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Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. (English) Zbl 1202.74014
Summary: The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that \(\Gamma \)-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples.

74A45 Theories of fracture and damage
74A15 Thermodynamics in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI
[1] Griffith, The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society London, Series A 221 pp 163– (1921)
[2] Irwin, Encyclopedia of Physics 6 pp 551– (1958)
[3] Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics 7 pp 55– (1962)
[4] Francfort, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids 46 pp 1319– (1998) · Zbl 0966.74060
[5] Bourdin, The Variational Approach to Fracture (2008) · Zbl 1176.74018
[6] Dal Maso, A model for the quasistatic growth of brittle fractures: existence and approximation results, Archive for Rational Mechanics and Analysis 162 pp 101– (2002) · Zbl 1042.74002
[7] Buliga, Energy minimizing brittle crack propagation, Journal of Elasticity 52 pp 201– (1999) · Zbl 0947.74055
[8] Bourdin, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids 48 pp 797– (2000) · Zbl 0995.74057
[9] Mumford, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics 42 pp 577– (1989) · Zbl 0691.49036
[10] Ambrosio, Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma\)-convergence, Communications on Pure and Applied Mathematics 43 pp 999– (1990) · Zbl 0722.49020
[11] Dal Maso, An Introduction to \(\Gamma\)-convergence (1993) · doi:10.1007/978-1-4612-0327-8
[12] Braides, Approximation of Free Discontinuity Problems (1998) · Zbl 0909.49001 · doi:10.1007/BFb0097344
[13] Braides, \(\Gamma\)-convergence for Beginners (2002)
[14] Hakim, Laws of crack motion and phase-field models of fracture, Journal of the Mechanics and Physics of Solids 57 pp 342– (2009) · Zbl 1421.74089
[15] Karma, Phase-field model of mode III dynamic fracture, Physical Review Letters 92 pp 8704.045501– (2001)
[16] Eastgate, Fracture in mode I using a conserved phase-field model, Physical Review E 65 (2002)
[17] Belytschko, Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, International Journal for Numerical Methods in Engineering 58 pp 1873– (2003) · Zbl 1032.74662
[18] Song, Cracking node method for dynamic fracture with finite elements, International Journal for Numerical Methods in Engineering 77 pp 360– (2009) · Zbl 1155.74415
[19] Gürses, A computational framework of three-dimensional configurational-force-driven brittle crack propagation, Computer Methods in Applied Mechanics and Engineering 198 pp 1413– (2009)
[20] Miehe, A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment, International Journal for Numerical Methods in Engineering 72 pp 127– (2007) · Zbl 1194.74444
[21] Miehe, A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization, International Journal of Fracture 145 pp 245– (2007) · Zbl 1198.74008
[22] Capriz, Continua with Microstructure (1989) · doi:10.1007/978-1-4612-3584-2
[23] Mariano, Multifield theories in mechanics of solids, Advances in Applied Mechanics 38 pp 1– (2001)
[24] Frémond, Non-smooth Thermomechanics (2002) · doi:10.1007/978-3-662-04800-9
[25] Kachanov, Introduction to Continuum Damage Mechanics (1986) · Zbl 0596.73091 · doi:10.1007/978-94-017-1957-5
[26] Frémond, Damage, gradient of damage and principle of virtual power, International Journal of Solids and Structures 33 pp 1083– (1996) · Zbl 0910.73051
[27] Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D: Nonlinear Phenomena 92 pp 178– (1996)
[28] Bittencourt, Quasi-automatic simulation of crack propagation for 2D LEFM problems, Engineering Fracture Mechanics 55 pp 321– (1996)
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