Phase-Field Fracture, SENT — the whole study as a term list

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A single-edge-notched specimen pulled in tension, with the crack regularized into a damage field \(d\in[0,1]\). What makes this worth its own tutorial is not the physics — that is the standard second-order AT2 model — but that none of it is driven from outside jno.fem. The coupled system, the irreversible history, the bound on the damage, the alternating solver and the non-uniform load path are all either a term in the list or a slot on fem.solve.

what	how
the stress is the energy derivative	sigma = jno.np.diff(psi, eps(u))
irreversible history \(H=\max_\tau \psi^+\)	Hs.evolves(maximum(Hs.i(-1), psi_p)) — on a coupled system
damage is a fraction	dm.bounds(0.0, 1.0) — an inequality, in the term list
the energy is non-convex in \((u,d)\) jointly	nonlinear=jno.solve.staggered([u, dm])
the load path is not uniform	tau=jno.solve.adaptive(limit=[(dm, 0.5)])
the reaction force	fem.eval(momentum, u_k)[grip]

The model

The elastic energy is split volumetric/deviatoric so only the tensile part drives damage and a closed crack still carries compression (Amor, Marigo & Maurini, JMPS 57 (2009) 1209–1229):

\[ \psi^+ = \tfrac12 K \langle \operatorname{tr}\varepsilon\rangle_+^2 + \mu\, \varepsilon_{\text{dev}}\!:\!\varepsilon_{\text{dev}}, \qquad \sigma = \frac{\partial}{\partial\varepsilon}\Big[ g(d)\,\psi^+ + \psi^- \Big]. \]

That last line is written literally — jno.np.diff differentiates the energy, so there is no hand-derived stress anywhere. Irreversibility comes from the history field \(H=\max_\tau\psi^+\), so a crack cannot heal (Miehe, Welschinger & Hofacker, IJNME 83 (2010) 1273–1311), and the damage equation is the standard AT2 form \((G_c/\ell)\,d - G_c\ell\,\Delta d = 2(1-d)H\).

The notch is cut, not painted on: a real slit removed from the geometry with jno.Shape, so the stress concentration at its tip is the mesh's own and the damage field needs no seeding.

plate = jno.Shape.rect(0.0, 0.0, 1.0, 1.0, size=h)
slit  = jno.Shape.rect(-0.01, 0.5 - w_slit, 0.5, 0.5 + w_slit, size=h)
dom   = (plate - slit).domain(tau=(0.0, 1.0, NOUT))     # tau: the pseudo-time LOAD path

Loading is displacement-controlled — u(*ct)[1] - DELTA * ct[-1], the grip displacement ramped in \(\tau\). That is not a stylistic choice: under load control a brittle specimen snaps at the peak and there is no branch left to follow, so a softening response can only be traced by prescribing the grip.

What it shows, and what it refuses to

Damage initiates at the notch tip and grows smoothly while the load rises. That part is stable, and the adaptive controller resolves it — the schedule comes back as [0, 0.143, 0.357, 0.679, 1.0], taking large steps while nothing happens and cutting as the tip loads up. The reaction rises to a peak and then softens, read straight off fem.eval.

Propagation across the ligament is not stable for this geometry. Ask for a finer resolution of the same path and the controller cannot deliver one:

tighter limit (0.25): refused, and named the unstable branch rather than returning a path

That is the correct answer, not a solver failure. Past the peak there is no nearby equilibrium, so no amount of load-step refinement finds one — verified separately: a 5× finer uniform grid gives the same jump, the crack tip going from \(x=0.05\) to \(x=1.0\) in a single increment either way. Following an unstable branch needs arc-length (dissipation-controlled) continuation, which jNO does not have. The alternative to reporting this is grinding to the step floor and returning a plausible-looking path that never happened.

Notes on the numbers

• The bound is solved, not clipped — damage stays in \([0,1]\) through the KKT conditions of the variational inequality, and is monotone because the history field is a running max.
• fem.block_index(dm) resolves the damage block: field order is first appearance in the term walk, and the degradation factor puts dm ahead of u. Never hardcode the index.
• The degradation floor \(\eta\) is still needed even with the bound: at \(d=1\) exactly, \((1-d)^2\) makes the displacement block singular. The bound keeps \(d\) in range; it does not make the operator well-posed.
• AT2 has no elastic threshold, so \(d>0\) everywhere from the first increment. Statements about localization have to be about substantial damage, not about where the field is nonzero.
• The two stragglers outside the crack band are the grip corners, where a fully clamped edge meets a free lateral edge and the elastic field is singular — a property of the boundary conditions rather than of the model.

For the fourth-order regularization on a biharmonic (Morley/Argyris) element, see Brittle Fracture — 4th-order phase-field; that model needs a \(C^1\)-ish space, which the load-path march does not yet carry.