)+e\left(1+{\kappa}_{2}\right)\right]\sin p{\pi}∼d_{1}\left(p\right)d_{4}\left(p\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left(1+{\kappa}_{1}\right)s_{2}(p)\right\}+s_{6}\left(p\right)\left[\sin p{\pi}∼s_{1}\left(p\right)+\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)\cos p{\pi}\sin 2p{\alpha}\right],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle s_{1}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)^{2}-2\left(1+{\kappa}_{1}\right)\left[e\left(1+{\kappa}_{2}\right)+2\left(1-e\right)\right]\nonumber\\\displaystyle \text{}&\text{}&\times\,\sin ^{2}p{\alpha}-\left[e\left(1+{\kappa}_{2}\right)+2\left(1-e\right)\right]^{2}∼d_{4}\left(p\right),\nonumber\\ \displaystyle s_{2}\left(p\right) & = & \displaystyle p\left(p-1\right)\sin ^{2}{\alpha}∼\sin p\left({\pi}-2{\alpha}\right)+∼\sin p{\pi}d_{4}\left(p\right)d_{6}\left(p\right),\nonumber\\ \displaystyle s_{3}\left(p\right) & = & \displaystyle 2\sin p{\alpha}∼\sin p\left({\pi}-{\alpha}\right)+p\sin 2{\alpha}∼\sin p\left({\pi}-2{\alpha}\right)-2p^{2}\sin ^{2}{\alpha}∼\cos p\left({\pi}-2{\alpha}\right),\nonumber\\ \displaystyle s_{4}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)\sin p{\pi}∼d_{3}\left(p\right)-2e\left(1+{\kappa}_{2}\right)\cos p{\pi}∼d_{1}\left(p\right),\nonumber\\ \displaystyle s_{5}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)\cos p{\pi}∼d_{3}\left(p\right)+2e\left(1+{\kappa}_{2}\right)\sin p{\pi}∼d_{1}\left(p\right),\nonumber\\ \displaystyle s_{6}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)∼d_{6}\left(p\right)+\left[e\left(1+{\kappa}_{2}\right)+2\left(1-e\right)\right]∼d_{1}\left(p\right),\nonumber\\ \displaystyle d_{1}\left(p\right) & = & \displaystyle \sin ^{2}p\left({\pi}-{\alpha}\right)-p^{2}\sin ^{2}{\alpha},\quad d_{2}\left(p\right)=p\sin 2{\alpha}-\sin 2p{\alpha},\nonumber\\ \displaystyle d_{3}\left(p\right) & = & \displaystyle p\sin 2{\alpha}-\sin 2p\left({\pi}-{\alpha}\right),\quad d_{4}\left(p\right)=p^{2}\sin ^{2}{\alpha}-\sin ^{2}p{\alpha},\nonumber\\ \displaystyle d_{5}\left(p\right) & = & \displaystyle p\sin 2{\alpha}+\sin 2p{\alpha},\quad d_{6}\left(p\right)=p\sin ^{2}{\alpha}-\sin ^{2}p\left({\pi}-{\alpha}\right).\nonumber\end{eqnarray}$

The destruction zone appearance changes the stress-strain state near the tip of the crack, which now will be characterized by the singularity index of stresses 𝜆_d1, which is determined as the least from the interval (−1, 0) root of the equation $\begin{eqnarray}\displaystyle D_{2}\left(-1-x\right)=0. & & \displaystyle\end{eqnarray}$ (12)

If, in the absence of the destruction zone, the crack tip opening displacement is equal to zero, then the appearance of the region of destruction results in the formation of nonzero shear displacement of the faces at the tip 𝛿 = lim_x→−0|〈u〉_y=0|, which is connected with the jump of a normal displacement 〈u _𝜃(0, 𝛼)〉 by the formula 𝛿 = 〈u _𝜃(0, 𝛼)〉∕sin𝛼. 〈u _𝜃(0, 𝛼)〉 is found during the problem-solving process and results in the next expression for the faces shear: $\begin{eqnarray}\displaystyle {\delta}=-\frac{2\left(1-{\nu}_{1}\right){\tau}_{1}}{G_{1}\sin {\alpha}}\frac{d}{\sqrt{{\pi}H\left(0\right)}}\mathop{\sum }_{i=1}^{2}\frac{f_{2i}d^{{\lambda}_{i}}K^{+}\left(-1-{\lambda}_{i}\right)}{J_{2}\left({\lambda}_{i}\right)}\frac{{\lambda}_{i}}{\left(1+{\lambda}_{i}\right)^{2}}. & & \displaystyle \nonumber\end{eqnarray}$ The faces shear 𝛿 brings the contribution in the common crack opening and must be taken into account when researching crack start conditions.

2.3. A numerical analysis of parameters of thin structure of interface crack tip at the prevailing shear loading

As it follows from the previous consideration, the parameters of a small scale plastic zone are fully determined by the elastic and nonelastic characteristics of the joined materials. The stress intensity coefficient K _II. K _II can be determined independently of the problem, which is examined in this part of the article, for any piece-homogeneous bodies of the suitable structure, and the value and configuration of the external loading as well by numerical or, as far as possible, analytical solution of the corresponding boundary-value problem of elasticity theory. As the determination of K _II was not the aim of this work, for the illustration of application of the above received solution of the problem on a plastic zone within the framework of complex model of interface crack and with the purpose of the analysis of its results, we shall take the advantage of the results by Ostryk’s research [36] on the contact zone at the tip of internal interface crack of length L in a piece-homogeneous plane at its loading on the infinities by tangential stress q (𝜏_xy → q < 0 at x, y → ∞).

The condition (5) brings about two values of angle of plastic zone slope, one of which corresponds to its propagation along the interface. It is consistent with the presence of two extremes in an angular dependence of tangential stress (3) in the investigated material (the upper one). Below we confine ourselves to consideration only of a lateral zone, as to the calculations of plastic zones on the interface within the framework of different models a large amount of publications are devoted to, in particular, [6,21,33,42,43].

The results of the calculations of the thin structure parameters of the tip of an interface crack (the slope angle 𝛼 of plastic zone to the interface, relative lengths s∕L of a contact zone according to [36], l∕L and d∕L of the whole plastic zone and of destruction zone) are presented in Table 1. For some values of the ratio of the shear moduli G ₁/G ₂ of the joined materials with equal Poisson’s ratios 𝜈₁ = 𝜈₂ = 0,3 at different values of the friction coefficient 𝜇 and external loading q = −0, 1𝜏_1s. At the calculations of the destruction zone parameters we assume that the tear resistance of the first material is equal to 𝜎₁ = 5𝜏_1s.

According to the calculations, in the accepted conditions of loading the length l of all plastic zones decrease at the small ratios of the shear modules of materials G ₁/G ₂, while at the close values of G ₁ and G ₂ it grows with the increase of friction between the crack faces, unlike the length d of the destruction zone, which decreases with the increase of friction at any G ₁/G ₂ < 1. The length of plastic zone appears almost on two orders less than the contact zone length, and the destruction zone length is far less than the whole zone length, which is why the use of the considered model is fully justified at the low loading.

At the small ratios G ₁/G ₂ the slope angle of plastic zone decreases with the increase of friction, while at the approaching G ₁ to G ₂ its increase is foreseen. In accordance with (5), the value of loading of piece-homogeneous body with an interface crack does not influence the orientation of zone, which thus will depend only on the elastic parameters of the joined materials and the friction coefficient between the faces. In the independence of slope angle of the plastic zone from the external loading the principal difference consists of the conclusions which follow from the obtained solution from the results of analogical research within the framework of an open interface crack as is studied by Kaminsky et al. [27]. At the same time, the values of slope angle of zone at the end of the crack without the account of the contact of faces, calculated at the pure shear by Kaminsky et al. [27], are close to the one found in this work.

In Table 1 the values of stress singularity indexes at the tip before (𝜆) and after the formation of initial plastic zone (𝜆₁), and also after the emergence of the destruction zone (𝜆_d1) are presented. The comparison of 𝜆 and 𝜆₁ shows that the formation of initial plastic zone causes the increase of the stress concentration in the area of r ≪ l (𝜆₁ < 𝜆), that gives the rise to the subsequent zone development by the formation in a destruction zone. But the appearance of destruction zone does not remove the concentration of stresses at the tip of a crack, but results only in its insignificant attenuation in the area of r ≪ d comparatively with an initial level: the least root 𝜆_d1 of equation (12) from an interval (−1, 0) is a little higher than singularity index 𝜆 before the formation of both areas (𝜆_d1 > 𝜆). Therefore, the development of plastic zone on this stage has not been completed and must be accompanied by the formation of new zones, in particular along the interface, as it is foreseen from the existence of the second maximum w (𝛼) at 𝛼 = 0 and the numerical simulation of small scale plastic zone by Shih and Asaro (41).

3. A model of interface crack in the conditions of the prevailing tensile loading

As was revealed by Comninou [12], Comninou and Schmueser [11], Gautesen and Dundurs [19] and others, at the predominance of tensile forces in the external loading of perpendicular to the plane of interface crack, the contact area of the faces without the account of plasticity of joined materials is extremely small, which allows to ignore it at the research of plastic zone. The calculation of lateral initial plastic zone was executed by Kaminsky et al. [27]. As the appearance of plastic zone changes the stress-strain state near the tip of the crack, it will influence the sizes of the contact area. This is why it is possible to expect the next structure of area near the tip within the framework of a complex model of an interface crack: from the tip, which is a stress concentrator, the thin lateral plastic zone spreads in more plastic material, which contains a very small area of destruction in the adjacent part to the crack tip and there is an area of faces contact, which has the sizes much smaller than the sizes of plastic zone, but much larger than the sizes areas of destruction. Their parameters in the plain strain conditions are determined in this part of the article on the condition that the length of the plastic zone is much smaller than the length of the crack.

3.1. The faces contact area and the destruction zone parameters calculation

Under the plain strain conditions we consider the problem of determination of the region of friction contact of the faces of an interface crack that lies on the rectilinear interface of two homogeneous isotropic elastic materials with shear moduli G ₁, G ₂ and Poisson’s ratios 𝜈₁, 𝜈₂. The contact zone is simulated by a notch whose faces interact according to the Coulomb law of dry friction. We also assume that compressive normal stresses act on the notch and that solely the tangential component of displacements may have jumps on the notch. The length of the contact zone and the contact stresses are determined in the problem solving process.

From the crack tip, the lateral plastic zone propagates because of the stress concentration in the plastic material with elastic constants G ₁, 𝜈₁. According to the Leonov-Panasyuk-Dugdale model we simulated this plastic zone by the straight line of discontinuities of tangential displacement, which originates from the crack tip under the angle 𝛼 to the interface. On the line of rupture the tangential stress is equal to the shear yield strength of the first material 𝜏_1s. Unlike Kaminsky et al. [27], we shall use the condition of a maximum energy dissipation speed as described in Section 2.1 (formula (5)) instead of the condition of a length zone maximum for determination of the zone orientation.

We study the case where the length of the contact zone s is much smaller than the crack length L, the length of the plastic zone l, and all other important sizes of the body. This enables us to consider the body as a piecewise-homogeneous plane containing a semi-infinite cut on the interface. A part of the faces of the cut adjacent to the tip is in contact with friction, and a semi-infinite line of discontinuity originates from the tip (Fig. 3).

A condition of infinity is formulated as a requirement to the possibility of sewing together the solution of the problem with the asymptotic solution of an analogical problem without the contact zone [27]. As a result we’ll come to the static problem of the elasticity theory with boundary conditions: $\begin{eqnarray}{\theta}=0,\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =\langle u_{r}\rangle =0;\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\theta}={\alpha},\quad \langle {\sigma}_{{\theta}}\rangle =\langle {\tau}_{r{\theta}}\rangle =0,\quad \langle u_{{\theta}}\rangle =0;\quad {\theta}={\alpha},\quad {\tau}_{r{\theta}}={\tau}_{1}\quad ({\tau}_{1}=\pm {\tau}_{1s}); & & \displaystyle\end{eqnarray}$ (13) $\begin{eqnarray}{\theta}=\pm {\pi},\quad r>s,\quad {\sigma}_{{\theta}}={\tau}_{r{\theta}}=0;\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle \pm {\pi},\quad r\lt s,\quad \langle {\sigma}_{{\theta}}\rangle =0,\quad {\tau}_{r{\theta}}=-{\mu}{\sigma}_{{\theta}},\quad \langle u_{{\theta}}\rangle =0;\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle {\theta} & = & \displaystyle \pm {\pi},\quad r\rightarrow \infty ,\quad \left\langle \frac{\partial u_{{\theta}}}{\partial r}\right\rangle \sim -\frac{2(1-{\nu}_{1})}{G_{1}}{\tau}_{1}\mathop{\sum }_{i}C_{i}r^{{\lambda}_{i}};\end{eqnarray}$ (15) C _i are the constants, which are found from the solution by Kaminsky et al. (27): $\begin{eqnarray}\displaystyle C_{i}=\frac{4\sin {\lambda}_{i}{\pi}}{D^{\prime }(-1-{\lambda}_{i})}\frac{(1+{\lambda}_{i})u(-1-{\lambda}_{i})J_{0}({\lambda}_{i})}{{\lambda}_{i}l^{{\lambda}_{i}}K^{+}(-1-{\lambda}_{i}){\tau}_{1}}Re\left[\frac{1-2i{\omega}}{1+2i{\omega}}\frac{F({\alpha})l^{-0.5+i{\omega}}}{1-2i{\omega}+2{\lambda}_{i}}\frac{K^{+}(-0,5-i{\omega})}{J_{0}(-0,5+i{\omega})}\right] & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle u(p) & = & \displaystyle (1+{\kappa}_{1})^{2}\sin (p({\pi}-{\alpha})-{\alpha})+2(1-e)^{2}u_{1}(p)+2(1+{\kappa}_{1})(1-e)u_{2}(p)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +e(1+{\kappa}_{2})[e(1+{\kappa}_{2})u_{3}(p)+2(1+{\kappa}_{1})u_{4}(p)+2(1-e)u_{5}(p)],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle u_{1}(p) & = & \displaystyle 2(p-1)\sin {\alpha}\sin p{\alpha}\sin p{\pi},\nonumber\\ \displaystyle u_{2}(p) & = & \displaystyle p\sin {\alpha}\cos p({\pi}+{\alpha})-\cos {\alpha}\sin p({\pi}-{\alpha})+2\sin {\alpha}\sin p{\alpha}\sin p{\pi},\nonumber\\ \displaystyle u_{3}(p) & = & \displaystyle 2p\sin {\alpha}\cos p({\pi}-{\alpha})-\sin (p({\pi}-{\alpha})+{\alpha}),\nonumber\\ \displaystyle u_{4}(p) & = & \displaystyle (p-1)\sin {\alpha}\cos p({\pi}+{\alpha}),u_{5}(p)=u_{1}(p)+p\sin {\alpha}\cos p({\pi}-{\alpha})-\cos {\alpha}\sin p({\pi}-{\alpha});\nonumber\\ \displaystyle J_{0}(x) & = & \displaystyle \exp \left[\frac{1}{2{\pi}}\int _{-\infty }^{+\infty }\frac{\ln G_{0}(it)}{it+(1+x)}dt\right],\quad G_{0}(p)=-\frac{D(p)\cos p{\pi}}{D_{0}(p)\sin ^{2}p{\pi}};\nonumber\\ \displaystyle D_{0}(p) & = & \displaystyle (e+{\kappa}_{1})^{2}+(1+e{\kappa}_{2})^{2}+2(e+{\kappa}_{1})(1+e{\kappa}_{2})\cos 2p{\pi};\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle D(p) & = & \displaystyle (1+e{\kappa}_{2})^{2}{\Delta}_{1}(p)+2(1-e)(1+e{\kappa}_{2}){\Delta}_{2}(p)+2(e+{\kappa}_{1})(1+e{\kappa}_{2}){\Delta}_{3}(p)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +2(1-e)(e+{\kappa}_{1}){\Delta}_{4}(p)+(e+{\kappa}_{1}){\Delta}_{5}(p),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\Delta}_{1}(p) & = & \displaystyle p\sin 2{\alpha}\sin p({\pi}-2{\alpha})-2p^{2}\sin ^{2}{\alpha}\cos p({\pi}-2{\alpha})+2\sin p{\alpha}\sin p({\pi}-{\alpha}),\nonumber\\ \displaystyle {\Delta}_{2}(p) & = & \displaystyle p^{2}\sin ^{2}{\alpha}\cos p(3{\pi}-2{\alpha})-p\sin 2{\alpha}\cos p(2{\pi}-{\alpha})\sin p({\pi}-{\alpha})+\cos p{\pi}\sin ^{2}p({\pi}-{\alpha}),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\Delta}_{3}(p) & = & \displaystyle -p^{2}\sin ^{2}{\alpha}\cos p({\pi}+2{\alpha})-p\sin 2{\alpha}\sin p{\alpha}\cos p({\pi}+{\alpha})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\sin p({\pi}-{\alpha})[\sin p(2{\pi}+{\alpha})-\sin p{\pi}\cos p({\pi}-{\alpha})],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}_{4}(p)=p^{2}\sin ^{2}{\alpha}\cos p({\pi}-2{\alpha})-p\sin 2{\alpha}\cos p{\alpha}\sin p({\pi}-{\alpha})+\cos p{\pi}\sin ^{2}p({\pi}-{\alpha}),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\Delta}_{5}(p)=\sin p{\pi}[\sin 2p({\pi}-{\alpha})-p\sin 2{\alpha}];\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle F({\alpha})=-ie^{\prime }KL^{-i{\omega}}F_{1}({\alpha}),\quad e^{\prime }=\frac{\sqrt{(e+{\kappa}_{1})(1+e{\kappa}_{2})}}{2\sqrt{2{\pi}}[(e+{\kappa}_{1})^{2}-(1+e{\kappa}_{2})^{2}]},\quad {\omega}=\frac{1}{2{\pi}}\ln \frac{1-{\beta}}{1+{\beta}},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle F_{1}({\alpha})=\left[a_{1}\cos ({\lambda}+2){\alpha}+a_{2}{\lambda}\cos {\lambda}{\alpha}-a_{3}\sin ({\lambda}+2){\alpha}-a_{4}{\lambda}\sin {\lambda}{\alpha}\right]_{{\lambda}=-0,5+i{\omega}},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle a_{1}=\frac{e+{\kappa}_{1}}{2}-1-e{\kappa}_{2}+\left[e+{\kappa}_{1}-\frac{1+e{\kappa}_{2}}{2}\right]\text{ch}∼2{\pi}{\omega}+i{\omega}\left[e+{\kappa}_{1}-(1+e{\kappa}_{2})\text{ch}∼2{\pi}{\omega}\right],\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle a_{2}=(1+e{\kappa}_{2})\text{ch}∼2{\pi}{\omega}-e-{\kappa}_{1},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle a_{3}=-{\omega}(1+e{\kappa}_{2})\text{sh}∼2{\pi}{\omega}-i\left[e+{\kappa}_{1}-\frac{1+e{\kappa}_{2}}{2}\right]\text{sh}∼2{\pi}{\omega},\quad a_{4}=-i(1+e{\kappa}_{2})\text{sh}∼2{\pi}{\omega},\nonumber\end{eqnarray}$ K = K ₁ + iK ₂ is a complex stress intensity factor, which is defined according to Rice [39] and characterizes a value and configuration of the external loading and is considered to be set after a problem specification; D ^′(p) = ∂D (p)∕∂p; 𝜆_i are the roots of the equation D (−1 − x) =0, which satisfies the condition Rex > −1. The requirement of the compressive character of normal stress on the crack faces (𝜎_𝜃(r, ±𝜋) < 0 at r < s) after formation of plastic zone is imposed on the stress intensity factor K, it is necessary for the faces contact. The other denotations determinations which are identical to the ones introduced in Section 2, are not given.

As well as in the previous section, the solution of the formulated boundary problem is found by the Wiener-Hopf method. From this solution the equation for zone length determination is obtained: $\begin{eqnarray}\displaystyle \mathop{\sum }_{i}\frac{C_{i}s^{{\lambda}_{i}}}{Q^{+}(-1-{\lambda}_{i})J_{1}({\lambda}_{i})}=0, & & \displaystyle\end{eqnarray}$ (16) where $\begin{eqnarray}\displaystyle Q^{+}(p) & = & \displaystyle \frac{{\Gamma}(1-p)}{{\Gamma}(-{\lambda}-p)}\quad (\text{Re}∼p\lt 0),\quad J_{1}(x)=\exp \left[\frac{1}{2{\pi}}\int _{-\infty }^{+\infty }\frac{\ln G(it)}{it+(1+x)}dt\right],\nonumber\\ \displaystyle G(p) & = & \displaystyle \frac{-2\sin p{\pi}D_{1}(p)}{[(e+{\kappa}_{1})+(1+e{\kappa}_{2})](\text{ctg}∼p{\pi}-{\mu}{\beta})D(p)};\nonumber\end{eqnarray}$ D ₁(p) is defined in the Appendix.

The obtained solution allows to find the components of the stress tensor near the crack tip in the joined materials as a power series for r: $\begin{eqnarray}\displaystyle {\sigma}_{mn}\left(r,{\theta}\right)={\sigma}_{mn}^{0}({\theta})+\mathop{\sum }_{k}g_{mn}({\theta},{\lambda}_{k}^{\prime })r^{{\lambda}_{k}^{\prime }}\quad (r\rightarrow 0), & & \displaystyle\end{eqnarray}$ (17) where ${\sigma}_{mn}^{0}({\theta}),g_{mn}({\theta},{\lambda}_{k}^{\prime })$ are the known functions, ${\lambda}_{k}^{\prime }$ are the roots of the equation $\begin{eqnarray}\displaystyle D_{1}\left(-1-x\right)=0, & & \displaystyle\end{eqnarray}$ (18) which satisfy the condition $Re{\lambda}_{k}^{\prime }>-1$ . In particular, the contact stress on the crack faces is equal to $\begin{eqnarray}\displaystyle \text{} & \text{} & {\sigma}_{{\theta}}(r,{\pi}) = \displaystyle (1+{\kappa}_{1}){\tau}_{1}\left[\mathop{\sum }_{k}\frac{D(-1-{\lambda}_{k}^{\prime })Q^{+}(-1-{\lambda}_{k}^{\prime })}{\sin {\lambda}_{k}^{\prime }{\pi}\cdot D_{1}^{\prime }(-1-{\lambda}_{k}^{\prime })}\mathop{\sum }_{i}\frac{C_{i}s^{{\lambda}_{i}}J_{1}({\lambda}_{k}^{\prime })}{Q^{+}(-1-{\lambda}_{i})J_{1}({\lambda}_{i})({\lambda}_{i}-{\lambda}_{k}^{\prime })}\left(\frac{r}{s}\right)^{{\lambda}_{k}^{\prime }}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad+\left.\mathop{\sum }_{k=1}^{\infty }\frac{(-1)^{k}D(-k)Q^{+}(-k)J_{1}(-1+k)}{{\pi}D_{1}(-k)}\mathop{\sum }_{i}\frac{C_{i}s^{{\lambda}_{i}}}{Q^{+}(-1-{\lambda}_{i})J_{1}({\lambda}_{i})(1-k+{\lambda}_{i})}\left(\frac{r}{s}\right)^{k-1}\right]\quad (r\leq s).\nonumber\end{eqnarray}$ The summation in the first term is carried out for all the roots ${\lambda}_{k}^{\prime }$ ( $Re{\lambda}_{k}^{\prime }>-1$ ) of the equation (18).

Numerical calculations show that the equation (18) has two real roots in an interval (−1,0). This means that the conservation of the stress concentration and the formation of the destruction zone is in the adjoining part to the crack tip of the plastic zone. A problem of the determination of the destruction zone parameters was considered above in Section 2.2, but in this solution it is necessary to take expression for the normal stress in the adjoining part of the plastic zone instead (10); this expression is received in the course of the solution of the problem investigated in this section. As a result we come to the analogical (11) equation for the destruction zone length determination $\begin{eqnarray}\displaystyle \mathop{\sum }_{k=1}^{2}\frac{F_{2k}d^{{\lambda}_{k}^{\prime }}K^{+}\left(-1-{\lambda}_{k}^{\prime }\right)J_{2}\left(0\right)}{\left(1+{\lambda}_{k}^{\prime }\right)K^{+}\left(-1\right)J_{2}\left({\lambda}_{k}^{\prime }\right)}=\frac{{\sigma}_{1}}{{\tau}_{1}}-F_{1}, & & \displaystyle\end{eqnarray}$ (19) in which $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{1}=-2(1+{\kappa}_{1})\frac{f(-1)Q^{+}(-1)J_{2}(0)}{{\pi}D_{2}(-1)}\mathop{\sum }_{i}\frac{C_{i}s^{{\lambda}_{i}}}{{\lambda}_{i}Q^{+}(-1-{\lambda}_{i})J_{2}({\lambda}_{i})},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle F_{2k}=2(1+{\kappa}_{1})\frac{f(-1-{\lambda}_{k}^{\prime })Q^{+}(-1-{\lambda}_{k}^{\prime })J_{2}({\lambda}_{k}^{\prime })}{\sin {\lambda}_{k}^{\prime }{\pi}D_{2}^{\prime }(-1-{\lambda}_{k}^{\prime })}\mathop{\sum }_{i}\frac{C_{i}s^{{\lambda}_{i}-{\lambda}_{k}^{\prime }}}{\left({\lambda}_{i}-{\lambda}_{k}^{\prime }\right)Q^{+}(-1-{\lambda}_{i})J_{2}({\lambda}_{i})},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle f(p)=f_{1}(p)+{\mu}f_{2}(p);\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{1}(p) & = & \displaystyle \left(1+{\kappa}_{1}\right)^{2}f_{11}-e^{2}\left(1+{\kappa}_{2}\right)^{2}f_{12}-\left(1-e\right)^{2}f_{13}+\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)f_{14}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\left(1+{\kappa}_{1}\right)\left(1-e\right)f_{15}+e\left(1+{\kappa}_{2}\right)\left(1-e\right)f_{16},\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{10}=p\sin {\alpha}\cos p({\pi}-{\alpha})-\cos {\alpha}\sin p({\pi}-{\alpha}),\quad f_{11}=\sin p{\pi}f_{10}, & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{12} & = & \displaystyle 2p^{3}\sin ^{3}{\alpha}\sin p{\alpha}-p^{2}\sin ^{2}{\alpha}\cos (p-1){\alpha}+p\sin {\alpha}[\cos p{\pi}\cos p{\alpha}\sin p({\pi}-2{\alpha})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\sin p{\alpha}(\sin ^{2}p({\pi}-{\alpha})+\sin ^{2}p{\alpha})]+\sin p{\alpha}\sin p({\pi}-{\alpha})\cos (p({\pi}-{\alpha})-{\alpha}),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{13} & = & \displaystyle 4(p^{2}\sin ^{2}{\alpha}-\sin ^{2}p{\alpha})f_{11},\nonumber\\ \displaystyle f_{14} & = & \displaystyle 2\sin p{\alpha}\cos p({\pi}+{\alpha})f_{10}-\cos (p+1){\alpha}[\sin ^{2}p({\pi}-{\alpha})-p^{2}\sin ^{2}{\alpha}],\nonumber\\ \displaystyle f_{15} & = & \displaystyle 4\sin ^{2}p{\alpha}f_{11},\quad f_{16}=2(p-1)\sin {\alpha}\sin p{\alpha}[\sin ^{2}p({\pi}-{\alpha})-f_{13};\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{2}(p) & = & \displaystyle \left(1+{\kappa}_{1}\right)^{2}f_{21}+e^{2}\left(1+{\kappa}_{2}\right)^{2}f_{22}+\left(1-e\right)^{2}f_{23}+\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)f_{24}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left(1+{\kappa}_{1}\right)\left(1-e\right)f_{25}+2e\left(1+{\kappa}_{2}\right)\left(1-e\right)f_{26},\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{21}=(p-1)\sin {\alpha}\sin p({\pi}-{\alpha})\sin p{\pi}, & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{22} & = & \displaystyle -2p^{3}\sin ^{3}{\alpha}\cos p{\alpha}-p^{2}\sin ^{2}{\alpha}\sin (p-1){\alpha}+p\sin {\alpha}\sin p({\pi}-{\alpha})\sin p{\pi}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\sin p{\alpha}\sin p({\pi}-{\alpha})\sin (p({\pi}-{\alpha})-{\alpha}),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{23}=-4(p^{2}\sin ^{2}{\alpha}-\sin ^{2}p{\alpha})f_{21}, & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{24} & = & \displaystyle p^{2}\sin ^{2}{\alpha}\sin (p+1){\alpha}+2p\sin {\alpha}\sin p{\alpha}\sin p({\pi}-{\alpha})\cos p({\pi}+{\alpha})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\sin p({\pi}-{\alpha})[\sin p{\alpha}\sin (p({\pi}-{\alpha})-{\alpha})+\sin {\alpha}\sin p({\pi}+2{\alpha})],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{25}=-4\sin ^{2}p{\alpha}f_{21}, & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle f_{26} & = & \displaystyle p^{3}\sin ^{3}{\alpha}[\cos p(2{\pi}-{\alpha})-2\cos p{\alpha}]+p^{2}\sin ^{2}{\alpha}[2\sin {\alpha}\sin p{\pi}\sin p({\pi}-{\alpha})-\cos {\alpha}\sin p{\alpha}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\,p\sin {\alpha}\sin p({\pi}-{\alpha})[\sin p{\pi}-\sin p{\alpha}\cos p({\pi}+{\alpha})]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\sin p{\alpha}\sin p({\pi}-{\alpha})[\cos {\alpha}\sin p({\pi}-{\alpha})-2\sin {\alpha}\sin p{\alpha}\sin p{\pi}];\nonumber\end{eqnarray}$ K ⁺(x) and J ₂(x) are defined in Section 2.2.

Like in Section 2.3 the shear opening of the crack is determined: $\begin{eqnarray}\displaystyle {\delta}=-\frac{2\left(1-{\nu}_{1}\right){\tau}_{1}}{G_{1}\sin {\alpha}}\frac{d}{\sqrt{{\pi}H\left(0\right)}}\mathop{\sum }_{k=1}^{2}\frac{F_{2k}d^{{\lambda}_{k}^{\prime }}K^{+}\left(-1-{\lambda}_{k}^{\prime }\right)}{J_{2}\left({\lambda}_{k}^{\prime }\right)}\frac{{\lambda}_{k}^{\prime }}{\left(1+{\lambda}_{k}^{\prime }\right)^{2}}. & & \displaystyle\end{eqnarray}$ (20)

3.2. A numerical analysis of thin structure parameters of interface crack tip at the prevailing tensile loading

In agreement with the above model, the determination of thin structure parameters of the crack tip is carried out in three stages. At first, according to Kaminsky et al. [27], the orientation of plastic zone and its length and stress-strain state near the tip of the crack after the appearance of zone is calculated. The obtained results are used in further calculations of the lengths of the contact zone and the stress field near the tip of the crack at the account of the faces contact according to the formulas (16) and (18). Then we determined the size of destruction zone, the crack opening and the stress singularity index on this stage from Eqs (19) and (20).

For definiteness, let’s assume that a piece-homogeneous plane with the interface crack of length L is loaded on infinity by homogeneous normal p and the tangential q stresses (𝜎_x → p > 0, 𝜏_xy → q < 0 at x, y →∞). The complex stress intensity factor is determined by the formulas of Rice [39]: $\begin{eqnarray}\displaystyle K=K_{1}+iK_{2},∼∼K_{1}=(p-2{\omega}q)\sqrt{{\pi}L}/\sqrt{2}\text{ch}∼{\pi}{\omega},∼∼K_{2}=(2{\omega}p+q)\sqrt{{\pi}L}/\sqrt{2}\text{ch}∼{\pi}{\omega}. & & \displaystyle \nonumber\end{eqnarray}$

The results of numerical calculations at 𝜈₁ = 𝜈₂ = 0,3, 𝜎₁ = 5𝜏_1s are presented in Table 2 for some separate parameters of the problem, which satisfy the requirements of small sizes of the contact zone and of the destruction zone. The external loading is characterized by a dimensionless parameter $f=\sqrt{p^{2}+q^{2}}$ / 𝜏_1s and configuration, set by a relation q∕p.

The calculations show that due to the formation of plastic zone the length s of the contact zone grows with the increase of the module of loading f, while in the absence of plastic zone its dependence on the module of loading is absent (see, for example, [4,5,28]). The size of contact zone is on a few orders less than the length l of plastic zone by Kaminsky et al. (27) but on a few orders larger than its own sizes in the absence of the plastic zone. At the selected parameters of loading, the size of contact zone is considerably larger than the length d of the destruction zone. With the approaching of the ratio G ₁/G ₂ to 1 the length of the contact zone substantially decreases and becomes less than the length of the destruction zone, so that the initial assumption of the model in this part of the article d ≪ s is violated. This is why the calculations in Table 2 are executed only for the small values G ₁/G ₂. At G ₁/G ₂ →1 the thin structure of the interface crack tip is foreseen to be of such a kind as: from the crack tip the plastic zone with a small scale destruction zone near the tip will spread in more plastic material, and near the tip will arise an extremely small contact of the crack faces. The model of the interface crack with such structure of the area near the tip is not considered in this article.

After the appearance of lateral plastic zone, the character of the dependence of contact zone sizes on the loading configuration set by a relation q∕p is kept. As well as in the absence of plastic zone [4,5,28], the length of contact zone increases with the increase of the tangential component q < 0 of the stress. This conclusion is applicable to the sizes of plastic zone and destruction zone as well. At the increase of the ratio of shear moduli of the joined materials the sizes of all plastic zone, of contact zone and destruction zone decrease. At the increase of friction between the faces the length of the contact zone increases, while the dependence of the length of the destruction zone on 𝜇 is found unmonotonous.

The simultaneous formation of the plastic, contact and destruction zones substantially changes the stress-strain state near the crack tip. The comparison of the singularity indexes of the stresses at the crack tip, which are caused by the plastic (𝜆₁), contact ${\lambda}_{1}^{\prime }$ ) zones and destruction zone (𝜆_d1) shows that at the approach to the crack tip at a distance of s ≪ r ≪ l the level of singularity firstly reduces (𝜆₁ > −1∕2), then increases at d ≪ r ≪ s ( ${\lambda}_{1}^{\prime }<{\lambda}_{1}$ ), and at the distances much less than the sizes of destruction zone (r ≪ d) reduces again to the lowest value ${\lambda}_{d1}>{\lambda}_{1}>{\lambda}_{1}^{\prime }$ .

4. Conclusions

The model of interface crack is developed and foresees the existence near the tip of the contact faces interactive by the law of dry friction and the lateral plastic zone with the small scale destruction zone in adjoining the crack tip part with the high level of tensile and shear deformations. Modelling a plastic zone under the plane strain conditions by the line of displacement rupture and using the Wiener-Hopf method, the equations for calculating the sizes of plastic zone, of the contact zone and of the destruction zone, the stress singularity indexes near the crack tip under the conditions of the prevailing shear or tensile loading were obtained.

The numerical analysis of the dependence of thin structure parameters of the interface crack tip on the external loading, friction and elastic characteristics of the joined materials is executed. It is revealed that the dependence of the sizes of the contact area of the crack faces on the friction coefficient between them is insignificant. It is established that due to the formation of lateral plastic area near the crack tip at the prevailing tensile loading the length of the contact zone area becomes dependent not only on the configuration but also on the module of loading. Herewith its value appears to be on a few orders more than when a plastic zone was absent. The length of the contact zone is extremely small for materials with the close values of the shear modules of the joined materials. The increase of the stress concentration is revealed due to the simultaneous appearance of lateral plastic zone and the contact of the faces. However, the concentration level near the crack tip decreases due to the formation of the destruction zone of material in the plastic zone part adjoining the tip.

Footnotes

Acknowledgements

Financial support was provided by the Ministry of Education and Science of Ukraine to the research project on “Construction of the solution of the problems about the cracks and plastic stripes near the angular points of elastic and elastic-plastic bodies” (grant no. 10113U000329), and is gratefully acknowledged by the authors.

Conflict of interest

None to report.

The solution of the Wiener-Hopf equation of the problem on the determination of small scale lateral plastic zone

We researched solutions of the formulated boundary-value problem (1)–(3) and present these below. In the first problem instead of the first condition in (2) we set (A.1) $\begin{eqnarray}\displaystyle {\theta}={\alpha},\quad r\lt l,\quad {\tau}_{r{\theta}}={\tau}_{1}-K_{\mathit{II}}F\left({\alpha}\right)∼r^{{\lambda}}, & & \displaystyle\end{eqnarray}$ and the stresses decrease as o (1/r) on the infinity. The second problem is an analogical problem without the rupture line. The solution is known [13] and thus it is enough to find the solution to the first problem only.

Applying the integral transformation of Mellin [44] to the equilibrium equations, deformations compatibility condition, Hook’s law and boundary conditions (1) and taking into account the second condition in (2) and condition (A.1), we come to the Wiener-Hopf functional equation of the first problem in a stripe −𝛿 ₁ < Re p < 𝛿₂ (𝛿₁, 𝛿₂ are small enough positive numbers): (A.2) $\begin{eqnarray}\displaystyle {\Phi}^{+}\left(p\right)+\frac{{\tau}_{1}}{p+1}+\frac{{\tau}_{2}}{p+{\lambda}+1}=\text{ctg}^{2}p{\pi}\cdot tg\left(p-{\lambda}\right){\pi}\cdot G\left(p\right){\Phi}^{-}\left(p\right), & & \displaystyle\end{eqnarray}$ $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Phi}^{+}\left(p\right)=\int _{1}^{\infty }{\tau}_{r{\theta}}\left({\rho}l,{\alpha}\right){\rho}^{p}d{\rho},\quad {\Phi}^{-}\left(p\right)=\frac{G_{1}}{2\left(1-{\nu}_{1}\right)}\int _{0}^{1}\left.\left\langle \frac{\partial u_{r}}{\partial r}\right\rangle \right|_{r={\rho}l\atop {\theta}={\alpha}}{\rho}^{p}d{\rho},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\tau}_{2}=-K_{\mathit{II}}F\left({\alpha}\right)l^{{\lambda}},\quad G\left(p\right)=\frac{\sin p{\pi}\cos \left(p-{\lambda}\right){\pi}\sqrt{1+{\mu}^{2}{\beta}^{2}}D_{1}\left(p\right)}{\cos ^{2}p{\pi}\left(1+e{\kappa}_{2}\right)\left(e+{\kappa}_{1}\right)\left(1+e{\kappa}_{2}+e+{\kappa}_{1}\right)D_{0}\left(p\right)},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle D_{0}\left(p\right)=\left(1-{\mu}^{2}{\beta}^{2}\right)+\left(1+{\mu}^{2}{\beta}^{2}\right)\cos 2p{\pi};\quad D_{1}\left(p\right)=D_{10}\left(p\right)+{\mu}D_{11}\left(p\right),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle D_{10}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)^{2}\left[\left(1+{\kappa}_{1}\right){\delta}_{1}\left(p\right)+e\left(1+{\kappa}_{2}\right){\delta}_{2}\left(p\right)+\left(1-e\right){\delta}_{3}\left(p\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle+\,\left(1+{\kappa}_{1}\right)\left[\left(1-e\right)^{2}{\delta}_{4}\left(p\right) +e^{2}\left(1+{\kappa}_{2}\right)^{2}{\delta}_{5}\left(p\right)\right]+e\left(1+{\kappa}_{2}\right)\left(1-e\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times\, \left[\left(1+e{\kappa}_{2}\right){\delta}_{6}\left(p\right)+\left(1+{\kappa}_{1}\right){\delta}_{7}\left(p\right)\right],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\delta}_{1}\left(p\right) & = & \displaystyle \sin ^{2}{\alpha}-\sin ^{2}p\left({\pi}-{\alpha}\right),\nonumber\\ \displaystyle {\delta}_{2}\left(p\right)&=&\displaystyle p\sin 2{\alpha}\sin 2p{\alpha}+2\left(\sin ^{2}{\alpha}\cos 2p{\alpha}+\sin ^{2}p{\alpha}-\sin ^{2}p{\pi}\right),\nonumber\\ \displaystyle {\delta}_{3}\left(p\right) & = & \displaystyle \sin 2p{\alpha}\left[p\sin 2{\alpha}-\sin 2p\left({\pi}-{\alpha}\right)\right]-4\sin ^{2}p{\alpha}{\delta}_{1}\left(p\right),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\delta}_{4}\left(p\right) & = & \displaystyle 4\left[\sin ^{2}{\alpha}\sin ^{2}p{\alpha}\right.+\sin p{\alpha}\sin p\left({\pi}-{\alpha}\right)\cos p{\pi}-\left.p^{2}\sin ^{2}{\alpha}\cos ^{2}p\left({\pi}-{\alpha}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -p\sin 2{\alpha}\left[\sin 2p{\alpha}\right.-\left.\sin 2p\left({\pi}-{\alpha}\right)\right],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle {\delta}_{5}\left(p\right) & = & \displaystyle \sin ^{2}{\alpha}\left(1-4p^{2}\sin ^{2}p{\alpha}\right)+p\sin 2{\alpha}\sin 2p{\alpha}-\sin p\left({\pi}-{\alpha}\right)\nonumber\\ \displaystyle \text{}&\text{}& \times\,\left[\sin p\left({\pi}+{\alpha}\right)+2\cos p{\pi}\sin p{\alpha}\right],\nonumber\\ \displaystyle {\delta}_{6}\left(p\right) & = & \displaystyle 2\cos p{\pi}\left[2\sin p\left({\pi}-{\alpha}\right)\sin p{\alpha}+p\sin 2{\alpha}\sin p\left({\pi}-2{\alpha} \right)\right. \nonumber\\ \displaystyle \text{}&\text{}& \left.-\,\displaystyle 2p^{2}\sin ^{2}{\alpha}\cos p\left({\pi}-2{\alpha}\right)\right],\nonumber\\ \displaystyle {\delta}_{7}\left(p\right) & = & \displaystyle 4\sin ^{2}{\alpha}\left[\left(1-p^{2}\right)\sin ^{2}p{\alpha}-p^{2}\cos ^{2}p\left({\pi}-{\alpha}\right)\right]+p\sin 2{\alpha}\sin 2p\left({\pi}-{\alpha}\right);\nonumber\\ \displaystyle D_{11}\left(p\right) & = & \displaystyle -\left(1+{\kappa}_{1}\right)\bar{D}_{11}\left(p\right)-2\left(1-e\right)\tilde{D}_{11}\left(p\right),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \bar{D}_{11}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)^{2}\bar{{\delta}}_{1}+e^{2}\left(1+{\kappa}_{2}\right)^{2}\bar{{\delta}}_{2}+4\left(1-e\right)^{2}\sin p{\pi}\bar{{\delta}}_{3}+2\left(1+{\kappa}_{1}\right)e\left(1+{\kappa}_{2}\right)\bar{{\delta}}_{4}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +2\left(1-e\right)\left(1+{\kappa}_{1}\right)\bar{{\delta}}_{5}+e\left(1+{\kappa}_{2}\right)\left(1-e\right)\bar{{\delta}}_{6},\nonumber\\ \displaystyle \tilde{D}_{11}\left(p\right) & = & \displaystyle \left(1+{\kappa}_{1}\right)^{2}\tilde{{\delta}}_{1}+\left[e^{2}\left(1+{\kappa}_{2}\right)^{2}+2\left(1-e\right)^{2}+3e\left(1+{\kappa}_{2}\right)\left(1-e\right)\right]\sin p{\pi}\tilde{{\delta}}_{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left(1+{\kappa}_{1}\right)\left[e\left(1+{\kappa}_{2}\right)\tilde{{\delta}}_{3}+2\left(1-e\right)\tilde{{\delta}}_{4}\right],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \bar{{\delta}}_{1}\left(p\right)=0,5\left[\sin 2{\alpha}-\sin 2p\left({\pi}-{\alpha}\right)\right], & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \bar{{\delta}}_{2}\left(p\right) & = & \displaystyle 2p^{2}\sin ^{2}{\alpha}\sin 2p{\alpha}+\left(1-2p\cos 2p{\alpha}\right)\sin {\alpha}\cos {\alpha}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\sin p\left({\pi}-{\alpha}\right)\left[\cos p\left({\pi}+{\alpha}\right)-2\sin p{\pi}\sin p{\alpha}\right],\nonumber\\ \displaystyle \bar{{\delta}}_{3}\left(p\right) & = & \displaystyle p\sin {\alpha}\left[2p\sin {\alpha}\sin p\left({\pi}-{\alpha}\right)\sin p{\alpha}-\cos p{\alpha}\sin \left(p\left({\pi}-{\alpha}\right)-{\alpha}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\sin p{\alpha}\left[\sin {\alpha}\cos \left(p\left({\pi}-{\alpha}\right)+{\alpha}\right)-\sin p\left({\pi}-{\alpha}\right)\right],\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \bar{{\delta}}_{4}\left(p\right)=-p\sin {\alpha}\left(\cos {\alpha}+\sin {\alpha}\sin 2p{\alpha}\right)+\sin {\alpha}\cos {\alpha}\cos 2p{\alpha}+2\sin p{\alpha}\sin p{\pi}\sin p\left({\pi}-{\alpha}\right), & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \bar{{\delta}}_{5}\left(p\right) & = & \displaystyle -p\sin {\alpha}\left[\sin {\alpha}\sin 2p{\alpha}+\sin p\left({\pi}-{\alpha}\right)\sin \left(p\left({\pi}-{\alpha}\right)+{\alpha}\right)\right]-\sin 2{\alpha}\sin ^{2}p{\alpha}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\sin p\left({\pi}-{\alpha}\right)\left[2\sin p{\alpha}\sin p{\pi}+\sin {\alpha}\sin \left(p\left({\pi}-{\alpha}\right)+{\alpha}\right)\right],\nonumber\\ \displaystyle \bar{{\delta}}_{6}\left(p\right) & = & \displaystyle 2p^{2}\sin ^{2}{\alpha}\sin p{\alpha}\left[2\cos p{\alpha}-\cos p\left(2{\pi}-{\alpha}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\,p\sin {\alpha}\left[2\cos {\alpha}\left(\sin ^{2}p{\alpha}-\sin ^{2}p\left({\pi}-{\alpha}\right)\right)-\sin p\left({\pi}+{\alpha}\right)\sin \left(p\left({\pi}-{\alpha}\right)-{\alpha}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\,\sin {\alpha}\left[\cos {\alpha}-\cos \left(p\left({\pi}-{\alpha}\right)+{\alpha}\right)\right]-4\sin p{\alpha}\sin p{\pi}\sin p\left({\pi}-{\alpha}\right);\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \tilde{{\delta}}_{1}\left(p\right) & = & \displaystyle \cos \left(p\left({\pi}-{\alpha}\right)+{\alpha}\right)\left[\cos {\alpha}\sin p\left({\pi}-{\alpha}\right)-p\sin {\alpha}\cos p\left({\pi}-{\alpha}\right)\right],\nonumber\\ \displaystyle \tilde{{\delta}}_{2}\left(p\right) & = & \displaystyle p\sin 2{\alpha}\sin p\left({\pi}-2{\alpha}\right)-2p^{2}\sin ^{2}{\alpha}\cos p\left({\pi}-2{\alpha}\right)+2\sin p{\alpha}\sin p\left({\pi}-{\alpha}\right),\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \tilde{{\delta}}_{3}\left(p\right) & = & \displaystyle p\sin {\alpha}\left[2\cos {\alpha}\sin ^{2}p{\alpha}-\right.\cos p\left({\pi}+{\alpha}\right)\left.\cos \left(p\left({\pi}-{\alpha}\right)-{\alpha}\right)\right]\nonumber\\\displaystyle \text{} & \text{} & \displaystyle +\,p^{2}\sin ^{2}{\alpha}\left(\sin 2p{\alpha}+\sin 2p{\pi}\right) +\sin p\left({\pi}-{\alpha}\right)\nonumber\\\displaystyle \text{} & \text{} & \displaystyle \times\,\left[\cos {\alpha}\cos \left(p\left({\pi}+{\alpha}\right)+{\alpha}\right)-2\sin p{\pi}\sin p{\alpha}\right],\nonumber\\ \displaystyle \tilde{{\delta}}_{4}\left(p\right) & = & \displaystyle 0,5p^{2}\sin ^{2}{\alpha}\left(\sin 2p{\alpha} +\sin 2p{\pi}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle+\,p\sin {\alpha}\left[\cos p{\pi}\sin p\left({\pi}-{\alpha}\right)\sin \left(p-1\right){\alpha}+2\cos {\alpha}\sin ^{2}p{\alpha}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\sin p\left({\pi}-{\alpha}\right)\sin p{\alpha}\left[\sin p{\pi}+\cos {\alpha}\sin \left(p{\pi}+{\alpha}\right)\right].\nonumber\end{eqnarray}$

Using the factorization of function G (p) by a formula (Gakhov (1966)): $\begin{eqnarray}G\left(p\right)=\frac{G^{+}\left(p\right)}{G^{-}\left(p\right)}\left(\text{Re}∼p=0\right),∼\exp \left[\frac{1}{2{\pi}i}\int _{-i\infty }^{i\infty }\frac{\ln G\left(z\right)}{z-p}dz\right]=\left\{\begin{array}{@{}l@{}}G^{+}\left(p\right)\quad \left(\text{Re}∼p\lt 0\right),\\ G^{-}\left(p\right)\quad \left(\text{Re}∼p>0\right),\end{array}\right.\end{eqnarray}$ and the presentation zctg $z{\pi}=\frac{{\Gamma}(1-z)}{{\Gamma}(1/2-z)}\frac{{\Gamma}(1+z)}{{\Gamma}(1/2+z)}$ [1], where 𝛤(z) is Euler’s gamma-function, we will reduce the equation (A.2) to the following kind (A.3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{{\Phi}^{+}\left(p\right)}{Q^{+}\left(p\right)G^{+}\left(p\right)}+\frac{{\tau}_{1}}{p+1}\left[\frac{1}{Q^{+}\left(p\right)G^{+}\left(p\right)}-\frac{1}{Q^{+}\left(-1\right)G^{+}\left(-1\right)}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\frac{{\tau}_{2}}{p+{\lambda}+1}\left[\frac{1}{Q^{+}\left(p\right)G^{+}\left(p\right)}-\frac{1}{Q^{+}\left(-{\lambda}-1\right)G^{+}\left(-{\lambda}-1\right)}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\frac{{\Phi}^{-}\left(p\right)\left(p-{\lambda}\right)Q^{-}\left(p\right)}{p^{2}G^{-}\left(p\right)}-\frac{{\tau}_{1}}{\left(p+1\right)Q^{+}\left(-1\right)G^{+}\left(-1\right)}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -\frac{{\tau}_{2}}{\left(p+{\lambda}+1\right)Q^{+}\left(-{\lambda}-1\right)G^{+}\left(-{\lambda}-1\right)}\quad \left(\text{Re}p=0\right),\end{eqnarray}$ where $Q^{\pm }(p)=\left[\frac{{\Gamma}(1\mp p)}{{\Gamma}(1/2\mp p)}\right]^{2}\frac{{\Gamma}(1\mp p\pm {\lambda})}{{\Gamma}(1/2\mp p\pm {\lambda})}$ .

The function in the left part of the equation (A.3) is analytical in the half-plane Re p < 0, and the function in its right part is analytical in the half-plane Re p > 0. Then, in accordance with the principle of analytical continuation, there must be a unified function, analytical in all plane of complex variable, which will be equal to the left and right parts (A.3) in corresponding half-planes. For its determination we will take into account that at the end of the rupture line according to the general statements about the behaviour of the stresses near the angular points of the elastic bodies [8,38] the asymptotic will be realized (A.4) $\begin{eqnarray}\displaystyle {\theta}={\alpha},\quad r\rightarrow l+0,\quad {\tau}_{r{\theta}}\sim \frac{k}{\sqrt{2{\pi}\left(r-l\right)}}, & & \displaystyle\end{eqnarray}$ where k is a stress intensity factor at the end of the rupture line which must be defined in the process of the solution of the problem, and we will apply the theorem of Abel’s type [35] to it and as a result we will receive the following: (A.5) $\begin{eqnarray}\displaystyle {\Phi}^{+}\left(p\right)\sim \frac{k}{\sqrt{-2pl}}\quad \left(p\rightarrow \infty ,\text{Re}∼p\lt 0\right). & & \displaystyle\end{eqnarray}$ Taking into account (A.5), we find that the left and right parts (A.3) tend to zero at p →∞. Then, according to Liouville’s theorem, a unified analytical function identically equals to zero which gives the solution of the equation (A.2) at once: $\begin{eqnarray}\displaystyle {\Phi}^{+}\left(p\right) & = & \displaystyle -Q^{+}\left(p\right)G^{+}\left(p\right)\left\{\frac{{\tau}_{1}}{p+1}\left[\frac{1}{Q^{+}\left(p\right)G^{+}\left(p\right)}-\frac{1}{Q^{+}\left(-1\right)G^{+}\left(-1\right)}\right]\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\frac{{\tau}_{2}}{p+{\lambda}+1}\left[\frac{1}{Q^{+}\left(p\right)G^{+}\left(p\right)}-\frac{1}{Q^{+}\left(-{\lambda}-1\right)G^{+}\left(-{\lambda}-1\right)}\right]\right\}\quad \left(\text{Re}∼p\lt 0\right),\nonumber\end{eqnarray}$ (A.6) $\begin{eqnarray}\displaystyle {\Phi}^{-}\left(p\right) & = & \displaystyle \frac{p^{2}G^{-}\left(p\right)}{\left(p-{\lambda}\right)Q^{-}\left(p\right)}\left[\frac{{\tau}_{1}}{\left(p+1\right)Q^{+}\left(-1\right)G^{+}\left(-1\right)}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\frac{{\tau}_{2}}{\left(p+{\lambda}+1\right)Q^{+}\left(-{\lambda}-1\right)G^{+}\left(-{\lambda}-1\right)}\right]\quad \left(\text{Re}∼p>0\right).\end{eqnarray}$ From (A.6) we find the asymptotic $\begin{eqnarray}\displaystyle {\Phi}^{+}\left(p\right)\sim \frac{-1}{\sqrt{-p}}\left[\frac{{\tau}_{1}}{Q^{+}\left(-1\right)G^{+}\left(-1\right)}+\frac{{\tau}_{2}}{Q^{+}\left(-{\lambda}-1\right)G^{+}\left(-{\lambda}-1\right)}\right]\quad \left(p\rightarrow \infty \right), & & \displaystyle \nonumber\end{eqnarray}$ comparing which with (A.5), we define the stress intensity factor at the end of the rupture line: (A.7) $\begin{eqnarray}\displaystyle k=-\sqrt{2l}\left[\frac{{\tau}_{1}}{Q^{+}\left(-1\right)G^{+}\left(-1\right)}+\frac{{\tau}_{2}}{Q^{+}\left(-{\lambda}-1\right)G^{+}\left(-{\lambda}-1\right)}\right]. & & \displaystyle\end{eqnarray}$ Equating the right part (A.7) to zero because of the requirement of the limitedness of stress meanings at the end of the plastic zone, we arrive at the expression (4) for its length.

References

Abramovitz

and Stegun

I.A.

, Handbook of Mathematical Function, New York, Dover Publications 1965.

Antipov

Y.A.

, An interface crack between elastic materials when there is dry friction, J. Appl. Math. Mech. (PMM) 59 (1995), 273–287.

Antipov

Y.A.

Avila-Pozos

Kolaczkovski

S.T.

and Movchan

A.B.

, Mathematical model of delamination cracks on imperfect interfaces, Int. J. Solids Struct. 38 (2001), 6665–6697.

Atkinson

, The interface crack with a contact zone (an analytical treatment), Int. J. Fract. 18 (1982), 161–177.

Atkinson

, The interface crack with a contact zone (the crack of finite length), Int. J. Fract. 19 (1982), 131–138.

Bakirov

V.F.

and Gol’dshtein

R.V.

, The Leonov–Panasyuk–Dugdale model for a crack at the interface of the joint of materials, J. Appl. Maths. Mech. 68 (2004), 153–161.

Cherepanov

G.P.

, Stresses in an inhomogeneous plate having slits, Notices of the USSR Academy of Sciences 1 (1962), 131–137.

Cherepanov

G.P.

, Mechanics of Brittle Fracture, New York, McGraw-Hill 1979.

Cherepanov

G.P.

, On the general theory of fracture, Mater. Sci. 22 (1986), 32–39.

10.

Comninou

and Dundurs

, Effect of friction on the interface crack loaded in shear, J. Elasticity 10 (1980), 203–212.

11.

Comninou

and Schmueser

, The interface crack in a combined tension-compression and shear field, ASME J. Appl. Mech. 46 (1979), 345–348.

12.

Comninou

, The interface crack, ASME J. Appl. Mech. 44 (1977), 631–636.

13.

Comninou

, Interface crack with friction in the contact zone, ASME J. Appl. Mech. 44 (1977), 780–781.

14.

Dugdale

D.S.

, Yielding of sheets containing slits, J. Mech. Phys. Solids 8 (1960), 100–104.

15.

Dundurs

, Discussion on “Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading”, ASME J. Appl. Mech. 32 (1965), 403–410.

16.

England

A.H.

, A crack between dissimilar media, ASME J. Appl. Mech. 32 (1965), 400–402.

17.

Erdogan

, Stress distribution in non-homogeneous elastic plane with cracks, ASME J. Appl. Mech. 30 (1963), 232–236.

18.

Gakhov

F.D.

, Boundary Value Problems, Oxford, Pergamon Press 1966.

19.

Gautesen

A.K.

and Dundurs

, The interface crack under combined loading, ASME J. Appl. Mech. 55 (1988), 580–586.

20.

Kamins’kyi

A.I.

Dudyk

M.V.

and Kipnis

L.A.

, Investigation of the process zone near the tip of an interface crack in the elastic body in shear within the framework of the complex model, J. Math. Sc. 220 (2017), 117–132.

21.

Kaminskii

A.A.

Kipnis

L.A.

and Kolmakova

V.A.

, Slip lines at the end of a cut at the interface of different media, Int. Appl. Mech. 31 (1995), 491–495.

22.

Kaminskii

A.A.

Kipnis

L.A.

and Kolmakova

V.A.

, On the Dugdaill model for a crack at the interface of different media, Int. Appl. Mech. 35 (1999), 58–63.

23.

Kaminskii

A.A.

and Nizhnik

S.B.

, Study of the laws governing the changes in the plastic zone at the crack tip and characteristics of the fracture toughness of metallic materials in relation to their structure (survey), Int. Appl. Mech. 31 (1995), 777–798.

24.

Kaminskii

A.A.

Usikova

G.I.

and Dmitrieva

E.A.

, Experimental study of the distribution of plastic strains near a crack tip during static loading, Int. Appl. Mech. 30 (1994), 892–897.

25.

Kaminskij

A.A.

and Kipnis

L.A.

, On a complex model of the pre-fracture zone at the end of a crack on the interface of elastic media, Dopovidi Natsional’noï Akademiï Nauk Ukraïny. Matematyka, Pryrodoznavstvo, Tekhnichni Nauky 2 (2010), 59–63, (In Russian), URI http://dspace.nbuv.gov.ua/handle/123456789/19595.

26.

Kaminskij

A.A.

and Kipnis

L.A.

, On the start of a crack on the interface of elastic media, Dopovidi Natsional’noï Akademiï Nauk Ukraïny. Matematyka, Pryrodoznavstvo, Tekhnichni Nauky 1 (2011), 38–43, (In Russian), URI http://dspace.nbuv.gov.ua/handle/123456789/36984.

27.

Kaminsky

A.A.

Dudik

M.V.

and Kipnis

L.A.

, On the direction of development of a thin prefracture process zone at the tip of an interfacial crack between dissimilar media, Int. Appl. Mech. 42 (2006), 136–144.

28.

Kaminsky

A.A.

Dudik

M.V.

and Kipnis

L.A.

, Small scale contact zone with the friction of the lips near the interfacial crack tip, Bulletin of Taras Shevchenko National University of Kyiv. Series Physics & Mathematics 1 (2014), 62–67, (In Ukrainian).

29.

Kohut

I.S.

and Kalyta

H.I.

, Evaluation of the sizes of the process zone for quasibrittle notched specimens, Mater. Sci. 44 (2008), 97–103.

30.

Leblond

J.B.

and Frelat

, Crack kinking from an initially closed, ordinary on interface crack, in the presence of friction, Eng. Fract. Mech. 71 (2004), 289–307.

31.

Leonov

M. Ya.

and Panasyuk

V.V.

, Development of the smallest cracks in a solid, Prykladna mekhanika 5,391–401, (in Ukrainian).

32.

Loboda

V.V.

, Analytical derivation and investigation of the interface crack models, Int. J. Solids Struct. 35 (1998), 4477–4489.

33.

Loboda

V.V.

and Sheveleva

A.E.

, Determining prefracture zones at a crack tip between two elastic orthotropic bodies, Int. Appl. Mech. 39 (2003), 566–572.

34.

Malyshev

B.M.

and Salganik

R.L.

, The strength of adhesive joints using the theory of crack, Inter. J. Fract. Mech. 1 (1965), 114–118.

35.

Noble

, Methods Based on the Wiener-Hopf Technique, Chelsea, New York, 1988.

36.

Ostryk

V.I.

, Friction contact of the edges of an interface crack under the conditions of tension and shear, Mater. Sci. 39 (2003), 214–224.

37.

Panasyuk

V.V.

and Savruk

M.P.

, Model for plasticity bands in elastoplastic failure mechanics, Mater. Sci. 28 (1992), 41–57.

38.

Parton

V.Z.

and Perlin

P.I.

, Mathematical Methods of the Theory of Elasticity, Moscow, Nauka 1981 , (In Russian).

39.

Rice

J.R.

, Elastic fracture mechanics concepts for interface crack, ASME J. Appl. Mech. 55 (1988), 98–103.

40.

Rice

J.R.

and Sih

G.C.

, Plane problems of cracks in dissimilar media, ASME J. Appl. Mech. 32 (1965), 418–423.

41.

Shih

C.F.

and Asaro

R.J.

, Elastic–plastic analysis of cracks on bimaterial interfaces: Part II – Structure of small-scale yielding fields, J. Appl. Mech. 56 (1989), 763–779.

42.

Tvergaard

, Infuence of plasticity on interface toughness in a layered solid with residual stresses, Int. J. Solids Struct. 40 (2003), 5769–5779.

43.

Tvergaard

and Hutchinson

J.H.

, On the toughness of ductile adhesive joints, J. Mech. Phys. Solids 44 (1996), 789–800.

44.

Uflyand

Ya.S.

, Survey of Articles on the Application of Integral Transforms in the Theory of Elasticity, North Carolina State University 1965.

45.

Williams

M.L.

, The stresses around a fault or crack in dissimilar media, Bull. Seism. Soc. Am. 49 (1959), 199–204.