Skip to main content
SpringerLink
Log in

A phase field model for ductile fracture considering the strain rate, stress triaxiality and Lode angle parameter

  • Research
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

Ductile materials exhibit rate-dependent behaviors when subjected to different loading rates, particularly during impact and explosion events. In order to investigate the high strain rate behaviors of metal materials, a phase field model considered the rate-dependent threshold for effective plastic work is proposed. And the presented model couples the influences of the stress triaxiality and Lode angle parameter on failure behaviors. Later, a single element is modeled to demonstrate the impacts of the model in predicting stress-strain relations under varying loading rates. To illustrate the impacts of the Lode angle parameter on load-displacement responses, rectangular notch specimens are used. Next, the presented model is employed to mimic the shear fracture of hat-shaped specimens at different strain rates based on the split Hopkinson pressure bar tests, and the model parameters are calibrated by comparing the strain waveforms between the simulations and experiments. The numerical results indicate the developed model is capable of accurately reproducing the shear ductile fracture of the hat-shaped specimens under high strain rates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  • Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 57:149–167

    Article  MathSciNet  Google Scholar 

  • Ambati M, Kruse R, De Lorenzis L (2016) A phase-field model for ductile fracture at finite strains and its experimental verification. Comput Mech 57:149–167

    Article  MathSciNet  Google Scholar 

  • Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57:1209–1229

    Article  Google Scholar 

  • Badnava H, Etemadi E, Msekh M (2017) A phase field model for rate-dependent ductile fracture. Metals 7(5):1–21

    Article  Google Scholar 

  • Bai Y, Wierzbicki T (2008) A new model of metal plasticity and fracture with pressure and Lode dependence. Int J Plast 24:1071–1096

    Article  Google Scholar 

  • Bai Y, Teng X, Wierzbicki T (2009) On the application of stress triaxiality formula for plane strain fracture testing. J Eng Mater Technol 131:1–10

    Article  Google Scholar 

  • Bardelcik A, Worswick MJ, Winkler S, Wells MA (2012) A strain rate sensitive constitutive model for quenched boron steel with tailored properties. Int J Impact Eng 50:49–62

    Article  Google Scholar 

  • Barenblatt G (1962) The mathematical theory of equilibrium of cracks in brittle fracture. Adv Mech Eng 7:55–129

    MathSciNet  Google Scholar 

  • Borden MJ, Hughes TJR, Landis CM, Anvari A, Lee IJ (2016) A phase field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng 312:130–166

    Article  MathSciNet  Google Scholar 

  • Børvik T, Hopperstad OS, Berstad T, Langseth M (2001) A computational model of viscoplasticity and ductile damage for impact and penetration. Eur J Mech A/Solids 20:685–712

    Article  Google Scholar 

  • Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48:797–826

    Article  MathSciNet  Google Scholar 

  • Bourdin B, Francfort G, Marigo JJ (2008) The variational approach to fracture. J Elast 91(1–3):5–148

    Article  MathSciNet  Google Scholar 

  • Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104

    Article  Google Scholar 

  • Edwards NJ, Song W, Cimpoeru SJ, Ruan D, Lu GX, Herzig N (2018) Mechanical and microstructural properties of 2024–T351 aluminum using a hat-shaped specimen at high strain rates. Mater Sci Eng A 720:203–213

    Article  Google Scholar 

  • Fang C, Guo X, Weng GJ, Li JH, Chen G (2021) Simulation of ductile fracture of zirconium alloys based on triaxiality dependent cohesive zone model. Acta Mech 232:3723–3736

    Article  Google Scholar 

  • Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

    Article  MathSciNet  Google Scholar 

  • Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond B Biol Sci 221:163–198

    Google Scholar 

  • Gu T, Wang Z (2022) A strain rate-dependent cohesive zone model for shear failure of hat-shaped specimens under impact. Eng Fract Mech 259:108145

    Article  Google Scholar 

  • Gultekin O, Dal H, Holzapfel GA (2018) Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: a rate-dependent anisotropic crack phase-field model. Comput Methods Appl Mech Eng 331:23–52

    Article  MathSciNet  Google Scholar 

  • Hai L, Li J (2021) A rate-dependent phase-field framework for the dynamic failure of quasi-brittle materials. Eng Fract Mech 252:107847

    Article  Google Scholar 

  • Han J, Matsubara S, Moriguchi S, Kaliske M, Terada K (2022) Crack phase-field model equipped with plastic driving force and degrading fracture toughness for ductile simulation. Comput Mech 69:151–175

    Article  MathSciNet  Google Scholar 

  • Hofacker M, Miehe C (2012) A phase field model for ductile to brittle failure model transition. Proc Appl Math Mech 12:173–174

    Article  Google Scholar 

  • Hu X, Huang X, Yao W, Zhang P (2021) Precise integration explicit phase field method for dynamic brittle fracture. Mech Res Commun 113:103698

    Article  Google Scholar 

  • Irwin GR (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 24:361–364

    Article  Google Scholar 

  • Kachanov LM (1958) Time of the rupture process under creep conditions. Izv Akad Nauk SSR Otd Tech 8:26–31

    Google Scholar 

  • Kolsky H (1947) An investigation of the Mechanical properties of materials at very high rates of loading. Proc Phys Soc Lond Sect B 62:676–700

    Article  Google Scholar 

  • Kumar S, Bhardwaj G (2018) A new enrichment scheme in XFEM to model crack growth behavior in ductile materials. Theor Appl Fract Mech 96:296–307

    Article  Google Scholar 

  • Kunh C, Noll T, Muller R (2016) On phase field modeling of ductile fracture. GAMM-Mitteilungen 1:35–54

    MathSciNet  Google Scholar 

  • Liu X, Yan S, Rasmussen KJR, Deierlein GG (2022) Experimental investigation of the effect of Lode angle on fracture initiation of steels. Eng Fract Mech 271:108637

    Article  Google Scholar 

  • Loew PJ, Peters B, Beex Lars AA (2019) Rate-dependent phase-field damage modeling of rubber and its experimental parameter identification. J Mech Phys Solids 127:266–294

    Article  MathSciNet  Google Scholar 

  • Ma YS, Sun DZ, Andrieux F, Zhang KS (2017) Influences of initial porosity, stress triaxiality and Lode parameter on plastic deformation and ductile fracture. Acta Mech Solida Sin 30:493–506

    Article  Google Scholar 

  • Meyer LW, Manwaring S (1986) Critical adiabatic shear strength of low alloyed steel under compressive loading. Metallurgical applications of shock wave and high-strain-rate phenomena. Marcel Dekker, New York, pp 657–674

    Google Scholar 

  • Miehe C, Welschinger F, Hofacker M (2010a) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Meth Eng 83:1273–1311

    Article  MathSciNet  Google Scholar 

  • Miehe C, Hofacker M, Welschinger F (2010b) A phase field model for rate-independent propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2765–2778

    Article  MathSciNet  Google Scholar 

  • Miehe C, Hofacker M, Schanzel LM, Aldakheel F (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to ductile failure criteria and crack propagation in thermos-elastic-plastic solids. Comput Methods Appl Mech Eng 294(9):486–522

    Article  MathSciNet  Google Scholar 

  • Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46(1):133–150

    Article  MathSciNet  Google Scholar 

  • Nguyen NV, Pham TH, Kim SE (2019) Strain rate-dependent behaviors of structural steel investigated using indentation and finite element analysis. Mech Mater 137:103089

    Article  Google Scholar 

  • Peirs J, Verleysen P, Van Paepegem W, Degrieck J (2011) Determining the stress-strain behavior at large strains from high strain rate tensile and shear experiments. Int J Impact Eng 38(5):406–415

    Article  Google Scholar 

  • Ran P, Lu X, Wang Z (2022) Investigation of the shear fracture behaviors of U71Mn at high strain rates using a shear-modified Gurson-Tvergaard-Needleman model. J Mater Eng Perform. https://doi.org/10.1007/s11665-022-07378-z

    Article  Google Scholar 

  • Samaniego C, Ulloa J, Rodríguez P, Houzeaux G, Vázquez M, Samaniego E (2021) A phase-field model for ductile fracture with shear bands: a parallel implementation. Int J Mech Sci 200:106424

    Article  Google Scholar 

  • Seabra MMR, Šuštraič P, Cesar de Sa JMA, Rodič T (2013) Damage driven crack initiation and propagation in ductile metals using XFEM. Comput Mech 52:161–179

    Article  MathSciNet  Google Scholar 

  • Smith CM, Deierlein GG, Kanvinde AM (2014) A stress-weighted damage model for ductile fracture initiation in structural steel under cyclic loading and generalized stress states. Tech Rep 187:10

    Google Scholar 

  • Tvergaard V, Hutchinson JW (1996) Effect of strain-dependent cohesive zone model on predictions of crack growth resistance. Int J Solids Struct 33(20–22):3297–3308

    Article  Google Scholar 

  • Ulmer H, Hofacker M, Miehe C (2013) Phase field modeling of brittle and ductile fracture. Proc Appl Math Mech 13:533–536

    Article  Google Scholar 

  • Wang T, Liu ZL, Cui YN, Ye X, Liu XM, Tian R, Zhuang Z (2020a) A thermos-elastic-plastic phase field model for simulating the evolution and transition of adiabatic shear band. Part I. Theory and model calibration. Eng Fract Mech 232:107028

    Article  Google Scholar 

  • Wang T, Liu ZL, Cui YN, Ye X, Liu XM, Tian R, Zhuang Z (2020b) A thermos-elastic-plastic phase field model for simulating the evolution and transition of adiabatic shear band. Part II. Dynamic collapse of thick-walled cylinder. Eng Fract Mech 231:107027

    Article  Google Scholar 

  • Wang T, Ye X, Liu Z, Liu X, Chu D, Zhuang Z (2020c) A phase-field model of thermo-elastic coupled brittle fracture with explicit time integration. Comput Mech 65:1305–1321

    Article  MathSciNet  Google Scholar 

  • Wang Y, Yang S, Chu D, Lu L, Liu Z (2023) Study of the mixed tensile-shear ductile fracture of impulsively loaded metal plates by developing a phase-field fracture model with stress triaxiality and Lode parameter dependence. Int J Fract. https://doi.org/10.1007/s10704-023-00695-x

    Article  Google Scholar 

  • Wilkins ML, Streit RD, Reaugh JE (1980) Cumulative-strain-damage model of ductile fracture: simulation and prediction of engineering fracture tests. Lawrence Livermore Laboratory, Livermore

    Book  Google Scholar 

  • Wu J, Wang Z (2022) Dynamic response and failure behavior of U71Mn using a hat-shaped specimen. J Mater Eng Perform 31:2193–2204

    Article  Google Scholar 

  • Wu J, Wang Z (2023) Comparative studies on shear failure behaviors of U71Mn rail steel at high strain rates using hat-shaped specimens. J Eng Mater Technol 145(1):011007

    Article  Google Scholar 

  • Yin B, Kaliske M (2020) A ductile phase-field model based on degrading the fracture toughness: theory and implementation at small strain. Comput Methods Appl Mech Eng 366:113068

    Article  MathSciNet  Google Scholar 

  • Yin B, Steinke C, Kaliske M (2019) Formulation and implementation of strain rate-dependent fracture toughness in context of the phase-model method. Int J Numer Meth Eng 121:233–255

    Article  Google Scholar 

  • Ziaei-Rad V, Shen Y (2016) Massive parallelization of the phase field formulation for crack propagation with time adaptivity. Comput Methods Appl Mech Eng 312:224–253

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by National Key R&D Program of China (Grant No. 2022YFB3401901) and Sichuan Science and Technology Program (Grant Nos. 2023NSFSC0394 and 2023NSFSC1988).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Tao Gu, Zhanjiang Wang & Pengfei Ran

Authors
  1. Tao Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhanjiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pengfei Ran
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

TG wrote the main manuscript text, and prepared Figures 1–14. ZW provided an approach of the modified model. PR prepared some experimental data. All authors reviewed the manuscript.

Corresponding author

Correspondence to Zhanjiang Wang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gu, T., Wang, Z. & Ran, P. A phase field model for ductile fracture considering the strain rate, stress triaxiality and Lode angle parameter. Int J Fract (2024). https://doi.org/10.1007/s10704-024-00770-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10704-024-00770-x

Keywords

Search

Navigation