Abstract
We consider a free-boundary and free-discontinuity energy connecting phase separation and fracture in an elastic material. The energy excludes the contribution of phase boundaries in the cracked region, providing a heuristic approximation of the interfacial energy in the current material configuration. Our primary result shows that the sharp energy may be recovered via Γ-convergence from a modified Cahn–Hilliard energy coupled with an Ambrosio–Tortorelli-type approximation of the (linear) elastic and fracture energy.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: EXC-2047/1 – 390685813
Award Identifier / Grant number: 211504053 – SFB 1060
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2108784
Award Identifier / Grant number: DMS-2136198
Funding statement: This work grew out of Solveig Wittig’s Master’s thesis at the University of Bonn. During much of the writing of the paper Kerrek Stinson was at the Institute for Applied Mathematics at the University of Bonn. Kerrek Stinson was supported by funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813, the DFG project 211504053 – SFB 1060; also by the NSF (USA) under awards DMS-2108784 and DMS-2136198.
References
[1] S. Almi and E. Tasso, A new proof of compactness in G(S)BD, Adv. Calc. Var. 16 (2023), no. 3, 637–650. 10.1515/acv-2021-0041Search in Google Scholar
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar
[3] L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math. 43 (1990), no. 8, 999–1036. 10.1002/cpa.3160430805Search in Google Scholar
[4] M. Z. Bazant, Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics, Acc. Chem. Res. 46 (2013), no. 5, 1144–1160. 10.1021/ar300145cSearch in Google Scholar PubMed
[5]
G. Bellettini, A. Coscia and G. Dal Maso,
Compactness and lower semicontinuity properties in
[6] A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Math. 1694, Springer, Berlin, 1998. 10.1007/BFb0097344Search in Google Scholar
[7] A. Braides, Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University, Oxford, 2002. 10.1093/acprof:oso/9780198507840.001.0001Search in Google Scholar
[8] M. Bresciani, M. Friedrich and C. Mora-Corral, Variational models with Eulerian–Lagrangian formulation allowing for material failure, Arch. Ration. Mech. Anal. 249 (2025), no. 1, Paper No. 4. 10.1007/s00205-024-02076-7Search in Google Scholar
[9] L. Bungert and K. Stinson, Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning, Calc. Var. Partial Differential Equations 63 (2024), no. 5, Paper No. 114. 10.1007/s00526-024-02721-9Search in Google Scholar
[10]
A. Chambolle and V. Crismale,
A density result in
[11]
A. Chambolle and V. Crismale,
Compactness and lower semicontinuity in
[12] A. Chambolle and V. Crismale, Equilibrium configurations for nonhomogeneous linearly elastic materials with surface discontinuities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 24 (2023), no. 3, 1575–1610. 10.2422/2036-2145.202006_002Search in Google Scholar
[13]
A. Chambolle and V. Crismale,
A general compactness theorem in
[14] S. Conti and B. Schweizer, Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance, Comm. Pure Appl. Math. 59 (2006), no. 6, 830–868. 10.1002/cpa.20115Search in Google Scholar
[15] R. Cristoferi and G. Gravina, Sharp interface limit of a multi-phase transitions model under nonisothermal conditions, Calc. Var. Partial Differential Equations 60 (2021), no. 4, Paper No. 142. 10.1007/s00526-021-02008-3Search in Google Scholar
[16] H. Dal and C. Miehe, Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains, Comput. Mech. 55 (2015), no. 2, 303–325. 10.1007/s00466-014-1102-5Search in Google Scholar
[17] G. Dal Maso, An introduction to Γ-convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser, Boston, 1993. 10.1007/978-1-4612-0327-8Search in Google Scholar
[18] G. Dal Maso, Generalised functions of bounded deformation, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1943–1997. 10.4171/jems/410Search in Google Scholar
[19] G. de Philippis, N. Fusco and A. Pratelli, On the approximation of SBV functions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 2, 369–413. 10.4171/rlm/768Search in Google Scholar
[20]
I. Fonseca and G. Leoni,
Modern Methods in the Calculus of Variations:
[21] M. Friedrich, A piecewise Korn inequality in SBD and applications to embedding and density results, SIAM J. Math. Anal. 50 (2018), no. 4, 3842–3918. 10.1137/17M1129982Search in Google Scholar
[22] M. Friedrich, Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials, Math. Eng. 2 (2020), no. 1, 75–100. 10.3934/mine.2020005Search in Google Scholar
[23] M. Friedrich and F. Solombrino, Quasistatic crack growth in 2d-linearized elasticity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 1, 27–64. 10.1016/j.anihpc.2017.03.002Search in Google Scholar
[24] H. Garcke, On a Cahn–Hilliard model for phase separation with elastic misfit, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 2, 165–185. 10.1016/j.anihpc.2004.07.001Search in Google Scholar
[25] C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math. 24 (2013), no. 2, 179–211. 10.1017/S095679251200037XSearch in Google Scholar
[26] C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes, Springer, Wiesbaden, 2014. 10.1007/978-3-658-05252-2Search in Google Scholar
[27] C. Heinemann and C. Kraus, A degenerating Cahn–Hilliard system coupled with complete damage processes, Nonlinear Anal. Real World Appl. 22 (2015), 388–403. 10.1016/j.nonrwa.2014.09.019Search in Google Scholar
[28] C. Heinemann, C. Kraus, E. Rocca and R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 177–247. 10.1007/s00205-017-1102-7Search in Google Scholar
[29] D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 619–655. 10.1007/s00205-009-0271-4Search in Google Scholar
[30] F. Iurlano, A density result for GSBD and its application to the approximation of brittle fracture energies, Calc. Var. Partial Differential Equations 51 (2014), no. 1–2, 315–342. 10.1007/s00526-013-0676-7Search in Google Scholar
[31] S. Y. Kholmatov and P. Piovano, Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain, Adv. Calc. Var. 17 (2024), no. 3, 673–725. 10.1515/acv-2022-0053Search in Google Scholar
[32] P. Li, Y. Zhao, Y. Shen and S.-H. Bo, Fracture behavior in battery materials, J. Phys. Energy 2 (2020), no. 2, Article ID 022002. 10.1088/2515-7655/ab83e1Search in Google Scholar
[33] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal. 98 (1987), no. 2, 123–142. 10.1007/BF00251230Search in Google Scholar
[34]
L. Modica and S. Mortola,
Un esempio di
[35]
D. T. O’Connor, M. J. Welland, W. K. Liu and P. W. Voorhees,
Phase transformation and fracture in single Li
[36] M. Šilhavý, Equilibrium of phases with interfacial energy: a variational approach, J. Elasticity 105 (2011), no. 1–2, 271–303. 10.1007/s10659-011-9341-6Search in Google Scholar
[37] K. Stinson, On Γ-convergence of a variational model for lithium-ion batteries, Arch. Ration. Mech. Anal. 240 (2021), no. 1, 1–50. 10.1007/s00205-020-01602-7Search in Google Scholar
[38] K. Stinson, Existence for a Cahn–Hilliard model for lithium-ion batteries with exponential growth boundary conditions, J. Nonlinear Sci. 33 (2023), no. 5, Paper No. 69. 10.1007/s00332-023-09927-9Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the asymptotic behavior of a diffraction problem with a thin layer
- Degenerate parabolic p-Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions
- Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV
- A Sobolev gradient flow for the area-normalised Dirichlet energy of H 1 maps
- An elliptic approximation for phase separation in a fractured material
- The L 1-relaxed area of the graph of the vortex map: Optimal upper bound
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- On nonexpansiveness of metric projection operators on Wasserstein spaces
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- Universality of renormalisable mappings in two dimensions: the case of polar convex integrands
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- Harnack inequality for degenerate elliptic equations with matrix weights
- Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory
- The Capacitary John–Nirenberg Inequality Revisited
Articles in the same Issue
- Frontmatter
- On the asymptotic behavior of a diffraction problem with a thin layer
- Degenerate parabolic p-Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions
- Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV
- A Sobolev gradient flow for the area-normalised Dirichlet energy of H 1 maps
- An elliptic approximation for phase separation in a fractured material
- The L 1-relaxed area of the graph of the vortex map: Optimal upper bound
- Compactness of Palais–Smale sequences with controlled Morse index for a Liouville-type functional
- Characterization of sets of finite local and non-local perimeter via non-local heat equation
- On nonexpansiveness of metric projection operators on Wasserstein spaces
- A continuous model of transportation in the Heisenberg group
- Universality of renormalisable mappings in two dimensions: the case of polar convex integrands
- Convergence of a heterogeneous Allen–Cahn equation to weighted mean curvature flow
- Harnack inequality for degenerate elliptic equations with matrix weights
- Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory
- The Capacitary John–Nirenberg Inequality Revisited