zbMATH — the first resource for mathematics

Phase field approximation of dynamic brittle fracture. (English) Zbl 1311.74106
Summary: Numerical methods that are able to predict the failure of technical structures due to fracture are important in many engineering applications. One of these approaches, the so-called phase field method, represents cracks by means of an additional continuous field variable. This strategy avoids some of the main drawbacks of a sharp interface description of cracks. For example, it is not necessary to track or model crack faces explicitly, which allows a simple algorithmic treatment. The phase field model for brittle fracture presented by C. Kuhn and R. Müller [“A continuum phase field model for fracture”, Eng. Fract. Mech. 77, No. 18, 3625–3634 (2010)] assumes quasi-static loading conditions. However dynamic effects have a great impact on the crack growth in many practical applications. Therefore this investigation presents an extension of the quasi-static phase field model for fracture from [loc. cit.] to the dynamic case. First of all Hamilton’s principle is applied to derive a coupled set of Euler-Lagrange equations that govern the mechanical behaviour of the body as well as the crack growth. Subsequently the model is implemented in a finite element scheme which allows to solve several test problems numerically. The numerical examples illustrate the capabilities of the developed approach to dynamic fracture in brittle materials.

74R10 Brittle fracture
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI
[1] Ambrosio, L; Tortorelli, VM, Approximation of functional depending on jumps by elliptic functional via \(γ \)-convergence, Commun Pure Appl Math, 43, 999-1036, (1990) · Zbl 0722.49020
[2] Amor, H; Marigo, JJ; Maurini, C, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, J Mech Phys Solid, 57, 1209-1229, (2009) · Zbl 1426.74257
[3] Bertram, A; Kalthoff, JF, Crack propagation toughness of rock for the range of low to very high crack speeds, Key Eng Mater, 251-252, 423-430, (2003)
[4] Borden, MJ; Verhoosel, CV; Scott, MA; Hughes, TJR; Landis, CM, A phase-field description of dynamic brittle fracture, Comput Meth Appl Mech Eng, 217-220, 77-95, (2012) · Zbl 1253.74089
[5] Bourdin, B, Numerical implementation of the variational formulation of quasi-static brittle fracture, Interfaces Free Boundaries, 9, 411-430, (2007) · Zbl 1130.74040
[6] Bourdin, B; Larsen, C; Richardson, C, A time-discrete model for dynamic fracture based on crack regularization, Int J Fract, 168, 133-143, (2011) · Zbl 1283.74055
[7] Braides A (2002) \(Γ \)-convergence for beginners. In: Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford · Zbl 1198.49001
[8] Chambolle, A, An approximation result for special functions with bounded deformation, J Math Pure Appl, 83, 929-954, (2004) · Zbl 1084.49038
[9] Das, S, Dynamic fracture mechanics in the study of the earthquake rupturing process: theory and observation, J Mech Phys Solid, 51, 1939-1955, (2003) · Zbl 1041.74539
[10] Fineberg, J; Gross, SP; Marder, M; Swinney, HL, Instability in the propagation of fast cracks, Phys Rev B, 45, 5146-5154, (1992)
[11] Francfort, GA; Marigo, JJ, Revisiting brittle fracture as an energy minimization problem, J Mech Phys Solid, 46, 1319-1342, (1998) · Zbl 0966.74060
[12] Freund LB (1990) Dynamic Fracture Mechanics. Cambridge University Press, Cambridge · Zbl 0712.73072
[13] Griffith, AA, The phenomena of rupture and flow in solids, Philos Trans R Soc Lond A, 221, 163-198, (1921)
[14] Gross D, Seelig T (2010) Fracture mechanics: with an introduction to micromechanics. Mechanical engineering series. Springer, Berlin · Zbl 1110.74001
[15] Gross, SP; Fineberg, J; Marder, M; McCormick, WD; Swinney, HL, Acoustic emissions from rapidly moving cracks, Phys Rev Lett, 71, 3162-3165, (1993)
[16] Gurtin, ME, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys D, 92, 178-192, (1996) · Zbl 0885.35121
[17] Hakim, V; Karma, A, Laws of crack motion and phase-field models of fracture, J Mech Phys Solid, 57, 342-368, (2009) · Zbl 1421.74089
[18] Hofacker, M; Miehe, C, Continuum phase field modeling of dynamic fracture: variational principles and staggered fe implementation, Int J Fract, 178, 113-129, (2012)
[19] Hofacker, M; Miehe, C, A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns, Int J Numer Methods Eng, 93, 276-301, (2013) · Zbl 1352.74022
[20] Hopkinson B (1921) The pressure of a blow. The Scientific Papers of Bertram Hopkinson pp 423-437
[21] Hopkinson, J, On the rupture of iron wire by a blow, Proc Manch Lit Philos Soc, 11, 40-45, (1872)
[22] Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications Inc, Mineola
[23] Irwin, GR, Analysis of stresses and strains near the end of a crack traversing a plate, J Appl Mech, 24, 361-364, (1957)
[24] Kalthoff, J, Modes of dynamic shear failure in solids, Int J Fract, 101, 1-31, (2000)
[25] Kalthoff, JF; Winkler, S, Failure mode transition of high rates of shear loading, Proc. I. Con. Imp. Load. Dyn. Beh. Mat., 1, 185-195, (1987)
[26] Katzav, E; Adda-Bedia, M; Arias, R, Theory of dynamic crack branching in brittle materials, Int. J. Fract., 143, 245-271, (2007) · Zbl 1197.74111
[27] Ravi-Chandar, K; Knauss, WG, An experimental investigation into dynamic fracture: iii on steady-state crack propagation and crack branching, Int J Fract, 26, 33-72, (1984)
[28] Krueger, R, Virtual crack closure technique: history, approach, and applications, Appl Mech Rev, 57, 109, (2004)
[29] Kuhn C (2013) Numerical and analytical investigation of a phase field model for fracture. Phd thesis, Technische Universität Kaiserslautern · Zbl 1253.74089
[30] Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fract Mech 77(18):3625-3634. Computational mechanics in fracture and damage: a special issue in honor of Prof. Gross
[31] Maso GD (1993) Introduction to \(Γ \)-convergence. Progress in nonlinear differential equations and their applications, Birkhäuser · Zbl 1352.74022
[32] Mello, M; Bhat, HS; Rosakis, AJ; Kanamori, H, Reproducing the supershear portion of the 2002 denali earthquake rupture in laboratory, Earth Planet Sci Lett, 387, 89-96, (2014)
[33] Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46(1):131-150 · Zbl 0955.74066
[34] Mumford, D; Shah, J, Optimal approximations by piecewise smooth functions and associated variational problems, Commun Pure Appl Math, 42, 577-685, (1989) · Zbl 0691.49036
[35] Ravi-Chandar, K; Knauss, W, An experimental investigation into dynamic fracture: I. crack initiation and arrest, Int J Fract, 25, 247-262, (1984)
[36] Ravi-Chandar, K; Knauss, W, An experimental investigation into dynamic fracture: ii. microstructural aspects, Int J Fract, 26, 65-80, (1984)
[37] Ravi-Chandar, K; Knauss, W, An experimental investigation into dynamic fracture: iii. on steady-state crack propagation and crack branching, Int J Fract, 26, 141-154, (1984)
[38] Ravi-Chandar, K; Knauss, W, An experimental investigation into dynamic fracture: iv. on the interaction of stress waves with propagating cracks, Int J Fract, 26, 189-200, (1984)
[39] Rosakis, AJ; Samudrala, O; Coker, D, Intersonic shear crack growth along weak planes, Mater Res Innov, 3, 236-243, (2000)
[40] Schlueter A, Willenbuecher A, Kuhn C (2013) GPU-accelerated crack path computation based on a phase field approach for brittle fracture. In: Proceedings of the 2nd young researcher symposium (YRS) 2013, pp 60-65. Fraunhofer Verlag, Kaiserslautern, Germany · Zbl 0966.74060
[41] Seelig, T; Gross, D, On the interaction and branching of fast running cracksâa numerical investigation, J Mech Phys Solids, 47, 935-952, (1999) · Zbl 0959.74058
[42] Sharon, E; Gross, SP; Fineberg, J, Local crack branching as a mechanism for instability in dynamic fracture, Phys Rev Lett, 74, 5096-5099, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.