The theoretical and experimental investigations demonstrate that the plastic zone formed near pointed stress concentrators in elastoplastic materials has a complex structure and contains a fairly developed plastic domain in which shear strains are predominant and a much smaller plastic process zone adjacent to the tip of the concentrator with high levels of both shear and tensile strains. With regard for these features and symmetry conditions, the “trident” model was proposed in [1] for the description of the plastic zone formed at the end of a mode I crack originating from the corner point of the interface of two media. According to this model, the zone develops in two stages. In the first stage, two narrow lateral plastic strips (modeled by segments of discontinuities of tangential displacements) symmetrically propagate from the crack tip at a certain angle to its initial direction in one of the materials of the joint. If the appearance of these strips does not remove the stress concentration near the crack tip, then, in the next stage, the plastic process zone modeled by a segment of discontinuities of normal displacements can be formed on the continuation of the crack (Fig. 1).

Fig. 1
figure 1

“Trident” model of plastic zone.

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The numerical analysis of the plastic zone performed within the framework of the “trident” model with regard for the two stages of its development is reduced to the solution of four problems (for the slopes of plastic strips β greater and smaller than a half of the opening angle of the interface α) by the Wiener–Hopf method. A similar problem for a plastic strip on the interface of two materials was solved in [2].

Parameters of the Initial Lateral Plastic Zone

Assume that, under the conditions of plane deformation in a piecewise homogeneous isotropic elastoplastic body, a mode I crack originates from a corner point of the interface of materials into the material with Young’s modulus E 1 and Poisson’s ratio ν 1.

Since the corner point of the interface of two materials playing the role of the origin of mode I crack is a stress concentrator with power singularity [3], this leads to the formation of a plastic zone in its neighborhood. We assume that, in the initial stage of development of the plastic zone, two narrow lateral plastic strips of the same length are formed in the material with elastic constants E i and ν i (i = 1, 2). Their length is much smaller than the crack length (here and in what follows, subscript і marks the quantities corresponding to the plastic strips in the і th material). According to the hypothesis of localization [4], we model these strips by two rectilinear segments of discontinuities of tangential displacements inclined at an angle β i to the direction of crack continuation. The tangential stresses on these segments are equal to the yield limit of the i th material τ is .

We compute the lengths of the lateral plastic strips according to the Wiener–Hopf method by analogy with the problem of plastic strips in the connecting material [5, 6] and arrive at the following expression:

$$ {l}_i={\left(\frac{\left|C\right|}{\tau_{is}}\right)}^{-1\kern0.1em /\kern0.1em \lambda }{R}_i\left({\beta}_i\right), $$
(1)

where C is a constant characterizing the intensity of external force field (it is specified by the conditions of the problem), λ in the least root, in the interval (−1, 0), of the equation [3]:

$$ {D}_0\left(-1-x\right)=0, $$
(2)
$$ \begin{array}{l}\begin{array}{c}\hfill {D}_0(p)=4{e}^2\left[{\kappa}_2 \sin 2p\alpha -p \sin 2\alpha \right]\left[{p}^2{ \sin}^2\alpha -{ \sin}^2p\left(\pi -\alpha \right)\right]\hfill \\ {}\hfill +e\left\{\left(1+{\kappa}_1\right)\left(1+{\kappa}_2\right) \sin 2p\pi -4\left[{\kappa}_2 \sin 2p\alpha -p \sin 2\alpha \right]\left[{p}^2{ \sin}^2\alpha -{ \sin}^2p\left(\pi -\alpha \right)\right]\right.\hfill \\ {}\hfill -\left(p \sin 2\alpha + \sin 2p\alpha \right)\left.\left[\left(1+{\kappa}_1\right)\left(1+{\kappa}_2\right)-4\left({p}^2{ \sin}^2\alpha +{\kappa}_1{ \sin}^2p\left(\pi -\alpha \right)\right)\right]\right\}\hfill \\ {}\hfill +\left(p \sin 2\alpha + \sin 2p\alpha \right)\left\{{\left(1+{\kappa}_1\right)}^2-4\left[{p}^2{ \sin}^2\alpha +{\kappa}_1{ \sin}^2p\left(\pi -\alpha \right)\right]\right\},\hfill \\ {}\hfill {\kappa}_i=3-4{\nu}_i,e=\frac{E_1}{E_2}\frac{1+{\nu}_2}{1+{\nu}_1};\hfill \\ {}\hfill {R}_i\left(\theta \right)={\left[\frac{\sqrt{\pi}\left|{F}_i\left(\theta \right)\right|\varGamma \left(1+\lambda \right){I}_i\left(0,\theta \right)}{2\varGamma \left(1.5+\lambda \right){I}_i\left(\lambda, \theta \right)}\right]}^{-1/\lambda },\hfill \\ {}\hfill {F}_i\left(\theta \right)=\left\{\begin{array}{cc}\hfill {F}_1\left(\theta \right),\hfill & \hfill \alpha \le \theta \le \pi, \hfill \\ {}\hfill {F}_2\left(\theta \right),\hfill & \hfill 0\le \theta \le \alpha; \hfill \end{array}\right.\hfill \end{array}\\ {}{\varDelta}_8=p \sin 2\beta \sin 2p\left(\alpha -\theta \right)-2{ \sin}^2p\left(\alpha -\theta \right) \cos 2\theta +2 \sin p\left(\alpha +\theta \right) \sin p\left(\alpha -\theta \right),\end{array} $$
$$ \begin{array}{c}\hfill {F}_1\left(\uptheta \right)=\frac{{\left(2\uppi \right)}^{\uplambda}}{{4\uplambda \upvarphi}_2}\left\{\left(\uplambda +2\right){\upvarphi}_1 \sin \left(\uplambda +2\right)\left(\uppi \hbox{-} \uptheta \right)\hbox{-} {\uplambda \upvarphi}_1 \sin \uplambda \left(\uppi \hbox{-} \uptheta \right)\right.\hfill \\ {}\hfill \left.\hbox{-} {\uplambda \upvarphi}_2 \cos \left(\uplambda +2\right)\left(\uppi \hbox{-} \uptheta \right)+{\uplambda \upvarphi}_2 \cos \uplambda \left(\uppi \hbox{-} \uptheta \right)\right\},\hfill \\ {}\hfill {F}_2\left(\uptheta \right)=\frac{{\hbox{-} \left(2\uppi \right)}^{\uplambda}}{4{\uplambda \upvarphi \upvarphi}_2 \cos \uplambda \upalpha}\left\{\left[\left(\uplambda +2\right){\uppsi}_1{\upvarphi}_1+{\uplambda \uppsi}_2{\upvarphi}_2\left] \cos \uplambda \upalpha \sin \right(\uplambda +2\right)\right.\uptheta \left.+\uplambda \left({\uppsi}_3{\upvarphi}_1+{\uppsi}_4{\upvarphi}_2\right) \sin \uplambda \uptheta \right\}\hfill \\ {}\hfill \upvarphi =\hbox{-} \left[\left(\uplambda +1\right) \sin 2\upalpha + \sin 2\left(\uplambda +1\right)\upalpha \right],\hfill \\ {}\hfill {\upvarphi}_1=\upvarphi \left\{\left(\mathrm{e}\hbox{-} 1\right)\uplambda \sin \left(\uplambda +2\right)\left(\uppi \hbox{-} \upalpha \right)\hbox{-} \left[\left(\mathrm{e}\hbox{-} 1\right)\left(\uplambda +2\right)\hbox{-} \mathrm{e}\left(1+{\upkappa}_2\right)+\left(1+{\upkappa}_1\right)\right]\right.\left. \sin \uplambda \left(\uppi \hbox{-} \upalpha \right)\right\}\hfill \\ {}\hfill \hbox{-} 2\mathrm{e}\left(1+{\upkappa}_2\right)\uplambda \cos \uppi \uplambda \sin \left(\uplambda +1\right)\upalpha \sin \upalpha, \hfill \\ {}\hfill {\upvarphi}_2=\upvarphi \left\{\left[\left(\mathrm{e}\hbox{-} 1\right)\left(\uplambda +2\right)\hbox{-} \mathrm{e}\left(1+{\upkappa}_2\right)+\left(1+{\upkappa}_1\right)\left] \cos \uplambda \right(\uppi \hbox{-} \upalpha \right)\right.\hfill \\ {}\hfill \hbox{-} \left.\left(\mathrm{e}\hbox{-} 1\right)\left(\uplambda +2\right) \cos \left(\uplambda +2\right)\left(\uppi \hbox{-} \upalpha \right)\right\}\hbox{-} 2\mathrm{e}\left(1+{\upkappa}_2\right) \sin \uppi \uplambda \left(\uplambda \sin \left(\uplambda +1\right)\upalpha \sin \upalpha + \cos \uplambda \upalpha \right),\hfill \\ {}\hfill {\uppsi}_1=\hbox{-} 2\uplambda \cos \left(\uppi \uplambda \hbox{-} \upalpha \right) \sin \upalpha +2 \sin \left(\uplambda +2\right)\left(\uppi \hbox{-} \upalpha \right) \cos \uplambda \upalpha, \hfill \\ {}\hfill {\uppsi}_2=\hbox{-} 2\uplambda \sin \left(\uppi \uplambda \hbox{-} \upalpha \right) \sin \upalpha +2 \cos \uplambda \uppi \hbox{-} 2 \cos \left(\uplambda +2\right)\left(\uppi \hbox{-} \upalpha \right) \cos \uplambda \upalpha, \hfill \\ {}\hfill {\uppsi}_3=\hbox{-} \upvarphi \cos \uplambda \left(\uppi \hbox{-} \upalpha \right)\hbox{-} 2 \sin \uppi \uplambda \left(\uplambda \sin \left(\uplambda +1\right)\upalpha \sin \upalpha + \cos \uplambda \upalpha \right),\hfill \\ {}\hfill {\uppsi}_4=\hbox{-} \upvarphi \sin \uplambda \left(\uppi \hbox{-} \upalpha \right)+2\uplambda \cos \uppi \uplambda \sin \left(\uplambda +1\right)\upalpha \sin \upalpha, \hfill \end{array} $$
$$ \begin{array}{c}\hfill {I}_i\left(x,\theta \right)= \exp \left[\frac{x+1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{ \ln {G}_i\left( it,\theta \right)}{t^2+{\left(x+1\right)}^2}dt}\right],\hfill \\ {}\hfill {G}_i\left(p,\uptheta \right)=\frac{D_i\left(p,\uptheta \right) \cos p\uppi}{D_0(p) \sin p\uppi},\hfill \\ {}\hfill {D}_1\left(p,\uptheta \right)=4\left(e-1\right)\left({\varDelta}_1{\varDelta}_4-{\varDelta}_2{\varDelta}_5\right)\left[e\left(1+{\kappa}_2\right) \sin 2p\upalpha -\left(e-1\right){\varDelta}_6\right]\hfill \\ {}\hfill -4\left(e-1\right)\left(1+{\kappa}_1\right){\varDelta}_6\left[ \sin 2p\left(\uptheta -\upalpha \right){\varDelta}_1+{ \sin}^2p\left(\uptheta -\upalpha \right){\varDelta}_5\right]\hfill \\ {}\hfill +e\left(1+{\kappa}_1\right)\left(1+{\kappa}_2\right)\left[4 \sin \left(p-1\right)\uptheta \sin \left(p+1\right)\uptheta {\varDelta}_1+\left({\varDelta}_6-{\varDelta}_3\right){\varDelta}_5\right]-{\left(1+{\kappa}_1\right)}^2{\varDelta}_5{\varDelta}_6,\hfill \\ {}\hfill {D}_2\left(p,\uptheta \right)=e\left(1+{\kappa}_1\right)\left(1+{\kappa}_2\right)\left[{\varDelta}_3\left({\varDelta}_4-{\varDelta}_5\right)+4\left({\varDelta}_1-{\varDelta}_2-{\varDelta}_7\right) \sin \left(p-1\right)\uptheta \sin \left(p+1\right)\uptheta \right]\hfill \\ {}\hfill +\left[4{\varDelta}_2 \sin \left(p-1\right)\uptheta \sin \left(p+1\right)\uptheta -{\varDelta}_3{\varDelta}_4\right]\Big[{\left(1+{\upkappa}_1\right)}^2+4\left(e-1\right)\left(1+{\kappa}_1\right){ \sin}^2p\left(\uppi -\upalpha \right)\hfill \\ {}\hfill -4{\left(e-1\right)}^2{\varDelta}_7\Big]+4{e}^2{\left(1+{\upkappa}_2\right)}^2{\varDelta}_7 \sin \left(p-1\right)\uptheta \sin \left(p+1\right)\uptheta +4\left(e-1\right)e\left(1+{\kappa}_2\right){\varDelta}_7{\varDelta}_8,\hfill \\ {}\hfill {\varDelta}_1={p}^2{ \sin}^2\uptheta -{ \sin}^2p\left(\uppi -\uptheta \right),\kern0.24em {\varDelta}_2={p}^2{ \sin}^2\left(\uptheta -\upalpha \right)-{ \sin}^2p\left(\uptheta -\upalpha \right),\hfill \\ {}\hfill {\varDelta}_3=p \sin 2\uptheta + \sin 2p\uptheta, \kern0.24em {\varDelta}_4=p \sin 2\left(\uptheta -\upalpha \right)+ \sin 2p\left(\uptheta -\upalpha \right),\hfill \\ {}\hfill {\varDelta}_5=p \sin 2\uptheta - \sin 2p\left(\uppi -\uptheta \right),\kern0.24em {\varDelta}_6=p \sin 2\upalpha + \sin 2p\upalpha, \hfill \\ {}\hfill {\varDelta}_7={p}^2{ \sin}^2\upalpha -{ \sin}^2p\left(\uppi -\upalpha \right),\hfill \\ {}\hfill {\varDelta}_8=p \sin 2\upbeta \sin 2p\left(\upalpha -\uptheta \right)-2{ \sin}^2p\left(\upalpha -\uptheta \right) \cos 2\uptheta +2 \sin p\left(\upalpha +\uptheta \right) \sin p\left(\upalpha -\uptheta \right),\hfill \end{array} $$

and Γ(p) is the Euler gamma-function.

As a criterion for choosing the direction of the propagation of plastic strips, it is customary to use the condition of maximum of the rate of energy dissipation [7], which yields the following conditions:

$$ \begin{array}{c}\hfill \frac{d{W}_i}{dt}=\frac{8{\tau}_{is}^2\left(1-{\nu}_i^2\right)}{\pi {E}_i\left(2+\lambda \right)}{\left(\frac{\left|C\right|\sqrt{\pi}\varGamma \left(\lambda +1\right)}{2{\tau}_{is}\varGamma \left(1.5+\lambda \right)}\right)}^{-2/\lambda }{w}_i\left({\beta}_i\right)\frac{1}{C}\frac{dC}{dt}\kern0.36em \to \kern0.36em \max, \hfill \\ {}\hfill {w}_i\left(\theta \right)={\left[\frac{\left|{F}_i\left(\theta \right)\right|{I}_i{\left(0,\theta \right)}^{1+\lambda }}{I_i\left(\lambda, \theta \right)}\right]}^{-2/\lambda }.\hfill \end{array} $$
(3)

We also determine the crack-tip opening displacement caused by the formation of plastic strips, which is expressed via the jump of tangential displacements in the strip determined in the course of solution of the problem:

$$ {\delta}_i^{pl}=2\left\langle \kern0.1em {u}_r\left(0,{\beta}_i\right)\right\rangle \kern0.1em \sin {\beta}_i=-\frac{16\left(1-{\nu}_i^2\right){\tau}_{is}{l}_i}{\pi {E}_i{I}_i\left(0,{\beta}_i\right)\sqrt{G_i\left(0,{\beta}_i\right)}}\frac{\lambda }{1+\lambda } \sin {\beta}_i. $$
(4)

The solution of the problem is used to find the principal terms of the expansions of stresses in asymptotic series in the vicinity of the crack tip after the formation of the plastic zone. To do this, we use the roots λ i (− 1 < λ i  < 0) of the equation

$$ {D}_i\left(-x-1,{\beta}_i\right)=0. $$
(5)

The difference between the roots of Eqs. (2) and (5) reveals the transformation of the stress–strain state in the neighborhood of the corner point caused by the formation of the initial plastic zone. In Fig. 2, we present the results of numerical analysis of the dependences of the stress-singularity indices on the opening angle of the interface of the media 2α in the absence of the plastic zone (λ, solid curves) and after the formation of the plastic zone in the first (λ 1, dotted lines) and second (λ 2, dashed lines) materials. This analysis shows that, for some opening angles, stress concentration disappears as a result of the formation of the plastic strip (λ i  = 0). At the same time, for the other angles, we observe its weakening after the formation of the plastic zone (λ i  > λ), except the case of propagation of the plastic strip in the second material where E 1 / E 2 < 1 and α < 45°.

Fig. 2
figure 2

Dependences of the stress-singularity indices on the opening angle of the interface of the media 2α (ν 1 = ν 2 = 0.25): (1) E 1 / E 2 = 0.2, (2) E 1 / E 2 = 5.

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Parameters of the Plastic Process Zone on the Continuation of the Crack

In the presence of stress concentration in the neighborhood of the corner point, the plastic zone develops as the load increases not only due to the increase in the geometric sizes of plastic strips but also as a result of the formation of a new strip. Due to symmetry conditions, it is assumed that a secondary plastic strip is formed on the crack continuation (plastic process zone). Its length d i is much smaller that the length of initial plastic strips. Modeling this narrow plastic process zone by the segment of discontinuities of normal displacements, where the normal stresses are equal to a given constant of the second material σ 2, we find its length by analogy with [2]:

$$ {d}_i={\left(\frac{\tau_{is}}{\sigma_2{\tilde{X}}_i}\right)}^{-1/{\lambda}_i}{l}_i $$
(6)
$$ \begin{array}{c}\hfill {\tilde{X}}_i=\frac{{\tilde{I}}_i\left({\lambda}_i\right){I}_i\left(0,{\beta}_i\right)}{{\tilde{I}}_i(0){I}_i\left({\lambda}_i,{\beta}_i\right)\left|{S}_i\left(-1-{\lambda}_i,{\beta}_i\right)\right|},\kern0.48em {\tilde{I}}_i(x)= \exp \left[\frac{x+1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{ \ln {\tilde{G}}_i(it)}{t^2+{\left(x+1\right)}^2}dt}\right],\hfill \\ {}\hfill {\tilde{G}}_1(p)=-\frac{2{D}_3\left(p,{\beta}_1\right) \cos p\pi }{D_1\left(p,{\beta}_1\right) \sin p\pi },\kern0.48em {\tilde{G}}_2(p)=-\frac{2{D}_4\left(p,{\beta}_2\right) \cos p\pi }{D_2\left(p,{\beta}_2\right) \sin p\pi },\hfill \\ {}\hfill {D}_3\left(p,\theta \right)=2\left(1+{\kappa}_1\right)e\left(1+{\kappa}_2\right)\left({\varDelta}_1{\varDelta}_{24}-{\varDelta}_5{\varDelta}_{26}\right)-{\left(1+{\kappa}_1\right)}^2{\varDelta}_5{\varDelta}_{25}\hfill \\ {}\hfill +4\left(e-1\right)\left(1+{\kappa}_1\right)\left({\varDelta}_1 \sin 2p\left(\alpha -\theta \right)-{\varDelta}_5{ \sin}^2p\left(\alpha -\theta \right)\right){\varDelta}_{25}\hfill \\ {}\hfill +\left({\varDelta}_1{\varDelta}_4-{\varDelta}_2{\varDelta}_5\right)\left[{e}^2{\left(1+{\kappa}_2\right)}^2-4\left(e-1\right)e\left(1+{\kappa}_2\right){ \sin}^2p\alpha -4{\left(e-1\right)}^2{\varDelta}_{25}\right],\hfill \\ {}\hfill {D}_4\left(p,\theta \right)=2\left(1+{\kappa}_1\right)e\left(1+{\kappa}_2\right)\left[\left({\varDelta}_1-{\varDelta}_2-{\varDelta}_7\right){\varDelta}_3+\left({\varDelta}_4-{\varDelta}_5\right){\varDelta}_{23}\right]\hfill \\ {}\hfill +2\left[{\varDelta}_2{\varDelta}_3-{\varDelta}_4{\varDelta}_{23}\right]\left\{{\left(1+{\kappa}_1\right)}^2+4\left(e-1\right)\left(1+{\kappa}_1\right){ \sin}^2p\left(\pi -\alpha \right)-4{\left(e-1\right)}^2{\varDelta}_7\right\}\hfill \\ {}\hfill +2{e}^2{\left(1+{\kappa}_2\right)}^2{\varDelta}_3{\varDelta}_7-4\left(e-1\right)e\left(1+{\kappa}_2\right){\varDelta}_7\left(2{\varDelta}_3{ \sin}^2p\left(\alpha -\theta \right)-2{\varDelta}_{23} \sin 2p\left(\alpha -\theta \right)\right),\hfill \end{array} $$
$$ \begin{array}{c}\hfill {S}_i\left(p,\uptheta \right)=\frac{f_i\left(p,\uptheta \right)}{D_i^{\prime}\left(p,\uptheta \right)}\frac{\lambda }{\lambda_i\left({\lambda}_i-\lambda \right)},\kern0.24em {D}_i^{\prime}\left(p,\uptheta \right)=\frac{\partial {D}_i\left(p,\uptheta \right)}{\partial p},\hfill \\ {}\hfill {f}_1\left(p,\uptheta \right)=\left\{2\left(e-1\right)\right.\left[2{\varDelta}_1{\varDelta}_9+{\varDelta}_5\left(p \sin \upalpha {\varDelta}_{11}+ \sin p\upalpha {\varDelta}_{12}\right)+2{\varDelta}_{10}\left(p \sin \upalpha {\varDelta}_{13}+ \sin p\upalpha {\varDelta}_{14}\right)\right]\hfill \\ {}\hfill +\left(1+{\upkappa}_1\right)\left({\varDelta}_5{\varDelta}_{15}+2{\varDelta}_{10}{\varDelta}_{16}\right)-e\left(1+{\upkappa}_2\right)\left.\left({\varDelta}_5{\varDelta}_{17}+2{\varDelta}_{10}{\varDelta}_{18}+2{\varDelta}_1 \sin \left(p+1\right)\upbeta \right)\right\}\left(1+{\upkappa}_1\right),\hfill \\ {}\hfill {f}_2\left(p,\uptheta \right)=e\left(1+{\upkappa}_1\right)\left(1+{\upkappa}_2\right)\left[ \sin \left(p+1\right)\uptheta \left({\varDelta}_2-{\varDelta}_1+{\varDelta}_7\right)+p \sin \uptheta {\varDelta}_{19}- \sin p\uptheta {\varDelta}_{20}\right]\hfill \\ {}\hfill +2e\left(1+{\upkappa}_2\right)\left(e-1\right){\varDelta}_7{\varDelta}_{21}-{e}^2{\left(1+{\upkappa}_2\right)}^2{\varDelta}_7 \sin \left(p+1\right)\uptheta \hfill \\ {}\hfill -\left(p-1\right)\left[{\left(1+{\upkappa}_1\right)}^2-4{\left(e-1\right)}^2{\varDelta}_7+4\left(1+{\upkappa}_1\right)\left(e-1\right){ \sin}^2p\left(\uppi -\upalpha \right)\right]{\varDelta}_{22},\hfill \\ {}\hfill {\varDelta}_9=p \sin \upalpha \cos \left[\left(p+1\right)\left(\upalpha -\uptheta \right)+p\upalpha \right]+ \sin p\upalpha \cos \left[p\left(\upalpha -\uptheta \right)-\uptheta \right],\hfill \end{array} $$
$$ \begin{array}{c}\hfill {\varDelta}_{10}=p{ \sin}^2\uptheta -{ \sin}^2p\left(\pi -\theta \right),\hfill \\ {}\hfill {\varDelta}_{11}=p \sin \left(\upalpha -\uptheta \right) \cos p\left(2\upalpha -\uptheta \right)- \sin p\left(\upalpha -\uptheta \right) \cos \left(p\upalpha +\upalpha -\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{12}=p \sin \left(\upalpha -\uptheta \right) \cos \left[p\left(\upalpha -\uptheta \right)-\upalpha \right]- \sin p\left(\upalpha -\uptheta \right) \cos \uptheta, \hfill \\ {}\hfill {\varDelta}_{13}=p \sin \left(\upalpha -\uptheta \right) \sin p\left(2\upalpha -\uptheta \right)+ \sin p\left(\upalpha -\uptheta \right) \sin \left(p\upalpha +\upalpha -\theta \right),\hfill \\ {}\hfill {\varDelta}_{14}=p \sin \left(\upalpha -\uptheta \right) \sin \left[p\left(\upalpha -\uptheta \right)-\upalpha \right]- \sin p\left(\upalpha -\uptheta \right) \sin \uptheta, \hfill \\ {}\hfill {\varDelta}_{15}=p \sin \upalpha \sin \left(p\uptheta +\upalpha -\uptheta \right)- \sin p\upalpha \sin \left(p\left(\upalpha -\uptheta \right)+\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{16}=p \sin \upalpha \cos \left(p\uptheta +\upalpha -\uptheta \right)+ \sin p\upalpha \cos \left(p\left(\upalpha -\uptheta \right)+\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{17}=p \sin \left(\upalpha -\uptheta \right) \sin \left(p\uptheta +\upalpha \right)- \sin p\left(\upalpha -\uptheta \right) \sin \left(p\upalpha +\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{18}=p \sin \left(\upalpha -\uptheta \right) \cos \left(p\uptheta +\upalpha \right)+ \sin p\left(\upalpha -\uptheta \right) \cos \left(p\upalpha +\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{19}=p \sin \upalpha \sin \left(p\uptheta +2\uptheta -\upalpha \right)- \sin p\left(\uppi -\upalpha \right) \sin p\left(\uppi +\upalpha -\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{20}=p \sin \upalpha \sin \left(\upalpha -\uptheta \right)- \sin p\left(\uppi -\upalpha \right) \sin \left[p\left(\uppi +\upalpha -2\uptheta \right)-\uptheta \right],\hfill \\ {}\hfill {\varDelta}_{21}=\left(p-1\right) \sin \uptheta \cos p\left(2\upalpha -\uptheta \right)+ \sin \left(p+1\right)\uptheta, \hfill \\ {}\hfill {\varDelta}_{22}=p \sin \upalpha \sin \left(\upalpha -\uptheta \right) \sin p\uptheta - \sin p\upalpha \sin p\left(\upalpha -\uptheta \right) \sin \uptheta, \hfill \\ {}\hfill {\varDelta}_{23}={p}^2{ \sin}^2\uptheta -{ \sin}^2p\uptheta, \kern0.24em {\varDelta}_{24}=p \sin \upalpha \cos \left(\upalpha -2\uptheta \right)+ \sin p\upalpha \cos p\left(\upalpha -2\uptheta \right),\hfill \\ {}\hfill {\varDelta}_{25}={p}^2{ \sin}^2\upalpha -{ \sin}^2p\upalpha, \kern0.24em {\varDelta}_{26}={p}^2 \sin \upalpha \cos \uptheta \sin \left(\uptheta -\upalpha \right)- \sin p\upalpha \cos p\uptheta \sin p\left(\uptheta -\upalpha \right).\hfill \end{array} $$

The solution of the problem enables us to determine the additional contribution to the crack-tip opening displacement caused by the formation of the plastic process zone modeled by the jump of normal displacements given by the formula

$$ \begin{array}{c}\hfill {\delta}_i^{pf}=-{u}_{\theta}\left(0,\pi \right)+{u}_{\theta}\left(0,-\pi \right)=-\frac{64\pi {\sigma}_2e\left(1+{\kappa}_2\right)\left(1-{\nu}_1^2\right){J}_i\sqrt{{\tilde{G}}_i(0)}{d}_i{\lambda}_i}{E_1{\tilde{J}}_i{\tilde{I}}_i(0)\left(1+{\lambda}_i\right)},\hfill \\ {}\hfill {J}_1={g}_1\left[e\left(1+{\kappa}_2\right){g}_2-\left(1+{\kappa}_1\right){g}_3\right],\kern0.24em {J}_2={g}_4\left[e\left(1+{\kappa}_2\right){g}_5-\left(1+{\kappa}_1\right){g}_2\right],\hfill \\ {}\hfill {g}_1=\left(\pi -\beta \right) \cos \beta - \sin \beta, \kern0.24em {g}_2= \cos \alpha \sin \left(\alpha -\beta \right)+\left(\alpha -\beta \right) \cos \beta, \hfill \\ {}\hfill {g}_3= \sin \alpha \cos \left(\alpha -\beta \right)+\alpha \cos \beta, \kern0.24em {g}_4= \sin \beta +\beta \cos \beta, \hfill \\ {}\hfill {g}_5= \sin \alpha \cos \left(\alpha -\beta \right)-\left(\pi -\alpha \right) \cos \beta; \hfill \\ {}\hfill {\tilde{J}}_1=\underset{p\to 0}{ \lim }{p}^{-3}{D}_3(p),\kern0.72em {\tilde{J}}_2=\underset{p\to 0}{ \lim }{p}^{-3}{D}_4(p).\hfill \end{array} $$
(7)

The total crack opening displacement δ i is determined as the sum of opening displacements caused by the formation of the lateral plastic zone (4) and the plastic process zone (7): δ i  = δ pl i  + δ pf i . Equating the crack opening displacement to its critical value, we can find the ultimate load corresponding to the onset of crack propagation.

After the formation of the plastic process zone, the stress field near the corner point is characterized by the roots of the equations D 3(−1 − x, β 1) = 0 or D 4(−1 − x, β 2) = 0 specifying the stress-singularity index for − 1 < Rex < 0. The numerical results show that these equations have no roots in the strip − 1 < Rex < 0 in broad ranges of the elastic parameters of the joined materials and opening angles of the interface. Hence, the stress concentration disappears at the corner point.

Analysis of the Accumulated Results

As follows from relations (1) and (6), plastic strips nonlinearly increase with the external load appearing in the expressions for their lengths via the factor C. Moreover, the lower the yield limit of the material σ i , the greater the length of the lateral plastic zone and the smaller the constant σ 2, the greater the length of plastic process zone. Formula (6) also implies that the sizes of plastic strips synchronously increase with the load.

If only one material in the analyzed piecewise homogeneous body is elastoplastic, then the lateral plastic strips propagate in it in the directions determined from the corresponding condition (3). However, if both materials of the joint are elastoplastic, then it is necessary to additionally compare the maximal rates of the dissipation of energy in the strips in each material. The ratios of these rates and the lengths of plastic strips are as follows:

$$ \begin{array}{c}\hfill \frac{d{W}_1/dt}{d{W}_2/dt}={\left(\frac{\tau_{2s}}{\tau_{1s}}\right)}^{-2\left(\lambda +1\right)/\lambda }Z,\kern0.36em \frac{l_1}{l_2}={\left(\frac{\tau_{2s}}{\tau_{1s}}\right)}^{-1/\lambda }X,\hfill \\ {}\hfill Z=\frac{1-\nu {}_1}{e\left(1-{\nu}_2\right)}{\left[\frac{\left|{F}_1\left({\beta}_1\right)\right|{I}_1{\left(0,{\beta}_1\right)}^{1+\lambda }{I}_2\left(\lambda, {\beta}_2\right)}{\left|{F}_2\left({\beta}_2\right)\right|{I}_2{\left(0,{\beta}_2\right)}^{1+\lambda }{I}_1\left(\lambda, {\beta}_1\right)}\right]}^{-2/\lambda };\hfill \\ {}\hfill X={\left[\frac{\left|{F}_1\left({\beta}_1\right)\right|{I}_1\left(0,{\beta}_1\right){I}_2\left(\lambda, {\beta}_2\right)}{\left|{F}_2\left({\beta}_2\right)\right|{I}_2\left(0,{\beta}_2\right){I}_1\left(\lambda, {\beta}_1\right)}\right]}^{-1/\lambda }.\hfill \end{array} $$
(8)

In Table 1, we present the results of numerical analysis of the dependences of the parameters of plastic zones on the opening angle of the interface of the media 2α for the ratios of Young’s moduli of the materials E 1 / E 2 = 0.2 and E 1 / E 2 = 5, equal Poisson’s ratios (ν 1 = ν 2 = 0.25), and equal yield limits of the joined materials (τ 1s  = τ 2s ).

Table 1 Parameters of the Process Zone within the Framework of the “Trident” Model
Full size table

It is easy to see that if E 1 < E 2 and the opening angles of the interface of two media are 2α ≤ 180°, then dW 1/dt > dW 2/dt (Z > 1) . According to the accepted energy criterion, this leads to the formation of two symmetric lateral plastic strips in the first material. At the same time, if 180° < 2α < 360°, then dW 1 / dt < dW 2 / dt and, hence, the formation of two lateral strips in the second material is predominant. If E 1 > E 2, and the opening angles 2α ≤ 40°, then dW 1 / dt > dW 2 / dt, i.e., two symmetric plastic strips are formed in the first material. For opening angles from 2α ≈ 60° up to a certain value 2α 1 ≃ 120°, which can be found from the condition dW 1 / dt = dW 2 / dt, the plastic strip propagates along the interface of the media either in the first or in the second material (β 1 = β 2 = α) depending on their yield limits. For the angles 2α > 120° and E 1 > E 2, we have dW 1 / dt < dW 2 / dt and, hence, we observe the formation of two symmetric lateral strips in the second material.

For τ 1s  ≠ τ 2s , the direction of propagation of the strips according to condition (8) also depends on the ratio of the yield limits of the materials. For the joint of identical materials, the slope of the plastic strips is equal to 75.8°, which is close to the value ≈ 72° obtained in [8]. Their difference is explained by the use of the condition of maximum of the strip lengths in [8]. The comparison of the ratios of strip lengths (X) and the rates of energy dissipation (Z) in different materials shows that the conditions of their maximum for some parameters of the joint (E 1/E 2 < 1, α ≃ 90° and E 1/E 2 > 1 α ≃ 70° − 90° ) may lead to different conclusions concerning the orientation of the strips.

The sizes of the plastic process zone are presented in Table 1 in the form of the factors \( {\tilde{X}}_i \) specifying the ratio of the lengths of the plastic process zone and the lateral plastic zone in the corresponding material (the missing values correspond to vanishing of the stress concentration in the stage of formation of the initial plastic strips). Note that the condition d i  < < l i accepted in the formulation of the problem requires the validity of the inequality \( {\tilde{X}}_i>>{\tau}_{is}\kern0.1em /\kern0.1em {\sigma}_2 \), which can be violated for small values of \( {\tilde{X}}_i \).

CONCLUSIONS

Under the conditions of plane deformation, within the framework of the “trident” model, we perform the numerical analysis of a small-scale plastic zone at the end of a mode I crack originating from the corner point of interface of two different elastoplastic materials. We reveal the possibility of prediction of the direction of propagation of lateral plastic strips. It is shown that the formation of the plastic process zone on the continuation of the crack after propagation of the initial lateral plastic strips removes stress concentration at the crack tip in a broad range of parameters of the investigated joint. Finally, we determine the crack-tip opening displacements, which enables one to find the ultimate loads.