Elements with (\eta_e > \eta_\texttol) are refined (bisected) and coarsening is applied where (\eta_e < 0.1,\eta_\texttol). This strategy concentrates degrees of freedom only where the crack evolves, keeping the global problem size modest. A monolithic coupling (solving (\mathbfu) and (\phi) simultaneously) is possible but computationally expensive. Instead, we adopt the staggered scheme (Miehe et al., 2010) that is unconditionally stable for quasi‑static loading:
The phase‑field approach was first introduced by Francfort & Marigo (1998) and later regularised by Bourdin, Francfort & Marigo (2000). Since then, a plethora of works (Miehe et al., 2010; Borden et al., 2012; Wu, 2018) have demonstrated its versatility for quasi‑static, dynamic, and fatigue fracture. However, practical adoption still requires a that guides the user from model formulation to implementation, parameter calibration, and verification.
The arc‑length parameter is updated each load step, ensuring a smooth equilibrium path through post‑peak regimes. | Component | Tool / Library | |-----------|----------------| | FEM core | deal.II (v9.5) | | Linear solver | PETSc (GMRES + ILU) | | Non‑linear solver | Newton‑Raphson with line‑search | | Mesh adaptivity | p4est (parallel refinement) | | Post‑processing | ParaView (VTK output) | Working Model 2d Crack-
[ \Delta W = \int_\Gamma_N \mathbft\cdot \Delta\mathbfu,\mathrmdS . \tag7 ]
The manuscript follows the conventional structure (Title, Abstract, Keywords, etc.) and includes all the essential elements (governing equations, numerical algorithm, validation, results, discussion, and references). Feel free to copy the LaTeX source into your favourite editor (Overleaf, TeXShop, etc.) and adapt the figures, tables, or code snippets to your own data. Authors : First Author ¹, Second Author ², Third Author ³ ¹ Department of Mechanical Engineering, University A, City, Country. ² Institute of Applied Mathematics, University B, City, Country. ³ Materials Science Division, Research Center C, City, Country. Instead, we adopt the staggered scheme (Miehe et al
Given uⁿ, φⁿ: 1. Update history field Hⁿ⁺¹ ← max(Hⁿ, ψ⁺(ε(uⁿ))) 2. Solve displacement problem → uⁿ⁺¹ (with φⁿ fixed) 3. Solve phase‑field problem → φⁿ⁺¹ (with uⁿ⁺¹ fixed) 4. Check convergence: ‖uⁿ⁺¹‑uⁿ‖ + ‖φⁿ⁺¹‑φⁿ‖ < ε_tol 5. If not converged → repeat steps 2‑4 The linearised systems are assembled using (e.g., via the Sacado package) to obtain consistent tangent operators. 3.4. Load Control & Arc‑Length For softening problems, displacement control can cause snap‑back. We implement an arc‑length (Riks) method that controls the total work increment:
Figure 1 : Load‑displacement response (phase‑field vs. LEFM). Figure 2 : Phase‑field contour at (F = 0.9F_c) (crack tip radius ≈ 3(\ell)). A DCB specimen (length 0.2 m, thickness 0.01 m) is subjected to a symmetric opening displacement. The energy release rate calculated from the phase‑field solution The arc‑length parameter is updated each load step,
[ \psi^+(\boldsymbol\varepsilon) ;\rightarrow; H(\mathbfx) . \tag4 ] 3.1. Finite‑Element Discretisation Both fields are approximated using quadratic Lagrange shape functions on an unstructured triangular mesh: