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arxiv: 2604.18666 · v1 · submitted 2026-04-20 · ❄️ cond-mat.mtrl-sci

Thermo-mechanically coupled phase-field fracture model considering elastocaloric effect of shape memory alloy

Shen Sun , Wei Tang , Weiwei He , Igor Polozov , Min Yi This is my paper

Pith reviewed 2026-05-10 03:44 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords phase-field modelshape memory alloyelastocaloric effectfracturethermo-mechanical couplingmartensitic transformationMn-Cu alloytoughening
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The pith

A phase-field model couples fracture in shape memory alloys with elastocaloric effect to show thermal toughening.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a phase-field fracture model for shape memory alloys that accounts for both cracking and the temperature changes during martensitic transformation along with the associated elastocaloric effect. It includes thermal strains from the cooling and eigenstrains from the phase change, plus an empirical rule that lowers thermal conductivity in fractured zones. Finite-element simulations of Mn-Cu SMA tensile tests show martensite nucleating at crack sites and spreading at 45 degrees, with the elastocaloric thermal expansion raising the load needed to propagate cracks. Larger values for the phase-transition kinetic parameter and orientation angle increase strength and temperature swings but reduce overall deformation capacity. The approach illustrates how the elastocaloric effect can be harnessed for coupled toughening and offers a route to more fracture-resistant elastocaloric devices.

Core claim

The thermo-mechanically coupled phase-field fracture model incorporates the thermal strain induced by the elastocaloric effect and the eigen strain induced by the phase transition. An empirical degradation function is adopted to describe the thermal conductivity decreasing with the fracture order parameter. The model is validated with the finite element method and tensile fracture properties of Mn-Cu SMA are simulated. It is found that the martensite variant nucleates at the stress concentration where the crack initiates, and commonly spreads with an angle of 45 degree. The thermal expansion strain caused by the eCE could strengthen the critical load capacity. A large kinetic parameter for相变

What carries the argument

Phase-field fracture order parameter coupled to thermo-mechanical equations that include elastocaloric thermal strain, martensitic eigenstrain, and an empirical degradation function for thermal conductivity.

If this is right

  • Martensite variant nucleates at the stress concentration where the crack initiates and spreads with an angle of 45 degree.
  • The thermal expansion strain caused by the eCE could strengthen the critical load capacity.
  • A large kinetic parameter for phase transition and the large orientation angle could enhance the strength and temperature change of eCE while the deformation capacity is reduced.
  • The phase-field model demonstrates its ability in the thermal-mechanically coupled toughening of SMA.
  • It also provides a possible fracture-resistance strategy by the utilization of eCE for elastocaloric devices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers of elastocaloric cooling devices could use the model to choose geometries and operating conditions that improve both cooling performance and crack resistance under load.
  • The same coupling framework may be extended to simulate fracture in other active materials that undergo phase transitions under combined thermal and mechanical fields.
  • Cyclic loading simulations with this model could predict long-term durability and fatigue life for SMA components in real devices.
  • Implementation in commercial finite-element packages would allow engineers to optimize SMA parts for both elastocaloric output and mechanical reliability.

Load-bearing premise

An empirical degradation function accurately captures the decrease in thermal conductivity as a function of the fracture order parameter, and the chosen kinetic parameter and orientation angle values generalize beyond the specific Mn-Cu tensile simulations shown.

What would settle it

Tensile fracture tests on Mn-Cu SMA that find no increase in critical load from elastocaloric thermal expansion or martensite spreading not at 45 degrees from crack initiation sites would falsify the central predictions.

Figures

Figures reproduced from arXiv: 2604.18666 by Igor Polozov, Min Yi, Shen Sun, Wei Tang, Weiwei He.

Figure 1
Figure 1. Figure 1: Fig.1. (a) Geometry and boundary conditions of the single [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

Modelling fracture behavior of the shape memory alloy (SMA) that interacts with martensitic transformation and the associated elastocaloric effect (eCE) still remains challenging. Herein, a thermo-mechanically coupled phase-filed fracture model considering elastocaloric effect of SMA is proposed to simulate the cracking process coupled with the non-isothermal martensitic transformation and the associated eCE. In the phase-field model, both the thermal strain induced by eCE and the eigen strain induced by the phase transition are considered. An empirical degradation function is adopted to describe the thermal conductivity decreasing with the fracture order parameter. The model is validated with the finite element method and tensile fracture properties of Mn-Cu SMA are simulated. It is found that the martensite variant nucleates at the stress concentration where the crack initiates, and commonly spreads with an angle of 45 degree. The thermal expansion strain caused by the eCE could strengthen the critical load capacity. A large kinetic parameter for phase transition and the large orientation angle could enhance the strength and temperature change of eCE while the deformation capacity is reduced. The phase-field model demonstrates its ability in the thermal-mechanically coupled toughening of SMA. It also provides a possible fracture-resistance strategy by the utilization of eCE for elastocaloric devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a thermo-mechanically coupled phase-field fracture model for shape memory alloys that incorporates the elastocaloric effect (eCE). The formulation accounts for thermal strains from eCE and eigenstrains from martensitic phase transitions. An empirical degradation function is adopted for the reduction of thermal conductivity with the fracture order parameter. The model is implemented and validated via the finite element method, then applied to simulate tensile fracture of Mn-Cu SMA. Reported outcomes include nucleation of martensite variants at stress concentrations with a characteristic 45-degree spread, an increase in critical load capacity due to thermal expansion strain from eCE, and the effects of a large kinetic parameter for phase transition and large orientation angle in enhancing strength and temperature change while reducing deformation capacity. The work concludes that the model demonstrates thermo-mechanical toughening of SMA and offers a possible fracture-resistance strategy via eCE for elastocaloric devices.

Significance. If the central claims hold after addressing the modeling choices, the work supplies a numerical framework for exploring coupled fracture, phase transformation, and caloric effects in SMAs. This could support design of more fracture-resistant elastocaloric devices by quantifying how eCE-induced thermal strains influence load capacity. The reported simulation outcomes (45-degree martensite spread and load strengthening) are physically plausible and illustrate the potential of non-isothermal coupling. The attempt to link phase-field fracture with eCE is a constructive step, though the overall significance remains moderate given the empirical elements and lack of experimental benchmarks for the coupled phenomena.

major comments (2)
  1. [Phase-field model formulation] The empirical degradation function adopted to describe thermal conductivity decreasing with the fracture order parameter (introduced in the phase-field model section) is presented without derivation from first principles, calibration against measured data, comparison to alternative forms (linear, power-law, etc.), or sensitivity analysis on its parameters. This choice directly controls the temperature field evolution, eCE magnitude, thermal strains, and the claimed increase in critical load capacity; without validation or robustness checks, the thermo-mechanical toughening demonstration rests on an untested modeling assumption.
  2. [Numerical simulations and parameter studies] In the simulation results for Mn-Cu tensile fracture, the kinetic parameter for phase transition and the orientation angle are stated to have large values that enhance strength and temperature change. The manuscript does not clarify whether these quantities are obtained from independent material characterization or adjusted to reproduce the observed load-displacement curves, which weakens the claim that the coupled model provides a general fracture-resistance strategy.
minor comments (2)
  1. [Abstract and validation subsection] The abstract states that the model is 'validated with the finite element method,' but the specific benchmarks, mesh convergence checks, or quantitative error metrics used in the validation are not detailed in the results section.
  2. [Throughout the model section] Notation for the phase-field order parameter, degradation functions, and the distinction between thermal and eigenstrains would benefit from a dedicated nomenclature table or explicit equation references in the main text to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and indicate the revisions we will make to improve the presentation and robustness of the work.

read point-by-point responses

  1. Referee: [Phase-field model formulation] The empirical degradation function adopted to describe thermal conductivity decreasing with the fracture order parameter (introduced in the phase-field model section) is presented without derivation from first principles, calibration against measured data, comparison to alternative forms (linear, power-law, etc.), or sensitivity analysis on its parameters. This choice directly controls the temperature field evolution, eCE magnitude, thermal strains, and the claimed increase in critical load capacity; without validation or robustness checks, the thermo-mechanical toughening demonstration rests on an untested modeling assumption.

    Authors: We acknowledge that the degradation function is empirical, as noted in the manuscript, and that further justification is warranted. The functional form was chosen to ensure a monotonic reduction in effective thermal conductivity as the fracture order parameter increases, reflecting the physical role of cracks as thermal barriers while maintaining bounded values between the intact and fully damaged states. Although derivation from first principles or direct calibration against Mn-Cu data for the coupled thermo-mechanical-fracture problem is not available in the literature, we will revise the manuscript to include a direct comparison of the adopted function against linear and power-law alternatives together with a sensitivity study on its parameters. These additions will quantify the influence on temperature evolution, elastocaloric temperature change, and the reported increase in critical load, thereby demonstrating the robustness of the thermo-mechanical toughening result. revision: yes

  2. Referee: [Numerical simulations and parameter studies] In the simulation results for Mn-Cu tensile fracture, the kinetic parameter for phase transition and the orientation angle are stated to have large values that enhance strength and temperature change. The manuscript does not clarify whether these quantities are obtained from independent material characterization or adjusted to reproduce the observed load-displacement curves, which weakens the claim that the coupled model provides a general fracture-resistance strategy.

    Authors: The kinetic parameter and orientation angle are representative values drawn from the SMA literature to illustrate the parametric sensitivity of the coupled fracture, phase transformation, and elastocaloric response. They are not fitted to any specific experimental load-displacement data in the present study; the simulations are intended to reveal qualitative trends rather than quantitative predictions for a particular specimen. We agree that this distinction should be stated explicitly. In the revised manuscript we will clarify the literature basis for these choices, emphasize that the model serves as a general framework for exploring eCE-based toughening strategies, and note that material-specific calibration would be required for quantitative device design. revision: yes

Circularity Check

0 steps flagged

No significant circularity: model assumptions and simulations are self-contained.

full rationale

The paper adopts an empirical degradation function for thermal conductivity as a modeling choice, implements a thermo-mechanically coupled phase-field formulation, and performs FEM-based simulations of Mn-Cu tensile fracture to demonstrate effects of the elastocaloric coupling and parameter variations. No load-bearing step reduces a claimed result to its inputs by construction, no fitted parameters are renamed as independent predictions, and no self-citation chain is invoked to justify uniqueness or ansatz choices. The reported behaviors (45° martensite spread, strengthening via thermal strain, influence of kinetic parameter and orientation angle) follow directly from the stated model equations and chosen inputs rather than tautological equivalence. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on the standard variational phase-field representation of fracture, the assumption that martensitic transformation can be described by a kinetic equation with a single scalar parameter, and an empirical (non-derived) function linking fracture order parameter to thermal conductivity. No new particles or forces are postulated.

free parameters (2)
  • kinetic parameter for phase transition
    Described as 'large' to enhance strength and temperature change; its value is chosen or fitted to produce the reported simulation outcomes.
  • parameters inside the empirical thermal-conductivity degradation function
    The function is stated to be empirical; its coefficients are not derived from independent measurements.
axioms (2)
  • standard math Phase-field order parameter smoothly approximates a sharp crack interface
    Invoked throughout the phase-field fracture formulation.
  • domain assumption Total strain decomposes additively into mechanical, thermal (eCE), and eigenstrain (phase transformation) contributions
    Central modeling choice stated in the abstract.

pith-pipeline@v0.9.0 · 5540 in / 1730 out tokens · 32666 ms · 2026-05-10T03:44:26.660091+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [1]

    Benchtop demonstration of an adaptive chevron completed using a new high-temperature shape-memory alloy

    Noebe R, Quackenbush T, Padula S. Benchtop demonstration of an adaptive chevron completed using a new high-temperature shape-memory alloy. 2005 R&T, TM-2006. 2006;214016:140-1

    work page 2005

  2. [2]

    Shape memory alloy (SMA) actuators: The role of material, form, and scaling effects

    Kim MS, Heo JK, Rodrigue H, Lee HT, Pané S, Han MW, et al. Shape memory alloy (SMA) actuators: The role of material, form, and scaling effects. Advanced Materials. 2023;35(33):2208517

    work page 2023

  3. [3]

    Shape morphing of aircraft wing: Status and challenges

    Sofla A, Meguid S, Tan K, Yeo W. Shape morphing of aircraft wing: Status and challenges. Materials & Design. 2010;31(3):1284-92

    work page 2010

  4. [4]

    Actuation performance of machined helical springs from NiTi shape memory alloy

    Wang J, Huang B, Gu X, Zhu J, Zhang W. Actuation performance of machined helical springs from NiTi shape memory alloy. International Journal of Mechanical Sciences. 2022;236:107744

    work page 2022

  5. [5]

    Design of shape memory alloy actuators for direct power by an automotive battery

    Leary M, Huang S, Ataalla T, Baxter A, Subic A. Design of shape memory alloy actuators for direct power by an automotive battery. Materials & Design. 2013;43:460-6

    work page 2013

  6. [6]

    Shape memory alloys for microsystems: A review from a material research perspective

    Bellouard Y . Shape memory alloys for microsystems: A review from a material research perspective. Materials Science and Engineering: A. 2008;481:582-9

    work page 2008

  7. [7]

    Highly dynamic shape memory alloy actuator for fast moving soft robots

    Huang X, Kumar K, Jawed MK, Mohammadi Nasab A, Ye Z, Shan W, et al. Highly dynamic shape memory alloy actuator for fast moving soft robots. Advanced Materials Technologies. 2019;4(4):1800540

    work page 2019

  8. [8]

    An overview of shape memory alloy -coupled actuators and robots

    Rodrigue H, Wang W, Han M -W, Kim TJ, Ahn S -H. An overview of shape memory alloy -coupled actuators and robots. Soft robotics. 2017;4(1):3-15

    work page 2017

  9. [9]

    Shape memory alloy thin films and heterostructures for MEMS applications: A review

    Choudhary N, Kaur D. Shape memory alloy thin films and heterostructures for MEMS applications: A review. Sensors and Actuators A: Physical. 2016;242:162-81

    work page 2016

  10. [10]

    Applications of shape memory alloys in offshore oil and gas industry: a review

    Song G, Patil D, Kocurek C, Bartos J. Applications of shape memory alloys in offshore oil and gas industry: a review. Earth and Space 2010: Engineering, Science, Construction, and Operations in Challenging Environments. 2010:1551-67

    work page 2010

  11. [11]

    A review of shape memory material’s applications in the offshore oil and gas industry

    Patil D, Song G. A review of shape memory material’s applications in the offshore oil and gas industry. Smart Materials and Structures. 2017;26(9):093002

    work page 2017

  12. [12]

    Shape memory alloy actuated accumulator for ultra -deepwater oil and gas exploration

    Patil D, Song G. Shape memory alloy actuated accumulator for ultra -deepwater oil and gas exploration. Smart Materials and Structures. 2016;25(4):045012

    work page 2016

  13. [13]

    Biomedical applications of shape memory alloys

    Petrini L, Migliavacca F. Biomedical applications of shape memory alloys. Journal of Metallurgy. 2011;2011(1):501483

    work page 2011

  14. [14]

    Effect of hydrogen on super-elastic behavior of NiTi shape memory alloy wires: experimental observation and diffusional -mechanically coupled constitutive model

    Jiang HM, Yu C, Kan Q, Xu B, Ma C, Kang G. Effect of hydrogen on super-elastic behavior of NiTi shape memory alloy wires: experimental observation and diffusional -mechanically coupled constitutive model. Journal of the Mechanical Behavior of Biomedical Materials. 2022;132:105276

    work page 2022

  15. [15]

    Specific mechanical properties of highly porous Ti -Zr-Mo-Sn shape memory alloy for biomedical applications

    Kim Y-w. Specific mechanical properties of highly porous Ti -Zr-Mo-Sn shape memory alloy for biomedical applications. Scripta Materialia. 2023;231:115433

    work page 2023

  16. [16]

    Fracture toughness of NiTi – Towards establishing standard test methods for phase transforming materials

    Haghgouyan B, Hayrettin C, Baxevanis T, Karaman I, Lagoudas DC. Fracture toughness of NiTi – Towards establishing standard test methods for phase transforming materials. Acta Materialia. 2019;162:226-38

    work page 2019

  17. [17]

    On the fracture toughness of pseudoelastic shape memory alloys

    Baxevanis T, Landis CM, Lagoudas DC. On the fracture toughness of pseudoelastic shape memory alloys. Journal of Applied Mechanics. 2014;81(4):041005. 16/18

    work page 2014

  18. [18]

    Experimental and numerical studies of NiTi dynamic fracture behaviors under the impact loading

    Cui Y , Zeng X, Tan VB, Zhang Z. Experimental and numerical studies of NiTi dynamic fracture behaviors under the impact loading. International Journal of Mechanical Sciences. 2022;235:107724

    work page 2022

  19. [19]

    Fracture mechanics and microstructure in NiTi shape memory alloys

    Gollerthan S, Young M, Baruj A, Frenzel J, Schmahl WW, Eggeler G. Fracture mechanics and microstructure in NiTi shape memory alloys. Acta Materialia. 2009;57(4):1015-25

    work page 2009

  20. [20]

    Fracture toughening mechanism of shape memory alloys due to martensite transformation

    Yi S, Gao S. Fracture toughening mechanism of shape memory alloys due to martensite transformation. International journal of solids and structures. 2000;37(38):5315-27

    work page 2000

  21. [21]

    Robertson SW, Ritchie RO. In vitro fatigue–crack growth and fracture toughness behavior of thin - walled superelastic Nitinol tube for endovascular stents: a basis for defining the effect of crack-like defects. Biomaterials. 2007;28(4):700-9

    work page 2007

  22. [22]

    On the fracture toughness of shape memory alloys

    Alsawalhi MY , Landis CM. On the fracture toughness of shape memory alloys. International Journal of Fracture. 2022;236(2):201-18

    work page 2022

  23. [23]

    Numerical modeling of an active elastocaloric regenerator refrigerator with phase transformation kinetics and the matching principle for materials selection

    Qian S, Yuan L, Yu J, Yan G. Numerical modeling of an active elastocaloric regenerator refrigerator with phase transformation kinetics and the matching principle for materials selection. Energy. 2017;141:744-56

    work page 2017

  24. [24]

    Phase field simulation to one -way shape memory effect of NiTi shape memory alloy single crystal

    Xu B, Kang G, Kan Q, Xie X, Yu C, Peng Q. Phase field simulation to one -way shape memory effect of NiTi shape memory alloy single crystal. Computational Materials Science. 2019;161:276- 92

    work page 2019

  25. [25]

    Phase-field modeling of martensitic microstructure in NiTi shape memory alloys

    Zhong Y , Zhu T. Phase-field modeling of martensitic microstructure in NiTi shape memory alloys. Acta materialia. 2014;75:337-47

    work page 2014

  26. [26]

    Three-dimensional field model and computer modeling of martensitic transformations

    Wang Y , Khachaturyan A. Three-dimensional field model and computer modeling of martensitic transformations. Acta materialia. 1997;45(2):759-73

    work page 1997

  27. [27]

    Phase-field simulation of domain structure evolution during a coherent hexagonal-to-orthorhombic transformation

    Wen Y , Wang Y , Chen L-Q. Phase-field simulation of domain structure evolution during a coherent hexagonal-to-orthorhombic transformation. Philosophical Magazine A. 2000;80(9):1967-82

    work page 2000

  28. [28]

    Cubic to tetragonal martensitic transformation in a thin film elastically constrained by a substrate

    Seol D, Hu S, Li Y , Chen L, Oh K. Cubic to tetragonal martensitic transformation in a thin film elastically constrained by a substrate. Metals and Materials International. 2003;9(3):221-6

    work page 2003

  29. [29]

    Phase field modeling of the tetragonal -to- monoclinic phase transformation in zirconia

    Mamivand M, Zaeem MA, El Kadiri H, Chen L -Q. Phase field modeling of the tetragonal -to- monoclinic phase transformation in zirconia. Acta Materialia. 2013;61(14):5223-35

    work page 2013

  30. [30]

    On the elastocaloric effect in CuAlBe shape memory alloys: A quantitative phase-field modeling approach

    Cissé C, Zaeem MA. On the elastocaloric effect in CuAlBe shape memory alloys: A quantitative phase-field modeling approach. Computational Materials Science. 2020;183:109808

    work page 2020

  31. [31]

    Phase field simulation on the super-elasticity, elastocaloric and shape memory effect of geometrically graded nano -polycrystalline NiTi shape memory alloys

    Xu B, Kang G. Phase field simulation on the super-elasticity, elastocaloric and shape memory effect of geometrically graded nano -polycrystalline NiTi shape memory alloys. International Journal of Mechanical Sciences. 2021;201:106462

    work page 2021

  32. [32]

    Three -dimensional, non-isothermal phase-field modeling of thermally and stress -induced martensitic transformations in shape memory alloys

    Cui S, Wan J, Zuo X, Chen N, Zhang J, Rong Y . Three -dimensional, non-isothermal phase-field modeling of thermally and stress -induced martensitic transformations in shape memory alloys. International Journal of Solids and Structures. 2017;109:1-11

    work page 2017

  33. [33]

    Effect of electric field orientation on ferroelectric phase transition and electrocaloric effect

    Li Z, Li J, Wu H -H, Li J, Wang S, Qin S, et al. Effect of electric field orientation on ferroelectric phase transition and electrocaloric effect. Acta Materialia. 2020;191:13-23

    work page 2020

  34. [34]

    Enhancing elastocaloric effect of NiTi alloy by concentration-gradient engineering

    Xu B, Wang C, Wang Q, Yu C, Kan Q, Kang G. Enhancing elastocaloric effect of NiTi alloy by concentration-gradient engineering. International Journal of Mechanical Sciences. 2023;246:108140

    work page 2023

  35. [35]

    From mechanical behavior and elastocaloric effect to microscopic mechanisms of gradient -structured NiTi alloy: A phase -field study

    Zhang Q, Chen J, Fang G. From mechanical behavior and elastocaloric effect to microscopic mechanisms of gradient -structured NiTi alloy: A phase -field study. International Journal of Plasticity. 2023;171:103809. 17/18

    work page 2023

  36. [36]

    Giant enhancement of elastocaloric effect by introducing microstructural holes

    Luo H, Tang W, Gong Q, Yi MJJoA, Compounds. Giant enhancement of elastocaloric effect by introducing microstructural holes. 2023;932:167636

    work page 2023

  37. [37]

    Thermodynamically consistent phase -field modeling of elastocaloric effect: Indirect vs direct method

    Tang W, Gong Q, Yi M, Xu B -XJIJoMS. Thermodynamically consistent phase -field modeling of elastocaloric effect: Indirect vs direct method. 2025;291:110134

    work page 2025

  38. [38]

    Revisiting brittle fracture as an energy minimization problem

    Francfort GA, Marigo J-J. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids. 1998;46(8):1319-42

    work page 1998

  39. [39]

    Numerical experiments in revisited brittle fracture

    Bourdin B, Francfort GA, Marigo J -J. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids. 2000;48(4):797-826

    work page 2000

  40. [40]

    Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

    Amor H, Marigo J -J, Maurini C. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. Journal of the Mechanics and Physics of Solids. 2009;57(8):1209-29

    work page 2009

  41. [41]

    A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

    Miehe C, Hofacker M, Welschinger F. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering. 2010;199(45-48):2765-78

    work page 2010

  42. [42]

    Regularized variational theories of fracture: a unified approach

    Freddi F, Royer-Carfagni G. Regularized variational theories of fracture: a unified approach. Journal of the Mechanics and Physics of Solids. 2010;58(8):1154-74

    work page 2010

  43. [43]

    Phase field modeling of directional fracture in anisotropic polycrystals

    Clayton J, Knap J. Phase field modeling of directional fracture in anisotropic polycrystals. Computational Materials Science. 2015;98:158-69

    work page 2015

  44. [44]

    Phase field modeling of fracture in anisotropic brittle solids

    Teichtmeister S, Kienle D, Aldakheel F, Keip M-A. Phase field modeling of fracture in anisotropic brittle solids. International Journal of Non-Linear Mechanics. 2017;97:1-21

    work page 2017

  45. [45]

    Phase-field modeling of crack branching and deflection in heterogeneous media

    Hansen-Dörr AC, Dammaß F, de Borst R, Kästner M. Phase-field modeling of crack branching and deflection in heterogeneous media. Engineering Fracture Mechanics. 2020;232:107004

    work page 2020

  46. [46]

    Multi -phase field approach to tensile fracture and compressive crushing in grained heterogeneous materials

    Lenarda P, Reinoso J, Paggi M. Multi -phase field approach to tensile fracture and compressive crushing in grained heterogeneous materials. Theoretical and Applied Fracture Mechanics. 2022;122:103632

    work page 2022

  47. [47]

    A mixed mode phase-field model of ductile fracture

    Huber W, Zaeem MA. A mixed mode phase-field model of ductile fracture. Journal of the Mechanics and Physics of Solids. 2023;171:105123

    work page 2023

  48. [48]

    Phase -field modeling of ductile fracture

    Ambati M, Gerasimov T, De Lorenzis L. Phase -field modeling of ductile fracture. Computational Mechanics. 2015;55(5):1017-40

    work page 2015

  49. [49]

    Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling

    Chu D, Li X, Liu Z. Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling. International Journal of Fracture. 2017;208(1):115-30

    work page 2017

  50. [50]

    Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method

    Svolos L, Bronkhorst CA, Waisman H. Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method. Journal of the Mechanics and Physics of Solids. 2020;137:103861

    work page 2020

  51. [51]

    Three-dimensional phase-field modeling of temperature-dependent thermal shock -induced fracture in ceramic materials

    Li D, Li P, Li W, Li W, Zhou K. Three-dimensional phase-field modeling of temperature-dependent thermal shock -induced fracture in ceramic materials. Engineering Fracture Mechanics. 2022;268:108444

    work page 2022

  52. [52]

    Investigation on crack initiation and propagation in hydraulic fracturing of bedded shale by hybrid phase -field modeling

    Liu J, Liang X, Xue Y , Fu Y , Yao K, Dou F. Investigation on crack initiation and propagation in hydraulic fracturing of bedded shale by hybrid phase -field modeling. Theoretical and Applied Fracture Mechanics. 2020;108:102651

    work page 2020

  53. [53]

    Phase-field modeling of hydraulic fracture

    Wilson ZA, Landis CM. Phase-field modeling of hydraulic fracture. Journal of the Mechanics and Physics of Solids. 2016;96:264-90. 18/18

    work page 2016

  54. [54]

    A framework to model the fatigue behavior of brittle materials based on a variational phase -field approach

    Carrara P, Ambati M, Alessi R, De Lorenzis L. A framework to model the fatigue behavior of brittle materials based on a variational phase -field approach. Computer Methods in Applied Mechanics and Engineering. 2020;361:112731

    work page 2020

  55. [55]

    A review on phase field models for fracture and fatigue

    Li P, Li W, Li B, Yang S, Shen Y , Wang Q, et al. A review on phase field models for fracture and fatigue. Engineering Fracture Mechanics. 2023;289:109419

    work page 2023

  56. [56]

    A phase field modeling approach of cyclic fatigue crack growth

    Schreiber C, Kuhn C, Müller R, Zohdi T. A phase field modeling approach of cyclic fatigue crack growth. International Journal of Fracture. 2020;225(1):89-100

    work page 2020

  57. [57]

    Study of transformation induced intergranular microcracking in tetragonal zirconia polycrystals with the phase field method

    Zhu J, Luo J. Study of transformation induced intergranular microcracking in tetragonal zirconia polycrystals with the phase field method. Materials Science and Engineering: A. 2017;701:69-84

    work page 2017

  58. [58]

    Ferroelastic toughening of single crystalline yttria-stabilized t'zirconia: A phase field study

    Sun Y , Luo J, Zhu J. Ferroelastic toughening of single crystalline yttria-stabilized t'zirconia: A phase field study. Engineering Fracture Mechanics. 2020;233:107077

    work page 2020

  59. [59]

    Transformation-induced fracture toughening in CuAlBe shape memory alloys: A phase-field study

    Cissé C, Zaeem MA. Transformation-induced fracture toughening in CuAlBe shape memory alloys: A phase-field study. International Journal of Mechanical Sciences. 2021;192:106144

    work page 2021

  60. [60]

    Phase field modelling of fracture and fatigue in Shape Memory Alloys

    Simoes M, Martinez -Paneda E. Phase field modelling of fracture and fatigue in Shape Memory Alloys. Computer Methods in Applied Mechanics and Engineering. 2021;373:113504

    work page 2021

  61. [61]

    A finite -strain phase -field description of thermomechanically induced fracture in shape memory alloys

    Hasan M, Zhang M, Baxevanis T. A finite -strain phase -field description of thermomechanically induced fracture in shape memory alloys. Shape Memory and Superelasticity. 2022;8(4):356-72

    work page 2022

  62. [62]

    Xiong J, Xu B, Kang G. Phase field simulation on the martensite transformation and reorientation toughening behaviors of single crystal NiTi shape memory alloy: effects of crystalline orientation and temperature. Engineering Fracture Mechanics. 2022;270:108585

    work page 2022

  63. [63]

    A thermo-mechanical phase -field fracture model: Application to hot cracking simulations in additive manufacturing

    Ruan H, Rezaei S, Yang Y , Gross D, Xu B -X. A thermo-mechanical phase -field fracture model: Application to hot cracking simulations in additive manufacturing. Journal of the Mechanics and Physics of Solids. 2023;172:105169

    work page 2023