Phase-field framework with constraints and its applications to ductile fracture in polycrystals and fatigue
Source abstract
Abstract
Abstract Modeling of ductile fracture in polycrystalline structures is challenging, since it requires integrated modeling of cracks, crystal plasticity, and grains. Here we extend the typical phase-field framework to the situations with constraints on the order parameters, and formulate two types of phase-field models on ductile fracture. The Type-I model incorporates three sets of order parameters, which describe the distributions of cracks, plastic strain, and grains, respectively. Crystal plasticity is employed within grain interiors accommodated by J2 plasticity at grain boundaries. The applications of the Type-I model to single crystals and bicrystals demonstrate the influences of grain orientations and grain boundaries on crack growth. In the Type-II model, J2 plasticity is assumed for the whole system and grain structures are neglected. Taking advantage of the efficiency of the fast Fourier transform, our Type-II model is employed to study low cycle fatigue. Crack closure and striation-like patterning of plastic strain are observed in the simulations. Crack growth rate is analyzed as a function of the J-integral, and the simulated fatigue life as a function of plastic strain agrees with the Coffin–Manson relation without a priori assumption.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Metallurgy and Material Forming · Aluminum Alloy Microstructure Properties · Metal Forming Simulation Techniques
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (62)
Show source-provided references
1. Anderson, T. L. Fracture mechanics: fundamentals and applications. (CRC press, 2017).10.1201/9781315370293 ↗
2. Ambati, M., Gerasimov, T. & De Lorenzis, L. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 55, 383–405 (2015).10.1007/s00466-014-1109-y ↗
3. Bourdin, B., Francfort, G. A. & Marigo, J.-J. The variational approach to fracture. J. Elast. 91, 5–148 (2008).10.1007/s10659-007-9107-3 ↗
4. Ambati, M., Gerasimov, T. & De Lorenzis, L. Phase-field modeling of ductile fracture. Comput. Mech. 55, 1017–1040 (2015).10.1007/s00466-015-1151-4 ↗
5. Amor, H., Marigo, J.-J. & Maurini, C. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J. Mech. Phys. Solids 57, 1209–1229 (2009).10.1016/j.jmps.2009.04.011 ↗
6. Borden, M. J., Verhoosel, C. V., Scott, M. A., Hughes, T. J. & Landis, C. M. A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Engrg. 217, 77–95 (2012).10.1016/j.cma.2012.01.008 ↗
7. Mesgarnejad, A., Imanian, A. & Karma, A. Phase-field models for fatigue crack growth. Theor. Appl. Fract. Mech. 103, 102282 (2019).10.1016/j.tafmec.2019.102282 ↗
8. Francfort, G. A. & Marigo, J.-J. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998).10.1016/S0022-5096(98)00034-9 ↗
9. Zhou, S., Rabczuk, T. & Zhuang, X. Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies. Adv. Eng. Softw. 122, 31–49 (2018).10.1016/j.advengsoft.2018.03.012 ↗
10. Miehe, C., Hofacker, M. & Welschinger, F. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Engrg. 199, 2765–2778 (2010).10.1016/j.cma.2010.04.011 ↗
11. Yin, B. & Kaliske, M. A ductile phase-field model based on degrading the fracture toughness: Theory and implementation at small strain. Comput. Methods Appl. Mech. Engrg. 366, 113068 (2020).10.1016/j.cma.2020.113068 ↗
12. Alessi, R., Marigo, J.-J., Maurini, C. & Vidoli, S. Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples. Int. J. Mech. Sci. 149, 559–576 (2018).10.1016/j.ijmecsci.2017.05.047 ↗
13. Kuhn, C., Noll, T. & Müller, R. On phase field modeling of ductile fracture. GAMM‐Mitteilungen 39, 35–54 (2016).10.1002/gamm.201610003 ↗
14. Alessi, R., Ambati, M., Gerasimov, T., Vidoli, S. & De Lorenzis, L. In Advances in computational plasticity 1-21 (Springer, 2018).10.1007/978-3-319-60885-3_1 ↗
15. Ambati, M., Kruse, R. & De Lorenzis, L. A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57, 149–167 (2016).10.1007/s00466-015-1225-3 ↗
16. Chen, H. et al. A two-set order parameters phase-field modeling of crack deflection/penetration in a heterogeneous microstructure. Comput. Methods Appl. Mech. Engrg. 347, 1085–1104 (2019).10.1016/j.cma.2019.01.014 ↗
17. Cheng, J., Tu, X. & Ghosh, S. Wavelet-enriched adaptive hierarchical FE model for coupled crystal plasticity-phase field modeling of crack propagation in polycrystalline microstructures. Comput. Methods Appl. Mech. Engrg. 361, 112757 (2020).10.1016/j.cma.2019.112757 ↗
18. Tu, X., Ray, A. & Ghosh, S. A coupled crystal plasticity FEM and phase-field model for crack evolution in microstructures of 7000 series aluminum alloys. Eng. Fract. Mech. 230, 106970 (2020).10.1016/j.engfracmech.2020.106970 ↗
19. Cheng, T.-L., Wen, Y.-H. & Hawk, J. A. Modeling elasto-viscoplasticity in a consistent phase field framework. Int. J. Plasticity 96, 242–263 (2017).10.1016/j.ijplas.2017.05.006 ↗
20. Cheng, T.-L., Wen, Y.-H. & Hawk, J. A. Diffuse interface approach to modeling crystal plasticity with accommodation of grain boundary sliding. Int. J. Plasticity 114, 106–125 (2019).10.1016/j.ijplas.2018.10.012 ↗
21. Guyer, J. E., Boettinger, W. J., Warren, J. A. & McFadden, G. B. Phase field modeling of electrochemistry. I. Equilibrium. Phys. Rev. E 69, 021603 (2004).10.1103/PhysRevE.69.021603 ↗
22. Pusztai, T., Bortel, G. & Gránásy, L. Phase field theory of polycrystalline solidification in three dimensions. EPL (Europhys. Lett.) 71, 131 (2005).10.1209/epl/i2005-10081-7 ↗
23. Ren, H., Zhuang, X., Anitescu, C. & Rabczuk, T. Multi-connected boundary conditions in solid mechanics and surgery theory. Computers Struct. 251, 106504 (2021).10.1016/j.compstruc.2021.106504 ↗
24. Wang, Y. U., Jin, Y. M. & Khachaturyan, A. G. Phase field microelasticity theory and simulation of multiple voids and cracks in single crystals and polycrystals under applied stress. J. Appl. Phys. 91, 6435–6451 (2002).10.1063/1.1471389 ↗
25. Ren, H., Zhuang, X. & Rabczuk, T. A nonlocal operator method for solving partial differential equations. Comput. Methods Appl. Mech. Engrg. 358, 112621 (2020).10.1016/j.cma.2019.112621 ↗
26. Rabczuk, T., Ren, H. & Zhuang, X. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers, Mater. Contin. 59, 31–55 (2019).10.32604/cmc.2019.04567 ↗
27. Ren, H., Zhuang, X., Trung, N.-T. & Rabczuk, T. Nonlocal operator method for the Cahn-Hilliard phase field model. Commun. Nonlinear Sci. Numer. Simula. 96, 105687 (2021).10.1016/j.cnsns.2020.105687 ↗
28. Prakash, A. & Lebensohn, R. Simulation of micromechanical behavior of polycrystals: finite elements versus fast Fourier transforms. Model. Simul. Mater. Sci. Eng. 17, 064010 (2009).10.1088/0965-0393/17/6/064010 ↗
29. Eisenlohr, P., Diehl, M., Lebensohn, R. A. & Roters, F. A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int. J. Plasticity 46, 37–53 (2013).10.1016/j.ijplas.2012.09.012 ↗
30. Manson, S. S. Thermal stress and low cycle fatigue. (McGraw Hill, New York, 1966).10.1115/1.3625225 ↗
31. Coffin, L. The effect of high vacuum on the low cycle fatigue law. Metall. Mater. Trans. B 3, 1777–1788 (1972).10.1007/BF02642561 ↗
32. Paris, P. C. A rational analytic theory of fatigue. trend Eng. 13, 9 (1961).
33. 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. Comput. Methods Appl. Mech. Engrg. 361, 112731 (2020).10.1016/j.cma.2019.112731 ↗
34. Alessi, R., Vidoli, S. & De Lorenzis, L. A phenomenological approach to fatigue with a variational phase-field model: The one-dimensional case. Eng. Fract. Mech. 190, 53–73 (2018).10.1016/j.engfracmech.2017.11.036 ↗
35. Ulloa, J., Wambacq, J., Alessi, R., Degrande, G. & François, S. Phase-field modeling of fatigue coupled to cyclic plasticity in an energetic formulation. Comput. Methods Appl. Mech. Engrg. 373, 113473 (2021).10.1016/j.cma.2020.113473 ↗
36. Haveroth, G., Vale, M., Bittencourt, M. & Boldrini, J. A non-isothermal thermodynamically consistent phase field model for damage, fracture and fatigue evolutions in elasto-plastic materials. Comput. Methods Appl. Mech. Engrg. 364, 112962 (2020).10.1016/j.cma.2020.112962 ↗
37. Seiler, M., Linse, T., Hantschke, P. & Kästner, M. An efficient phase-field model for fatigue fracture in ductile materials. Eng. Fract. Mech. 224, 106807 (2020).10.1016/j.engfracmech.2019.106807 ↗
38. Laird, C. & Veaux, deLa R. Additional evidence for the plastic blunting process of fatigue crack propagation. Metall. Trans. A 8, 657–664 (1977).10.1007/BF02676989 ↗
39. Laird, C. In Physical Metallurgy (eds. R. W. Cahn & P. Haasent) (Elsevier Science BV, 1996).
40. Cline, D. Variational principles in classical mechanics. (University of Rochester River Campus Librarie, 2017).
41. Liberzon, D. Calculus of variations and optimal control theory: a concise introduction. (Princeton university press, 2011).10.2307/j.ctvcm4g0s ↗
42. Allen, S. M. & Cahn, J. W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979).10.1016/0001-6160(79)90196-2 ↗
43. Chen, L.-Q. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002).10.1146/annurev.matsci.32.112001.132041 ↗
44. Lemaitre, J. & Chaboche, J.-L. Mechanics of solid materials. (Cambridge university press, 1994).
45. Nye, J. F. Physical properties of crystals: their representation by tensors and matrices. (Oxford university press, 1985).
46. Kalidindi, S. R., Bronkhorst, C. A. & Anand, L. Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solids 40, 537–569 (1992).10.1016/0022-5096(92)80003-9 ↗
47. Chowdhury, H., Naumenko, K. & Altenbach, H. Aspects of power law flow rules in crystal plasticity with glide-climb driven hardening and recovery. Int. J. Mech. Sci. 146, 486–496 (2018).10.1016/j.ijmecsci.2017.06.019 ↗
48. Bourdin, B., Larsen, C. J. & Richardson, C. L. A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 168, 133–143 (2011).10.1007/s10704-010-9562-x ↗
49. Wu, J.-Y., Nguyen, V. P., Zhou, H. & Huang, Y. A variationally consistent phase-field anisotropic damage model for fracture. Comput. Methods Appl. Mech. Engrg. 358, 112629 (2020).10.1016/j.cma.2019.112629 ↗
50. Steinke, C. & Kaliske, M. A phase-field crack model based on directional stress decomposition. Computational Mech. 63, 1019–1046 (2019).10.1007/s00466-018-1635-0 ↗
51. Wang, Y. U. Computer modeling and simulation of solid-state sintering: a phase field approach. Acta materialia 54, 953–961 (2006).10.1016/j.actamat.2005.10.032 ↗
52. Mughrabi, H. Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metall. 31, 1367–1379 (1983).10.1016/0001-6160(83)90007-X ↗
53. Mughrabi, H. A two-parameter description of heterogeneous dislocation distributions in deformed metal crystals. Mater. Sci. Eng. 85, 15–31 (1987).10.1016/0025-5416(87)90463-0 ↗
54. Wang, Y. U., Jin, Y. M. & Khachaturyan, A. G. Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid. J. Appl. Phys. 92, 1351–1360 (2002).10.1063/1.1492859 ↗
55. Ren, H., Zhuang, X., Anitescu, C. & Rabczuk, T. An explicit phase field method for brittle dynamic fracture. Computers Struct. 217, 45–56 (2019).10.1016/j.compstruc.2019.03.005 ↗
56. Reid, C. N. Deformation geometry for materials scientists: international series on materials science and technology. Vol. 11 (Elsevier, 2016).
57. Potirniche, G. & Daniewicz, S. Analysis of crack tip plasticity for microstructurally small cracks using crystal plasticity theory. Eng. Fract. Mech. 70, 1623–1643 (2003).10.1016/S0013-7944(02)00204-7 ↗
58. Rabbolini, S., Pataky, G., Sehitoglu, H. & Beretta, S. Fatigue crack growth in Haynes 230 single crystals: an analysis with digital image correlation. Fatigue Fract. Engng. Mater. Struct. 38, 583–596 (2015).10.1111/ffe.12261 ↗
59. Earmme, Y., Johnson, W. C. & Lee, J. Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: linear and power-law strain hardening. Metall. Mater. Trans. A 12, 1521–1530 (1981).10.1007/BF02643699 ↗
60. Khachaturyan, A. G. Theory of structural transformations in solids. (Courier Corporation, 2013).
61. Jin, Y. M., Wang, Y. U. & Khachaturyan, A. G. Three-dimensional phase field microelasticity theory of a multivoid multicrack system in an elastically anisotropic body: model and computer simulations. Philos. Mag. 83, 1587–1626 (2003).10.1080/1478643031000080735 ↗
62. Lamba, H. The J-integral applied to cyclic loading. Eng. Fract. Mech. 7, 693–703 (1975).10.1016/0013-7944(75)90025-9 ↗