6. Determination of the SSS of samples loaded with concentrated forces at the boundary (compression test)

The methods of studied crack fracture resistance based on sample compression in experimental practice are widely used.

The direct application of the abovementioned variant of the method of integral equations for the case when the concentrated forces act on the boundary of the domain is associated with significant errors, because unknown functions in the vicinity of the points of application of forces have a singularity. Due to this, it is necessary to separate a singular part in the solution for a more precise solution of this type of task.

## 6.1 Determining a singular part of the solution of this problem

Let us consider a point of the plate boundary z0, in which the concentrated force (X, Y) is applied. A singular part of the solution (of Lekhnitskii potentials) will be the same as in the half-plane, whose boundary is tangent to the plate at the point of action of the concentrated force. Let us mark this angle through φ and potentials for the half-plane through Φ0ð Þ z<sup>1</sup> , Ψ0ð Þ z<sup>2</sup> .

Let us consider at first a half-plane y<0, which is loaded with force (X, Y) at an arbitrary point x<sup>0</sup> on the boundary. The Lekhnitskii potentials for this half-plane will be [17]

$$\Phi\_0 = \frac{A'}{z\_1 - \varkappa\_0}, \Psi\_0 = \frac{B'}{z\_2 - \varkappa\_0}, 0$$

where

$$A' = -\frac{X + s\_2 Y}{2\pi i (s\_1 - s\_2)},\\ B' = \frac{X + s\_1 Y}{2\pi i (s\_1 - s\_2)}\ .$$

It can be shown [17] that the half-plane also corresponds to the Φ0, Ψ<sup>0</sup> potentials, whose boundary passes through the point ð Þ x0; 0 and is inclined at an arbitrary angle under the action of the same force. Let us consider a semicircle with the center at point ð Þ x0; 0 of radius ρ0, which belongs to the half-plane. It is easy to show that the principal vector of all forces applied to the arc of the semicircle is equal to ð Þ �X; �Y . This proves that the case of loading by the concentrated force of the halfplane corresponds to the potentials Φ0, Ψ0.

Let us now consider a bounded plate occupying the domain D. The self-balanced concentrated forces Xj; Yj � �, jð Þ <sup>¼</sup> <sup>1</sup>; …; <sup>J</sup> : are applied to the boundary of this domain at points zj ¼ xj þ iyj . Let us represent the complex potentials in the form

$$\Phi(\mathbf{z}\_1) = \Phi\_0(\mathbf{z}\_1) + \Phi\_\Delta(\mathbf{z}\_1),\\\Psi(\mathbf{z}\_1) = \Psi\_0(\mathbf{z}\_1) + \Psi\_\Delta(\mathbf{z}\_1),\\\tag{16}$$

$$\Phi\_0(\mathbf{z}\_1) = \sum\_{j=1}^J \frac{A\_j'}{\mathbf{z}\_1 - \mathbf{z}\_{1j}},\\\Psi\_0(\mathbf{z}\_2) = \sum\_{j=1}^J \frac{B\_j'}{\mathbf{z}\_2 - \mathbf{z}\_{2j}},$$

where z1<sup>j</sup> ¼ xj þ s1yj , z2<sup>j</sup> ¼ xj þ s2yj . Here the coefficients A<sup>0</sup> j , B<sup>0</sup> <sup>j</sup> are determined based on expressions for A<sup>0</sup> , B<sup>0</sup> by the substitution of X and Y on Xj and Yj, respectively. By substituting formulas (16) into boundary conditions, we obtain the boundary problem for obtaining the introduced complex potentials at ð Þ x; y ∈ L:

$$\begin{aligned} (1+is\_1)z\_1'\Phi\_\Delta(z\_1) + (1+i\overline{s\_1})\overline{z\_1'}\overline{\Phi\_\Delta(z\_1)} +\\ + (1+is\_2)z\_2'\Psi\_\Delta(z\_2) + (1+i\overline{s\_2})\overline{z\_2'}\Psi\_\Delta(z\_2) = -i(X+iY)\_{0^\circ} \end{aligned} \tag{17}$$

where L is the boundary of domain D:

$$i(X+iY)\_0 = (\mathbf{1}+i\mathbf{i}\_1)\mathbf{z}\_1'\Phi\_0(\mathbf{z}\_1) + (\mathbf{1}+i\mathbf{i}\_1')\overline{\mathbf{z}\_1'\Phi\_0(\mathbf{z}\_1)} + (\mathbf{1}+i\mathbf{i}\_2)\mathbf{z}\_2'\Psi\_0(\mathbf{z}\_2) + (\mathbf{1}+i\mathbf{i}\_2')\overline{\mathbf{z}\_2'\Psi\_0(\mathbf{z}\_2)}.$$

It is easy to show that the right-hand side of formula (17) is a continuous and limited function, and therefore the introduced complex potentials with an index Δ are continuous and limited in the vicinity of the points of application of forces. In this regard, the above-developed numerical algorithm based on BIEM can be used to determine these potentials.

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