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A coupled phase field formulation for modelling fatigue cracking in lithium-ion battery electrode particles

W. Ai, B. Wu, E. Martínez-Pañeda

arXiv:2206.12727v1physics.chem-phcond-mat.mtrl-scics.CEphysics.app-ph

TL;DR

Particle-cracking models have not accounted for fatigue damage across realistic particle geometries and non-idealized stress states. The paper develops a multiphysics phase-field fatigue model and simulates particles over multiple cycles. It reports lower critical thresholds when fatigue is included, with reductions of 90%, 20%, and 60% for critical C-rate, particle radius, and initial crack size, respectively.

  • Problem

    High-fidelity modelling of particle cracking remains challenging because of its multiphysics nature, and a framework for fatigue growth in realistic geometries and non-idealized stress states is needed.

    Source: §1 · p. 3

  • Method

    The paper develops a multiphysics phase-field fatigue model and simulates cylindrical, spherical, and realistic electrode particles under multiple cycles.

    Source: §1 · p. 4, §4.4 · p. 24

  • Results

    90%, 20%, and 60% reductions in critical C-rate, particle radius, and initial crack size, respectively, are reported when fatigue damage is considered.

    Source: §4.4 · p. 24

  • Takeaways & Limitations

    The reported critical values support accounting for fatigue cracks in the design of battery control systems.

    Source: §4.4 · p. 24

    The reported critical values are specific to the material properties and conditions considered.

    Source: §4.4 · p. 24

Abstract

from arXiv · show

Electrode particle cracking is one of the main phenomena driving battery capacity degradation. Recent phase field fracture studies have investigated particle cracking behaviour. However, only the beginning of life has been considered and effects such as damage accumulation have been neglected. Here, a multi-physics phase field fatigue model has been developed to study crack propagation in battery electrode particles undergoing hundreds of cycles. In addition, we couple our electrochemo-mechanical formulation with X-ray CT imaging to simulate fatigue cracking of realistic particle microstructures. Using this modelling framework, non-linear crack propagation behaviour is predicted, leading to the observation of an exponential increase in cracked area with cycle number. Three stages of crack growth (slow, accelerating and unstable) are observed, with phenomena such as crack initialisation at concave regions and crack coalescence having a significant contribution to the resulting fatigue crack growth rates. The critical values of C-rate, particle size and initial crack length are determined, and found to be lower than those reported in the literature using static fracture models. Therefore, this work demonstrates the importance of considering fatigue damage in battery degradation models and provides insights on the control of fatigue crack propagation to alleviate battery capacity degradation.

1. Introduction

Particle cracking contributes to battery capacity fade, but existing approaches have not captured fatigue crack growth across realistic particle geometries and loading regimes. This work introduces a multiphysics phase-field fatigue model to investigate particle cracking and its dependence on operating and geometric conditions.

1. Introduction

  • Particle fracture reduces active material, while fresh crack surfaces support side reactions such as SEI growth and lithium inventory loss.Particle cracking is observed during cycling, and its propagation rate can increase with cycle number.Source: §1 · pp. 2–3
  • High-fidelity particle-cracking models remain challenging because the problem couples multiple physical processes.Earlier particle phase-field studies and cohesive-zone models considered single charge or discharge steps, despite cyclic loading and fatigue damage.Source: §1 · p. 3
  • Models imply multiple, identical and equally spaced microcracks at the surfaces of idealised electrode particles, and refer to the particles’ crack-growth behaviour.Simplified models can capture battery-capacity degradation up to 1 C currents, but fail at high C-rates, where electrode-particle microstructure plays a more dominant role.Source: §1 · p. 3
  • The proposed framework targets realistic particle geometries, non-idealized stress states, and the full range of fatigue behaviour.The introduction identifies these capabilities as a needed modelling framework.Source: §1 · pp. 3–4
  • The study presents a multiphysics phase-field model for fatigue crack propagation in electrode particles and examines C-rate, particle size, and crack geometry.The authors aim to connect particle-level fatigue insights with battery cell-level performance.Source: §1 · p. 4

2. Phase field modelling of fatigue cracking in electrode particles

The formulation couples lithium diffusion and chemical strain with phase-field fracture and a fatigue-damage model. Fatigue damage degrades fracture toughness over cyclic loading.

2. Phase field modelling of fatigue cracking in electrode particles

  • The formulation combines lithium diffusion and associated chemical strains, phase-field fracture, and variational fatigue damage.These components form the paper’s coupled theory for fatigue cracking in electrode particles.Source: §2 · p. 4

2.1. Lithium diffusion and chemical strains

  • Lithium diffusion is governed by a no-source conservation equation, and concentration-dependent chemical strain enters the stress–strain constitutive model.The formulation includes the hydrostatic-stress contribution to lithium transport.Source: §2.1 · pp. 4–5

2.2. A phase field description of fracture

  • The phase field represents intact and fully cracked regions as d = 0 and d = 1, with crack evolution following Griffith energy balance.The hybrid formulation uses stress equilibrium and a phase-field evolution law; a volumetric-deviatoric split prevents damage under compression.Source: §2.2 · pp. 5–6

2.3. Fatigue damage model

  • Fatigue damage accumulates through a degradation function that lowers fracture toughness; the approach recovers Paris’ law in the appropriate regime and conditions.The fatigue threshold is αT = Gc/(12l), above which fatigue damage can start.Source: §2.3 · p. 7

3. Methods: Numerical implementation

The numerical implementation uses finite elements to discretize the coupled concentration, displacement, and phase fields. Their governing equations are solved in a monolithic, fully coupled manner.

3. Methods: Numerical implementation

  • The governing equations are formulated in weak form and yield a discrete system of residual vectors.The formulation includes the coupled diffusion, mechanical, and fracture equations.Source: §3 · pp. 7–9
  • Finite elements discretize lithium concentration, displacement, and phase field using nodal shape-function approximations.The numerical implementation is described for two and three dimensions.Source: §3 · p. 8
  • The coupled equations are solved monolithically rather than with staggered schemes, which can require many load increments to capture equilibrium in fatigue problems.Newton–Raphson is used to obtain the solution.Source: §3 · p. 9
  • The implementation uses an implicit BDF scheme and the PARDISO solver in COMSOL Multiphysics.The model combines the Structural Mechanics, Transport of Diluted Species, and Helmholtz Equation modules.Source: §3 · p. 9

4. Numerical experiments

The numerical experiments examine fatigue cracking under cyclic flux loading in idealized cylindrical and spherical particles and realistic microstructures. They assess how C-rate, particle size, initial crack length, stress, lithium transport, fatigue damage, and crack interactions shape cracking thresholds and cycle-dependent growth, combining modelling with imaging for realistic particles.

4.1. Cylindrical electrode particle

  • Mesh sensitivity analysis shows only a 2% deviation in energy release rate (ERR) from the result obtained using 34,208 elements.The simulations cover 300 cycles in 2D problems and 140 cycles in the 3D case study.Source: §4 · p. 11, §4.1 · p. 12
  • Lithium accumulates near the crack tip because hydrostatic stress drives diffusion there, facilitating cracking particularly during the final stage of stable growth.The concentration and hydrostatic-stress distributions are shown at the end of discharge.Source: §4.1 · p. 13
  • Cracking progresses through stable growth, rapid propagation to the particle core, and renewed stable growth after the unstable crack arrests.The unstable event occurs during charging: outer tensile stresses drive propagation, which arrests near the compressive region; later tensile stresses at the core drive stable growth.Source: §4.1 · p. 13
  • Under 3 C cycling, the cylindrical particle shows initial stable crack growth, unstable cracking after 10 cycles, arrest near the centre, and subsequent stable growth.The energy release rate initially rises with cycle number; the unstable crack arrests as it approaches the particle centre.Source: §4.1 · pp. 12–13, Figure 2 · p. 14
  • Fatigue damage creates a heterogeneous toughness distribution that increases as the unstable crack propagates; lithiation also contributes fatigue damage in the particle core.Crack growth occurs during delithiation, while lithiation cycles damage the core.Source: §4.1 · p. 14
  • Under the conditions and lifespan considered, unstable crack growth occurred after 10 cycles at 3 C versus more than 130 cycles at 1 C; the critical C-rate was 0.5.Source: §4.1 · pp. 15–16
  • For 5 µm cylindrical particles cycled at 1 C for 300 cycles, initial cracks of 0.1 µm or larger are critical; this threshold is smaller than reported when fatigue is neglected.Cracks above the threshold reach approximately the same cracked-domain extent, but unstable cracking increases as initial crack size decreases; the most critical case is just above threshold.Source: §4.1 · p. 16
  • At 1 C with initial crack ratio a0/R0 = 0.02, the cylindrical-particle critical radius is 0.4 µm; larger particles have higher energy release rates and larger cracked surfaces.The number of cycles before unstable growth increases with particle size, while particles below the critical radius show no fatigue cracking under the tested loading and cycle conditions.Source: §4.1 · p. 17

4.2. Spherical electrode particle

  • For particles with R0 = 5 µm and a0/R0 = 0.02 over N = 300 cycles, rates at or below 2 C show negligible damage, while 5 C causes unstable cracking after approximately 50 cycles.Source: §4.2 · p. 18
  • Unstable cracking occurs only for the largest particle radius, R0 = 10 µm, with fast crack growth before 100 cycles; at R0 ≤ 6 µm, toughness degradation is negligible and no cracking occurs.Source: §4.2 · pp. 19–20

4.3. Realistic electrode particle

  • Realistic-particle crack area exhibits an exponential increase with cycle number, while cracks block diffusion and Li accumulates at crack fronts and in blocked regions.The discussion describes four stages overall: slow, accelerating, unstable growth, then arrest in low-stress regions followed by stable growth driven by fatigue damage and Li accumulation at crack tips.Source: Figure 6 · p. 21, §4.3 · p. 23, §4.4 · p. 24
  • In the realistic particle, cracked area grows slowly, accelerates, then grows rapidly; side cracks nucleate after about 80 cycles and later coalesce.The study identifies concave regions as crack-nucleation sites, and crack coalescence accompanies rapid area growth; the particle approaches end of life as cracks merge.Source: §4.3 · pp. 21–22

4.4. Analysis of the results and discussion

  • Higher particle radius and C-rate enhance fatigue cracking through larger concentration gradients and volume-change stresses; sufficiently short initial defects do not grow, but shorter cracks above the threshold grow more unstably.The trends reflect chemical strains, fatigue damage accumulation, and Li transport ahead of crack tips.Source: §4.4 · p. 24
  • Compared with Klinsman et al.’s static, monotonic-fracture results for one discharge process, accounting for fatigue damage reduces critical C-rate, particle radius, and initial crack size by 90%, 20%, and 60%, respectively.Source: §4.4 · p. 24
  • Complex cracking phenomena, including interactions between multiple cracks, occur early in the fracture of realistic particles.The results indicate that life predictions require a more comprehensive approach.Source: §4.4 · p. 25
  • Paris-law semi-empirical models based on idealisations such as equally spaced short cracks are unsuitable for analysing and designing fatigue-resistant electrodes.Source: §4.4 · p. 25
  • The results emphasise physically motivated models combined with advanced imaging techniques for fatigue analysis.Source: §4.4 · p. 25

5. Conclusions

The framework models fatigue cracking over hundreds of charge/discharge cycles and predicts crack-growth regimes, parameter thresholds, and behavior in realistic particle geometries. The conclusions also identify particle-to-electrode linking as future work.

5. Conclusions

  • The framework combines stress-driven extended Fickian lithium diffusion, intercalation-induced volumetric strains, phase-field cracks, and fatigue degradation to study particles over hundreds of cycles.Cracked-domain evolution is interpreted through lithium transport, chemical strains, and fatigue damage mechanics.Source: §5 · p. 25
  • Fatigue susceptibility increases with C-rate and particle size, while reducing an above-threshold initial crack length produces more unstable cracking.Source: §5 · p. 26
  • Critical C-rate, particle-size, and initial-crack-length values are significantly smaller than literature values based on half-cycle analysis without fatigue damage.Source: §5 · p. 26
  • Cracked-domain growth exhibits four regimes: slow, accelerating, fast unstable, and Li diffusion-driven stable growth after crack arrest.The authors state that this behavior cannot be captured by simple models.Source: §5 · p. 26
  • Coupling X-CT image analysis enables realistic particle geometries, whose fatigue behavior involves complex multiple-crack phenomena.Source: §5 · p. 26
  • The framework provides a platform for assessing design strategies against particle cracking and examining regimes of fast capacity degradation; linking particle- and electrode-level behavior remains future work.The authors also point to image-based modeling for studying electrode fragmentation and next-generation electrode design.Source: §5 · p. 26

CRediT author contribution statement

The contribution statement assigns methodology, software, investigation, data curation, and formal analysis to Weilong Ai, with project, funding, and writing roles distributed among the listed authors.

CRediT author contribution statement

Appendix A. Path dependence of the J-integral

The appendix reformulates the J-integral for a diffusing electrochemical solid and tests whether the modified integral depends on its integration path. Measurements from three paths differ by less than 4%.

Appendix A. Path dependence of the J-integral

  • The modified J-integral includes elastic strain energy and solute-diffusion energy contributions in a 2D electrochemical solid.The reformulation is attributed to.Source: Appendix A · p. 27
  • Path independence was tested on a cylindrical particle with two cracks under 5 C discharge using three integration paths.Source: Appendix A · p. 27
  • The three J-integral measurements were almost identical throughout the particle lifespan, with largest differences below 4%.Source: Appendix A · p. 27

Appendix B. Additional details of numerical implementation

The appendix describes formation of the consistent stiffness matrices by differentiating residuals with respect to nodal variables, then presents the linearised finite-element system.

Appendix B. Additional details of numerical implementation

  • Consistent stiffness matrices are obtained by differentiating the residuals in Eq. (14) with respect to nodal variables.Source: Appendix B · p. 28
  • The subscripts i and j on bold variables refer to contributions from nodes i and j, not matrix components.Source: Appendix B · p. 28
  • The linearised finite-element system is then stated, with its remaining terms given afterward.Source: Appendix B · p. 28
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