The DDT is a method to analyze a medium containing multiple cracks. In this technique, the dislocations are distributed in the locations of the cracks and the stress field, and fracture parameters such as stress intensity factors (SIFs) are determined for the cracked medium [17]. Stress fields of dislocations usually contain singularities that result in singular integral equations in the DDT. This is usually the case in classical elasticity. In the framework of the gradient elasticity, the singularity of the stress (and hyperstress) components is different from classical elasticity. For instance, Gourgiotis and Georgiadis [22] have given the solution for a screw dislocation in the couple stress theory and have solved governing integral equations in DDT with cubic singularity.

As mentioned earlier, a great difference in the generalized elasticity is the non-singular stress fields, which will alleviate the solution of the resulting integral equations. In this section, the DDT is modified for the class of graded materials in the framework of strain gradient elasticity. The stress fields presented in the previous section [Eqs. (4), (7), and (10)] are used in this section in DDT.

The graded plane, with a varying shear modulus *μ*=*μ*_{0} exp(2*γy*), is assumed to contain a screw dislocation situated at a point with coordinates (*η*, *ζ*). The line of the dislocation is assumed to be parallel to the *x*-axis:

$$\mu \mathrm{(}y\mathrm{)}={\mu}_{0}\mathrm{exp}\mathrm{(}2\gamma y\mathrm{)}.\text{\hspace{1em}(11)}$$(11)

The stress components at a point with coordinates (*x*, *y*) due to the dislocation at (*η*, *ζ*) may be obtained by making the conversions *x*→(*x-η*), *y*→*y-ζ* in Eqs. (4), (7), and (10).

The movable orthogonal coordinate system *n, t* is chosen such that the origin may move on the crack while the *t*-axis remains tangent to the crack surface. The antiplane traction on the surface of the *k*th crack, in terms of stress components in the Cartesian coordinates *x, y*, becomes

$$\begin{array}{l}{\sigma}_{nz}^{i}\mathrm{(}{x}_{k},\text{\hspace{0.17em}}{y}_{k}\mathrm{)}={\sigma}_{yz}^{i}\mathrm{cos}\mathrm{(}{\theta}_{k}\mathrm{)}\text{-}{\sigma}_{xz}^{1}\mathrm{sin}\mathrm{(}{\theta}_{k}\mathrm{)},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ k\in \{1,2,\mathrm{...},N\},\text{\hspace{0.17em}}i=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}2,\end{array}\text{\hspace{1em}(12)}$$(12)

where $${\theta}_{k}\mathrm{(}s\mathrm{)}={\mathrm{tan}}^{\text{-}1}\mathrm{(}{{\beta}^{\prime}}_{k}\mathrm{(}s\mathrm{)}/{{\alpha}^{\prime}}_{k}\mathrm{(}s\mathrm{)}\mathrm{)}$$ is the angle between the *x*- and *t*-axes and prime denotes differentiation with respect to the argument. Additionally, *i*=0, 1, 2 denote the components in the classical, first strain gradient, and second strain gradient elasticity, respectively. A crack is constructed by a continuous distribution of dislocations. Suppose dislocations with unknown density *B*_{z} are distributed on the infinitesimal segment $$\sqrt{{\mathrm{(}{{\alpha}^{\prime}}_{j}\mathrm{)}}^{2}+{\mathrm{(}{{\beta}^{\prime}}_{j}\mathrm{)}}^{2}}dt$$ at the surface of the *j*th crack where -1≤*t*≤1 and prime denotes differentiation with respect to the relevant argument. The antiplane traction on the surface of the *k*th crack due to the presence of the above-mentioned distribution of dislocations on all *N* cracks yields integral equations for the dislocation densities:

$$\begin{array}{l}{\sigma}_{nz}^{i}\mathrm{(}{\alpha}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\beta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)}=\\ \text{}{\displaystyle \sum _{j=1}^{N}{\displaystyle {\int}_{-1}^{1}{K}_{i}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}j\mathrm{)}\sqrt{{[{{\alpha}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}+{[{{\beta}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}}{B}_{j}^{i}\mathrm{(}t\mathrm{)}dt}},\\ \text{}\text{}\text{-}1\le s\le 1,\text{\hspace{0.17em}}k\in \{1,2,\mathrm{...},N\},\text{\hspace{0.17em}}i=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}2,\end{array}\text{\hspace{1em}(13)}$$(13)

while the kernels of the integral are

$$\begin{array}{c}{K}_{0}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}j\mathrm{)}\equiv {K}_{0}\mathrm{(}{\alpha}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\beta}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\alpha}_{j}\mathrm{(}t\mathrm{)},\text{\hspace{0.17em}}{\beta}_{j}\mathrm{(}t\mathrm{)}\mathrm{)}\\ =\frac{{\mu}_{0}\gamma {e}^{\gamma {y}_{kj}}}{2\pi}\left[\frac{{x}_{kj}}{r}{K}_{1}\mathrm{(}\gamma {r}_{kj}\mathrm{)}\right]\mathrm{cos}\mathrm{(}{\theta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)}\\ +\frac{{\mu}_{0}\gamma {e}^{\gamma {y}_{kj}}}{2\pi}\left[\mathrm{(}\frac{{y}_{kj}}{r}{K}_{1}\mathrm{(}\gamma {r}_{kj}\mathrm{)}\text{-}{K}_{0}\mathrm{(}\gamma {r}_{kj}\mathrm{)}\mathrm{)}\right]\mathrm{sin}\mathrm{(}{\theta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)},\end{array}$$

$$\begin{array}{c}{K}_{1}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}j\mathrm{)}\equiv {K}_{1}\mathrm{(}{\alpha}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\beta}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\alpha}_{j}\mathrm{(}t\mathrm{)},\text{\hspace{0.17em}}{\beta}_{j}\mathrm{(}t\mathrm{)}\mathrm{)}={K}_{0}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}j\mathrm{)}\text{-}\frac{{\mu}_{0}{e}^{\gamma {y}_{kj}}}{2\pi}\left[\frac{{x}_{kj}}{{r}_{kj}}\frac{\sqrt{1+{\gamma}^{2}{l}^{2}}}{l}{K}_{1}\mathrm{(}\frac{\sqrt{1+{\gamma}^{2}{l}^{2}}}{l}{r}_{kj}\mathrm{)}\right]\mathrm{cos}\mathrm{(}{\theta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)}\\ +\frac{{\mu}_{0}{e}^{\gamma {y}_{kj}}}{2\pi}\left[\text{-}\frac{{y}_{kj}}{{r}_{kj}}\frac{\sqrt{1+{\gamma}^{2}{l}^{2}}}{l}{K}_{1}\mathrm{(}\frac{\sqrt{1+{\gamma}^{2}{l}^{2}}}{l}{r}_{kj}\mathrm{)}+\gamma {K}_{0}\mathrm{(}\frac{\sqrt{1+{\gamma}^{2}{l}^{2}}}{l}{r}_{kj}\mathrm{)}\right]\mathrm{sin}\mathrm{(}{\theta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)},\end{array}$$

$$\begin{array}{c}{K}_{2}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}j\mathrm{)}\equiv {K}_{2}\mathrm{(}{\alpha}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\beta}_{k}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\alpha}_{j}\mathrm{(}t\mathrm{)},\text{\hspace{0.17em}}{\beta}_{j}\mathrm{(}t\mathrm{)}\mathrm{)}\\ ={K}_{0}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}j\mathrm{)}+\left\{\frac{{\mu}_{0}{e}^{\gamma {y}_{kj}}}{2\pi}\frac{1}{{c}_{1}^{2}\text{-}{c}_{2}^{2}}\frac{{x}_{kj}}{r}[-{c}_{1}^{2}{\kappa}_{1}{K}_{1}\mathrm{(}{\kappa}_{1}{r}_{kj}\mathrm{)}+{c}_{2}^{2}{\kappa}_{2}{K}_{1}\mathrm{(}{\kappa}_{2}{r}_{kj}\mathrm{)}]\right\}\mathrm{cos}\mathrm{(}{\theta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)}\\ \text{-}\left\{\frac{{\mu}_{0}{e}^{\gamma {y}_{kj}}}{2\pi}\frac{1}{{c}_{1}^{2}\text{-}{c}_{2}^{2}}\left[\text{-}{c}_{1}^{2}\gamma {K}_{0}\mathrm{(}{\kappa}_{1}{r}_{kj}\mathrm{)}+{c}_{2}^{2}\gamma {K}_{0}\mathrm{(}{\kappa}_{2}{r}_{kj}\mathrm{)}+{c}_{1}^{2}\frac{{\kappa}_{1}{y}_{kj}}{r}{K}_{1}\mathrm{(}{\kappa}_{1}{r}_{kj}\mathrm{)}\text{-}{c}_{2}^{2}\frac{{\kappa}_{2}{y}_{kj}}{r}{K}_{1}\mathrm{(}{\kappa}_{2}{r}_{kj}\mathrm{)}\right]\right\}\mathrm{sin}\mathrm{(}{\theta}_{k}\mathrm{(}s\mathrm{)}\mathrm{)},\end{array}\text{\hspace{1em}(14)}$$(14)

where *x*_{kj}=*α*_{k}-*α*_{j}, *y*_{kj}=*β*_{k}-*β*_{j}, and $${r}_{kj}=\sqrt{{x}_{kj}^{2}+{y}_{kj}^{2}.}$$ Equation (13) consists of Cauchy singular integral equations for *i*=0 (classical elasticity) and non-singular integral equations for *i*=1, 2 (strain gradient elasticity).

Consider a plane with the following loadings:

$${\sigma}_{yz}^{\infty}={\sigma}_{0},\text{\hspace{0.17em}}{\sigma}_{xz}^{\infty}=0.\text{\hspace{1em}(15)}$$(15)

The graded plane in the absence of cracks under this loading is in a state of pure antiplane shear with the following stress field:

$${\sigma}_{yz}\mathrm{(}x,\text{\hspace{0.17em}}y\mathrm{)}={\sigma}_{0}.\text{\hspace{1em}(16)}$$(16)

Substituting the above stress field in Eq. (12), the traction in the location of cracks in the plane in the absence of cracks yields

$${\sigma}_{nz}\mathrm{(}{x}_{k},\text{\hspace{0.17em}}{y}_{k}\mathrm{)}={\sigma}_{0}\mathrm{cos}\mathrm{(}{\theta}_{k}\mathrm{)}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}k\in \{1,2,\mathrm{...},N\}.\text{\hspace{1em}(17)}$$(17)

By virtue of Bueckner superposition principle, the traction in Eq. (17) with the opposite sign should be substituted in the left-hand side of Eq. (13).

Employing the definition of dislocation density function, the equation for the crack opening displacement across the *j*th crack becomes

$${w}_{j}^{\text{-}}\mathrm{(}s\mathrm{)}\text{-}{w}_{j}^{+}\mathrm{(}s\mathrm{)}={\displaystyle {\int}_{\text{-}1}^{s}\sqrt{{[{{\alpha}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}+{[{{\beta}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}}{B}_{j}^{i}\mathrm{(}t\mathrm{)}dt}.\text{\hspace{1em}(18)}$$(18)

Furthermore, the displacement field is single valued out of crack surfaces. Thus, the dislocation density for an embedded crack is subjected to the following closure requirement:

$${\int}_{\text{-}1}^{1}\sqrt{{[{{\alpha}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}+{[{{\beta}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}}{B}_{j}^{i}\mathrm{(}t\mathrm{)}dt}=0,\text{\hspace{1em}(19)$$(19)

where *i*=0, 1, 2, and *j*∈{1, 2, …, *N*} is the crack number. To obtain the dislocation density for cracks, the integral Eqs. (13) and (19) are to be solved simultaneously. In a cracked plane, the dislocation density obtained from Eqs. (13) and (19) can be used to determine the stress distribution in the plane. To evaluate the stress distribution on a curve such as *α*_{0}(*s*), *β*_{0}(*s*), these coordinates and the obtained densities from Eqs. (13) and (19) should be substituted into Eq. (13), which yields

$$\begin{array}{l}{\sigma}_{nz}^{i}\mathrm{(}{\alpha}_{0}\mathrm{(}s\mathrm{)},\text{\hspace{0.17em}}{\beta}_{0}\mathrm{(}s\mathrm{)}\mathrm{)}=\\ \text{\hspace{1em}}{\displaystyle \sum _{j=1}^{N}{\displaystyle {\int}_{\text{-}1}^{1}{K}_{i}\mathrm{(}s,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}j\mathrm{)}\sqrt{{[{{\alpha}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}+{[{{\beta}^{\prime}}_{j}\mathrm{(}t\mathrm{)}]}^{2}}{B}_{j}^{i}\mathrm{(}t\mathrm{)}dt}},\text{\hspace{0.17em}}\\ \text{}\text{-}1\le s\le 1,\text{\hspace{0.17em}}i=0,1,2\end{array}\text{\hspace{1em}(20)}$$(20)

which should be superposed with stress fields (16). The system of integral Eqs. (13) and (19) is an ill-posed problem. In the case of classical elasticity, this system of equations is Cauchy singular, whereas in the cases of gradient elasticity, it is non-singular. In the following section, proper methods are selected to solve the singular and non-singular equations.

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