Physics, Units, and Formulation¶

This page summarizes the continuum model used by PhAST and maps the main symbols to the YAML and fluent Python interfaces. It is intended as the bridge between the paper formulation, the public examples, and the executable solver configuration.

For sparse-direct and autograd-enabled linear-solve details, see Sparse and Matrix-Free Solves.

Variational Fracture Model¶

Sharp Griffith fracture minimizes elastic energy plus crack-surface energy:

where \(\mathbf{u}\) is displacement, \(\Gamma\) is the crack set, \(G_c\) is the critical energy release rate, and \(\mathcal{H}^{n-1}\) measures crack surface. Phase-field fracture replaces the sharp crack by a scalar damage field \(d \in [0, 1]\), where \(d=0\) is intact and \(d=1\) is fully damaged. The damaged set can be interpreted as a regularized crack band whose width is controlled by \(\ell_0\); in practical meshes, the visible diffuse band is typically spread over a few elements around the crack centreline.

PhAST uses the regularized energy

The degradation function is

with a small residual stiffness \(\eta\) to avoid singular stiffness rows in fully damaged zones. The length scale \(\ell_0\) controls the width of the diffuse crack band.

The equivalent crack surface density is often written separately as

Governing Equations¶

For quasistatic brittle fracture, the displacement field satisfies mechanical equilibrium with degraded tensile energy:

with

The usual mechanical boundary conditions are

For AT2 with history-field irreversibility, the damage equation has the common linearized form

where

The natural phase-field boundary condition is

For explicit dynamics, PhAST solves the momentum balance

with damage updated through the configured phase-field substep cadence.

AT1 and AT2 Crack-Density Terms¶

The regularization choice enters through \(w(d)\) and \(c_w\):

The AT1 threshold is often written as

In PhAST, choose the model with material.overrides.pf_model: AT1 or material.overrides.pf_model: AT2.

For AT1, the damage update is a bound-constrained variational problem. The lower bound must be enforced during the iterative solve; a post-solve clamp is not equivalent to the active-set solve.

Energy Splits¶

Damage should not degrade compressive strain energy. PhAST therefore splits the elastic energy into tensile and compressive parts:

and applies \(g(d)\) only to \(\psi^+\). The public configuration key is material.overrides.energy_split.

The distinction is visible in bending, impact, and branching examples, where compressive regions should not create artificial damage. The public dynamic gallery includes crack curvature and branching examples such as examples/dynamic/B2_kalthoff_winkler/ and examples/dynamic/B7_dynamic_crack_branching_comsol/.

For a picture-first explanation of these terms, see the visual glossary.

Mixed-mode impact	Dynamic branching

Plane Strain and Plane Stress¶

Two-dimensional examples use either plane strain or plane stress. The main difference is the effective elastic operator. In plane strain, \(\varepsilon_{zz}=0\) and the Lamé parameter is

In plane stress, \(\sigma_{zz}=0\) and the in-plane reduction uses

The public fracture examples choose the assumption through their material and workflow settings. Miehe-style spectral split studies are usually documented as plane strain, while thin PMMA dynamic examples are typically treated with plane-stress settings and an Amor-style split.

Staggered Solver Loop¶

The coupled fracture problem is solved by alternate minimization: freeze the damage field, solve mechanics, then freeze mechanics and solve damage. The iteration stops when the damage update is sufficiently small.

        flowchart TD
    A[Start load or time step] --> B[Freeze damage d]
    B --> C[Solve mechanics for displacement u]
    C --> D[Update tensile history field H]
    D --> E[Freeze displacement u]
    E --> F[Solve damage equation for d]
    F --> G[Apply irreversibility and bounds]
    G --> H{norm_inf(d_new - d_old) < tolerance?}
    H -- no --> B
    H -- yes --> I[Store fields, histories, and visuals]
    I --> J[Advance to next step]
    

In notation, the convergence check is

where \(\varepsilon_{\mathrm{stag}}\) maps to the configured staggered tolerance.

Matrix-Free Operator View¶

PhAST’s public fracture path applies finite-element operators through gather-compute-scatter tensor kernels rather than assembling a global stiffness matrix for every operator application. For a linearized internal force \(\mathbf{f}_{\mathrm{int}}\), the Krylov product can be viewed as

or, around a nonlinear reference state \(\bar{\mathbf{u}}\),

This is the matrix-free sense used throughout the repository: the solver materializes the action of the weak-form operator on a vector, while avoiding a persistent global stiffness matrix unless an optional sparse-direct pathway or preconditioner setup explicitly requires one.

Symbol to Configuration Map¶

Unit System¶

The public examples use a consistent mm-N-MPa-s convention unless a local README states otherwise:

Mesh resolution should be judged relative to \(\ell_0\). A common rule of thumb is

near the expected crack path, with local refinement around notches, holes, or branching regions.

References¶

• Bourdin, B., Francfort, G. A., and Marigo, J.-J. (2000). Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids.

• Amor, H., Marigo, J.-J., and Maurini, C. (2009). Regularized formulation of the variational brittle fracture with unilateral contact. International Journal for Numerical Methods in Engineering.

• Miehe, C., Welschinger, F., and Hofacker, M. (2010). Thermodynamically consistent phase-field models of fracture. International Journal for Numerical Methods in Engineering.

• Kumar, A., Francfort, G. A., and Lopez-Pamies, O. (2020). Revisiting nucleation in the phase-field approach to brittle fracture. Journal of the Mechanics and Physics of Solids.

• PhaseFieldX documentation: https://phasefieldx.readthedocs.io/