## 4 Computational and Mathematical Aspects
### Failure Criteria
#### 4.4.2 Cubic Tensor Polynomial  
A key point for a successful application of the proposed methodology is the use of a reliable failure criterion for the modeling of the masonry failure. To this end, a tensor polynomial has been used for the modeling of the masonry failure. Specifically, according to this criterion, the masonrymaterial is assumed to exhibit a distinct anisotropic nature and the failure surface can be described by the following equation:  
\[f(\sigma)=F_{i}\sigma_{i}+F_{\text{ij}}\sigma_{i}\sigma_{j}+F_{\text{ijk}}\sigma_ {i}\sigma_{j}\sigma_{k}+\cdots\left\{\begin{array}{c}\langle\ 1\ \text{no failure}\\ =1\ \text{failure}\\ \rangle\ 1\ \text{exceeded}\ \text{failure}\end{array}\right. \tag{22}\]  
where \(i\), \(j\), \(k\) = 1, 2,..., 6. \(F_{i}\), \(F_{ij}\) and \(F_{ijk}\) are (strength) tensors of the second, fourth and sixth rank, respectively.  
Based on the above equation, restricting the analysis to a plane stress state and assuming that a cubic formulation is a reasonably accurate representation of the failure surface and by taking into consideration the symmetry and anisotropic nature of the material [129; 171; 172], the masonry failure surface, known as the cubic tensor failure criterion, can be expressed by Equation (23).  
\[\left.\begin{array}{c}f\big{(}\sigma_{x},\sigma_{y},\tau\big{)}=2.27\sigma_{ x}+9.87\sigma_{y}+0.573\sigma_{x}^{2}+1.32\sigma_{y}^{2}+\,6.25\tau^{2}-0.30 \sigma_{x}\sigma_{y}+\\ \\ 0.009585\sigma_{x}^{2}\sigma_{y}+\,0.003135\sigma_{x}\sigma_{y}^{2}+0.28398 \sigma_{x}\tau^{2}+\,0.4689\sigma_{y}\tau^{2}=1\end{array}\right\} \tag{23}\]  
By using the dimensionless terms \(\left(\overline{\sigma}_{x}=\frac{\sigma_{x}}{f_{\text{av}}^{0^{\prime}}},\ \overline{\sigma}_{y}=\frac{\sigma_{y}}{f_{\text{av}}^{0^{\prime}}},\ \overline{\tau}=\frac{\tau}{f_{\text{av}}^{0^{\prime}}}\right)\) and taking into account that \(f_{\text{av}}^{90^{\prime}}=7.555\) Mpa, the above equation can be written in the form of:  
\[\left.\begin{array}{c}17.15\ \overline{\sigma}_{x}+74.57\ \overline{\sigma}_{y}+32.71\ \overline{\sigma}_{x}^{2}+75.34\ \overline{\sigma}_{y}^{2}+\,356.74\ \overline{\tau}^{2}+\\ -17.12\ \overline{\sigma}_{x}\overline{\sigma}_{y}+4.13\ \overline{\sigma}_{x}^{2}\ \overline{\sigma}_{y}+1.35\ \overline{\sigma}_{x}\ \overline{\sigma}_{y}^{2}+122.46\ \ \overline{\sigma}_{x}\overline{\tau}^{2}+\\ +202.20\ \overline{\sigma}_{y}\overline{\tau}^{2}=1\end{array}\right\} \tag{24}\]  
Figure 7: Kupfer and Gerstle isotropic failure criterion.  
Eliminating all third-order terms in Equation (24), a simplified failure criterion is derived [129; 171; 172]:  
\[\left.\begin{array}{l}17.15\ \overline{\sigma}_{x}+74.57\ \overline{\sigma}_{y}+32.71\ \overline{\sigma}_{x}^{2}+75.34\ \overline{\sigma}_{y}^{2}+356.74\ \overline{\tau}^{2}\\ -25.91\ \overline{\sigma}_{x}\overline{\sigma}_{y}=1\end{array}\right\} \tag{25}\]  
This simple form of the criterion has already been used [173; 174; 175; 176]. In Figures 9 and 10, graphical representations of cubic and simplified failure criterion in dimensionless principal and normal stress terms are depicted. Detailed description about the cubic tensor polynomial masonry failure criterion can be found in the works of [PERSON] [129] and [PERSON] [171; 172].  
Figure 8: [PERSON] modified isotropic failure criterion.  
Figure 9: Dimensionless failure surface of masonry material using the cubic tensor polynomial failure criterion: (**a**) non-dimensionless normal stress terms; and (**b**) non-dimensionless principal stress terms.