Phase-field primer for mechanics engineers¶

This document is a compact review for engineers who already know finite elements, linear elasticity, and small-strain plasticity, but have not used phase-field fracture before. It explains why the model in phast looks the way it does and points to the public references at the end of this page.

Variational fracture in one minute¶

Griffith’s 1921 fracture criterion balances elastic strain energy against a surface energy proportional to the crack area. Francfort and Marigo (1998) recast the criterion as a global energy minimisation: find a displacement field u and a crack set Gamma minimising

where \(\psi\) is the elastic strain energy density, \(G_c\) is the critical energy release rate, and \(\mathcal{H}^{n-1}\) measures the \((n-1)\)-dimensional crack surface.

The discrete-crack-set problem is intractable on a fixed mesh. Bourdin, Francfort and Marigo (2000) proposed a regularised form that approximates the sharp crack \(\Gamma\) by a smooth scalar damage field \(d \in [0, 1]\) with a length scale \(\ell_0\). As \(\ell_0 \to 0\) the regularised energy gamma-converges to the sharp-crack energy. In phast the regularised energy is

\(g(d) = (1-d)^2 + \eta_{\mathrm{residual}}\) is the degradation function; \(w(d)\) is the local dissipation density; \(c_w\) is a normalisation constant. The Euler-Lagrange system is two coupled PDEs (mechanics and damage) which the staggered solver alternates between.

AT1 vs AT2¶

The two standard regularisations differ in w(d) and c_w:

AT2 is mathematically simpler – the damage equation is linear in \(d\) for fixed history field \(\mathcal{H}\), so a single CG solve does the job. The downside: at any non-zero strain a tiny amount of damage develops everywhere, because there is no elastic threshold. Most papers therefore enforce a post-hoc nucleation threshold or a pre-existing notch.

AT1 has a true elastic phase: damage stays at zero until the local driving energy \(\mathcal{H}\) exceeds \(\mathcal{H}_{c,0}=3G_c/(16\ell_0)\). This matches the intuition of “no damage until the strength is reached” but the damage sub-problem is now constrained (\(d \geq 0\)), so production AT1 runs use projected CG (bounds_method='projected_cg'). A post-clamp after an unconstrained solve is not a valid replacement for the AT1 active-set solve.

In a YAML config, switch with material.overrides.pf_model: AT1 or AT2. AT1 is the right choice when you care about nucleation without a pre-crack (Ambati et al. 2015, Bleyer et al. 2017); AT2 is the right choice for propagation from an existing notch (Borden et al. 2012).

The history field

enforces irreversibility (a node cannot heal). Combined with the damage-bound constraint \(d_{\mathrm{new}} \geq d_{\mathrm{old}}\), this gives the monotone crack growth observed in experiments.

Staggered minimisation loop¶

The coupled equations are solved by alternate minimization. PhAST freezes damage while solving mechanics, updates the tensile history field, then freezes mechanics while solving damage.

        flowchart TD
    A[Load or time step] --> B[Freeze damage d]
    B --> C[Solve mechanics for u]
    C --> D[Update history H]
    D --> E[Freeze u]
    E --> F[Solve damage for d]
    F --> G[Project bounds and irreversibility]
    G --> H{norm_inf damage update < tolerance?}
    H -- no --> B
    H -- yes --> I[Advance step and write outputs]
    

The stopping criterion is

Energy splits – why we don’t degrade psi directly¶

If g(d) multiplies the full strain energy psi(eps), cracks can close under compression and develop on the compressive side of a bend – both unphysical. The fix is to split

where only psi+ (the “damaging” part) is degraded. phast ships five splits, all in fem_operators.py:

amor is a safe starting point. spectral is what most published dynamic-fracture benchmarks use (Borden 2012, Bleyer 2017). For isotropic Mode I loading with no compressive zones, isotropic is faster and gives the same answer.

Plane-stress spectral is supported as a reduced 2D in-plane strain-spectral projection. It is not a fully condensed 3D plane-stress spectral decomposition with damage-dependent out-of-plane strain. For mature validated paths, use plane-strain spectral for Miehe-style principal-strain splits or plane-stress amor for thin PMMA-style dynamic benchmarks.

The four parameters that matter¶

The mesh size h near the crack must satisfy roughly h <= l0 / 2 to resolve the diffuse damage band. If you double l0, you can halve the element count – but the apparent fracture toughness changes slightly (the Gc of the regularised model is not exactly Gc of the sharp-crack model unless the mesh is fine enough).

In mathematical form:

References¶

• Bourdin et al. (2000, 2011) for regularized variational fracture and time-discrete dynamic fracture.

• Borden et al. (2012) for phase-field dynamic brittle-fracture benchmarks.

• Ambati et al. (2015) for a review of phase-field brittle fracture and energy splits.

• Bleyer, Roux-Langlois, and Molinari (2017) for dynamic branching and velocity-toughening studies.

Once you have the theory in mind, head to Setting up new problems to translate it into a PhAST model and durable YAML configuration.

For a picture-first companion, see the visual glossary.