A general phase-field model for fatigue failure in brittle and ductile solids. (English) Zbl 1468.74056

Summary: In this work, the phase-field approach to fracture is extended to model fatigue failure in high- and low-cycle regime. The fracture energy degradation due to the repeated externally applied loads is introduced as a function of a local energy accumulation variable, which takes the structural loading history into account. To this end, a novel definition of the energy accumulation variable is proposed, allowing the fracture analysis at monotonic loading without the interference of the fatigue extension, thus making the framework generalised. Moreover, this definition includes the mean load influence of implicitly. The elastoplastic material model with the combined nonlinear isotropic and nonlinear kinematic hardening is introduced to account for cyclic plasticity. The ability of the proposed phenomenological approach to naturally recover main features of fatigue, including Paris law and Wöhler curve under different load ratios is presented through numerical examples and compared with experimental data from the third author et al.’s previous work [“Microstructure influence on fatigue behaviour of nodular cast iron”, Mater. Sci. Eng. A 556, 88–99 (2012; doi:10.1016/j.msea.2012.06.062)]. Physical interpretation of additional fatigue material parameter is explored through the parametric study.


74R20 Anelastic fracture and damage
74R10 Brittle fracture
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
Full Text: DOI


[1] Suresh, S., Fatigue of materials (1998), Cambridge: Cambridge University Press, Cambridge
[2] Stephens, RI; Fatemi, A.; Stephens, RR; Fuchs, HO, Metal fatigue in engineering (2000), New York: Wiley, New York
[3] Pineau, A.; McDowell, DL; Busso, EP; Antolovich, SD, Failure of metals II: fatigue, Acta Mater, 107, 484-507 (2016)
[4] Paris, PC; Gomez, MP; Anderson, WE, A rational analytic theory of fatigue, Trend Eng, 13, 9-14 (1961)
[5] Forman, RG; Mettu, SR; Ernst, HA; Saxena, A.; McDowell, DL, Behavior of surface and corner cracks subjected to tensile and bending loads in Ti-6Al-4V alloy, Fracture mechanics: 22nd symposium, 519-546 (1992), Philadelphia: American Society for Testing and Materials, Philadelphia
[6] Lemaitre, J., A Course on Damage Mechanics (1996), Berlin Heidelberg: Springer, Berlin Heidelberg · Zbl 0852.73003
[7] Schutz, W., A history of fatigue, EngFractMech, 54, 2, 263-300 (1996)
[8] Basquin, OH, The exponential law of endurance tests, Proc Am Soc Test Mater, 10, 625-630 (1910)
[9] Maierhofer, J.; Pippan, R.; Ganser, HP, Modified NASGRO equation for physically short cracks, Int J Fatigue, 59, 200-207 (2014)
[10] Borges, MF; Neto, DM; Antunes, FV, Numerical simulation of fatigue crack growth based on accumulated plastic strain, TheoretApplFractMech, 108, 10 (2020)
[11] Xu, K.; Qiao, G-Y; Pan, X-Y; Chen, X-W; Liao, B.; Xiao, F-R, Simulation of fatigue properties for the weld joint of the X80 weld pipe before and after removing the weld reinforcement, Int J Press Vessels Pip, 187, 104164 (2020)
[12] Brod, M.; Just, G.; Dean, A.; Jansen, E.; Koch, I.; Rolfes, R.; Gude, M., Numerical modelling and simulation of fatigue damage in carbon fibre reinforced plastics at different stress ratios, Thin Wall Struct, 139, 219-231 (2019)
[13] Schijve, J., Fatigue of structures and materials (2009), Dordrecht: Springer, Dordrecht · Zbl 1419.74023
[14] Alliche, A., Damage model for fatigue loading of concrete, Int J Fatigue, 26, 9, 915-921 (2004) · Zbl 1107.74335
[15] Di Pisa, C.; Aliabadi, MH, Fatigue crack growth analysis of assembled plate structures with dual boundary element method, EngFractMech, 98, 200-213 (2013)
[16] Abdul-Baqi, A.; Schreurs, PJG; Geers, MGD, Fatigue damage modeling in solder interconnects using a cohesive zone approach, Int J Solids Struct, 42, 3-4, 927-942 (2005) · Zbl 1086.74516
[17] Abali, BE, Computational study for reliability improvement of a circuit board, MechAdv Mater Modern Process, 3, 1-11 (2017)
[18] Cisilino, AP; Aliabadi, MH, Dual boundary element assessment of three-dimensional fatigue crack growth, Eng Anal Boundary Elem, 28, 9, 1157-1173 (2004) · Zbl 1070.74051
[19] Francfort, GA; Marigo, JJ, Revisiting brittle fracture as an energy minimization problem, J MechPhys Solids, 46, 8, 1319-1342 (1998) · Zbl 0966.74060
[20] Griffith, AA, The phenomena of rupture and flow in solids, Philos Trans R SocLondSer A, 221, 582-593, 163-198 (1921) · Zbl 1454.74137
[21] Bourdin, B.; Francfort, GA; Marigo, JJ, The variational approach to fracture, J Elast, 91, 1-3, 5-148 (2008) · Zbl 1176.74018
[22] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Int J Numer Meth Eng, 83, 10, 1273-1311 (2010) · Zbl 1202.74014
[23] Kuhn, C.; Schluter, A.; Mueller, R., On degradation functions in phase field fracture models, Comput Mater Sci, 108, 374-384 (2015)
[24] Seles, K.; Jurcevic, A.; Tonkovic, Z.; Soric, J., Crack propagation prediction in heterogeneous microstructure using an efficient phase-field algorithm, TheoretApplFractMech, 100, 289-297 (2019)
[25] Teichtmeister, S.; Kienle, D.; Aldakheel, F.; Keip, MA, Phase field modeling of fracture in anisotropic brittle solids, Int J Non-Linear Mech, 97, 1-21 (2017)
[26] Ambati, M.; Gerasimov, T.; De Lorenzis, L., A review on phase-field models of brittle fracture and a new fast hybrid formulation, ComputMech, 55, 2, 383-405 (2015) · Zbl 1398.74270
[27] Seleš, K.; Lesičar, T.; Tonković, Z.; Sorić, J., A phase field staggered algorithm for fracture modeling in heterogeneous microstructure, Key Eng Mater, 774, 632-637 (2018)
[28] Heider, Y.; Sun, WC, A phase field framework for capillary-induced fracture in unsaturated porous media: drying-induced vs hydraulic cracking, Comput Methods ApplMechEng, 359, 26 (2020) · Zbl 1441.74205
[29] Pillai, U.; Heider, Y.; Markert, B., A diffusive dynamic brittle fracture model for heterogeneous solids and porous materials with implementation using a user-element subroutine, Comput Mater Sci, 153, 36-47 (2018)
[30] Guillen-Hernandez, T.; Quintana-Corominas, A.; Garcia, IG; Reinoso, J.; Paggi, M.; Turon, A., In-situ strength effects in long fibre reinforced composites: a micro-mechanical analysis using the phase field approach of fracture, TheoretApplFractMech, 108, 16 (2020)
[31] Hansen-Dorr, AC; Dammass, F.; de Borst, R.; Kastner, M., Phase-field modeling of crack branching and deflection in heterogeneous media, EngFractMech, 232, 22 (2020)
[32] Kuhn, C.; Muller, R., A continuum phase field model for fracture, EngFractMech, 77, 18, 3625-3634 (2010)
[33] Miehe, C.; Aldakheel, F.; Raina, A., Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory, Int J Plast, 84, 1-32 (2016)
[34] Ambati, M.; Gerasimov, T.; De Lorenzis, L., Phase-field modeling of ductile fracture, ComputMech, 55, 5, 1017-1040 (2015) · Zbl 1329.74018
[35] Aldakheel, F.; Wriggers, P.; Miehe, C., A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling, ComputMech, 62, 4, 815-833 (2018) · Zbl 1459.74024
[36] Alessi, R.; Ambati, M.; Gerasimov, T.; Vidoli, S.; De Lorenzis, L.; Oñate, E.; Peric, D.; Souza Neto, E.; Chiumenti, M., Comparison of phase-field models of fracture coupled with plasticity, Advances in computational plasticity. Computational methods in applied sciences (2018), Cham: Springer, Cham
[37] Aldakheel, F.; Hudobivnik, B.; Wriggers, P., Virtual element formulation for phase-field modeling of ductile fracture, Int J MultiscaleComputEng, 17, 2, 181-200 (2019) · Zbl 1468.74085
[38] Krueger, M.; Dittmann, M.; Aldakheel, F.; Haertel, A.; Wriggers, P.; Hesch, C., Porous-ductile fracture in thermo-elasto-plastic solids with contact applications, ComputMech, 65, 4, 941-966 (2020) · Zbl 1462.74143
[39] Yin, B.; Kaliske, M., A ductile phase-field model based on degrading the fracture toughness: Theory and implementation at small strain, Comput Methods ApplMechEng, 366, 23 (2020) · Zbl 1442.74024
[40] Kienle, D.; Aldakheel, F.; Keip, MA, A finite-strain phase-field approach to ductile failure of frictional materials, Int J Solids Struct, 172, 147-162 (2019)
[41] Kuhn, C.; Noll, T.; Müller, R., On phase field modeling of ductile fracture, GAMM-Mitteilungen, 39, 1, 35-54 (2016) · Zbl 1397.74027
[42] Noll, T.; Kuhn, C.; Olesch, D.; Müller, R., 3D phase field simulations of ductile fracture, GAMM-Mitteilungen, 43, 2 (2020)
[43] Zi, G.; Song, JH; Budyn, E.; Lee, SH; Belytschko, T., A method for growing multiple cracks without remeshing and its application to fatigue crack growth, Model Simul Mater SciEng, 12, 5, 901-915 (2004)
[44] Hosseini, ZS; Dadfarnia, M.; Somerday, BP; Sofronis, P.; Ritchie, RO, On the theoretical modeling of fatigue crack growth, J MechPhys Solids, 121, 341-362 (2018)
[45] Branco, R.; Antunes, FV; Costa, JD, A review on 3D-FE adaptive remeshing techniques for crack growth modelling, EngFractMech, 141, 170-195 (2015)
[46] Boldrini, JL; de Moraes, EAB; Chiarelli, LR; Fumes, FG; Bittencourt, ML, A non-isothermal thermodynamically consistent phase field framework for structural damage and fatigue, Comput Methods ApplMechEng, 312, 395-427 (2016) · Zbl 1439.74023
[47] Caputo, M.; Fabrizio, M., Damage and fatigue described by a fractional derivative model, J ComputPhys, 293, 400-408 (2015) · Zbl 1349.74030
[48] Amendola, G.; Fabrizio, M.; Golden, JM, Thermomechanics of damage and fatigue by a phase field model, J Therm Stresses, 39, 5, 487-499 (2016)
[49] Schreiber, C.; Kuhn, C.; Muller, R.; Zohdi, T., A phase field modeling approach of cyclic fatigue crack growth, Int J Fract, 225, 1, 89-100 (2020)
[50] Alessi, R.; Vidoli, S.; De Lorenzis, L., A phenomenological approach to fatigue with a variational phase-field model: the one-dimensional case, EngFractMech, 190, 53-73 (2018)
[51] 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 ApplMechEng, 361, 29 (2020) · Zbl 1442.74195
[52] Seiler, M.; Linse, T.; Hantschke, P.; Kastner, M., An efficient phase-field model for fatigue fracture in ductile materials, EngFractMech, 224, 15 (2020)
[53] Aldakheel F, Schreiber C, Müller R, Wriggers P (2021) Phase-field modeling of fatigue crack propogation in brittle materials. Accepted as a part of Springer-Book
[54] Ulloa, J.; Wambacq, J.; Alessi, R.; Degrande, G.; Francois, S., Phase-field modeling of fatigue coupled to cyclic plasticity in an energetic formulation, Comput Methods ApplMechEng, 373, 37 (2021) · Zbl 07337757
[55] Seleš, K.; Lesičar, T.; Tonković, Z.; Sorić, J., A residual control staggered solution scheme for the phase-field modeling of brittle fracture, EngFractMech, 205, 370-386 (2018)
[56] Canzar, P.; Tonkovic, Z.; Kodvanj, J., Microstructure influence on fatigue behaviour of nodular cast iron, Mater SciEngStruct Mater Prop Microstruct Process, 556, 88-99 (2012)
[57] Miehe, C.; Schanzel, LM; Ulmer, H., Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids, Comput Methods ApplMechEng, 294, 449-485 (2015) · Zbl 1423.74838
[58] Pham, K.; Amor, H.; Marigo, JJ; Maurini, C., Gradient damage models and their use to approximate brittle fracture, Int J Damage Mech, 20, 4, 618-652 (2011)
[59] Kuhn, C.; Schluter, A.; Muller, R., On degradation functions in phase field fracture models, Comput Mater Sci, 108, 374-384 (2015)
[60] Aldakheel, F., Mechanics of nonlocal dissipative solids: gradient plasticity and phase field modeling of ductile fracture, Stuttgart: InstitutfürMechanik (Bauwesen) (2016), Lehrstuhl I: Universität Stuttgart, Lehrstuhl I
[61] Noii, N.; Aldakheel, F.; Wick, T.; Wriggers, P., An adaptive global-local approach for phase-field modeling of anisotropic brittle fracture, Comput Methods ApplMechEng, 361, 112744 (2020) · Zbl 1442.74213
[62] Kastner, M.; Hennig, P.; Linse, T.; Ulbricht, V.; Naumenko, K.; Assmus, M., Phase-field modelling of damage and fracture-convergence and local mesh refinement, Advanced methods of continuum mechanics for materials and structures, 307-324 (2016), New York: Springer, New York
[63] Chaboche, JL, Constitutive equations for cyclic plasticity and cyclic viscoplasticity, Int J Plast, 5, 3, 247-302 (1989) · Zbl 0695.73001
[64] Amor, H.; Marigo, JJ; Maurini, C., Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, J MechPhys Solids, 57, 8, 1209-1229 (2009) · Zbl 1426.74257
[65] Gerasimov, T.; De Lorenzis, L., A line search assisted monolithic approach for phase-field computing of brittle fracture, Comput Methods ApplMechEng, 312, 276-303 (2016) · Zbl 1439.74349
[66] Seleš K (2018) Abaqus code for a residual control staggered solution scheme for the phase-field modeling of brittle fracture. https://data.mendeley.com/datasets/p77tsyrbx2/4
[67] Cojocaru, D.; Karlsson, AM, A simple numerical method of cycle jumps for cyclically loaded structures, Int J Fatigue, 28, 12, 1677-1689 (2006)
[68] Canzar P (2012) Experimental and numerical modelling of fatigue behavior of nodular cast iron. Soctoral thesis (in croatian), Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb
[69] ASTM Standard E647 (2002) Standard Test Method for Measurement of Fatigue Crack Growth Rates
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