2. The integral representations for anisotropic plates with holes and cracks

We consider a plate which is weakened with a system hole with boundaries L1, …, LJ (j = 1, …, J), and cracks are placed along curves Γ<sup>k</sup> ð Þ k ¼ 1; …;K . The L<sup>0</sup> is the outer boundary of plates. Assume (Figure 1) that a plate is loaded with concentrated forces (Xj, Yj), j = 1, ..., M acting at the points (aj, bj); tractions ð Þ XT; YT are applied to the crack edges, which are accepted the same on its opposite edges; and tractions ð Þ XL; YL are applied to the boundaries of the holes and plate.

#### 2.1 Governing equations

Let us start from the Lekhnitskii complex potentials Φð Þ z<sup>1</sup> , Ψð Þ z<sup>2</sup> , where zj ¼ x þ sjy and sj, j ¼ 1, 2 are roots with positive imaginary part of the characteristic equation ΔðÞ¼ s 0 [10]:

where

$$\mathbf{a}\mathbf{a}\mathbf{d} \tag{1} \\ \qquad \qquad \Delta(\mathfrak{s}) = a\_{11}\mathfrak{s}^{4} - 2a\_{16}\mathfrak{s}^{3} + (2a\_{12} + a\_{66})\mathfrak{s}^{2} - 2a\_{26}\mathfrak{s} + a\_{22} \tag{1}$$

αij are elastic compliances which are included in the Hooke's law [10]:

$$\mathfrak{e}\_{\mathbf{x}} = \mathfrak{a}\_{11}\mathfrak{e}\_{\mathbf{x}} + \mathfrak{a}\_{12}\mathfrak{e}\_{\mathbf{y}} + \mathfrak{a}\_{16}\mathfrak{tau}\_{\mathbf{xy}}\\\mathfrak{e}\_{\mathbf{y}} = \mathfrak{a}\_{12}\mathfrak{o}\_{\mathbf{x}} + \mathfrak{a}\_{22}\mathfrak{o}\_{\mathbf{y}} + \mathfrak{a}\_{26}\mathfrak{tau}\_{\mathbf{xy}}\\\chi\_{\mathbf{xy}} = \mathfrak{a}\_{16}\mathfrak{o}\_{\mathbf{x}} + \mathfrak{a}\_{26}\mathfrak{o}\_{\mathbf{y}} + \mathfrak{a}\_{66}\mathfrak{tau}\_{\mathbf{xy}}$$

where εx, εy, γxy are strains and σx, σy, τxy are stresses.

Consider an arbitrary path Γ, which belongs to the domain D occupied by the plate, and select a positive direction of traversal (Figure 2).

Then introduce in consideration the stress vectors q<sup>Γ</sup> ! at the plane tangent to the curve. The normal to it is located right relative to the selected direction of traversal. The projections ð Þ XΓ; Y<sup>Γ</sup> of stress vectors q<sup>Γ</sup> ! and derivatives of displacements ð Þ <sup>u</sup>; <sup>v</sup> with respect to an arc coordinate at the curve through Lekhnitskii complex potentials are determined by the formula [17]:

Figure 1. Scheme of the problem.

Determination of Stresses in Composite Plates with Holes and Cracks Based on Singular Integral… DOI: http://dx.doi.org/10.5772/intechopen.87718

Figure 2. qΓ ! is the stress vector at plane AB.

$$Y\_{\Gamma} = -2\text{Re}\left[\Phi(\mathbf{z}\_1)\mathbf{z}\_1' + \Psi(\mathbf{z}\_2)\mathbf{z}\_2'\right],\ \mathbf{X}\_{\Gamma} = 2\text{Re}\left[\mathbf{s}\_1\Phi(\mathbf{z}\_1)\mathbf{z}\_1' + \mathbf{s}\_2\Psi(\mathbf{z}\_2)\mathbf{z}\_2'\right],\tag{2}$$

$$u' = 2\text{Re}\left[p\_1\Phi(z\_1)z\_1' + p\_2\Psi(z\_2)z\_2'\right], \ v' = 2\text{Re}\left[q\_1\Phi(z\_1)z\_1' + q\_2\Psi(z\_2)z\_2'\right],\tag{3}$$

where u<sup>0</sup> ¼ du=ds, v<sup>0</sup> ¼ dv=ds and z<sup>0</sup> <sup>j</sup> ¼ dx=ds þ sjdy=ds, where ds is a differential of arc at Γ.

The stress vectors qΓð Þ¼ z X<sup>Γ</sup> þ iY<sup>Γ</sup> at path Γ are determined using the formulas (2) by the formula:

$$q\_{\Gamma} = (\mathfrak{s}\_1 - i)\mathfrak{z}\_1' \Phi(\mathfrak{z}\_1) + (\overline{\mathfrak{s}\_1} - i)\overline{\mathfrak{z}\_1'} \overline{\Phi(\mathfrak{z}\_1)} + (\mathfrak{s}\_2 - i)\mathfrak{z}\_2' \Psi(\mathfrak{z}\_2) + (\overline{\mathfrak{s}\_2} - i)\overline{\mathfrak{z}\_2'} \overline{\Psi(\mathfrak{z}\_2)}.\tag{4}$$

Assume that the vectors ð Þ X; Y and ð Þ u; v are known at path Γ. Then based on Eqs. (2) and (3) at Γ one has [12, 15]

$$\Phi(\mathbf{z}\_1) = \frac{-v' + \mathbf{s}\_1 u' + p\_1 X + q\_1 Y}{\Delta\_1 \mathbf{z}\_1'}, \quad \Psi(\mathbf{z}\_2) = \frac{-v' + \mathbf{s}\_2 u' + p\_2 X + q\_2 Y}{\Delta\_2 \mathbf{z}\_2'} \tag{5}$$

where Δ<sup>j</sup> ¼ Δ<sup>0</sup> sj � �, j <sup>¼</sup> <sup>1</sup>, 2.

#### 2.2 Integral equations for anisotropic bounded plate with holes and cracks

Let us write a general solution of the problem based on [12, 15] through the Lekhnitskii potentials in the form

$$\begin{aligned} \Phi(\mathbf{z}\_1) &= \int [\boldsymbol{u}' \Phi\_1(\mathbf{z}\_1, t\_1) + \boldsymbol{v}' \Phi\_2(\mathbf{z}\_1, t\_1)] ds \\ &+ \int [\boldsymbol{g}'\_1 \Phi\_1(\mathbf{z}\_1, t\_1) + \boldsymbol{g}'\_2 \Phi\_2(\mathbf{z}\_1, t\_1)] ds + \Phi\_S(\mathbf{z}\_1) + \Phi\_\Delta(\mathbf{z}\_1), \end{aligned} \tag{6}$$

$$\begin{aligned} \Psi(\mathbf{z}\_2) &= \int\_L [\boldsymbol{u}' \Psi\_1(\mathbf{z}\_2, t\_2) + \boldsymbol{v}' \Psi\_2(\mathbf{z}\_2, t\_2)] ds \\ &+ \int \left[ \boldsymbol{g}'\_1 \Psi\_1(\mathbf{z}\_2, t\_2) + \boldsymbol{g}'\_2 \Psi\_2(\mathbf{z}\_2, t\_2) \right] ds + \Psi\_S(\mathbf{z}\_2) + \Psi\_\Delta(\mathbf{z}\_2), \end{aligned}$$

where L ¼ L<sup>0</sup> þ L<sup>1</sup> þ … þ LJ, Γ ¼ Γ<sup>1</sup> þ Γ<sup>2</sup> þ … þ ΓK, s is an arc coordinate, and ΦΔð Þ z<sup>1</sup> and ΨΔð Þ z<sup>2</sup> are the known functions, which are determined by the following formulas:

$$\Phi\_{\Delta}(\mathbf{z}\_{1}) = \int\_{L} [\mathbf{X}\_{L}\Phi\_{3}(\mathbf{z}\_{1},t\_{1}) + \mathbf{Y}\_{L}\Phi\_{4}(\mathbf{z}\_{1},t\_{1})]d\mathbf{s},\\\Psi\_{\Delta}(\mathbf{z}\_{1}) = \int\_{L} [\mathbf{X}\_{L}\Psi\_{3}(\mathbf{z}\_{1},t\_{1}) + \mathbf{Y}\_{L}\Psi\_{4}(\mathbf{z}\_{1},t\_{1})]d\mathbf{s},\tag{7}$$

$$\Phi\_j = \frac{A\_j}{t\_1 - x\_1}, \Psi\_j = \frac{B\_j}{t\_2 - x\_2},\tag{8}$$

$$A\_1 = -\frac{\dot{i}s\_1}{2\pi\Delta\_1},\ A\_2 = \frac{\dot{i}}{2\pi\Delta\_1},\ A\_3 = -\frac{\dot{i}p\_1}{2\pi\Delta\_1},\ \ A\_4 = -\frac{\dot{i}q\_1}{2\pi\Delta\_1},$$

$$B\_1 = -\frac{\dot{i}s\_2}{2\pi\Delta\_2},\ \ B\_2 = \frac{\dot{i}}{2\pi\Delta\_2},\ B\_3 = -\frac{\dot{i}p\_2}{2\pi\Delta\_2},\ \ B\_4 = -\frac{\dot{i}q\_2}{2\pi\Delta\_2}.$$

Here, u<sup>0</sup> , v<sup>0</sup> are the values of the derivatives of the displacements with respect to the arc coordinate at the boundary of the plate and holes, g<sup>1</sup> ¼ u<sup>þ</sup> � u�, g<sup>2</sup> ¼ v<sup>þ</sup> � v� are the displacements discontinuity at the cracks, u�, v� are limit values of displacements in the approach to the section at the left and the right relative to the selected direction, and the potentials ΦS, Ψ<sup>S</sup> correspond to the concentrated forces and have the form [12]:

$$\Phi\_{\rm S}(\mathbf{z}\_{1}) = \frac{i}{2\pi\Delta\_{1}}\sum\_{j=1}^{M} \left(p\_{1}\mathbf{X}\_{j} + q\_{1}\mathbf{Y}\_{j}\right) \frac{\mathbf{1}}{\mathbf{z}\_{1} - \mathbf{z}\_{1j}},\\\Psi\_{\rm S}(\mathbf{z}\_{2}) = \frac{i}{2\pi\Delta\_{2}}\sum\_{j=1}^{M} \left(p\_{2}\mathbf{X}\_{j} + q\_{2}\mathbf{Y}\_{j}\right) \frac{\mathbf{1}}{\mathbf{z}\_{2} - \mathbf{z}\_{2j}},\tag{9}$$

where in zkj ¼ aj þ skbj, j = 1, 2, and k = 1, 2.

Note that when the boundary is traction free, then ΦΔ ¼ ΨΔ ¼ 0.

Let us substitute the potentials (6) into the formulas (4) for projections of stress vectors determined at the boundaries path L and Γ. Using Plemelj-Sokhotski formula, we obtain a system of integral equations [12, 15]:

$$\begin{split} \int\_{L} [u'(s)\mathbf{Q}\_1(\mathbf{Z},T) + v'(s)\mathbf{Q}\_2(\mathbf{Z},T)]ds \\ + \int\_{\Gamma} [\mathbf{g}'\_1(s)\mathbf{Q}\_1(\mathbf{Z},T) + \mathbf{g}'\_2(s)\mathbf{Q}\_2(\mathbf{Z},T)]ds = \mathbf{Q}(\mathbf{Z}), \mathbf{Z} \in L \cup \Gamma, \end{split} \tag{10}$$

where Qjð Þ Z; T are stress vectors qL at point Z with coordinates ð Þ x; y . L ∪ Γ, the stress vector is determined by the formula (4) accordingly through complex potentials Φjð Þ z1; t<sup>1</sup> , Ψjð Þ z2; t<sup>2</sup> , j ¼ 1, 2; T is a point with coordinates ð Þ ξ; η , which belongs to the contour L ∪ Γ; Q Zð Þ¼ QLð Þ� Z QSð Þ� Z QΔð Þ Z with Z ∈L and Q Zð Þ¼ QTð Þ� Z QSð Þ� Z QΔð Þ Z with Z ∈ Γ; QL ¼ XL þ iYL; and Qm ¼ Xm þ iYm , where Xm ¼ 2Re s1Φmð Þ z<sup>1</sup> z<sup>0</sup> <sup>1</sup> þ s2Ψmð Þ z<sup>2</sup> z<sup>0</sup> 2 � �, Ym ¼ �2Re <sup>Φ</sup>mð Þ <sup>z</sup><sup>1</sup> <sup>z</sup><sup>0</sup> <sup>1</sup> þ Ψmð Þ z<sup>2</sup> z<sup>0</sup> 2 � � and

$$m = \mathfrak{S}, \mathfrak{A}.$$

Using the results [12], we obtained that the unknown functions u<sup>0</sup> , v<sup>0</sup> at the boundary of each of the holes Lj, j ¼ 0, 1, …, J in representation (6) are defined up to a summand u~<sup>0</sup> ¼ �ωjdy=ds, ~v<sup>0</sup> ¼ ωjdx=ds, where ω<sup>j</sup> are arbitrary constants. At numerical solution of the problem, the constants ωj, j ¼ 0, …, J are to be necessarily fixed. In addition, to ensure the displacement continuity condition, it is necessary to impose the following conditions on unknown functions:

Determination of Stresses in Composite Plates with Holes and Cracks Based on Singular Integral… DOI: http://dx.doi.org/10.5772/intechopen.87718

$$\begin{aligned} \int\_{L\_j} u' ds &= 0, \int\_{L\_j} v' ds = 0, j = 0, \dots, J; \\ \int\_{\Gamma\_j} g'\_1 ds &= 0, \int\_{\Gamma\_j} g'\_2 ds = 0, j = 1, \dots, K \end{aligned} \tag{11}$$

Let us consider a problem-solving equation (10) for the case of one hole and a crack. Let us assume that the contour on which the crack is placed is described parametrically in the form x ¼ αΓð Þτ , y ¼ βΓð Þτ , � 1≤τ ≤ 1, and the equation of the boundary hole is described in the form x ¼ αLð Þθ , y ¼ βLð Þθ , 0≤θ <2π.

Let us assume the representation for the displacement discontinuity at the cracks:

$$\mathbf{g}'\_1 \mathbf{s}' = \frac{d\mathbf{g}\_1}{d\tau} = \frac{U\_\Gamma(\tau)}{\sqrt{1 - \tau^2}}, \quad \mathbf{g}'\_2 \mathbf{s}' = \frac{d\mathbf{g}\_2}{d\tau} = \frac{V\_\Gamma(\tau)}{\sqrt{1 - \tau^2}}.$$

Let us replace the integrals with Lobatto-type quadrature formulas [15], and the integrals at the boundaries of the holes replaced by the quadrature of a rectangle, which, for periodic functions, are Gauss quadrature-type formulas [12]. Then we obtain the system of equations:

$$\frac{1}{2}H\sum\_{k=1}^{N\_O} \left( q\_{\nu k}^{(1)} U\_k^L + q\_{\nu k}^{(2)} V\_k^L \right) + \sum\_{m=1}^{N\_\Gamma} \mathcal{C}\_m \left( q\_{\nu m}^{(1)} U\_m^\Gamma + q\_{\nu m}^{(2)} V\_m^\Gamma \right) = q\_\nu, \nu = \mathbf{1}, \dots, N\_O + N\_\Gamma - \mathbf{1}, \tag{12}$$

where q ð Þj <sup>ν</sup><sup>k</sup> <sup>¼</sup> Qj <sup>Z</sup>ν; <sup>T</sup><sup>L</sup> k � �, pð Þ<sup>j</sup> <sup>ν</sup><sup>m</sup> <sup>¼</sup> Qj <sup>Z</sup>ν; <sup>T</sup><sup>Γ</sup> m � �, q<sup>ν</sup> <sup>¼</sup> QLð Þ� <sup>Z</sup><sup>ν</sup> QSð Þ� <sup>Z</sup><sup>ν</sup> <sup>Q</sup>Δð Þ <sup>Z</sup><sup>ν</sup> , U<sup>L</sup> <sup>k</sup> <sup>¼</sup> <sup>u</sup><sup>0</sup> xL <sup>k</sup> ; y<sup>L</sup> k � �s 0 k, V<sup>L</sup> <sup>k</sup> <sup>¼</sup> <sup>v</sup><sup>0</sup> <sup>x</sup><sup>L</sup> <sup>k</sup> ; y<sup>L</sup> k � �s 0 <sup>k</sup>, U<sup>Γ</sup> <sup>m</sup> ¼ g<sup>0</sup> <sup>1</sup> x<sup>Γ</sup> <sup>m</sup>; y<sup>Γ</sup> m � �s 0 m, V<sup>Γ</sup> <sup>m</sup> ¼ g<sup>0</sup> <sup>2</sup> x<sup>Γ</sup> <sup>m</sup>; y<sup>Γ</sup> m � �s 0 m, xL <sup>k</sup> <sup>¼</sup> <sup>α</sup>Lð Þ <sup>θ</sup><sup>k</sup> , y<sup>L</sup> <sup>k</sup> <sup>¼</sup> <sup>β</sup>Lð Þ <sup>θ</sup><sup>k</sup> , <sup>θ</sup><sup>k</sup> <sup>¼</sup> Hk, <sup>~</sup>θ<sup>n</sup> <sup>¼</sup> <sup>θ</sup><sup>n</sup> � <sup>H</sup>=2, H <sup>¼</sup> <sup>2</sup>π=NO, xT <sup>m</sup> <sup>¼</sup> <sup>α</sup>Γð Þ <sup>τ</sup><sup>m</sup> , yT <sup>m</sup> ¼ βΓð Þ τ<sup>m</sup> , τ<sup>m</sup> ¼ � cosð Þ πNð Þ m � 1 , m ¼ 1, …, NΓ; Cm ¼ π<sup>N</sup> at <sup>m</sup> 6¼ 1 and <sup>m</sup> 6¼ <sup>N</sup>Γ; C<sup>1</sup> <sup>¼</sup> CN<sup>Γ</sup> <sup>¼</sup> <sup>0</sup>, <sup>5</sup>πN; <sup>π</sup><sup>N</sup> <sup>¼</sup> <sup>π</sup> <sup>N</sup>Γ�<sup>1</sup> ; T<sup>L</sup> <sup>k</sup> is a point with coordinates xL <sup>k</sup> ; y<sup>L</sup> � �, T<sup>Γ</sup> <sup>m</sup> is a point with coordinates x<sup>Γ</sup> <sup>m</sup>; y<sup>Γ</sup> � �, Z<sup>ν</sup> is a point with coordinates

k m x~L <sup>ν</sup> ; ~y<sup>L</sup> ν � � with 1≤ν≤ NL, and is a point with coordinates x~<sup>Γ</sup> <sup>ν</sup>�NL ; <sup>~</sup>y<sup>Γ</sup> ν�NL � � with NL <sup>&</sup>lt;ν<sup>≤</sup> NL <sup>þ</sup> <sup>N</sup><sup>Γ</sup> � 1, where <sup>x</sup>~<sup>L</sup> <sup>k</sup> <sup>¼</sup> <sup>α</sup><sup>L</sup> <sup>~</sup>θ<sup>k</sup> � �, ~y<sup>L</sup> <sup>k</sup> <sup>¼</sup> <sup>β</sup><sup>L</sup> <sup>~</sup>θ<sup>k</sup> � �, <sup>~</sup>θ<sup>n</sup> <sup>¼</sup> <sup>θ</sup><sup>n</sup> � <sup>H</sup>=2, <sup>x</sup><sup>~</sup> Γ m <sup>¼</sup> <sup>α</sup><sup>Γ</sup> <sup>~</sup>τmÞ, <sup>~</sup>y<sup>Γ</sup> <sup>m</sup> <sup>¼</sup> <sup>β</sup><sup>Γ</sup> <sup>~</sup>τmÞ, <sup>~</sup>τ<sup>m</sup> ¼ � cos½ � <sup>π</sup>Nð Þ <sup>m</sup> � <sup>0</sup>; <sup>5</sup> � � .

We obtain the additional equation of system (12) from condition (11)

$$\sum\_{k=1}^{N\_\Gamma} \mathbf{C}\_k \left( U\_k^\Gamma + iV\_k^\Gamma \right) = \mathbf{0}.\tag{13}$$

Analogously to [12], we should remove three equations with 1≤ν≤ NL and add the following three equations to the received incomplete system:

$$\sum\_{k=1}^{N} U\_k^L = 0, \sum\_{k=1}^{N} V\_k^L = 0, U\_m^L = 0, 1 \le m \le N \tag{14}$$

The first two equations follow from the displacements continuity conditions (9). The last equation is obtained when fixing an arbitrary constant ω.

The system of Eqs. (12)–(14) is generalized in the case of hole and crack system in the same way as it was done in [12].
