Fracture Energy Release Rate Formulation

• The paper introduces a weighted-variational surrogate model that reduces computational cost while accurately simulating brittle fracture.
• The paper models the critical energy release rate as a spatially random Gaussian field to capture microstructural variability in crack propagation.
• The paper validates the surrogate model against high-fidelity simulations and experiments, ensuring robust predictions for probabilistic fracture analysis.

Fracture energy release rate formulation establishes the fundamental link between the mechanical energy stored in a body and the dissipative cost required to propagate cracks or defects. In brittle fracture, the primary criterion determining crack advance is whether the elastic energy released per unit extension of crack surpasses the material’s intrinsic fracture toughness. Quantitative formulation of the energy release rate (ERR) is at the core of both classical sharp-interface approaches and modern regularized numerical methods. The precise characterization of ERR is critical in connecting microstructural, rate-dependent, or stochastic material behavior to macroscopic crack evolution.

1. Surrogate Formulation for Brittle Fracture Simulation

The paper develops a weighted-variational (W–V) surrogate model to approximate the standard hybrid phase-field formulation for brittle fracture in two-dimensional settings, facilitating efficient simulation of energy release rate phenomena. The phase-field approach typically requires fine discretization in space and time to resolve the diffuse crack, with computational demands that grow quickly for large domains or many realizations. The W–V surrogate modifies the total energy functional by a scale factor ξ, allowing the use of coarser meshes $h$, larger displacement increments $\Delta u$, and a regularization length $\ell$ tuned to $h$. The surrogate is defined as

$$E^*_{\Delta u, \ell, h}(u, \phi) = \xi \cdot E_\ell(u, \phi; \Delta u, \ell, h)$$

with $\xi$ determined through reference runs of the high-accuracy hybrid phase-field model. Surrogate calibration ensures that the force-displacement response and crack trajectories closely track high-fidelity solutions despite much reduced computational cost. This enables efficient large-scale stochastic analyses—including Monte Carlo studies with many fracture simulations—where the computational budget might otherwise be prohibitive.

2. Modeling the Energy Release Rate as a Spatial Random Field

A distinguishing feature is the explicit modeling of the critical energy release rate $G_c$ as a non-homogeneous, spatially random field to reflect microstructural variability. The paper employs a stationary Gaussian Process with a Matérn covariance function:

$$C(r) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu} \frac{r}{\ell}\right)^\nu K_\nu\left(\sqrt{2\nu} \frac{r}{\ell}\right)$$

where $r = \|x - x'\|$, $\sigma^2$ is the variance, and $K_\nu$ is the modified Bessel function of the second kind. The parameter $\nu>0$ governs the field’s smoothness, and $\ell$ is the correlation length. For $\nu=3/2$ a closed-form spectral representation simplifies the generation of $G_c$ fields. The resulting $G_c(x)$ is sampled independently for each Monte Carlo realization, inducing stochasticity in the spatial resistance to crack growth, and allowing investigation of uncertainty propagation from material randomness to macroscopic fracture behavior.

3. Numerical Implementation and Calibration

Both the baseline hybrid phase-field model and the W–V surrogate are implemented with the FEniCS finite element software. The hybrid model uses a staggered approach: alternate minimization of (a) linear momentum for displacement $u$ and (b) fracture/damage field $\phi$. The surrogate model adopts a similar alternate minimization with the weighted energy but uses larger $\Delta u$ and coarser $h$. Key computational steps:

• Select $\ell$, $h$, $\Delta u$ and simulate the reference problem via hybrid phase-field
• Calibrate scaling factor $\xi$ to align key force/displacement or energy response metrics between the surrogate and reference solutions
• Run the W–V surrogate with the calibrated parameters for each random $G_c$ field realization

Tabulated comparison between hybrid phase-field and weighted-variational (W–V) methods:

Model	Discretization	Calibration	Execution Time
Hybrid phase-field	Fine $h$, small $\Delta u$	None	High
W–V surrogate	Coarse $h$, large $\Delta u$	Calibrated scale ξ	Reduced (90%)

This acceleration is critical to enable large ensemble simulations for stochastic studies (Blanco-Cocom et al., 2021).

4. Monte Carlo Analysis and Statistical Response

With the surrogate model, the energy release rate field $G_c(x)$ is varied randomly across Monte Carlo realizations. Each run produces a unique crack path and force-displacement response. Statistics of the output include:

• Probability distributions of fracture trajectories
• Histograms and quantiles of peak load, critical displacement, and post-peak softening
• Occurrence of incomplete or branching cracks at high $G_c$ variance (100% perturbation)

In cases of modest randomness ($\approx$10% $G_c$ perturbations), the distributions of crack paths and mechanical response produced by the surrogate and hybrid models align closely. At extreme variability, richer stochastic crack patterns are observed only through extensive simulation enabled by the surrogate’s efficiency. The output provides insight into the probabilistic character of failure in heterogeneous brittle materials.

5. Experimental Validation and Realistic Applications

Both the hybrid phase-field model and the surrogate are validated against laboratory single-crack experiments on cement mortar plates. Experiments involve mixed-mode loading of notched specimens, with the numerical models reproducing specimen geometry and loading conditions. Key comparisons:

• Simulated crack paths and fracture initiation locations agree well with experimental fracture surfaces
• Force-displacement curves from simulations recover the initiation peak and softening behavior observed in tests

This confirms that the surrogate, when properly calibrated, is not only computationally efficient but also predictive for real-world brittle fracture phenomena. Consequently, the W–V approach is well-suited for practical stochastic structural analysis where repeated large-scale simulations are required.

6. Implications and Outlook

By treating the energy release rate as a spatially random Gaussian field and leveraging an efficiently calibrated variational surrogate, the approach enables uncertainty quantification in fracture mechanics with high computational throughput. This modeling paradigm makes possible large-scale probabilistic characterization of fracture—quantifying variability in crack growth, structural capacity, and load-response in the presence of inherent material randomness. The convergence between surrogate and phase-field results, together with experimental agreement, underscores the method’s robustness and flexibility. As stochastic mechanics, machine learning surrogates, and high-performance computing evolve, such calibrated variational surrogates will become increasingly vital in bridging fine-scale randomness and macroscopic fracture risk in complex structures.