Skip to main content
SpringerLink
Log in

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

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

Abstract

A thermal–mechanical coupled phase field fracture model is developed to study the complex dynamic crack propagation path in brittle material under thermal shock loading. By introducing a global continuum phase-field variable to describe the diffusive crack, the coupling between heat transfer, deformation and fracture is conveniently realized. A novel elastic energy density function is proposed to drive the evolution of phase-field variable in a more realistic way. The three-field coupling equations are efficiently solved by adopting a staggered time integration scheme. The coupled phase field fracture model is verified by comparing with three classical examples and is then applied to study the fracture of disk specimens under central thermal shock. The simulations reproduce the three different types of crack paths observed in experiments. It is found that the crack grows through the heating area straightly at lower heating body flux, while branches into two at higher heating body flux loading. The crack branching prefers to occur in the heating area with larger heating radius and prefers to occur outside the heating area with smaller heating radius. Interestingly, the crack branches when propagation speed is at its lowest point, and it always occurs close to the compression region. It is shown that the heterogeneous stress field induced by temperature inhomogeneity may have a strong influence on the crack branching under the thermal shock loading.

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

Similar content being viewed by others

References

  • Akatsu T, Takashima H, Shinoda Y, Wakai F, Wakayama S (2016) Thermal-shock fracture and damage resistance improved by whisker reinforcement in alumina matrix composite. Int J Appl Ceram Technol 13:653–661

    Article  Google Scholar 

  • Awaji H, Honda S, Yamamoto N, Endo T, Hirosaki N (2002) Thermal shock strength and thermal shock fracture toughness of ceramics. Fract Mech Ceram 13:363–379

    Article  Google Scholar 

  • Bahr H-A, Weiss H-J (1988) Multiple crack propagation in a strip caused by thermal shock. Theor Appl Fract Mech 10:219–226

    Article  Google Scholar 

  • Belytschko T, Liu WK, Moran B, Elkhodary KI (2014) Nonlinear finite elements for continua and structures, 2nd edn. Wiley, Chichester

    Google Scholar 

  • Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95

    Article  Google Scholar 

  • Bourdin B (2007) Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound 9:411–430

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bourdin B, Larsen CJ, Richardson CL (2010) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168:133–143

    Article  Google Scholar 

  • Bourdin B, Marigo JJ, Maurini C, Sicsic P (2014) Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys Rev Lett 112:014301

    Article  Google Scholar 

  • Chen HS, Yu X, Guo YZ, Fang DN (2013) Dynamic fracture toughness of piezoelectric ceramics. J Am Ceram Soc 96:2036–2038

    Article  Google Scholar 

  • Corson F, Adda-Bedia M, Henry H, Katzav E (2009) Thermal fracture as a framework for quasi-static crack propagation. Int J Fract 158:1–14

    Article  Google Scholar 

  • Geyer JF, Nemat-Nasser S (1982) Experimental investigation of thermally induced interacting cracks in brittle solids. Int J Solids Struct 18:349–356

    Article  Google Scholar 

  • Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93:105504

    Article  Google Scholar 

  • Hesch C, Weinberg K (2014) Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int J Numer Methods Eng 99:906–924

    Article  Google Scholar 

  • Hofacker M, Miehe C (2012) Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. Int J Fract 178:113–129

    Article  Google Scholar 

  • Honda S, Suzuki T, Nishikawa T, Awaji H, Akimune Y, Hirosaki N (2002) Estimation of thermal shock properties for silicon nitride having high thermal conductivity. J Ceram Soc Jpn 110:38–43 (in Japanese)

  • Honda S, Ogihara Y, Kishi T, Hashimoto S, Wamoto YI (2009) Estimation of thermal shock resistance of fine porous alumina by infrared. J Ceram Soc Jpn 117:1208–1215

    Article  Google Scholar 

  • Jenkins DR (2005) Optimal spacing and penetration of cracks in a shrinking slab. Phys Rev E Stat Nonlinear Soft Matter Phys 71:056117

    Article  Google Scholar 

  • Jiang CP, Wu XF, Li J, Song F, Shao YF, Xu XH, Yan P (2012) A study of the mechanism of formation and numerical simulations of crack patterns in ceramics subjected to thermal shock. Acta Mater 60:4540–4550

    Article  Google Scholar 

  • Kalthoff JF (2000) Modes of dynamic shear failure in solids. Int J Fract 101:1–31

    Article  Google Scholar 

  • Kalthoff J, Winkler S (1988) Failure mode transition at high rates of shear loading. Impact Load Dyn Behav Mater 1:185–195

    Google Scholar 

  • Karma A, Lobkovsky AE (2004) Unsteady crack motion and branching in a phase-field model of brittle fracture. Phys Rev Lett 92:245510

    Article  Google Scholar 

  • Katzav E, Adda-Bedia M, Arias R (2007) Theory of dynamic crack branching in brittle materials. Int J Fract 143:245–271

    Article  Google Scholar 

  • Klinsmann M, Rosato D, Kamlah M, McMeeking RM (2016) Modeling crack growth during Li insertion in storage particles using a fracture phase field approach. J Mech Phys Solids 92:313–344

    Article  Google Scholar 

  • Kuhn C, Müller R (2009) Phase field simulation of thermomechanical fracture. PAMM 9:191–192

    Article  Google Scholar 

  • Kuhn C, Muller R (2011) A new finite element technique for a phase field model of brittle fracture. J Theor Appl Mech 44:1115–1133

    Google Scholar 

  • Li J, Song F, Jiang C (2015) A non-local approach to crack process modeling in ceramic materials subjected to thermal shock. Eng Fract Mech 133:85–98

    Article  Google Scholar 

  • Liu G, Li Q, Msekh MA, Zuo Z (2016) Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model. Comput Mater Sci 121:35–47

    Article  Google Scholar 

  • Maurini C, Bourdin B, Gauthier G, Lazarus V (2014) Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach. Int J Fract 184:75–91

    Article  Google Scholar 

  • McAuliffe C, Waisman H (2015) A unified model for metal failure capturing shear banding and fracture. Int J Plast 65:131–151

    Article  Google Scholar 

  • McAuliffe C, Waisman H (2016) A coupled phase field shear band model for ductile–brittle transition in notched plate impacts. CMAME 305:173–195

    Google Scholar 

  • Menouillard T, Belytschko T (2010) Analysis and computations of oscillating crack propagation in a heated strip. Int J Fract 167:57–70

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Miehe C, Schänzel LM, Ulmer H (2015) 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 Appl Mech Eng 294:449–485

    Article  Google Scholar 

  • Miehe C, Dal H, Schänzel LM, Raina A (2016) A phase-field model for chemo-mechanical induced fracture in lithium-ion battery electrode particles. Int J Numer Methods Eng 106:683–711

    Article  Google Scholar 

  • Msekh MA, Sargado JM, Jamshidian M, Areias PM, Rabczuk T (2015) Abaqus implementation of phase-field model for brittle fracture. Comput Mater Sci 96:472–484

    Article  Google Scholar 

  • Qu Z, Cheng X, He R, Pei Y, Zhang R, Fang D (2016) Rapid heating thermal shock behavior study of CVD ZnS infrared window material: numerical and experimental study. J Alloys Compd 682:565–570

    Article  Google Scholar 

  • Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26:141–154

    Article  Google Scholar 

  • Rokhi MM, Shariati M (2013) Implementation of the extended finite element method for coupled dynamic thermoelastic fracture of a functionally graded cracked layer. J Braz Soc Mech Sci Eng 35:69–81

    Article  Google Scholar 

  • Ronsin O, Perrin B (1998) Dynamics of quasistatic directional crack growth. Phys Rev E 58:7878–7886

    Article  Google Scholar 

  • Ronsin O, Heslot F, Perrin B (1995) Experimental study of quasistatic brittle crack propagation. Phys Rev Lett 75:2352–2355

    Article  Google Scholar 

  • Schlüter A, Willenbücher A, Kuhn C, Müller R (2014) Phase field approximation of dynamic brittle fracture. Comput Mech 54:1141–1161

    Article  Google Scholar 

  • Shao Y, Zhang Y, Xu X, Zhou Z, Li W, Liu B, Jin ZH (2011) Effect of crack pattern on the residual strength of ceramics after quenching. J Am Ceram Soc 94:2804–2807

    Article  Google Scholar 

  • Sharon E, Fineberg J (1999) Confirming the continuum theory of dynamic brittle fracture for fast cracks. Nature 397:333–335

    Article  Google Scholar 

  • Sicsic P, Marigo J-J, Maurini C (2014) Initiation of a periodic array of cracks in the thermal shock problem: a gradient damage modeling. J Mech Phys Solids 63:256–284

    Article  Google Scholar 

  • Song JH, Wang H, Belytschko T (2007) A comparative study on finite element methods for dynamic fracture. Comput Mech 42:239–250

    Article  Google Scholar 

  • Spatschek R, Brener E, Karma A (2011) Phase field modeling of crack propagation. Philos Mag 91:75–95

    Article  Google Scholar 

  • Tang SB, Zhang H, Tang CA, Liu HY (2016) Numerical model for the cracking behavior of heterogeneous brittle solids subjected to thermal shock. Int J Solids Struct 80:520–531

    Article  Google Scholar 

  • Tarasovs S, Ghassemi A (2014) Self-similarity and scaling of thermal shock fractures. Phys Rev E Stat Nonlinear Soft Matter Phys 90:012403

    Article  Google Scholar 

  • Yoffe EH (1951) LXXV. The moving griffith crack. London Edinb Dublin Philos Mag J Sci 42(330):739–750

    Article  Google Scholar 

  • Zamani A, Eslami MR (2010) Implementation of the extended finite element method for dynamic thermoelastic fracture initiation. Int J Solids Struct 47:1392–1404

    Article  Google Scholar 

  • Zhang XH, Han JC, He XD, Yan C, Wang BL, Xu Q (2004) Ablation-resistance of combustion synthesized \(\text{ TiB }_{2}\)-Cu cermet. J Am Ceram Soc 88:89–94

    Article  Google Scholar 

  • Zhang X, Krischok A, Linder C (2016) A variational framework to model diffusion induced large plastic deformation and phase field fracture during initial two-phase lithiation of silicon electrodes. Comput Methods Appl Mech Eng 312:51–77

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China, under Grant No. 11302115, Tsinghua University Initiative Scientific Research Program, Chinese 1000-talents Plan for Young Researchers, Foundation of National Key Laboratory of Science and Technology on Computational Physics, Science Challenge Program, under Grant No. JCKY2016212A502.

Author information

Authors and Affiliations

  1. Applied Mechanics Laboratory, Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing, 100084, China

    Dongyang Chu, Xiang Li & Zhanli Liu

Authors
  1. Dongyang Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhanli Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhanli Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chu, D., Li, X. & Liu, Z. Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling. Int J Fract 208, 115–130 (2017). https://doi.org/10.1007/s10704-017-0220-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-017-0220-4

Keywords

Search

Navigation