Abstract
Domain switching in the vicinity of a crack tip is known as one of the major aspects of local nonlinear behavior of ferroelectrics, and it plays an important role in the fracture behavior. In the present paper, a fracture model based on a phase field continuum and a damage variable is presented to study the fracture behavior of ferroelectrics and its interaction with the domain structures. In this model the energy of fracture is regularized by the damage variable. When the damage variable equals one, it represents undamaged material. In this case, the energy reduces to the phase field potential with the spontaneous polarization being an order parameter, and the system of equations becomes the same as that of a conventional phase field continuum. When the damage variable becomes zero, it represents a crack region, and the potential becomes the energy density stored in the crack medium. The evolution of the damage variable is governed by a Ginzburg-Landau type equation. In this way, the fracture model can simulate the fracture behavior such as crack growth, kinking and formation, with no a priori assumption on fracture criteria and predefined crack paths. The model is implemented in a 2D Finite Element Method in combination with implicit time integration and non-linear Newton iteration. As example, the fracture model is used to simulate the fracture of an edge crack in a ferroelectric single crystal under mechanical mode-I loading. In the simulation crack propagation, kinking and formation are observed. In particular, the results show the interaction between the domain structure evolution and the crack propagation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aranson IS, Kalatsky VA, Vinokur VM (2000) Continuum field description of crack propagation. Phys Rev Lett 85(1): 118–121
Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48: 797–826
Chen YH, Hasebe N (2005) Current understanding on fracture behaviors of ferroelectric/piezoelectric materials. J Intell Mater Sys Struc 16(7–8): 673–687
Eastgate LO, Sethna JP, Rauscher M, Cretegny T (2002) Fracture in mode i using a conserved phase-field model. Phys Rev E71: 036117
Francfort GA, Marigo J (1998) Revising brittle fracture as an energy minimization problem. J Mech Phys Solids 35(7): 1319–1342
Hughes T (2000) The finite element method. Dover, Mineola, New York
Kamlah M (2001) Ferroelectric and ferroelastic piezoceramics—modeling of electromechanical hysteresis phenomena. Cont Mech Thermodyn 13: 219–268
Kuhn C, Mueller R (2008) A phase field model for fracture. PAMM 8(1): 10223–10224
Kuhn C, Mueller R, A continuum phase field model for fracture. Eng Fract Mech, (Special Issue on CMFD, submitted)
Mueller R, Gross D, Schrade D, Xu BX (2007) Phase field simulation of domain structure in ferroelectric materials within the context of inhomogeneity evolution. Int J Frac 147(1–4): 173–180
Mueller R, Xu BX, Schrade D, Gross D Modelling of domain structure evolution in ferroelectric materials. Int J Fract, (Special Issue on IUTAM2009, accepted)
Schneider GA (2007) Influence of electric field and mechanical stresses on the fracture of ferroelectrics. Ann Rev Mater Res 37: 491–538
Schrade D, Mueller R, Xu BX, Gross D (2007) Domain evolution in ferroelectric materials: a continuum phase field model and finite element implementation. Comput Meth Appl Mech Eng 196: 4365–4374
Schrade D, Xu BX, Mueller R, Gross D (2009) On phase field simulations of ferroelectrics: Parameter identification and verification. Proceedings of the ASME 2008 smart materials, adaptive structures and intelligent systems (SMASIS2008), pp 301–308
Schrade D, Mueller R, Gross D (2010) Parameter identification in phase field models for ferroelectrics. Proc Appl Math Mech 9(1): 369–370
Song YC, Soh AK, Ni Y (2007) Phase field simulation of crack tip domain switching in ferroelectrics. J Phys D Appl Phys 40: 1175–1182
Spatschek R, Hartmann M, Brener EA, Mueller-Krumbhaar H (2006) Phase field modeling of fast crack propagation. Phys Rev Lett 96: 015502
Xu BX, Schrade D, Mueller R, Gross D (2009) Micromechanical analysis of ferroelectric structures by a phase field method. Comput Mater Sci 45: 832–836
Xu BX, Schrade D, Gross D, Mueller R (2010) Phase field simulation of domain structures in cracked ferroelectrics. Int J Fract. Special Issue: Broberg symposium. doi:10.1007/s10704-010-9471-z
Wang J, Zhang TY (2007) Simulations of polarization switching-induced toughening in ferroelectric ceramics. Acta Mate 55: 2465–2477
Zhang TY, Zhao M, Tong P (2002) Fracture of piezoelectric ceramics. Adv Appl Mech 38: 147–289
Zhang TY, Gao CF (2004) Fracture behaviors of piezoelectric materials. Theor Appl Frac Mech 41: 339–379
Author information
Authors and Affiliations
Corresponding author
Additional information
The generous support by the Alexander von Humboldt Foundation is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Xu, BX., Schrade, D., Gross, D. et al. Fracture simulation of ferroelectrics based on the phase field continuum and a damage variable. Int J Fract 166, 163–172 (2010). https://doi.org/10.1007/s10704-010-9520-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-010-9520-7