About the influence of the elastoplastic properties of the adhesive on the value of the $${\varvec{J}}$$-integral in the DCB sample

On the basis of the general variational formulation of the problem of the deformation of two bodies connected by a thin layer, a system of differential equations of equilibrium of the double-cantilever beam is obtained, taking into account the shear deformations of the cantilevers, both in the interface section and in the free section, taking into account also the elastoplastic properties of the layer. In this work, we use the connection representation of the J-integral in terms of the energy product and the energy product of dissipation. For purely elastic deformation, on the basis of the analytical solution of the system, an expression is obtained for the stress state of an extremely thin layer connecting the cantilevers, which is dependent on the material properties of both the layer and the cantilevers. The obtained expression for the elastic energy flux is compared with the known ones. The energy product at the top of the layer is found, the value of which depends only on the material properties of the consoles. With the elastoplastic behavior of the layer, the energy product of dissipation was found, which turned out to be dependent on the yield stress of the adhesive. The energy product in this case is proportional to the layer thickness. For adhesives with pronounced plastic properties, taking into account the dissipative mechanism of energy release leads to fundamental differences in the J-integral in comparison with the elastic calculation. The dependences of the DCB sample compliance with subcritical growth of the plastic deformation region in the adhesive are plotted.

Keywords: Energy product; Energy product of dissipation; Interaction layer; Linear parameter; Elastoplastic

Introduction

One of the approaches to the experimental study of critical streams of elastic energy at the crack tip, the $G_{\mathit{IC}}$ , is Irwin's compliance method (Irwin and Kies [14]; Kanninen and Popelar [17]; Broek [6]). The method consists in experimentally measuring the compliance of a sample with different crack lengths and then calculating the derivative of the compliance along the crack length. If the sample compliance is obtained in the form of an analytical expression depending on the crack length, then, to calculate the $G_{\mathit{IC}}$ , it is sufficient to measure the critical external load and the crack length. As a rule, obtaining analytical representations for bodies of finite dimensions involves the introduction of a number of simplifying hypotheses. In this case, the results obtained depend directly on the considered mathematical model of the problem. Among the widely used experimental samples in fracture mechanics, a double-cantilever beam (DCB sample) stands out. In this sample, a section of unconnected cantilevers is highlighted, considered as a crack in the form of a mathematical section. The crack length can be adjusted based on the ISO15024 standard ([15]). This procedure was used in Banea et al. ([2]) and Lopes et al. ([19]). The compliance of the cantilevers and, accordingly, $G_{\mathit{IC}}$ in the classical approach of Irwin Kanninen and Popelar ([17]) is found from the theory of bending of the beam. In De Moura et al. ([8]) and Banea et al. ([3]), to refine the experimental results, it is proposed to take into account the shear deformations during bending of the free sections of the cantilevers. However, the proposed models do not take into account the deformation at the junction of the cantilevers, including the elastoplastic deformation of the adhesive layer, and its effect on $G_{\mathit{IC}}$ at the crack tip.

Let us note the solutions based on two-dimensional (2D) elasticity for the separation of bodies connected along the boundary (Ustinov and Idrisov [27]; Ustinov et al. [28]), among which we single out the solutions for the DCB sample (Andrews and Massabò [1]; Kanninen [16]; Li et al. [18]). To solve these problems, the apparatus of the linear theory of elasticity is used to determine the corresponding stress intensity factors at the dead-end point of the mathematical section. The real mechanical characteristics of the bonding adhesives are not considered in the corresponding problems.

In this paper, a crack-like defect is considered in the model of the interaction layer (Glagolev et al. [12]; Glagolev and Markin [11]; Berto et al. [4]; Glagolev et al. [13]; Berto et al. [5]). In this model, the crack is represented as a physical section with an indefinite end shape and a material layer on its continuation. It is shown in Glagolev and Markin ([11]), Berto et al. ([4]), Glagolev et al. ([13]) and Berto et al. ([5]) that the product of the free energy accumulated at the end of the layer by its thickness (the energy product or the EP) is equivalent to $G_{\mathit{IC}}$ with the elastic behaviour of the material. In the case of elastoplastic behaviour of the layer, to determine $G_{\mathit{IC}}$ it is necessary to take into account the term defined in Berto et al. ([5]) as the energy product of dissipation. On the basis of the general variational formulation (Glagolev et al. [12]; Glagolev and Markin [11]; Berto et al. [4]), a system of differential equations of equilibrium is obtained in the work, taking into account the shear deformations of the cantilevers, both at the interface with the layer and in the free area. From the solution of this system, an expression for the EP is obtained, which supplements the known elastic solutions for the flow of elastic energy into the crack tip. Taking into account elastoplastic properties of the adhesive layer of the solution of the corresponding elastic-plastic problem shows that the energy product of dissipation greatly changes the calculated value of $G_{\mathit{IC}}$ as compared to a purely elastic model.

Formulation of the opening mode problem

Figure 1 shows DCB sample $\ell +a$ , consisting of three bodies. Plates 1 and 2 with the same thickness h along the length $\ell$ are connected by an interaction layer 3 with thickness ${\delta }_0$ . The material of the plates and layer is assumed to be linearly elastic, with different moduli of elasticity and zero Poisson's ratios. The right end of the sample is rigidly fixed against horizontal and vertical displacements, while a vertical symmetric load with intensity $\mathit{P}$ acts on the left ends of the consoles. The rest of the sample surface is free from external load.

Graph: Fig. 1 Model of the DCB sample

To describe the interaction of layer 3 with bodies 1 and 2, we will apply the concept of an "interaction layer" developed in Berto et al. ([4]), Glagolev et al. ([13]) and Berto et al. ([5]). In this case, the equilibrium conditions for bodies 1 and 2 are written in a variational form for body 1:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \underset{S_1}{\int }\mathit{\sigma }··\delta \mathit{\epsilon }ds+\underset{\ell }{\int }{\overline{\sigma }}_{22}\delta u_2^{+}d{x}_1+\underset{\ell }{\int }{\overline{\sigma }}_{12}\delta u_1^{+}d{x}_1}\hfill \\ +0.5{\delta }_0\left({\displaystyle \underset{\ell }{\int }{\overline{\sigma }}_{11}\frac{\partial \delta u_1^{+}}{\partial x_1}d{x}_1+\underset{\ell }{\int }{\overline{\sigma }}_{12}\frac{\partial \delta u_2^{+}}{\partial x_1}d{x}_1}\right)\hfill \\ {\displaystyle =\underset{L_1}{\int }{\mathit{P}}^1·\delta \mathit{u}dl}\hfill \end{array}\end{array}$$

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and body 2:

2 $$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \underset{S_2}{\int }\mathit{\sigma }··\delta \mathit{\epsilon }ds-\underset{\ell }{\int }{\overline{\sigma }}_{22}\delta u_2^{-}d{x}_1-\underset{\ell }{\int }{\overline{\sigma }}_{12}\delta u_1^{-}d{x}_1}\hfill \\ +0.5{\delta }_0\left({\displaystyle \underset{\ell }{\int }{\overline{\sigma }}_{11}\frac{\partial \delta u_1^{-}}{\partial x_1}d{x}_1+\underset{\ell }{\int }{\overline{\sigma }}_{12}\frac{\partial \delta u_2^{-}}{\partial x_1}d{x}_1}\right)\hfill \\ {\displaystyle =\underset{L_2}{\int }{\mathit{P}}^2·\delta \mathit{u}dl,}\hfill \end{array}\end{array}$$

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where $S_1$ , $S_2$ —areas of bodies 1 and 2; $\mathit{\sigma }$ , $\mathit{\epsilon }$ —stress and strain tensors; $\overline{\mathit{\sigma }}$ , $\overline{\mathit{\epsilon }}$ —average stress and strain tensors of the layer with components:

3 $$\begin{array}{cc}\hfill {\overline{\sigma }}_{21}\left(x_1\right)=& {\displaystyle {\overline{\sigma }}_{12}\left(x_1\right)=\frac{1}{{\delta }_0}\underset{-0.5{\delta }_0}{\overset{0.5{\delta }_0}{\int }}{\sigma }_{21}\left(x_1,,,x_2\right)d{x}_2,}\hfill \\ \hfill {\overline{\sigma }}_{22}\left(x_1\right)=& {\displaystyle \frac{1}{{\delta }_0}\underset{-0.5{\delta }_0}{\overset{0.5{\delta }_0}{\int }}{\sigma }_{22}\left(x_1,,,x_2\right)d{x}_2,{\overline{\sigma }}_{11}\left(x_1\right)}\hfill \\ \hfill =& {\displaystyle \frac{1}{{\delta }_0}\underset{-0.5{\delta }_0}{\overset{0.5{\delta }_0}{\int }}{\sigma }_{11}\left(x_1,,,x_2\right)d{x}_2,}\hfill \\ \hfill {\overline{\epsilon }}_{22}\left(x_1\right)=& \left(\frac{u_2^{+}\left(x_1\right)-u_2^{-}\left(x_1\right)}{{\delta }_0}\right),{\overline{\epsilon }}_{11}\left(x_1\right)\hfill \\ \hfill =& 0.5\left(\frac{\partial u_1^{+}\left(x_1\right)}{\partial x_1},+,\frac{\partial u_1^{-}\left(x_1\right)}{\partial x_1}\right),\hfill \end{array}$$

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4 $$\begin{array}{cc}\hfill {\overline{\epsilon }}_{21}\left(x_1\right)=& {\overline{\epsilon }}_{12}\left(x_1\right)\hfill \\ \hfill =& 0.5(\frac{u_1^{+}\left(x_1\right)-u_1^{-}\left(x_1\right)}{{\delta }_0}\hfill \\ \hfill & +0.5\left(\frac{\partial u_2^{+}\left(x_1\right)}{\partial x_1},+,\frac{\partial u_1^{-}\left(x_1\right)}{\partial x_1}\right)),\hfill \end{array}$$

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where $u_k^{\pm }$ —are the components of the vectors of displacements of the upper and lower boundaries of the layer, respectively; $k=1,2$ ; $L_1,L_2$ boundary of application of external load for body 1 and 2; $··$ is double scalar multiplication; $·$ is scalar multiplication. Rigid adhesion is postulated between the boundaries of region 3 and regions 1 and 2:

$$\begin{array}{c}\hfill {\mathit{u}}^{+}=\mathit{u}\left(x_1,,,{\delta }_0/2\right);{\mathit{u}}^{-}=\mathit{u}\left(x_1,,,-{\delta }_0/2\right)x_1\in \left[0,;,\ell \right].\end{array}$$

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For the plate material, we assume the constitutive relations in the form of Hooke's law:

5 $$\begin{array}{c}\hfill {\sigma }_{\mathit{ij}}=E{\epsilon }_{\mathit{ij}},\end{array}$$

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where E—is the modulus of elasticity.

We assume that the stress state of the layer for a given type of loading is determined by one component of the average stress tensor. For the material of the interaction layer 3, the constitutive relations are taken in the form:

6 $$\begin{array}{c}\hfill {\overline{\sigma }}_{22}=E_3{\overline{\epsilon }}_{22};{\overline{\sigma }}_{11}={\overline{\sigma }}_{12}=0,\end{array}$$

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where $E_3$ —is the modulus of elasticity of the layer material.

Relations (6) correspond to the representation of the Prandtl type of elastic bonds (Prandtl and Knauss [23]; Entov and Salganik [10]; Salganik et al. [24]). By symmetry, the displacement field of the projection objectives satisfies the conditions $u_1^1\left(x_1,,,x_2\right)=$ $=u_1^2\left(x_1,,,x_2\right)=u_1\left(x_1,,,x_2\right)$ , $u_2^1\left(x_1,,,x_2\right)=-u_2^2\left(x_1,,,x_2\right)=$ $=u_2\left(x_1,,,x_2\right)$ , and the vector of the distributed external load— ${\mathit{P}}^1=-{\mathit{P}}^2=\mathit{P}$ . Thus, it is sufficient to confine ourselves to considering body 1. In view of the fact that the interaction layer satisfies the conditions (6), the system of Eqs. (1) and (2) is converted to a variational equation:

7 $$\begin{array}{c}\hfill {\displaystyle \underset{S_1}{\int }\mathit{\sigma }··\delta \mathit{\epsilon }ds+\underset{\ell }{\int }{\overline{\sigma }}_{22}\delta u_2^{+}d{x}_1=\underset{L_1}{\int }\mathit{P}·\delta \mathit{u}dl.}\end{array}$$

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The solution of system (5)–(7) is reduced to determining the displacement field $u\left(x_1,,,x_2\right)$ in body 1 (see Fig. 1), taking into account the boundary conditions at its ends:

8 $$\begin{array}{cc}& {\left(u_1,\left(x_1,,,x_2\right)\right|}_{x{}_1=\ell }=0,\hfill \end{array}$$

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9 $$\begin{array}{cc}& {\left(u_2,\left(x_1,,,x_2\right)\right|}_{x{}_1=\ell }=0,\hfill \end{array}$$

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10 $$\begin{array}{cc}& {\left({\sigma }_{11}\right|}_{x{}_1=-a}=0,\hfill \end{array}$$

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11 $$\begin{array}{cc}& {\left({\sigma }_{12}\right|}_{x{}_1=-a}=-P.\hfill \end{array}$$

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To simplify the task, we assume that the displacement field in body 1 is defined as follows:

12 $$\begin{array}{cc}& u_1\left(x_1,,,x_2\right)=u_1^{+}\left(x_1\right)-\phi \left(x_1\right)\left(x_2,-,{\delta }_0/2\right),\hfill \end{array}$$

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13 $$\begin{array}{cc}& u_2\left(x_1,,,x_2\right)=u_2^{+}\left(x_1\right).\hfill \end{array}$$

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The parameter $\phi$ included in representation (12) has the geometric meaning of a small angle of rotation of the material normal to the plane $x_2={\delta }_0/2$ in body 1. According to distribution (12) and (13), the deformations in the console will be determined in the form:

14 $$\begin{array}{cc}& {\epsilon }_{11}\left(x_1,,,x_2\right)=\frac{d{u}_1^{+}\left(x_1\right)}{d{x}_1}-{\phi }^{\prime }\left(x_1\right)\left(x_2,-,{\delta }_0/2\right),\hfill \end{array}$$

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15 $$\begin{array}{cc}& {\epsilon }_{12}\left(x_1,,,x_2\right)={\epsilon }_{12}\left(x_1\right)=0.5\left(\frac{d{u}_2^{+}\left(x_1\right)}{d{x}_1},-,\phi ,\left(x_1\right)\right),\hfill \end{array}$$

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16 $$\begin{array}{cc}& {\epsilon }_{22}\left(x_1,,,x_2\right)=0,\hfill \end{array}$$

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Expressions (14) and (15), like the theory of Timoshenko and Goodier ([25]) and Timoshenko and Woinowsky-Krieger ([26]) and work of Mattei and Bardella ([20]), Panettieri et al. ([21]) and Panteghini and Bardella ([22]) take into account shear deformations and rotations of the normals in the body.

Consider the work of internal stresses for body 1, taking into account the specified deformation fields (15) and (16):

17 $$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \underset{S_1}{\int }\mathit{\sigma }··\delta \mathit{\epsilon }ds}\hfill \\ {\displaystyle =\underset{-a}{\overset{-0}{\int }}\underset{{\delta }_0/2}{\overset{h+{\delta }_0/2}{\int }}\left({\sigma }_{11},\left(\frac{d\delta u_1^{+}}{d{x}_1},-,\left(x_2\right)\right)\right)}\hfill \\ \left(\left(\left(-,{\delta }_0/2\right),\frac{d\delta \phi }{d{x}_1}\right),+,{\sigma }_{12},\left(\frac{d\delta u_2^{+}}{d{x}_1},-,\delta ,\phi \right)\right)d{x}_{1}d{x}_2\hfill \\ {\displaystyle +\underset{+0}{\overset{\ell }{\int }}\underset{{\delta }_0/2}{\overset{h+{\delta }_0/2}{\int }}\left({\sigma }_{11},\left(\frac{d\delta u_1^{+}}{d{x}_1},-,\left(x_2,-,{\delta }_0/2\right),\frac{d\delta \phi }{d{x}_1}\right)\right)}\hfill \\ \left(+,,{\sigma }_{12},\left(\frac{d\delta u_2^{+}}{d{x}_1},-,\delta ,\phi \right)\right)d{x}_{1}d{x}_2.\hfill \end{array}\end{array}$$

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Let us introduce the generalised forces into consideration:

18 $$\begin{array}{cc}& {\displaystyle Q_{11}\left(x_1\right)=\underset{{\delta }_0/2}{\overset{h+{\delta }_0/2}{\int }}{\sigma }_{11}d{x}_2,}\hfill \end{array}$$

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19 $$\begin{array}{cc}& {\displaystyle Q_{12}\left(x_1\right)=\underset{{\delta }_0/2}{\overset{h+{\delta }_0/2}{\int }}{\sigma }_{12}d{x}_2,}\hfill \end{array}$$

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and a generalised moment:

20 $$\begin{array}{c}\hfill {\displaystyle M_{11}\left(x_1\right)=\underset{{\delta }_0/2}{\overset{h+{\delta }_0/2}{\int }}{\sigma }_{11}\left(x_2,-,{\delta }_0/2\right)d{x}_2.}\end{array}$$

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We integrate by parts a series of terms on the right side of (17) taking into account (18)–(20):

21 $$\begin{array}{cc}\hfill {\displaystyle \underset{m}{\overset{n}{\int }}{Q}_{1k}\frac{d\delta u_k^{+}}{d{x}_1}d{x}_1=}& {\left(Q_{1k},\delta ,u_k^{+}\right|}_{x_1=m}^{x_1=n}\hfill \\ \hfill =& {\displaystyle -\underset{m}{\overset{n}{\int }}\frac{d{Q}_{1k}}{d{x}_1}\delta u_k^{+}d{x}_1,}\hfill \end{array}$$

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22 $$\begin{array}{cc}\hfill {\displaystyle \underset{m}{\overset{n}{\int }}{M}_{11}\frac{d\delta \phi }{d{x}_1}d{x}_1=}& {\left(M_{11},\delta ,\phi \right|}_{x_1=m}^{x_1=n}\hfill \\ \hfill & {\displaystyle -\underset{m}{\overset{n}{\int }}\frac{d{M}_{11}}{d{x}_1}\delta \phi d{x}_1,}\hfill \end{array}$$

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where m,n—are the corresponding limits of integration along the coordinate $x_1$ .

Consider the right side of (8). Find the stress vector at the left end of the console $\mathit{P}=-{\mathit{e}}_1·\mathit{\sigma }=P{\mathit{e}}_2$ . The work of the external load is equal to:

23 $$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \underset{L_1}{\int }P·\delta udl={\left({\displaystyle \underset{h+{\delta }_0/2}{\overset{{\delta }_0/2}{\int }}P\delta u_2^{+}\left(-,d,x_2\right)}\right|}_{x_1=-a}}\hfill \\ ={\left(P,h,\delta ,u_2^{+}\right|}_{{}_{x_1=-a}}={\left(Q_2,\delta ,u_2^{+}\right|}_{x_1=-a}.\hfill \end{array}\end{array}$$

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Substituting (17), (21)–(23) in (8) and equating the terms with equal variations, we arrive at two systems of differential equations for the site $x_1\in \left[-,a,;,0\right)$ :

24 $$\begin{array}{c}\hfill \frac{d{M}_{11}}{d{x}_1}-Q_{12}=0,\frac{d{Q}_{11}}{d{x}_1}=0,\frac{d{Q}_{12}}{d{x}_1}=0,\end{array}$$

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for sites $x_1\in \left(0,;,\ell \right]$ :

25 $$\begin{array}{c}\hfill \frac{d{M}_{11}}{d{x}_1}-Q_{12}=0,\frac{d{Q}_{11}}{d{x}_1}=0,\frac{d{Q}_{12}}{d{x}_1}={\overline{\sigma }}_{22},\end{array}$$

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with conjugation conditions:

26 $$\begin{array}{cc}& {\left(u_1^{+}\right|}_{x_1=-0}={\left(u_1^{+}\right|}_{x_1=+0},\hfill \end{array}$$

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27 $$\begin{array}{cc}& {\left(\phi \right|}_{x_1=-0}={\left(\phi \right|}_{x_1=+0},\hfill \end{array}$$

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28 $$\begin{array}{cc}& {\left(u_2^{+}\right|}_{x_1=-0}={\left(u_2^{+}\right|}_{x_1=+0},\hfill \end{array}$$

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29 $$\begin{array}{cc}& {\left(M_{11}\right|}_{x_1=-0}={\left(M_{11}\right|}_{x_1=+0},\hfill \end{array}$$

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30 $$\begin{array}{cc}& {\left(Q_{12}\right|}_{x_1=-0}={\left(Q_{12}\right|}_{x_1=+0},\hfill \end{array}$$

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31 $$\begin{array}{cc}& {\left(Q_{11}\right|}_{x_1=-0}={\left(Q_{11}\right|}_{x_1=+0},\hfill \end{array}$$

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and natural boundary conditions at the left end:

32 $$\begin{array}{cc}& {\left(Q_{12}\right|}_{x_1=-a}=-Q_2,\hfill \end{array}$$

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33 $$\begin{array}{cc}& {\left(Q_{11}\right|}_{x_1=-a}=0,\hfill \end{array}$$

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34 $$\begin{array}{cc}& {\left(M_{11}\right|}_{x_1=-a}=0.\hfill \end{array}$$

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On the right end from (8) and (9), taking into account (12) and (13), we consider the boundary conditions:

35 $$\begin{array}{cc}& {\left(u_1^{+}\right|}_{x_1=\ell }=0,\hfill \end{array}$$

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36 $$\begin{array}{cc}& {\left(\phi \right|}_{x_1=\ell }=0,\hfill \end{array}$$

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37 $$\begin{array}{cc}& {\left(u_2^{+}\right|}_{x_1=\ell }=0.\hfill \end{array}$$

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Taking into account (14)–(16) and the conditions of plane deformation ( ${\epsilon }_{33}=0$ ), defining relations (5), we write in the form:

38 $$\begin{array}{cc}& {\sigma }_{11}=E\left(\frac{d{u}_1^{+}\left(x_1\right)}{d{x}_1},-,{\phi }^{\prime },\left(x_1\right),\left(x_2,-,{\delta }_0/2\right)\right),\hfill \end{array}$$

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39 $$\begin{array}{cc}& {\sigma }_{12}=\frac{E}{2}\left(\frac{d{u}_2^{+}\left(x_1\right)}{d{x}_1},-,\phi ,\left(x_1\right)\right).\hfill \end{array}$$

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Let us write the expressions for the generalised forces (18) and (19) and the moment (20) taking into account (38) and (39):

40 $$\begin{array}{cc}& Q_{11}\left(x_1\right)=E\left(h,\frac{d{u}_1^{+}}{d{x}_1},-,\frac{h^2}{2},{\phi }^{\prime }\right),\hfill \end{array}$$

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41 $$\begin{array}{cc}& Q_{12}\left(x_1\right)=\frac{\mathit{Eh}}{2}\left(\frac{d{u}_2^{+}}{d{x}_1},-,\phi \right),\hfill \end{array}$$

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42 $$\begin{array}{cc}& M_{11}\left(x_1\right)=E\left(\frac{h^2}{2},\frac{d{u}_1^{+}}{d{x}_1},-,\frac{h^3}{3},{\phi }^{\prime }\right).\hfill \end{array}$$

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As a result, the problem (24)–(37) and (6), considering (40)–(42), becomes closed relative to the three unknown functions: $u_1^{+}\left(x_1\right)$ , $u_2^{+}\left(x_1\right)$ , $\phi \left(x_1\right)$ .

Elastic problem solution

Consider the solution to the problem at the interface between the layer and the console. From the system of Eq. (24) we carry out the transfer of the boundary conditions (32)–(34) to the point $x_1=0$ :

43 $$\begin{array}{cc}& {\left(Q_{12}\right|}_{x_1=0}=-Q_2,\hfill \end{array}$$

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44 $$\begin{array}{cc}& {\left(Q_{11}\right|}_{x_1=0}=0,\hfill \end{array}$$

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45 $$\begin{array}{cc}& {\left(M_{11}\right|}_{x_1=0}=-Q_{2}a.\hfill \end{array}$$

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Assuming $\frac{E_3}{E}\ge \frac{6{\delta }_0}{h}$ , we write the general solution (25) for the site $x_1\in \left(0,;,\ell \right]$ in the form:

46 $$\begin{array}{cc}\hfill u_1^{+}=& \frac{h}{2}\phi +k1x+k2;u_2^{+}=k3e^{{\lambda }_1{x}_1}+k4e^{{\lambda }_2{x}_1}\hfill \\ \hfill & +k5e^{{\lambda }_3{x}_1}+k6e^{{\lambda }_4{x}_1};\hfill \\ \hfill \phi =& k3\left({\lambda }_1,-,\frac{K1}{{\lambda }_1}\right)e^{{\lambda }_1{x}_1}+k4\left({\lambda }_2,-,\frac{K1}{{\lambda }_2}\right)e^{{\lambda }_2{x}_1}\hfill \\ \hfill & +k5\left({\lambda }_3,-,\frac{K1}{{\lambda }_3}\right)e^{{\lambda }_3{x}_1}+k6\left({\lambda }_4,-,\frac{K1}{{\lambda }_4}\right)e^{{\lambda }_4{x}_1},\hfill \\ \hfill \end{array}$$

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where $k1-k6$ are the constants of integration; $K1=\frac{4}{h{\delta }_0}\frac{E_3}{E}$ ; ${\lambda }_1=\sqrt{\frac{2}{h{\delta }_0}\frac{E_3}{E}\left(1,+,\sqrt{1-\frac{6{\delta }_{0}E}{h{E}_3}}\right)}$ ; ${\lambda }_2=-\sqrt{\frac{2}{h{\delta }_0}\frac{E_3}{E}\left(1,+,\sqrt{1-\frac{6{\delta }_{0}E}{h{E}_3}}\right)}$ ; ${\lambda }_3=\sqrt{\frac{2}{h{\delta }_0}\frac{E_3}{E}\left(1,-,\sqrt{1-\frac{6{\delta }_{0}E}{h{E}_3}}\right)}$ ; ${\lambda }_4=-\sqrt{\frac{2}{h{\delta }_0}\frac{E_3}{E}\left(1,-,\sqrt{1-\frac{6{\delta }_{0}E}{h{E}_3}}\right)}$ .

Satisfaction of solutions (46) with conditions (43)–(45) and (35)–(37) leads to a system of linear equations with respect to the integration constants $k1-k6$ . Note that, for $\frac{\ell }{\sqrt{h{\delta }_0}}\to \infty$ , the determinant of the system tends to zero and the system becomes poorly defined. In this case, consider its analytical solution. From (46) and conditions (35)–(37) we obtain $k1=k2=k3=k5=0$ , and in this case the condition (44) executed identically. Thus, for finding two constants of integration, we use the conditions (43) and (45), which lead to the following system of linear equations:

47 $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{k4}{{\lambda }_2}+\frac{k6}{{\lambda }_4}=-\frac{Q_2{\delta }_0}{2E_3}\hfill \\ k4{\lambda }_2\left({\lambda }_2,-,\frac{K1}{{\lambda }_2}\right)+k6{\lambda }_4\left({\lambda }_4,-,\frac{K1}{{\lambda }_4}\right)\hfill \\ =\frac{12Q_{2}a}{E{h}^3}.\hfill \end{array}\right)\end{array}$$

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Note that ${\lambda }_2\left({\lambda }_2,-,\frac{K1}{{\lambda }_2}\right)=-{\lambda }_4^2$ ,

${\lambda }_4\left({\lambda }_4,-,\frac{K1}{{\lambda }_4}\right)=-{\lambda }_2^2$ . From the system (47) we find:

48 $$\begin{array}{cc}& k4=\frac{Q_2}{\left({\lambda }_2,-,{\lambda }_4\right)}\left(\frac{12a}{E{h}^3{\lambda }_4},-,{\lambda }_2^2,\frac{{\delta }_0}{2E_3}\right),\hfill \end{array}$$

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49 $$\begin{array}{cc}& k6=-\frac{Q_2}{\left({\lambda }_2,-,{\lambda }_4\right)}\left(\frac{12a}{E{h}^3{\lambda }_2},-,{\lambda }_4^2,\frac{{\delta }_0}{2E_3}\right).\hfill \end{array}$$

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From (46), taking into account (48) and (49), we find the vertical displacement at the top of the layer:

50 $$\begin{array}{c}\hfill \begin{array}{c}{\left(u_2^{+}\right|}_{x_1=0}=k4+k6=Q_2\left(\frac{12a}{E{h}^3{\lambda }_2{\lambda }_4}\right)\hfill \\ \left(-,\left({\lambda }_2,+,{\lambda }_4\right),\frac{{\delta }_0}{2E_3}\right).\hfill \end{array}\end{array}$$

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Consider (50) under the condition $\frac{{\delta }_0}{h}\ll 1$ :

$$\begin{array}{c}\hfill {\lambda }_2=-2\sqrt{\frac{E_3}{h{\delta }_{0}E}};{\lambda }_4=-\sqrt{\frac{6}{h^2}};\end{array}$$

Graph

${\left(u_2^{+}\right|}_{x_1=0}=\frac{Q_2\sqrt{2}}{\sqrt{E{E}_3}}\sqrt{\frac{{\delta }_0}{h}}\left(\sqrt{3},\frac{a}{h},+,\frac{1}{\sqrt{2}}\right)$ . From (3) and (6) and the last expression we find the values of the stresses in the top of the layer:

51 $$\begin{array}{c}\hfill {\left({\overline{\sigma }}_{22}\right|}_{x_1=0}=\frac{Q_{2}2\sqrt{2}}{\sqrt{{\delta }_{0}h}}\sqrt{\frac{E_3}{E}}\left(\sqrt{3},\frac{a}{h},+,\frac{1}{\sqrt{2}}\right).\end{array}$$

Graph

In Berto et al. ([5]), the following expression was obtained for the J-integral with elastoplastic properties of the adhesive:

52 $$\begin{array}{c}\hfill J={\delta }_o{\rho }_0\phi +{\delta }_o{\rho }_{0}w,\end{array}$$

Graph

where ${\rho }_0\phi$ —is the change in the specific free energy; ${\rho }_{0}w$ —change in specific dissipation. Let us consider the change in the J-integral under elastic deformation, when ${\rho }_{0}w=0$ . In this case, the value of the J-integral determines the expression for the energy product of the layer (Glagolev and Markin [11]). For a uniaxial stress state in a layer: $J=2\gamma =\frac{{\overline{\sigma }}_{22}^2}{2E_3}{\delta }_0$ .

Taking into account (51), we obtain:

53 $$\begin{array}{c}\hfill 2\gamma =\frac{12Q_2^2}{\mathit{Eh}}\left({\left(\frac{a}{h}\right)}^2,+,\sqrt{\frac{2}{3}},\left(\frac{a}{h}\right),+,\frac{1}{6}\right).\end{array}$$

Graph

Let us compare the obtained representation (53) with the known expressions. Here is the expression for the elastic energy flux obtained in Banea et al. ([2]),taking into account the shear deformations in the free consoles of the DCB sample:

54 $$\begin{array}{c}\hfill G_{\mathit{IC}}=\frac{6Q_2^2}{h}\left(\frac{2}{E_f},{\left(\frac{a_{\mathit{eq}}}{h}\right)}^2,+,\frac{1}{5G}\right),\end{array}$$

Graph

where $E_f$ is the corrected elastic modulus; G is the shear modulus; $a_{\mathit{eq}}$ is the equivalent crack length.

Assuming $E_f\approx E$ , $a_{\mathit{eq}}\approx a$ , $G=0.5E$ from (54), we obtain:

55 $$\begin{array}{c}\hfill G_{\mathit{IC}}=\frac{12Q_2^2}{\mathit{Eh}}\left(2,{\left(\frac{a}{h}\right)}^2,+,\frac{1}{5}\right).\end{array}$$

Graph

The absence of shear deformations in free consoles gives the classical expression for the elastic energy flux into the crack tip for a DCB sample (Kanninen and Popelar [17]):

56 $$\begin{array}{c}\hfill G_{\mathit{IC}}^{\bullet }=\frac{12{\left(Q_2,a\right)}^2}{E{h}^3}.\end{array}$$

Graph

In Bruno and Greco ([7]), an expression for the elastic energy flux was obtained in the framework of the theory of plates. At $v=0$ , the expression for the elastic energy flux takes the form:

57 $$\begin{array}{c}\hfill G_I^b=\frac{12Q_2^2}{\mathit{Eh}}\left({\left(\frac{a}{h}\right)}^2,+,\sqrt{\frac{2}{2.5}},\left(\frac{a}{h}\right),+,\frac{1}{5}\right).\end{array}$$

Graph

The 2D solution, considered in Andrews and Massabò ([1]), Kanninen ([16]) and Li et al. ([18]), gives the following expression:

58 $$\begin{array}{c}\hfill G_I^a=\frac{12Q_2^2}{\mathit{Eh}}\left({\left(\frac{a}{h}\right)}^2,+,2,·,0.673,\left(\frac{a}{h}\right),+,{(0.673)}^2\right).\\ \hfill \end{array}$$

Graph

In Fig. 2 shows the dependence of the elastic energy flux ${\overline{G}}_I$ , referred to $G_I^a$ , on a/h. Graph 1 corresponds to formula (58), graph 2—to formula (57), graph 3—to formula (56), graph 4—to formula (55), graph 5—to formula (53). It can be seen from the above dependencies that at $\frac{a}{h}>4$ , the difference between the calculation results using formulas (58), (57), (53) is less than 12%, and the difference between (57) and (53) is less than 2%. Thus, the use of simplifying hypotheses (12), (13) when describing the displacement field of the console gives a rather close result to solution (57).

Graph: Fig. 2 Dependence of the relative flux of elastic energy on a/h

Comparing expressions (53), (57), (58) and (55), we find that taking into account the shear deformation of coupled cantilevers introduces a significant correction to the expression of the elastic energy flux in the form of a term that is linear with respect to the crack length.

A significant difference (up to 25% for solutions (55), (56)) is explained by the fact that solution (58) is a 2D solution of the linear theory of elasticity without restrictions on the distribution of the displacement field.

According to the data of Lopes et al. ([19]), for the sample with the following geometric and mechanical characteristics: $a=0.055$ m, $h=0.0127$ m, $b=0.025$ m, $E=2·{10}^{11}$ Pa, where b is the sample thickness, let us estimate the value of $G_I=G_{\mathit{IC}}$ at the moment of crack initiation under a critical external load. In Lopes et al. ([19]), the experimental values of the external load during crack initiation in the adhesive were $P_{\mathit{cr}}=1.1$ kN for Araldite AV138 resin, $P_{\mathit{cr}}=1.5$ kN for Araldite 2015 resin, and $P_{\mathit{cr}}=3.1$ kN for Sikaforce 7752 resin.

Following the data of works (Banea et al. [2]; Lopes et al. [19]), we give in Table 1 the mechanical characteristics of a number of adhesives according to the manufacturer's data and experimental data from Lopes et al. ([19]) for the CCM method.

Table 1 Mechanical properties of adhesives

Mechanical properties	Araldite AV138	Araldite 2015	Sikaforce 7752
$E3$ (GPa)	4.9	1.85	0.49
$σy$ (MPa)	36.49	12.63	3.24
$σf$ (MPa)	39.45	21.63	11.48
$εf$ (%)	1.21	4.77	19.18
$GIC$ (N/m)	200	430	2360
$GICCCM$ (N/m)	$149±9$	$985±272$	$4238±677$

The following designations are adopted in Table 1: $G_{\mathit{IC}}$ —critical energy flow; ${\sigma }_e$ is the yield strength; ${\sigma }_f$ —tensile strength; ${\epsilon }_f$ —ultimate strain.

In this case $Q_2=Q_{\mathit{cr}}=\frac{P_{\mathit{cr}}}{b}$ , where $Q_{\mathit{cr}}$ is the critical value of the external load. Table 2 presents the results of calculation by the formulas (53), (57) and (58).

Table 2 Calculated values of the energy flux into the crack tip with linear elastic behaviour of the adhesive

Calculated characteristics	Araldite AV138	Araldite 2015	Sikaforce 7752
$2γ$ (N/m)	205	382	1631
$GICb$ (N/m)	209	388	1658
$GICa$ (N/m)	229	425	1819

The calculated values according to the formula (58) for the Araldite 2015 adhesive practically coincided with the manufacturer's data. For Sikaforce 7752, the calculation gives a value less than the declared value, and for Araldite AV138 it gives a higher value. When considering the experimental data of the elastic energy flux by the CCM method from Table 1, the difference in the results is significant.

The more pronounced the plastic properties of the adhesive are, the greater is the difference in the conservative calculation given as a result of the elastic model. For the adhesives Araldite 2015 and Sikaforce 7752, the calculation of the critical value of the J-integral using formula (58), according to Table 2, gives a result 2.3 times lower than the data obtained using the CCM method. In this case, the behavior model of the adhesive can be significant.

Let us consider the construction of the J-integral when approximating the displacement field in the console (12), (13) due to taking into account the elastoplastic properties of the adhesive.

Elastoplastic problem solution

For the material of the interaction layer 3, the constitutive relations at the stage of reversible deformation, when ${\overline{\sigma }}_{22}\le {\sigma }_0$ , are taken in the form (6), and at the stage of plastic flow:

59 $$\begin{array}{c}\hfill {\overline{\sigma }}_{22}={\sigma }_0;{\overline{\sigma }}_{11}={\overline{\sigma }}_{12}=0,\end{array}$$

Graph

where ${\sigma }_0$ is the yield stress of the layer material.

The stage of elastic deformation continues until the interaction layer reaches the yield stress ${\overline{\sigma }}_{22}={\sigma }_0$ at the point x = 0. From formula (51) we find the value of the external load at this moment $Q_{2p}=\frac{{\sigma }_0\sqrt{{\delta }_{0}h}}{2\sqrt{2}\left(\sqrt{3},\frac{a}{h},+,\frac{1}{\sqrt{2}}\right)}\sqrt{\frac{E}{E_3}}$ .

It follows from this formula that the stage of purely reversible deformation in the absence of dissipation is possible at a finite value of the interaction layer thickness. In the case of degeneration into a mathematical cut, the stage of elastic deformation, as in the Dugdale model (Dugdale [9]), is absent.

We assume that the plastic flow is realized in the adhesive region of length ${\ell }_p$ . Equilibrium conditions (25) for the section a of the console $x_1\in \left(0,;,{\ell }_p\right)$ can be written taking into account (59) in the form:

60 $$\begin{array}{c}\hfill \frac{d{M}_{11}}{d{x}_1}-Q_{12}=0,\frac{d{Q}_{11}}{d{x}_1}=0,\frac{d{Q}_{12}}{d{x}_1}={\sigma }_0.\end{array}$$

Graph

We carry out the transfer of the boundary conditions (43)–(45) to a point $x_1={\ell }_p$ taking into account (57):

61 $$\begin{array}{cc}& {\left(Q_{12}\right|}_{x_1={\ell }_p}={\sigma }_0{\ell }_p-Q_2,\hfill \end{array}$$

Graph

62 $$\begin{array}{cc}& {\left(Q_{11}\right|}_{x_1={\ell }_p}=0,\hfill \end{array}$$

Graph

63 $$\begin{array}{cc}& {\left(M_{11}\right|}_{x_1={\ell }_p}=\frac{{\sigma }_0{\ell }_p^2}{2}-Q_2\left(a,+,{\ell }_p\right).\hfill \end{array}$$

Graph

Here and below, $Q_2=Q_{2p}$ for ${\ell }_p=0$ .

Using the general solution (46) and the conditions: $k1=k2=k3=k5=0$ from (61)–(63) we obtain a solution in the elastic region $x\ge {\ell }_p$ in the following form:

64 $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{k4}{{\lambda }_2}{e}^{{\lambda }_2{\ell }_p}+\frac{k6}{{\lambda }_4}{e}^{{\lambda }_4{\ell }_p}=\left({\sigma }_0,{\ell }_p,-,Q_2\right)\frac{{\delta }_0}{2E_3}\hfill \\ k4{\lambda }_2\left({\lambda }_2,-,\frac{K1}{{\lambda }_2}\right)e^{{\lambda }_2{\ell }_p}+k6{\lambda }_4\left({\lambda }_4,-,\frac{K1}{{\lambda }_4}\right)e^{{\lambda }_4{\ell }_p}\hfill \\ =\frac{12}{E{h}^3}\left(Q_2,\left(a,+,{\ell }_p\right),-,\frac{{\sigma }_0{\ell }_p^2}{2}\right).\hfill \end{array}\right)\end{array}$$

Graph

We write solution (64) in the form:

65 $$\begin{array}{cc}\hfill k4=& \frac{e^{-{\lambda }_2{\ell }_p}}{\left({\lambda }_2,-,{\lambda }_4\right)}(\frac{12a}{E{h}^3{\lambda }_4}(Q_2\left(a,+,{\ell }_p\right)\hfill \\ \hfill & -\frac{{\sigma }_0{\ell }_p^2}{2})+{\lambda }_2^2\left({\sigma }_0,{\ell }_p,-,Q_2\right)\frac{{\delta }_0}{2E_3}),\hfill \end{array}$$

Graph

66 $$\begin{array}{cc}\hfill k6=& -\frac{e^{-{\lambda }_4{\ell }_p}}{\left({\lambda }_2,-,{\lambda }_4\right)}(\frac{12}{E{h}^3{\lambda }_2}(Q_2\left(a,+,{\ell }_p\right)\hfill \\ \hfill & -\frac{{\sigma }_0{\ell }_p^2}{2})+{\lambda }_4^2\left({\sigma }_0,{\ell }_p,-,Q_2\right)\frac{{\delta }_0}{2E_3}).\hfill \end{array}$$

Graph

From (46), (65) and (66), we write the expression for the vertical displacement at a point $x_1={\ell }_p$ :

67 $$\begin{array}{cc}\hfill u_2^{+}\left({\ell }_p\right)=& \frac{1}{\sqrt{E{E}_3}}\sqrt{\frac{{\delta }_0}{h}}(\frac{\sqrt{6}}{h}(Q_2\left(a,+,{\ell }_p\right)\hfill \\ \hfill & -\frac{{\sigma }_0{\ell }_p^2}{2})-\left({\sigma }_0,{\ell }_p,-,Q_2\right))\hfill \end{array}$$

Graph

and taking into account (3) and (6) the stresses in the layer:

68 $$\begin{array}{cc}\hfill {\overline{\sigma }}_{22}\left({\ell }_p\right)& =\frac{2}{\sqrt{{\delta }_{0}h}}\sqrt{\frac{E_3}{E}}(\frac{\sqrt{6}}{h}(Q_2\left(a,+,{\ell }_p\right)\hfill \\ \hfill & -\frac{{\sigma }_0{\ell }_p^2}{2})-\left({\sigma }_0,{\ell }_p,-,Q_2\right)).\hfill \end{array}$$

Graph

From (68) we obtain the expression for the length of the plasticity zone:

69 $$\begin{array}{c}\hfill {\ell }_p=\frac{1}{\sqrt{6}}(\sqrt{1+6{\left(\frac{Q_2}{{\sigma }_{0}h}\right)}^2+\frac{12Q_{2}a}{{\sigma }_0{h}^2}-\sqrt{6}\sqrt{\frac{{\delta }_{0}E}{h{E}_3}}}\\ \hfill -1+\frac{\sqrt{6}{Q}_2}{{\sigma }_{0}h})h,\end{array}$$

Graph

where at ${\ell }_p=0,Q_2=Q_{2p}$ . When ${\delta }_0=0,Q_{2p}=0$ .

The values of the horizontal displacement and the angle of rotation at the point $x_1={\ell }_p$ are found by the formulas (46):

$$\begin{array}{c}\hfill \phi \left({\ell }_p\right)=k4\left({\lambda }_2,-,\frac{K1}{{\lambda }_2}\right)e^{{\lambda }_2{\ell }_p}+k6\left({\lambda }_4,-,\frac{K1}{{\lambda }_4}\right)e^{{\lambda }_4{\ell }_p};\end{array}$$

Graph

$u_1^{+}\left({\ell }_p\right)=\frac{h}{2}\phi \left({\ell }_p\right)$ , where k4, k6 are determined according to (65) and (66).

Let us write solution (60) in the plastic flow region $0\le x_1\le {\ell }_p$ . Taking into account (40)–(42) and boundary conditions (43)–(45), we obtain

70 $$\begin{array}{c}\hfill \begin{array}{c}{u}_1^{+}=\frac{h}{2}\phi +c_1;u_2^{+}=-\frac{{\sigma }_0{x}_1^4}{2E{h}^3}+\frac{2Q_2}{E{h}^3}{x}_1^3\hfill \\ +\left(\frac{2{\sigma }_0}{\mathit{Eh}},+,\frac{12Q_{2}a}{E{h}^3}\right)\frac{x_1^2}{2}+\left(c_2,-,\frac{2Q_2}{\mathit{Eh}}\right)x_1+c_3;\hfill \\ \phi =-\frac{2{\sigma }_0{x}_1^3}{E{h}^3}+\frac{6Q_2}{E{h}^3}{x}_1^2+\frac{12Q_{2}a}{E{h}^3}{x}_1+c_2.\hfill \end{array}\end{array}$$

Graph

where $c_1,c_2,c_3$ —are the constants of integration.

The integration constants are found from the conditions of continuity of functions $u_1^{+},u_2^{+},\phi$ at the point $x_1={\ell }_p$ .

$$\begin{array}{c}\hfill \begin{array}{c}{c}_1=0;\hfill \\ c_2=\phi \left({\ell }_p\right)+\frac{2{\sigma }_0{\ell }_p^3}{E{h}^3}-\frac{6Q_2{\ell }_p^2}{E{h}^3}-\frac{12Q_{2}a{\ell }_p}{E{h}^3};\hfill \\ c_3=u_2^{+}\left({\ell }_p\right)-\frac{3{\sigma }_0{\ell }_p^4}{2E{h}^3}+\frac{4Q_2}{E{h}^3}{\ell }_p^3-\left(\frac{2{\sigma }_0}{\mathit{Eh}},-,\frac{12Q_{2}a}{E{h}^3}\right)\frac{{\ell }_p^2}{2}\hfill \\ -\left(\phi ,\left({\ell }_p\right),-,\frac{2Q_2}{\mathit{Eh}}\right){\ell }_p.\hfill \end{array}\end{array}$$

Graph

For this model we will assume the value of the yield stress as the average relative to the limits of elasticity and strength:

71 $$\begin{array}{c}\hfill {\sigma }_0=\frac{{\sigma }_y+{\sigma }_f}{2}.\end{array}$$

Graph

Under a uniaxial stress state of the layer ${\rho }_0{\phi }_k=0.5{\sigma }_0{\epsilon }_{22}^e$ , ${\rho }_{0}w={\sigma }_0\left({\epsilon }_{22},-,{\epsilon }_{22}^e\right)$ , where ${\epsilon }_{22}^e$ —is the elastic limit, ${\epsilon }_{22}$ —deformation in a state of destruction. Taking into account (3) expression (52) takes the form:

72 $$\begin{array}{c}\hfill J=2{\sigma }_0\left(u_2,-,0.5,u_2^e\right),\end{array}$$

Graph

where $u_2$ is the limiting value of displacement at a point $x_1=0$ ; $u_2^e=\frac{{\sigma }_0{\delta }_0}{2E_3}$ —the limiting value of elastic displacement at a point $x_1=0$ . Hence it follows that the stage of elastic deformation can be taken into account only at a certain final value of the thickness of the interaction layer.

To find (72) from the critical external load $Q_2=P_e/b$ and a fixed value of ${\delta }_0$ we find ${\ell }_p$ . From (70) and (67) we obtain the value $u_2=c_3$ .

Based on the mechanical properties from Table 1 in Fig. 3, the dependence of the $J_C$ -integral on the decimal logarithm of the ratio ${\delta }_0/h$ for the considered adhesives is plotted. At a critical external load, we have $J=J_C$ .

Graph 1 is plotted for Araldite AV138, Graph 2 for Araldite 2015, and Graph 3 for Sikaforce 7752.

Graph: Fig. 3 Dependence of the JC-integral on the relative thickness of the layer

From Fig. 3, it can be seen that at small relative layer thicknesses, convergence of the values of the $J_C$ -integral takes place, which practically remain unchanged at $\frac{{\delta }_0}{h}<{10}^{-5}$ . In Table 3, we place the calculated data of the $J_C$ integral and the length of the plastic zone (69) at $\frac{{\delta }_0}{h}={10}^{-10}$ .

Table 3 Values of the $J_C$ -integral and length of the plastic zone ${\ell }_p$

Calculated characteristics	Araldite AV138	Araldite 2015	Sikaforce 7752
$ℓpm$	0.009	0.019	0.058
$JCNm$	107	409	3892

The use of the plasticity condition in the form (59) for the yield point determined by expression (71) taking into account the data in Table 1 leads to low resulting values of the $J_C$ integral for the adhesive and Araldite AV 138 as compared with the purely elastic singular solution (53). This is due to the fact that, for thin layers in the vicinity of a crack-like defect, the plastic region, according to (69), is formed at an arbitrarily small load. For thin layers, we assume ${\sigma }_y=0$ . In this case, according to (71), the yield stress in the model constitutive relation (59) should be considered in the form:

73 $$\begin{array}{c}\hfill {\sigma }_0=\frac{{\sigma }_f}{2}.\end{array}$$

Graph

The calculated data of the $J_C$ -integral with the yield point (73) are given in Table 4.

Table 4 Values of the $J_C$ -integral and length of the plastic zone ${\ell }_p$

Calculated characteristics	Araldite AV138	Araldite 2015	Sikaforce 7752
$ℓpm$	0.018	0.026	0.070
$JCNm$	172	511	4551

From analysis of the data in Tables 3 and 4 it follows that the use of expression (73) in the elastoplastic model leads to an increase in the values of the length of the plastic zone and the $J_C$ -integral in comparison with condition (71).

From the results of Table 1, Table 2 and Table 4 the values of the $J_C$ -integral and EP for the Araldite AV138 adhesive correspond to the manufacturer's and the experimental data. In this case, a singular elastic solution and an elastoplastic solution with finite stresses lead to practically the same criterial characteristic for different mechanisms of its formation. In expression (53), the accumulated specific free energy is considered, and in (72), the specific dissipation prevails in the final infinitesimal cross-section of the layer. In the adhesives Araldite 2015 and Sikaforce 7752, which have more pronounced plastic properties, the dissipative mechanism of the formation of the $J_C$ -integral gives values that are closer to the experiment as compared to a purely elastic solution.

Let us consider the influence of the subcritical growth of the plastic zone of the adhesive on the dependence of the wedging force on the vertical displacement of the left end of the console. From (65) in the case ${\overline{\sigma }}_{22}({\ell }_p)={\sigma }_0$ we have:

74 $$\begin{array}{c}\hfill Q_2=\frac{{\sigma }_0\left(\left(\frac{\sqrt{6}}{2h},{\ell }_p^2,+,{\ell }_p\right),+,\frac{1}{2},\sqrt{\frac{{\delta }_{0}hE}{E_3}}\right)}{1+\frac{\sqrt{6}}{h}(a+{\ell }_p)}\end{array}$$

Graph

Figure 4 shows the dependence of the relative value (74) on the relative length of the plastic zone ${\overline{\ell }}_p$ of the considered adhesives at $\frac{{\delta }_0}{h}={10}^{-10}$ . The length of the plastic zone is referred to the critical value ${\ell }_p$ for the Sikaforce 7752 adhesive. The wedging force is referred to the critical value $Q_2$ for the Sikaforce 7752 adhesive. The numbering of the graphs corresponds to Fig. 3. Further, the numbering of the graphs remains unchanged.

Graph: Fig. 4 Dependence of the wedging force on the length of the plastic zone in the adhesive layer

From system (24), taking into account the boundary conditions (32)–(34) and the values $u_2^{+}(0)$ , $u_1^{+}(0)$ , $\phi (o)$ found from (67), we obtain the expression for the vertical displacement at the left end of the console:

75 $$\begin{array}{c}\hfill {\left(u_2^{+}\right|}_{x_1=-a}=\frac{2Q_{2}a}{\mathit{Eh}}\left(4,{\left(\frac{a}{h}\right)}^2,+,1\right)-c_{2}a+c_3.\end{array}$$

Graph

Figure 5 shows the dependence of the relative value (75) ${\overline{u}}_2^{+}$ on the relative length of the plastic zone ${\overline{\ell }}_p$ of the considered adhesives. The vertical displacement is referred to the critical value (73) for Sikaforce 7752 adhesive.

Graph: Fig. 5 Dependence of vertical displacement on the length of the plastic zone in the adhesive layer

In Fig. 6 shows the dependence ${\overline{Q}}_2$ on ${\overline{u}}_2^{+}$ at subcritical growth of the plastic zone.

Graph: Fig. 6 Dependence of the wedging force on the vertical displacement with the growth of the plastic zone in the layer

From Fig. 6 we see that for adhesives with pronounced plastic properties, taking into account the plastic deformation zone introduces a significant nonlinearity into the dependence of the DCB sample compliance. For the Araldite AV138 adhesive with a relatively short subcritical plastic deformation zone, the compliance graph is close to a linear dependence.

Figure 7 shows the dependence of the relative value (70) $\overline{J}$ on the relative length of the plastic zone ${\overline{\ell }}_p$ of the considered adhesives. The J-integral value (70) is referred to the critical value $J_C$ for the Sikaforce 7752 adhesive.

Graph: Fig. 7 Dependence of the J-integral on the length of the plastic zone in the adhesive layer

In contrast to elastic change, the irreversible components of the J-integrals depend on the properties of the adhesive. Moreover, their subcritical growth has significantly nonlinear dependences for adhesives with pronounced plastic properties. Figure 8 shows the dependence $\sqrt{\overline{J}}$ on ${\overline{Q}}_2$ .

Graph: Fig. 8 Dependence J¯ on external forces Q¯2

In the elastic model, which gives expressions (53)–(58), the dependence $\sqrt{\overline{J}}$ on ${\overline{Q}}_2$ is linear. In this case, the mechanical properties of the adhesive do not affect its behavior. For the elastoplastic model, each yield point corresponds to its own curve of dependence $\sqrt{\overline{J}}$ on ${\overline{Q}}_2$ .

Conclusion

The analysis of the deformation of the DCB sample is carried out based on the consideration of the interaction layer between the consoles. Two models of adhesive behaviour were used-ideally elastic and ideally elastoplastic.

In the framework of these models, expressions for the J-integral are obtained. For the elastic behavior of the layer material, the value of the J-integral does not depend on the mechanical properties of the adhesive. For ideally elastoplastic behavior of the adhesive layer, the main contribution to the formation of the J-integral is made by dissipation in the layer and its value depends on the yield stress of the adhesive.

It is shown from the calculation results that, for adhesives with pronounced elastoplastic properties, the calculation of the $J_C$ -integral with allowance for the dissipative component gives a result that is closer to the experimental values as compared to the elastic solution. To experimentally find the value of the $J_C$ -integral based on the proposed solution, it is sufficient to determine the crack initiation load in the adhesive, without analyzing the sample compliance due to crack growth.

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By F. Berto; V. V. Glagolev; L. V. Glagolev and A. A. Markin

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