## 2 Methods
### Elemental Formulation
#### 2.2.2 Cracking Element  
On the other hand, for the cracking element, its unknown vector is defined as \(\mathbf{U}^{(e)}=\left[\mathbf{u}_{1}\cdots\mathbf{u}_{n},\zeta_{n},\zeta_{t} \right]^{T}\). Its \(B\) matrix is extended to  
\[\mathbf{B}^{(e)}=\left[\
abla\mathbf{N}^{(e)},\mathbf{B}_{\zeta}\right], \tag{9}\]  
where \(\mathbf{N}^{(e)}\) is the original shape function, and  
\[\mathbf{B}_{\zeta}^{(e)}=-\left(l_{c}^{(e)}\right)^{-1}\left[\begin{array}{ ccc}n_{x}^{(e)}\cdot n_{x}^{(e)}&n_{x}^{(e)}\cdot l_{x}^{(e)}\\ n_{y}^{(e)}\cdot n_{y}^{(e)}&n_{y}^{(e)}\cdot l_{y}^{(e)}\\ 2\,n_{x}^{(e)}\cdot n_{y}^{(e)}&n_{x}\cdot l_{y}^{(e)}+n_{y}^{(e)}\cdot l_{x}^ {(e)}\end{array}\right], \tag{10}\]  
where the element-dependent parameter \(l_{c}^{(e)}\) is obtained as \(l_{c}^{(e)}=V^{(e)}\) / \(A^{(e)}\), \(V^{(e)}\) denotes the volume of element \(e\) and \(A^{(e)}\) stands for the surface area of an equivalent crack parallel to the real crack. In fact, \(l_{c}\) corresponds to the classic characteristic length [77; 78]. Here, the determination of \(A^{(e)}\) for the 8-node quadrilateral (Q8) and 6-node triangular (T6) elements is slightly different insofar as the equivalent crack passes through the center point of Q8 but through the midpoint of one edge of T6; see Figure 2. More details can be found in [56; 57].  
Its elemental stiffness matrix \(\mathbf{K}^{(e)}\) can be obtained as  
\[\mathbf{K}_{j-1}^{(e)}=\int\left(\mathbf{B}^{(e)}\right)^{T}\,\mathbf{C}^{(e) }\,\mathbf{B}^{(e)}\,d\left(e\right)+\left[\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}&A^{(e)}\,\mathbf{D}^{(e)}\end{array}\right]. \tag{11}\]  
Figure 2: Relationships between \(l_{c}\), \(V^{(e)}\) and \(A^{(e)}\) of Q8 and T6.  
The crack orientation (**n**) and crack openings will change during the iteration. \(\mathbf{K}_{j-1}^{(e)}\) of the cracking element is not constant. Its residual vector at iteration step \(j-1\), \(\mathbf{R}_{j-1}^{(e)}\) is obtained as  
\[\mathbf{R}_{j-1}^{(e)}=\left[\begin{array}{c}\mathbf{F}^{(e)}\\ -A^{(e)}\;\mathbf{T}^{(e)}\end{array}\right]-\mathbf{S}_{j-1}^{(e)}\;\left( \mathbf{U}_{i-1}+\Delta\mathbf{U}_{j-1}\right), \tag{12}\]  
where \(\mathbf{T}^{(e)}=\left[T_{n}^{(e)},T_{t}^{(e)}\right]^{T}\) (see Equation (2)) and \(\mathbf{S}_{j-1}\) is a designed unsymmetrical matrix  
\[\mathbf{S}_{j-1}=\left[\begin{array}{cc}\int\left(\
abla\mathbf{N}^{(e)} \right)^{T}\mathbf{C}^{(e)}\;\left(\widehat{\
abla\mathbf{N}^{(e)}}\right)d( e)&\int\left(\
abla\mathbf{N}^{(e)}\right)^{T}\mathbf{C}^{(e)}\;\mathbf{B}_{ \zeta}^{(e)}d(e)\\ V^{(e)}\left(\mathbf{B}_{\zeta}^{(e)}\right)^{T}\mathbf{C}^{(e)}\;\left( \widehat{\
abla\mathbf{N}^{(e)}}\right)&V^{(e)}\left(\mathbf{B}_{\zeta}^{(e)} \right)^{T}\mathbf{C}^{(e)}\;\mathbf{B}_{\zeta}^{(e)}\end{array}\right], \tag{13}\]  
where \(\widehat{\
abla\mathbf{N}^{(e)}}\) is the value of \(\
abla\mathbf{N}^{(e)}\) at the center point of element \(e\).  
In summary, the elemental formulations of an uncracked element and cracking element are very similar. Once an uncracked element becomes a cracking element, its cracks can be captured by replacing its elemental stiffness matrix and residual vector, provided in Equations (7) and (8), with the new form provided in Equations (11) and (12). Formally, a quadrilateral cracking element is similar to a 9-node quadrilateral element (transformed from Q8), and a triangular cracking element is similar to a 7-node triangular element (transformed from T6). The displacement degrees of freedom of the center point are now used to represent the normal and shear crack openings.