Virtual Fracture Experiments

• Virtual Fracture Experiments are computational procedures using numerical models that simulate crack initiation, propagation, and coalescence in materials.
• They integrate techniques like finite element, FDEM, phase-field, and graph-based methods to replicate complex fracture behaviors.
• These experiments enable rigorous validation against physical tests and are applied in geomechanics, biomechanics, and composite design.

Virtual fracture experiments are computational procedures that quantitatively simulate the initiation, propagation, and coalescence of cracks in solids and composites using mathematically rigorous numerical models. These experiments serve both as surrogates for physical testing and as tools to investigate mechanisms that are inaccessible or impractical to measure in situ. Approaches span continuum finite elements, discrete/combined element techniques, phase-field and cohesive-zone models, virtual image-based quantification, and, more recently, data-driven and machine-learning-based frameworks. The objective is to predict load-displacement response, fracture paths, energy dissipation, and failure modes at resolutions and scenarios relevant to engineering, geomechanics, biomechanics, and materials science.

1. Foundational Numerical Frameworks for Virtual Fracture

Virtual fracture experiments are grounded in advanced finite element (FE), boundary element (BE), and hybrid discrete-continuum methods. In standard FE-based formulations, fracture is accommodated by local criteria (stress, strain, energy release rates) and adaptively modifying the computational mesh to insert new surfaces as cracks evolve. Notable frameworks include:

• Nonlinear Finite Element Dynamical Frameworks: Employing Green–Lagrange strain measures and Lamé-parameter-based constitutive relations, these models simulate both brittle and ductile fracture by explicit evolution of tetrahedral meshes upon satisfying fracture criteria. Node duplication and local re-meshing enable arbitrary crack path formation and realistic fragment generation (O'Brien et al., 2023).
• Combined Finite-Discrete Element Method (FDEM): FDEM enables capturing the transition from localized damage to macro-fragmentation. It incorporates interfacial damage models (mode-I and mode-II cohesive laws), explicit time integration, and contact algorithms to resolve coalescing cracks and fragment assembly in heterogeneous materials (e.g., fracture coalescence in granite) (Euser et al., 2018).
• Virtual Element Method (VEM) and Extended VEM (X-VEM): VEM generalizes finite elements to arbitrary polygonal/polyhedral meshes, enabling mesh flexibility in crack-tracking and adaptive refinement. X-VEM further introduces crack-tip enrichments analogous to XFEM, achieving mesh-independent singularity representation while maintaining computational robustness for crack-tip field and SIF extraction (Benvenuti et al., 2021).
• Remeshing-Free Graph-Based FEM: This approach recasts hyperelastic strain energy exclusively in terms of mesh edge lengths and propagates fracture by decoupling edge forces upon local damage thresholds. The fixed-size global system matrix circumvents the computational explosion traditionally associated with remeshing, providing scalable simulation on high-resolution geometries (Mandal et al., 2021).

2. Variational and Phase-Field Formulations

Modern virtual fracture models leverage variational methods that regularize sharp crack interfaces and obviate a priori crack-path assumptions. Key developments include:

• Gradient Damage / Phase-Field Models: A scalar phase-field variable $d(\boldsymbol{x}) \in [0,1]$ (undamaged to fully broken) spatially diffuses the otherwise sharp crack discontinuity. The energy functional couples degradation of the elastic energy with a regularized crack energy (e.g., Ambrosio–Tortorelli type), yielding the coupled system for displacement and damage fields. Phase-field models naturally handle complex crack nucleation, branching, and merging phenomena, and can incorporate anisotropy, finite strains, and microstructural considerations (Hug et al., 2022, Lee et al., 2024, Moreno-Mateos et al., 2024).
• Configurational Force and J-Integral Methods: In highly deformable soft materials, virtual experiments utilize hyperelastic calibrations fitted to full-field experiments, coupled to configurational force evaluation at crack tips for virtual toughness extraction. This framework is particularly suited to elastomers under multiaxial loading (Moreno-Mateos et al., 26 May 2025).
• Variational Multiscale Cohesive Methods (VMCM): These methods explicitly enrich the finite element solution near the crack to both impose traction-separation relations and allow for crack path objectivity. VMCM distinguishes between bulk and microstructural scales, permitting objective tracking of bridging zones and size-dependent fracture resistance in fiber-reinforced composites (Rudraraju et al., 2015).

3. Virtual Fracture Experiment Design and Workflow

A typical virtual fracture experiment comprises the following essential steps:

1. Geometry and Meshing: Generation of specimen geometries—ranging from simplified laboratory analogs (three-point bend bars, notched plates, composite coupons) to biological or heterogeneous natural media (bones, ores)—and discretization using appropriate mesh types (tetrahedral, hexahedral, polygonal).
2. Material Parameterization: Assignment of constitutive parameters (Young’s modulus, Poisson’s ratio, yield strength, fracture toughness, critical energy release rate, anisotropy tensors), often spatially mapped from ex vivo imaging (e.g., CT-based density-to-stiffness for bone) or phase segmentation of composites (Hug et al., 2022, Wilhelm et al., 2024).
3. Boundary and Loading Conditions: Application of relevant boundary constraints and load histories, including displacement or force control, or impact/velocity-loading scenarios for dynamic splitting tests (O'Brien et al., 2023, Mandal et al., 2021).
4. Fracture Criteria and Evolution Laws: Specification of criteria for damage initiation (tensile/compressive, Mohr–Coulomb, energy-based, SIF threshold). For phase-field models, this involves an irreversibility condition (history variable), while in graph-based or FDEM frameworks it is local (per element/edge/interface) (Euser et al., 2018, Lee et al., 2024).
5. Numerical Solution and Adaptive Control: Use of time integration schemes (explicit/implicit Newmark, central difference, Newton–Raphson for steady/quasi-static cases), mesh refinement strategies (superconvergent patch recovery, enrichment radius control), and staggered or monolithic solvers for coupled field problems (Benvenuti et al., 2021, Hug et al., 2022).
6. Output Analysis and Validation: Virtual fracture experiments extract force-displacement curves, crack path, SIFs, damage evolution snapshots, fragment statistics (size, shape distribution), and stress/strain fields. Validation against experimental benchmarks includes regression metrics (e.g., $R^2$, RMSE), comparison of failure loads, and qualitative match of fracture topologies (Hug et al., 2022, Moreno-Mateos et al., 26 May 2025).

4. Novel Data-Driven and Image-Based Virtual Fracture Experiments

Recent advances extend virtual fracture experiments to include data-driven, learning-based, and imaging methodologies:

• Generative Neural Approaches: Techniques such as DeepFracture employ BEM-generated fracture patterns as ground truth for supervised learning of conditional generative models. Latent impulse codes encode collision parameters, and the network outputs volumetric geometric-segmentation signed distance fields (GS-SDF) for fragment prediction. This enables runtime prediction of realistic brittle fracture patterns under arbitrary collision histories, integrated with physics engines for interactive animation (Huang et al., 2023).
• Virtual Reassembly from Tomography: Algorithms align and reassemble 3D CT-imaged fragments of post-mortem fractured composites or minerals, using rigid registration (mask-normalized cross-correlation, FFT optimization), hierarchical matching, and subsequent voxelwise analysis of intergranular vs. transgranular crack mechanisms. Entropy-based local neighborhood analysis quantifies fracture mode, providing data-driven insight into microstructural failure modes relevant for process engineering and recycling optimization (Wilhelm et al., 2024).

5. Size Effects, Anisotropy, and Complex Microstructures

Virtual fracture experiments enable detailed study of scaling effects, fracture anisotropy, and microstructural toughening mechanisms:

• Size-Dependent Fracture: Using phase-field/gradient-damage formulations with an intrinsic length scale $\ell$, virtual experiments reproduce the transition from flaw-sensitive to flaw-insensitive rupture as $a/\ell$ varies, matching experimental data for a wide range of elastomeric materials. The ratio $\Gamma/W^*$ calibrates the process zone size, and the models capture quantitative trends in normalized rupture stretch (Lee et al., 2024).
• Anisotropy and Microstructural Programming: Phase-field models with stretch-activated fracture energy and an explicit dependence on microstructural directionality (via crosslinking degree or inclusions) allow virtual programming of crack path and work-of-fracture. Computational testbeds model composite unit cells with tailored toughness and guide/bias crack propagation, supporting design of toughened soft materials (Moreno-Mateos et al., 2024).
• Bridging Zones in Composites: The VMCM reveals specimen-size dependence of bridging zones and resultant fracture resistance in laminated fiber-reinforced composites, challenging classical assumptions of material-invariant toughness and enabling objective simulation of large crack bridging and its evolution (Rudraraju et al., 2015).

6. Computational Efficiency and Remeshing Strategies

A core bottleneck in large-scale virtual fracture simulation is mesh management:

• Remeshing and Mesh-Independence: Classic FEM approaches require element splitting and adaptive remeshing (with associated matrix resizing), while methods such as graph-based FEM (Mandal et al., 2021) and VEM/X-VEM (Benvenuti et al., 2021) maintain fixed-sized system matrices or exploit mesh-agnostic spaces. These paradigms promote computational tractability for detailed virtual fracture experiments involving millions of DOFs or real-time integration with interactive or data-driven processes.
• Efficient Quadrature and Singularity Treatment: Integration of crack-tip singularities and discontinuities (e.g., $r^{-1/2}$ fields) is handled in VEM/X-VEM by reducing integrals to one-dimensional edge quadrature, ensuring both accuracy and efficiency on general meshes (Benvenuti et al., 2021).

7. Validation, Applications, and Outlook

Virtual fracture experiments are rigorously validated through direct comparison to laboratory test data, including load-displacement curves, full-field strain (e.g., DIC), fragment size statistics, and crack topologies. Applications span from geomechanics (granite fracture coalescence (Euser et al., 2018), mineral liberation (Wilhelm et al., 2024)) to biomechanics (CT-based bone failure (Hug et al., 2022)), polymer science (elastomer toughness (Lee et al., 2024, Moreno-Mateos et al., 26 May 2025, Moreno-Mateos et al., 2024)), and computational graphics (real-time destruction for animation (O'Brien et al., 2023, Huang et al., 2023)).

Future virtual experiments are trending toward multi-physics integration, high-throughput virtual libraries for composite design, and seamless fusion of experimental data, mechanistic modeling, and data-driven generative architectures. These developments position virtual fracture as a central tool for both scientific discovery and engineering design in complex materials and structures.