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.
The first equation is the for a degraded material. The second is a reaction‑diffusion equation governing the evolution of the crack field. Irreversibility is enforced by a history field (H(\mathbfx) = \max_t\le t\psi^+(\boldsymbol\varepsilon(\mathbfx,t))) so that the tensile energy term never decreases:
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:
The load‑displacement curve obtained with the phase‑field model matches the analytical LEFM prediction for the critical stress intensity factor (K_IC= \sqrtE G_c). The computed (F_c= 4.58) kN is within 2 % of the analytical value. The crack path follows the straight line of the notch, confirming the absence of mesh bias. Working Model 2d Crack-
The regularisation length (\ell) controls the width of the diffusive crack zone ((\approx 3\ell)). When (\ell\to0), (\Pi) (\Gamma)-converges to the classical Griffith functional. Stationarity of (\Pi) with respect to admissible variations (\delta\mathbfu) and (\delta\phi) yields the coupled Euler‑Lagrange equations :
[ \mathbfu^h(\mathbfx) = \sum_i=1^N_n \mathbfN_i(\mathbfx) , \mathbfu i, \qquad \phi^h(\mathbfx) = \sum i=1^N_n N_i(\mathbfx) , \phi_i, \tag5 ]
All source files are provided in the supplementary material (GitHub repository github.com/YourGroup/2DPhaseFieldCrack ). 4.1. Benchmark 1 – Single‑Edge Notched Tension (SENT) Geometry : rectangular plate (L=1.0) m, (H=0.5) m, notch length (a_0=0.2) m. Material : (E=30) GPa, (\nu=0.2), (G_c=2.7) kJ/m(^2). Parameters : (\ell = 2.5,h_\min) (where (h_\min) is the smallest element size after refinement). ² Institute of Applied Mathematics, University B, City,
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:
[ \eta_e = \int_\Omega_e \ell |\nabla\phi^h|^2 ,\mathrmdV . \tag6 ]
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 second is a reaction‑diffusion equation governing the
[ \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:
Corresponding author : first.author@univa.edu A robust computational framework for simulating quasi‑static fracture in brittle solids is presented. The model couples linear elasticity with a regularized phase‑field description of cracks, yielding a fully variational formulation that naturally captures crack nucleation, branching, and interaction without explicit tracking of the crack surface. The governing equations are derived from the minimisation of the total free energy, leading to a coupled system of a displacement‑balance equation and a diffusion‑type phase‑field evolution equation. An adaptive finite‑element discretisation with a staggered solution scheme is implemented in 2‑D. Benchmark problems—including the single‑edge notched tension test, the double‑cantilever beam, and a complex multi‑crack interaction case—demonstrate excellent agreement with analytical solutions and experimental data. Sensitivity analyses reveal the influence of the regularisation length, fracture energy, and load‑control strategies on crack paths. The presented workflow constitutes a “working model” that can be readily extended to anisotropic, heterogeneous, or dynamic fracture problems.
where (N_n) is the number of nodes. Quadratic interpolation is essential to resolve the steep gradients of (\phi) within the diffusive crack zone. A goal‑oriented error estimator based on the phase‑field gradient is used:
[ G = \frac{P^2
[ \Delta W = \int_\Gamma_N \mathbft\cdot \Delta\mathbfu,\mathrmdS . \tag7 ]