Abstract
This section describes a phase field model for fracture. For the brittle version of the model the discretisation with finite elements is discussed. Higher order elements and elements based on exponential shape functions, that capture the one dimensional solution behavior, are addressed. For the exponential shape functions special attention is given to the quadrature rule, which plays an important role for the efficiency and accuracy. Furthermore, an adaptive strategy that combines standard bi-linear and exponential elements with higher accuracy is proposed. To extend the physical aspects of the fracture phase field model the existing model is extended to ductile fracture introducing plastic deformation. Depending on the hardening behavior different fracture modes are obtained and discussed.
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References
G.A. Francfort, J.J. Margio, Revisiting brittle fracture as an energy minimization program. J. Mech. Phys. Solids, 1319–1342 (1998)
B. Bourdin, Numerical implementation of the variational formulatoin of quasi-static brittle fracture. Interfaces Free Bound. 9(3), 411–430 (2007)
A.A. Griffith, The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. A 221, 163–198 (1921)
D. Gross, Th. Seelig, Fracture Mechanics (Springer, 2018)
C. Kuhn, Numerical and analytical Investigation of a Phase Field Model for Fracture. Ph.D. thesis. Technische Universität Kaiserslautern (2013)
C. Kuhn, R. Müller, Interpretation of parameters in phase field models for fracture. Proc. Appl. Math. Mech. 12, 161–162 (2012)
C. Kuhn, R. Müller, A new finite element technique for a phase field model of brittle fracture. J. Theor. Appl. Mech., 1115–1133 (2011)
O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 7th edn. (Butterworth-Heinemann, 2013)
T. Hidetosi, M. Mori, Double exponential formulas for numerical integration. RIMS, Kyoto Univ., 721–741 (1974)
R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods (Oxford University Press, 2013)
C. Kuhn, R. Müller, A discussion of fracture mechanisms in heterogeneous materials by means of configurational forces in a phase field fracture model. Comput. Methods Appl. Mech. Eng. 312, 95–116 (2016)
B. Bourdin, C. Larsen, C. Richardson, A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract., 133–143 (2011)
M. Hofacker, C. Miehe, A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int. J. Numer. Meth. Eng., 276–301 (2013)
C. Kuhn, R. Müller, A continuum phase field model for fracture. Eng. Fract. Mech. 77(18), 3625–3634 (2010)
S. Nagaraja, M. Elhaddad, M. Ambati, S. Kollmannsberger, L.D. Lorenzis, E. Rank, Comput. Mech. 63(6), 1283–1300 (2019)
M.J. Borden et al., 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 (2016)
T. Noll, C. Kuhn, R. Müller, Investigation of a phase field model for elasto-plastic fracture. Proc. Appl. Math. Mech. 16, 157–158 (2016)
T. Noll, C. Kuhn, R. Müller, A monolithic solution scheme for a phase field model of ductile fracture. Proc. Appl. Math. Mech. 17, 75–78 (2017)
T. Noll, C. Kuhn, R. Müller, Modeling of ductile fracture by a phase field approach. Proc. Appl. Math. Mech. 18, 1–2 (2018)
H. Amor, J.-J. Marigo, C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids 57(8), 1209–1229 (2009)
J.C. Simo, T.C.R. Hughes, Computational Inelasticity (Springer, New York, 1998)
B. Bourdin, G.A. Francfort, J.J. Marigo, Numerical experimentsin revisited brittle fracture. J. Mech. Phys. Solids, 797–82 (2000)
C. Kuhn, T. Noll, R. Müller, On phase field modeling of ductile fracture. GAMM-Mitteilungen, 35–54 (2016)
C. Kuhn, A. Schlüter, R. Müller, On degradation functions in phase field fracture models. Comput. Mater. Sci. 108, 374–384 (2015)
M. Borden et al., A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217–220, 77–95 (2012)
T. Noll, C. Kuhn, D. Olesch, R. Müller, 3D phase field simulations of ductile fracture. GAMM-Mitteilungen, 43-2 (2019)
H. Yuan, W. Brocks, Quantification of constraint effects in elastic-plastic crack front fields. J. Mech. Phys. Solids 46(2), 219–241 (1998)
D. Fernández Zúñiga, J.F. Kalthoff, A. Fernández Canteli, J. Grasa, M. Doblaré, Three dimensional finite element calculations of crack tip plastic zones and KIC specimen size requirements, in 17th European Conference on Fracture: Multilevel Approach to Fracture of Materials, Components and Structures (2005)
S. Kudari, K. Kodancha, Effect of specimen thickness on plastic zone, in 17th European Conference on Fracture 2008: Multilevel Approach to Fracture of Materials, Components and Structures 1, 530–538 (2008)
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Kuhn, C., Noll, T., Olesch, D., Müller, R. (2022). Phase Field Modeling of Brittle and Ductile Fracture. In: Schröder, J., Wriggers, P. (eds) Non-standard Discretisation Methods in Solid Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-030-92672-4_11
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