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Finite Element Modeling of Fracture in Brittle Foods with Physically Consistent Parameter Calibration: A Practical Case Study on Cooled Dark Chocolate

Patrick Kurzeja, Felix Rörentrop, Kai Langenfeld, Jörn Mosler

arXiv:2610.37124v1cs.CE

TL;DR

Food fracture can be unwanted or deliberately induced, making peak load, failure deformation, and fragment geometry relevant product properties. This work calibrates sharp-interface and phase-field fracture models using cooled dark chocolate and an ordered, practical parameter-identification recipe. The results show that phase-field width creates trade-offs among structural response and crack-path predictions, so calibration objectives must be specified.

  • Problem

    Food fracture may damage quality during production and transport or be deliberately induced during cutting, grinding, and mastication, yet its product properties require reliable prediction.

    Source: §1.1 · p. 2, §6.2 · p. 20

  • Method

    The study assesses sharp-interface and phase-field models for cooled dark chocolate and identifies parameters in stages, choosing the framework according to whether the crack path is known.

    Source: §6.1 · p. 19, §7 · p. 20

  • Results

    13.9 % relative deviation in crack length at lφ = 10 mm versus 33.1 % at lφ = 1 mm; phase-field width also trades off peak displacement and crack-path predictions.

    Source: §5.2 · pp. 17–18

  • Takeaways & Limitations

    The phase-field width should be chosen for the application’s objective, because peak force, peak displacement, and crack path can favor different values within practical mesh bounds.

    Source: §5.2 · p. 18

    Finite phase-field width introduces curvature effects that vanish only in the idealized lφ → 0 limit, while the width must also be resolved by the mesh.

    Source: §2.4.4 · p. 8, §4.2.2 · p. 16

Abstract

from arXiv · show

Fracture in food can be undesired during production and transportation but also intended during cutting, grinding or mastication. The associated peak load, the deformation at failure and the resulting fragment geometry are all product properties. This work condenses the modeling of such fracture into a physically rigorous recipe that considers the practical constraints imposed by finite element simulations where necessary and applies idealized theoretical assessments where applicable. Cooled dark chocolate serves as a reproducible model material. Two well-established modeling frameworks for Griffith-type brittle fracture are assessed at an affordable spatial discretization: a sharp-interface and a diffuse phase-field approach. The assessment covers the choice of numerical framework, its discretization and a staged identification of the model parameters. Each parameter is identified from the loading stage that governs it: elastic parameters from pre-crack tension and compression, critical energy release rate from three-point-bending peak force, and the phase-field width last. The phase-field width interferes with the structural response, crack curvature and stress fields, depending on its lower bound set by the affordable spatial resolution. Elastic slope, peak force and crack path therefore cannot be met practically by a single parameter set. The objective must be fixed for the calibration of the model parameters and reported together with the numerical parameters and the boundary value problem.

1. Introduction

Food fracture can damage quality and increase waste, yet it is also deliberately induced in processes such as cutting and mastication. This work develops a practical, physically rigorous recipe for modeling brittle food fracture, assessing two frameworks and new cooled-dark-chocolate test data.

1.1. Mechanical damage and fracture in foods: motivation and classification

  • Accurate damage predictions matter because food fracture can impair quality and shelf-life, contribute to waste, and also occur intentionally during processing or consumption.Brittle damage is associated with elastic deformation and can occur with little noticeable macroscopic deformation.Source: §1.1 · p. 2

1.2. Scope: Practical and physically rigorous modeling recipe for fracture in brittle foods

  • The recipe assesses sharp-interface and phase-field frameworks under computationally affordable spatial resolution rather than relying only on idealized asymptotic limits.Cooled dark chocolate is the reproducible model material for the case study.Source: §1.2 · pp. 2–3
  • The modeling targets are elastic slope, peak force, and crack path, which relate respectively to sub-critical deformation, fracture thresholds, and fragment geometry.Peak force and elastic slope together determine peak displacement, important in displacement-controlled processes such as grinding and kneading.Source: §1.2 · p. 3
  • Cooled dark chocolate supports reproducible, low-effort casting into non-standard geometries, remelting for reuse, and tests of both frameworks against new experimental data.The data set comprises tension, compression, three-point-bending, and plate-bending tests.Source: §1.2 · p. 3

2. Numerical frameworks and model parameters

The paper compares discrete sharp-interface and diffuse phase-field descriptions of Griffith-type brittle fracture. In the sharp-interface formulation, crack growth is modeled through incremental energy minimization, with the critical energy release rate governing resistance to crack extension.

2.1. Griffith-type fracture and sharp vs. diffuse interface approaches

  • Sharp-interface models represent cracks as discrete surfaces, while diffuse models represent them as finite-width damage bands; the paper assesses one framework of each type.Other approaches, including XFEM, eigenerosion, adaptive methods, and cohesive zone models, offer different combinations of flexibility and numerical complexity.Source: §2.1 · p. 4

2.2. Sharp-interface approach and the critical energy release rate

  • The sharp-interface energy combines elastic strain energy, fracture energy, and the potentials of body forces and surface tractions.The crack measure is crack length in 2D and crack surface area in 3D.Source: §2.2 · p. 5
  • The critical energy release rate gc is the energy required to create unit crack surface area and governs resistance to crack growth and structural responses such as peak force.For linear elastic materials, gc is related to fracture toughness through Irwin’s relation; measured values may differ by loading mode, while these frameworks use a single mode-independent gc for the tension-dominated cracking considered.Source: §2.2 · p. 5
  • Incremental energy minimization advances displacement and crack set while enforcing crack irreversibility; along a prescribed path, crack growth occurs when the energy release rate reaches gc.The implementation places cracks along finite-element edges and uses local energy minimization.Source: §2.2 · p. 5

2.3. Phase-field approach and phase-field width

The phase-field approach replaces a discrete crack with a damage field that can propagate independently of mesh topology. Its finite width regularizes the crack representation but adds a parameter and a mesh-resolution requirement.

2.3. Phase-field approach and phase-field width

  • The phase-field variable φ is zero in intact material and one in fully broken material, with intermediate values smearing the crack over a finite-width band.Because the field is not fixed to element edges, cracks can ideally propagate and branch independently of mesh topology, making the approach suitable when the path is unknown in advance.Source: §2.3 · p. 6
  • The phase-field energy degrades elastic stiffness through f(φ) and replaces crack-surface measure with a volume integral of crack-surface density.Residual stiffness in broken material keeps the finite-element equations regular.Source: §2.3 · p. 6
  • The width lφ sets the transition-band width and regularizes the problem, but is introduced as a numerical parameter that ideally vanishes for practically sharp cracks.The mesh must be fine enough to resolve the band, making lφ an additional fracture-related parameter.Source: §2.3 · pp. 6–7
  • Incremental minimization solves for displacement and phase fields while the constraint φn−1 ≤ φn enforces irreversibility.The displacement fields remain continuous because the displacement jump is replaced by the degraded band.Source: §2.3 · p. 6

2.4. Modeling challenges

The section outlines practical challenges for modeling brittle fracture in food, including crack initiation and numerical effects that complicate representing crack paths and fields.

2.4.1. Crack nucleation (sharp interface and phase field)

  • A finite-width phase field can nucleate cracks in intact material at a threshold set by its numerical width, so both frameworks use an imposed initial crack.The imposed crack is a geometric imperfection in the sharp-interface model and a prescribed damage field in the phase-field model.Source: §2.4.1 · p. 7

2.4.2. Free-discontinuity problems (sharp interface)

  • Sharp-interface crack geometries are usually tied to element edges, making the method most suitable when the crack path is known in advance.The crack extent is then determined along the prescribed path.Source: §2.4.2 · p. 7

2.4.3. Mesh-dependent sharp-interface limit (phase field)

  • The phase-field width should approach zero but remain larger than the element size, so testing the sharp-interface limit requires refining both parameters together.Source: §2.4.3 · p. 7

2.4.4. Curvature interference (phase field)

  • At finite phase-field width, crack curvature enters the growth conditions and is an unintended effect when regularization is meant only to smear the crack.This curvature effect vanishes only in the idealized limit as the width approaches zero.Source: §2.4.4 · p. 8

2.4.5. Stress field interference (phase field)

  • The finite-width damage band blunts the crack-tip process zone, and stress variations below its scale cannot effectively steer crack growth.A hole, notch, or free edge within a few phase-field widths can interact with the crack at a distance.Source: §2.4.5 · p. 8

3. Experimental setup and cooled dark chocolate as a practical case study

The experiments establish elastic parameters for cooled dark chocolate and provide a practical basis for assessing brittle-fracture models. The study standardizes the material and testing conditions to reduce variability.

3.1. Available data basis for chocolate

  • Prior chocolate measurements vary widely with composition, temperature, and loading rate, and do not directly cover cooled dark chocolate.The authors therefore derive parameters from additional experiments and check them against literature data.Source: §3.1 · pp. 8–9

3.2. Sample production

  • The 70% dark chocolate was prepared through a standardized casting and cooling process, with rejected specimens and storage conditions controlled to support reproducible material behavior.Specimens were refrigerated before testing; air-pocketed specimens were discarded.Source: §3.2 · pp. 9–10

3.3. Experiments

  • Tension, compression, three-point bending, and plate bending serve successive recipe steps, from estimating elastic properties to observing a freely evolving crack path.Figure 3 depicts the specimen geometries and test results, while the force signals were corrected for machine and clamping effects.Source: §3.3 · p. 10, Figure 3 · p. 12

3.4. Reliability and uncertainties

  • The model treats chocolate as ideally brittle and isotropic, neglects viscous effects at the slow test rate, and relies on equal, negligible warming across specimens.The measured peak force in three-point bending was 0.8293 (±0.0586) N; the authors state the derived values suit this methodological assessment, with deviations to consider elsewhere.Source: §3.4 · pp. 10–11

3.5. Derived elastic parameters

  • Simulations use E = 25 MPa and ν = 0.4, based on tension and compression estimates, with the modulus independently checked against the three-point-bending elastic response.The compression estimate was ν = 0.4017 ± 0.0163, and the tension estimate was E = 24.844 ± 6.039 MPa.Source: §3.5 · p. 11

4. Three-point-bending – Comparison of numerical frameworks for straight cracks

Three-point bending compares sharp-interface and phase-field models for a straight crack, calibrating fracture parameters against the experimental peak force while tracking effects on the structural response. At finite mesh size, phase-field width and fracture toughness must be calibrated together, and matching peak force alone does not guarantee a full-response match.

4.1. Sharp-interface solutions

  • Local energy minimization is the cheapest sharp-interface strategy and matches the target elastic slope, peak force, and prescribed crack path.The calibrated critical energy release rate is 0.0043 N/mm; global minimization predicts fracture below half the experimental force, while arc-length control resolves snap-back at the local solution’s peak force.Source: §4.1 · pp. 13–14, Figure 5 · p. 14

4.2.1. Numerical calibration and convergence for lφ and h

  • At fixed mesh size, lφ = 0.0184 mm gives a good phase-field match, while widths that are too large or too small introduce distinct errors.Large widths regularize and smooth cracking; small widths that the mesh cannot resolve overshoot peak force. The affordable h = 0.0056 mm, lφ = 0.0184 mm combination lies in the energy-minimizing valley and is 1.9% from the experimental peak force.Source: §4.2.1 · p. 14, Figure 6 · p. 15

4.2.2. Mixed calibration with lφ and gc at finite mesh size h

  • At finite mesh size, lφ co-determines structural response, so gc and lφ must be treated as joint fracture-related calibration parameters.Their peak-force effects oppose one another: gc follows exponent 1/2, while lφ follows exponent −0.173 in this finite-mesh specimen, reflecting its intermediate stress concentration.Source: §4.2.2 · p. 15
  • Choosing an admissible lφ and rescaling gc can preserve the experimental peak force without changing mesh size, but peak-force matching alone does not reproduce the full response.lφ must also remain small enough to retain an instantaneous fracture profile; otherwise the force peak smooths and biases elastic slope and peak displacement, which matters in displacement-controlled processes.Source: §4.2.2 · pp. 15–16

5. Plate-bending – phase-field width and crack path interference

A notched, holed chocolate plate was bent to test a freely evolving crack against phase-field simulations. Larger phase-field widths better matched the experimental crack path in this setup, although the preferred width depends on the calibration target.

5. Plate-bending – phase-field width and crack path interference

  • The experiment addressed freely evolving cracks, unlike the prior three-point-bending case, where symmetry and a stress singularity practically predetermined the path.Such unconstrained paths also matter for food transport, including chocolate, chips, and eggs.Source: §5 · p. 16
  • The phase-field method was selected for its mesh-independent formulation and suitability for freely evolving cracks, while its finite width can affect curvature and stress fields.Other parameters were held fixed so their recalibration would not mask or bias the width-related interference.Source: §5 · p. 16

5.1. Setup

  • A 100 mm × 100 mm × 5 mm plate of cooled dark chocolate with 70 % cocoa at 3 °C was cast, drilled, and notched, then loaded to induce cracking at 5.0–7.5 N.One half was clamped, and a calibrated weight was lowered slowly; the mesh was refined around the pre-crack and on the right side of the drilled hole.Source: §5.1 · p. 17

5.2. Results and discussion

  • All four specimens produced similar straight cracks, averaging 18.597 ± 1.687 mm in length, with r2 det = 0.983 for straightness.The notch and hole reliably governed crack evolution; observed deviations also reflected fabrication inaccuracies in notch-hole distance.Source: §5.2 · p. 17
  • 13.9% relative crack-length deviation at lφ = 10 mm rose to 33.1% at lφ = 1 mm, with the largest width closest to experiment.The overall crack shape showed the same agreement trend, though a perfect match was precluded by quasi-static modeling, numerical approximations, and experimental inaccuracies.Source: §5.2 · p. 17, Figure 9 · p. 18, Figure 10 · p. 18

6. Modeling recipe for brittle fracture in foods by the practical example of cooled dark chocolate

The recipe orders framework choice, discretization, and parameter identification because numerical and material parameters interact at finite mesh size. Its calibration targets and checks are chosen for the intended food process and final boundary-value problem.

6. Modeling recipe for brittle fracture in foods by the practical example of cooled dark chocolate

  • The recipe orders modeling decisions because numerical and material parameters are not independent at finite mesh size.It is designed for computational resources typically available in applied settings.Source: §6 · p. 19

6.1. The six steps

  • Choose a sharp-interface framework for a known crack path and phase field for a free path, then select element size and an initial crack with computational cost and propagation in mind.The phase-field mesh must resolve lφ; an initial crack makes propagation, and hence gc, the governing mechanism.Source: §6.1 · p. 19
  • Identify E and ν before fracture and cross-check them in an independent configuration; prefer alternative boundary conditions when clamping is unreliable.Source: §6.1 · p. 19
  • For fixed geometry, crack path, and stiffness, identify gc from a known-path specimen by rescaling one simulation because Fpeak ∝√gc.The scaling also holds for the phase-field approach.Source: §6.1 · p. 19
  • For phase field, choose lφ within its admissible range, rescale gc to peak force, and check the full force-displacement profile, including peak displacement.Report the regularization-dependent gc with lφ and h; verify the final parameter set against the final boundary-value problem for crack-path compatibility or interference.Source: §6.1 · p. 19

6.2. Implications for food engineering

  • Food processes set the calibration objective: peak force matters for force-controlled handling, while peak displacement matters more for grinding, kneading, and partly chewing.Peak forces of a few newtons may warrant suitable load cells and signal leveling with sample-free reference runs.Source: §6.2 · pp. 19–20
  • A standardized chocolate fabrication route kept geometric scatter around 1% and peak-force reproducibility within 7%.Chocolate permits specimens with varied geometric features, but standardization is a prerequisite for that reproducibility.Source: §6.2 · p. 20
  • In food applications, crack path can be a product property that determines fragment geometry, processability, aroma release, and perceived quality.Scoring lines, inclusions, pores, and layered structures can make paths partly predictable, favoring sharp-interface modeling in those cases.Source: §6.2 · p. 20

7. Conclusion and outlook

The study assesses sharp-interface and phase-field fracture modeling with cooled dark chocolate and develops a practical calibration recipe. It finds that the best framework and parameter choices depend on crack-path information and the calibration objective, so results must be reported with their numerical setting and boundary-value problem.

7. Conclusion and outlook

  • The sharp-interface approach reproduced bending-test elastic slope, peak force, and straight crack path with the lowest computational effort, but suits paths known in advance best.Phase field is more suitable for free-discontinuity problems but requires lφ as an additional parameter.Source: §7 · p. 20
  • At finite mesh size, calibrated parameters are tied to the numerical setting and regularization, and phase-field width co-determines structural response.The critical energy release rate could not simply be transferred to the phase-field model, and idealized scaling relations require scrutiny for foods with notches, pores, or sharp corners.Source: §7 · p. 20
  • Peak force, peak displacement, and crack path can favor different phase-field widths, so a parameter set is valid only for its stated objective and boundary-value problem.Both the objective and boundary-value problem must be reported for non-idealized limit cases.Source: §7 · pp. 20–21
  • Inclusions, pores, layered structures, temperature, and loading-rate sensitivity motivate extensions such as thermal, plasticity, heterogeneity, anisotropy, curvature, and dynamic formulations.Alternative tests, including splitting tensile tests, may be preferable when clamping is unreliable, though they have a non-uniform stress state.Source: §7 · p. 21

Appendix A. Numerical implementation

The implementation uses Matlab finite elements and lists the simulation parameters in an appendix table.

Appendix A. Numerical implementation

  • The simulations use a finite-element implementation in Matlab, with parameter values as summarized in Table A.1 unless otherwise noted.Source: Appendix A · p. 30
  • Table A.1 summarizes the material and numerical parameters used for simulations unless otherwise specified.Source: Table A.1 · p. 30

Appendix A.1. Sharp interface

The sharp-interface implementation represents cracks along element edges and selects crack configurations by minimizing total energy. The bending setup uses a prescribed crack path along the symmetry line.

Appendix A.1. Sharp interface

  • Cracks follow element edges with no traction transfer; duplicating nodes permits displacement jumps without adding degrees of freedom, while elements remain elastic.Source: Appendix A.1 · p. 30
  • For each candidate crack set, the method solves elastic equilibrium and selects the configuration with the lowest total energy.Quadratic bulk-energy scaling means only one finite-element computation is required per crack configuration.Source: Appendix A.1 · p. 30
  • The two-dimensional bending model uses 31k quadrilateral and triangular elements, with square bilinear elements of edge length h = 0.0056 mm near the symmetry line.Source: Appendix A.1 · p. 30
  • Symmetry-line element edges define the prescribed crack path; plane-stress conditions and a 0.25 mm pre-crack made from disconnected elements complete the setup.Source: Appendix A.1 · p. 30

Appendix A.2. Phase field

The phase-field implementation uses linear discretization and a history-field approach to enforce irreversibility. Boundary conditions prevent damage at the bending supports and loading region, while the plate mesh balances accuracy and calculation time.

Appendix A.2. Phase field

  • Both displacement and phase fields use linear shape functions, and irreversibility is enforced with the history-field approach of Miehe et al..The incremental problem is solved for a local solution using the algorithm in [70].Source: Appendix A.2 · p. 30
  • Pre-cracks are prescribed by fully damaging the field, while phase-field boundary conditions prevent damage at the three-point-bending supports and loading region.Source: Appendix A.2 · p. 30
  • For plate bending, two layers of linear hexahedral elements through the thickness, about 7k total, compromise between accuracy and calculation time.Source: Appendix A.2 · p. 30
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