# Three Dimensional CS-FEM Phase-Field Modeling Technique for Brittle Fracture in Elastic Solids
## 2 Methods
### Implementation in ABAQUS UEL  
We implemented the PhaseField-CSFEM in ABAQUS (ABAQUS 2017, Dassault Systemes, Providence, RI, USA) and solved it using a staggered scheme. Equations (13)-(17) contain the coupled phase and displacement fields which can be solved using a monolithic approach [26; 28], however we decoupled and solved them separately using the SFEM in ABAQUS UEL. In the literature, the staggered scheme has presented quite a few stability advantages over the monolithic solution [25; 27]. The decoupled governing equations are:  
\[r_{c}=\int_{\Omega}((\frac{\bar{S}c}{l_{c}}c-2(1-c)H](N^{c})^{T}+g_{c}l_{c}(B^ {c})^{T}\
abla c)\mathbf{d}\mathbf{\Omega} \tag{32}\]  
Figure 2: A 3D H8 element subdivided into eight smoothing cells: (**a**) subdivision into smoothing cells by joining the midpoints of the edges and the faces; (**b**) face area of a smoothing cell with its outward normal and length; and (**c**) entire face of the element containing surfaces of the four smoothing cells. where \(r_{c}\) is the residual force vector and \(N^{c}\) and \(B^{c}\) are the shape function and the strain gradient matrix for 3D eight-node hexahedral elements, respectively. The phase-field component in the decoupled stiffness matrix is given by:  
\[K_{c}=\int_{\Omega}([\frac{G_{c}}{l_{c}}+2H](N^{c})^{T}(N^{c})+G_{c}l_{c}(B^{c} )^{T}(B^{c}))\mathbf{d\Omega} \tag{33}\]  
Similarly, the displacement-field contribution in the stiffness matrix is given by \(B^{u}\) matrix calculated from Equation (23),  
\[K_{u}=\int_{\Omega}[(1-c)^{2}+d](B^{u})^{T}\mathbf{C}(B^{u})d\mathbf{\Omega} \tag{34}\]  
and the internal force(residual) vector is  
\[r_{u}=\int_{\Omega}[(1-c)^{2}+d](B^{u})^{T}\mathbf{\sigma_{0}}d\mathbf{\Omega} \tag{35}\]  
where \(\mathbf{\sigma_{0}}\) is the stress calculated by Equation (30). Thus, the following equation is solved using the modified Newton-Raphson scheme  
\[\left[ \begin{array}{c}u\\ c\end{array} \right]_{t+\delta t}=\left[ \begin{array}{c}u\\ c\end{array} \right]_{t}-\left[ \begin{array}{cc}K_{u}&0\\ 0&K_{c}\end{array} \right]_{t}\left[ \begin{array}{c}r_{u}\\ r_{c}\end{array} \right]_{t} \tag{36}\]  
The term \(H\) in Equations (32) and (33) is a history field which is equal to the strain energy from the previous step. In the staggered scheme, in the first iteration of every load step, the history variable is updated via \(H_{n+1}=\psi_{n}\) and the displacement field (Equations (34) and (35)) is solved by updating the value of the phase-field from the previous load step (\(c_{n}\)). The history variable satisfies certain properties [25] and ensures that no penalty term is necessary to enforce the irreversibility of the crack field. In this formulation, the \(B^{u}\) matrix in Equations (34) and (35) is calculated based on the smoothing cells, using CS-FEM-H8. The non-linear system of equations (Equation (36)) is solved via an incremental iterative approach using a Newton-Raphson scheme. The continuum mechanics equations are solved by the more accurate CS-FEM [2] and the phase-field equations are solved by the standard FEM. In the UEL, the user needs to specify the tangent stiffness matrix \(AMATRX\) and the internal force vector \(RHS\), which the solver calls for each element. The properties are calculated at the center of the smoothing domain for each subcell. In the staggered scheme of solutions [27], we use the _common_ block in the UEL to facilitate the transfer of variables. Through the _common_ block, the UEL allows a user to write formulations of multiple elements in a single code. We define three elements, the first two being the phase-field and the displacement elements and the third a dummy element written to facilitate post-processing in the ABAQUS viewer. ABAQUS is unaware of the inherent shape functions used in the element formulation(UEL) and thus is unable to extrapolate the results on its own. We write a user material (UMAT), where the results stored as solution variables \(STATEV\) are transferred from the UEL.