Calibration and Validation of a Phase-Field Model of Brittle Fracture within the Damage Mechanics Challenge

TL;DR

The paper develops and calibrates a phase-field model of brittle fracture for an orthotropic, geo-architected gypsum to predict failure in a notched TPB beam under mixed-mode loading. A two-stage calibration first estimates elastic parameters from plane-wave and unconfined compression tests, then optimizes the remaining elastic parameter and fracture toughness $\mathcal{G}_{\text{c}}$ by matching four TPB load–deflection curves, with the length scale fixed at $\ell = 0.625$ mm and an AT1 degradation. The calibrated model accurately reproduces TPB experiments and successfully performs a blind prediction of the DMC test, yielding close agreement in peak force, post-peak behavior, and crack paths (with $\Delta s<2$ mm generally). These results demonstrate that phase-field fracture models, when paired with a judicious two-stage calibration, can reliably capture mixed-mode fracture in heterogeneous, anisotropic materials and support predictive design in geo-architected systems.

Abstract

In the context of the Damage Mechanics Challenge, we adopt a phase-field model of brittle fracture to blindly predict the behavior up to failure of a notched three-point-bending specimen loaded under mixed-mode conditions. The beam is additively manufactured using a geo-architected gypsum based on the combination of bassanite and a water-based binder. The calibration of the material parameters involved in the model is based on a set of available independent experimental tests and on a two-stage procedure. In the first stage an estimate of most of the elastic parameters is obtained, whereas the remaining parameters are optimized in the second stage so as to minimize the discrepancy between the numerical predictions and a set of experimental results on notched three-point-bending beams. The good agreement between numerical predictions and experimental results in terms of load-displacement curves and crack paths demonstrates the predictive ability of the model and the reliability of the calibration procedure.

Figures (18)

• Figure 1: Geometry and loading conditions of the notched beams for (a) the DMC test and (b) the characterization tests. All the beams have the same width $W =$ 12.7 mm, height $H =$ 25.4 mm and length $L =$ 76.2 mm; the midspan is indicated with a dashed-dotted line.
• Figure 1: (a) Crack volume (black) defined by the phase-field iso-surface with $d=0.95$ (all elements with $d<0.1$ are hidden for visibility). (b) Definition of the numerical crack surface from averaging the minimum (green) and maximum (red) crack surface coordinates with $d>0.95$. (c) Post-processing of the crack surface asperity measurements (colored) and average experimental crack surface (black), data from dmc_calibration2.
• Figure 1: QR code for accessing the AR renders of the models.
• Figure 2: (a) Schematic representation of the domain $\Omega$ with prescribed traction $\bar{\boldsymbol{t}}$ on the Neumann boundary $\partial \Omega_{\text{N}}$ and prescribed displacement $\bar{\boldsymbol{u}}$ on the Dirichlet boundary $\partial \Omega_{\text{D}}$. The optimal profile of the phase-field variable along a coordinate perpendicular to the axis of the regularized crack $\Gamma_\ell$ is shown in (b) in comparison to the sharp crack representation which is recovered for a length-scale parameter $\ell \rightarrow 0$.
• Figure 3: Meshes for the (a) HC, (b) HB, (c) HA, (d) H45 and (e) DMC tests.