Irreversibility in Fracture Theory

• Irreversibility in fracture is defined as the condition that once a crack forms, it cannot heal or migrate, ensuring a monotonic evolution of damage.
• Variational inequalities and complementarity conditions mathematically implement the irreversibility constraint in phase-field and damage models.
• Numerical methods such as projection schemes, penalty methods, and slack-variable approaches enforce stability and energy balance during irreversible crack propagation.

Irreversibility in fracture theory is the principle that, once a crack is formed, its history cannot be reversed: cracks may nucleate and grow, but cannot shrink, heal, or migrate in the material reference configuration. This condition is foundational in both sharp-interface (Griffith–Francfort–Marigo) and regularized (phase-field, damage) approaches to brittle and cohesive fracture. Irreversibility is mathematically realized as a one-sided evolution constraint—most commonly imposing monotonicity of the crack set or damage field—and is rigorously justified by thermodynamics, variational inequalities, and stability considerations.

1. Thermodynamic Origins and Irreversibility in Brittle Fracture

The physical basis of the irreversibility condition in fracture arises from the second law of thermodynamics. In the inverse-deformation framework, the elastic energy is re-expressed in terms of an inverse map $h$, whose derivative $H = h'(y)$ (“inverse strain”) is nonnegative, preserving orientation and preventing mass interpenetration. Regularization introduces a surface energy density, and the admissible configuration space is restricted by the unilateral constraint $H(y)\geq0$. Considering a quasistatic, time-parameterized family of equilibrium configurations, entropy production arguments show that material motion of the crack front (i.e., changing its position within the reference configuration) leads to negative entropy production, thus violating the second law. Explicitly, the dissipation rate $D$ computed as

$D = -\epsilon^2\llbracket H'' \rrbracket V,$

where $V$ is the material velocity of the crack front, must be nonnegative. This yields the admissibility condition $\phi V \geq 0$, where $\phi$ is the configurational force. Detailed analysis confirms that, for both end and internal cracks under the nonnegativity and orientation-preservation constraints, the only admissible motion is $V = 0$: cracks cannot migrate nor heal. Thus, the crack set $\Gamma(t)$ must satisfy

$$\Gamma(s) \subset \Gamma(t) \quad \forall\, 0 \leq s \leq t,$$

which is the non-healing irreversibility condition (Gupta, 4 Dec 2025).

2. Mathematical Formulation: Variational Inequalities and Complementarity

Irreversibility is realized in fracture models as a unilateral constraint on the internal variable representing damage or the phase field. In phase-field formulations, the damage parameter $\phi(x,t)$ (with $\phi=0$ intact, $\phi=1$ fully broken) is required to be monotonically nondecreasing in time at every material point: $$\dot \phi(x,t) \geq 0 \quad \text{a.e. in} \;\Omega.$$ In incremental, time-discrete schemes, this translates to the pointwise constraint

$\phi^{n+1}(x) \geq \phi^{n}(x).$

This produces a constrained minimization or saddle point problem for the total energy, whose optimality conditions are variational inequalities of the form: $$\delta_{\phi} E(u,\phi; \psi - \phi) \geq 0 \quad \forall\, \psi \in V, \; \psi \geq \phi^n,$$ where $V$ is the admissible space. These inequalities can be alternatively reformulated via Karush–Kuhn–Tucker (KKT) conditions, introducing Lagrange multipliers for the inequality constraints, leading to complementarity relations (stationarity, primal/dual feasibility, and complementary slackness). These conditions guarantee evolution only in allowable, energetically permissible directions, enforcing irreversibility exactly (Wambacq et al., 2020, Maggiorelli et al., 14 Feb 2025, Gupta, 4 Dec 2025, Wu, 2024, Crismale et al., 2018).

3. Irreversibility in Phase-Field and Damage Models

In phase-field models, the core irreversibility constraint is encoded through variational inequalities at each incremental step and retained in the evolution by construction. The most general continuous formulation is

$$\dot \phi(x,t) \geq 0 \,,\qquad h(\phi,u) \leq 0 \,,\qquad \dot \phi \, h(\phi,u)=0,$$

where $h(\phi,u)$ is the driving force for fracture evolution. This can be stated in the Kuhn–Tucker form:

• $\dot\phi \geq 0$ (damage cannot heal)
• $h \leq 0$ (no energy descent in forbidden directions)
• $\dot\phi \, h = 0$ (complementarity at points of evolution)

Numerical and analytical realizations employ staggered gradient-flows, projected solvers, interior-point barrier methods, slack–variable approaches, or penalization. Monotonicity at the discrete level is maintained via suitable projection or active-set strategies, such as projected successive over-relaxation (PSOR), bound-constrained conjugate gradients, and monolithic Newton-type solvers with embedded complementarity (Greco et al., 28 Jan 2025, Bharali et al., 2022, Walloth et al., 2021, Wambacq et al., 2020, Gerasimov et al., 2018).

Table: Common Numerical Enforcement Techniques for Irreversibility in Phase-Field Models

Method	Enforcement Principle	Typical Reference
Variational Inequality	Primal inequality constraints, KKT	(Gupta, 4 Dec 2025, Maggiorelli et al., 14 Feb 2025)
Penalty Methods	Add large penalty for violation	(Gerasimov et al., 2018)
Slack Variable	Reformulate as equality via slack	(Bharali et al., 2022)
Interior Point	Barrier for complementarity in primal-dual system	(Wambacq et al., 2020)
Projection Schemes	Pointwise projection in update step	(Greco et al., 28 Jan 2025, Bharali et al., 2022)

4. Integration with Stability and Energy Balance

The irreversibility constraint fundamentally restricts admissible variations in stability analysis. Local stability is determined by the positivity of the second variation (Hessian) restricted to a cone of perturbations consistent with the irreversibility (typically nonnegative variations in the phase-field variable). In sharp-interface and inverse-deformation fracture, the test variations must also freeze crack locations, explicitly setting variation to zero at crack faces and disallowing any perturbation that would shift the crack set or reduce the broken region. Thus, the stability condition is always tested in a convex cone of admissible directions: $$V_h = \{ \eta \in H^2 : \eta'(y) \geq 0 \text{ on broken set},\, \eta(y_c)=0 \text{ at crack faces} \},$$ and only within this subspace is the positivity of the quadratic form checked (Gupta, 4 Dec 2025, Terzi et al., 2024).

Energy dissipation is strictly nonnegative, and the time-derivative of the total energy incorporates the one-sided nature of crack growth. In gradient-flow settings, dissipation is active only during forward (irreversible) evolution, and energy release rates are constrained by the maximal toughness supplied by the irreversibility inequality, exactly matching the Griffith criterion at equilibrium (Kimura et al., 2023, Maggiorelli et al., 14 Feb 2025).

5. Physical, Atomistic, and Cohesive-Zone Interpretations

At the atomistic scale, irreversibility is realized as a non-decreasing memory of bond damage: once a bond's maximal extension exceeds a rupture threshold, it remains permanently damaged. The continuum limit recovers the inclusion of crack sets, $K(s) \subset K(t)$, in the Francfort–Marigo energetic formulation (Friedrich et al., 2024). In cohesive-zone and fatigue-dominated models, irreversibility may be extended to the total variation of the crack opening, ensuring energy dissipation for all opening/closing cycles (i.e., fatigue-induced fracture), with the cumulative variable being monotone non-decreasing in time (Crismale et al., 2018, Bonacini et al., 2020).

6. Model-Specific Application and Numerical Implementation

In classical phase-field models (AT1/AT2), irreversibility is imposed either globally (damage irreversibility) or restricted to the crack set (“crack-set irreversibility”) to improve $\Gamma$-convergence and physical fidelity in heterogeneous media. Associated and non-associated cohesive-zone phase-field models employ the condition $\dot d \geq 0$ with Kuhn–Tucker complementarity, with careful attention to the scaling of degradation and dissipation functions to prevent distortion of the traction-separation law under the irreversibility constraint (Wu, 2024). In micromorphic and coupled plasticity-fracture models, irreversibility of the local damage field is enforced via projection, max-operations, or local Kuhn–Tucker updates, with global regularization penalizing deviation from the regularized field [2(Auth et al., 2024, Bharali et al., 2022)].

7. Summary and Significance

The irreversibility condition in fracture theory, underpinned by thermodynamics, is rigorously embedded as a one-sided monotonicity constraint on the evolving crack, damage, or phase-field variable. Its mathematical realization via variational inequalities, complementarity, and projection restricts the energy landscape, defining the correct cone of admissible variations and governing equilibrium stability, energy balance, and the pathwise evolution of cracks. This condition is crucial for the physical fidelity of fracture simulations and for ensuring that energetic and topological features of crack growth are correctly predicted across discrete, regularized, and multiscale models (Gupta, 4 Dec 2025, Maggiorelli et al., 14 Feb 2025, Friedrich et al., 2024, Sato, 2023, Wu, 2024).