# Matrix Method of Defect Analysis for Structures with Areas of Considerable Stiffness Differences
## 2 Materials and Methods
### Stiffness Matrix Formulation for Four-Node Element
This was obtained by adopting general integration limits in the form of \(\xi_{1}\) to \(\xi_{2}\) for the \(\xi\) axis and \(\eta_{1}\) to \(\eta_{2}\) for the \(\eta\) axis, and the values of the shape function at the beginning and the end of the sub-area for the \(\xi\) axis as \(\text{m}_{1}{}^{\prime}\), \(\text{m}_{2}{}^{\prime}\) and for the \(\eta\) axis as \(\text{m}_{1}{}^{\prime\prime}\), \(\text{m}_{2}{}^{\prime\prime}\). Deriving the stiffness matrix of a single component area defined in this way is quite complicated, as it requires the substitution of eight parameters. In order to simplify the final form of the stiffness matrix, the parameters \(\text{m}_{1}{}^{\prime}\), \(\text{m}_{2}{}^{\prime}\), \(\text{m}_{1}{}^{\prime\prime}\), \(\text{m}_{2}{}^{\prime\prime}\) are made dependent on the integration limits \(\xi_{1}\), \(\xi_{2}\), \(\eta_{1}\), \(\eta_{2}\), developing the extension of the integration range. The extension idea of the broken shape functions is illustrated in Figure 5b. Figure 5: Definition of multi-area two-dimensional element: (**a**) broken shape functions; (**b**) extension of the broken shape functions. The extension of the integration range changes the boundaries, and the final form of the shape functions again takes the form of linear functions. For example, for a two-dimensional element with nine component areas, where the sub-area \(\mathrm{K_{5}}\) has much lower stiffness than the others, extension of the integration range leads to a substitute element with a changed range of sub-areas, which is schematically shown in Figure 5b. The applied method significantly simplifies the final explicit form of the stiffness matrix of the component area and eliminates potential over-stiffening related to the use of broken functions. It is assumed that the structures analyzed using multi-area elements contain multi-coherent areas and are made of a material that is non-homogeneous. A non-homogeneous material at different points of the area considered has different properties (material characteristics), i.e., it contains various types of defects, inclusions etc. In the case of the material matrix of a two-dimensional multi-area element, the Young's modulus was divided into one assigned to the horizontal direction \(E_{1}\) and one assigned to the vertical direction \(E_{2}\). For the same purpose, \(G_{1}\) and \(G_{2}\) values were also used. This notation can be described as the identification of the shape functions' directions. The separation is due to the shape function distribution, separately on the horizontal and vertical directions of the multi-area element. In this case, the material matrix takes the following form:  
\[\mathbf{D_{e}}=\frac{E}{(1-v^{2})} \begin{bmatrix}E_{11}&E_{11}\cdot v&0&0\\ E_{22}\cdot v&E_{22}&0&0\\ 0&0&G_{1}&G_{2}\end{bmatrix} \text{ and }\ E_{11}=\frac{E_{1}}{1-v^{2}}\,\ \ E_{22}=\frac{E_{2}}{1-v^{2}}\, \tag{7}\]  
where \(E_{1}\)--[PERSON]'s modulus on the direction of the axis \(\xi\),  
\(E_{2}\)--[PERSON]'s modulus on the direction of the axis \(\eta\),  
\(G_{1}\)--shear modulus on the direction of the axis \(\xi\),  
\(G_{2}\)--shear modulus on the direction of the axis \(\eta\). For such assumptions, the stiffness matrix of a single component sub-area \(\mathbf{K_{e}^{k}}\) is determined from the relationship where two strain matrices are introduced \(\mathbf{B_{e}^{k}}\) and \(\mathbf{B}\mathbf{B_{e}^{k}}\). \[\mathbf{K_{e}^{k}}=t_{k}\cdot\iint\limits_{\Omega_{k}}\mathbf{B_{e}^{kT}} \cdot\mathbf{D_{e}^{k}}\cdot\mathbf{B}\mathbf{B_{e}^{k}}\cdot\mathrm{d}\Omega _{k}=t_{k}\cdot\int\limits_{\eta_{1}}^{\eta_{1+1}^{*}}\int\limits_{\xi_{1}^{*} }^{\eta_{1+1}^{*}}\mathbf{B_{e}^{k}}\cdot\mathrm{d}\mathbf{\xi}^{*}\mathrm{d} \eta^{*} \tag{8}\]  
The strain matrices were determined from the standard transformation. \[ \begin{array}{c}\mathbf{B_{e}^{k}}=\mathbf{L_{1}N}\\ \mathbf{B}\mathbf{B_{e}^{k}}=\mathbf{L_{2}N}\\ \end{array}  \tag{9}\]  
The differential operators take the following form:  
\[\mathbf{L_{1}}=