# American Institute of Mathematical Sciences

February  2013, 6(1): 63-99. doi: 10.3934/dcdss.2013.6.63

## Computational aspects of quasi-static crack propagation

 1 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany 2 Humboldt-Universität zu Berlin, Institut für Mathematik, Rudower Chaussee 25，12489 Berlin, Germany

Received  May 2011 Revised  August 2011 Published  October 2012

The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization.
We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.
Citation: Dorothee Knees, Andreas Schröder. Computational aspects of quasi-static crack propagation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 63-99. doi: 10.3934/dcdss.2013.6.63
##### References:
 [1] M. Costabel and M. Dauge, Crack singularities for general elliptic systems, Math. Nachrichten, 235 (2002), 29-49. doi: 10.1002/1522-2616(200202)235:1<29::AID-MANA29>3.0.CO;2-6. [2] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. [3] P. Destuynder and M. Djaoua, Sur une interpretation mathématique de l'intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci., 3 (1981), 70-87. doi: 10.1002/mma.1670030106. [4] P. Destuynder, M. Djaoua and S. Lescure, On a numerical method for fracture mechanics, Singularities and constructive methods for their treatment, Proc. Conf., Oberwolfach 1983, Lect. Notes Math., 1121 (1985), 69-84. [5] C. Ebmeyer and J. Frehse, Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains, Math. Nachrichten, 203 (1999), 47-74. [6] Messoud A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167. [7] Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput., 28 (1974), 963-971. doi: 10.1090/S0025-5718-1974-0391502-8. [8] G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. [9] A. Giacomini and M. Ponsiglione, Discontinuous finite element approximation of quasistatic crack growth in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 77-118. [10] M. Giaquinta and S. Hildebrandt, "Calculus of Variations I," Springer-Verlag, Berlin, Heidelberg, 1996. [11] P. Grisvard, Singularité en élasticité, Arch. Ration. Mech. Anal., 107 (1989), 157-180. doi: 10.1007/BF00286498. [12] D. Gross, "Bruchmechanik," Springer Verlag, Berlin, Heidelberg, 1996. [13] A. M. Khludnev and J. Sokolowski, Griffith formulae for elasticity systems with unilateral conditions in domains with cracks, Eur. J. Mech., A/Solids, 19 (2000), 105-119. doi: 10.1016/S0997-7538(00)00138-8. [14] A. M. Khludnev, V, A. Kovtunenko and A. Tani, Evolution of a crack with kink and non-penetration, J. Math. Soc. Japan, 60 (2008), 1219-1253. [15] N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," Philadelphia, PA: SIAM, 1988. doi: 10.1137/1.9781611970845. [16] D. Knees, "A Short Survey on Energy Release Rates," Oberwolfach Report, Mini-Workshop Mini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Organizers: G. Dal Maso, C. J. Larsen, C. Ortner, 2011. [17] D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci., 31 (2008), 501-528. [18] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci., 18 (2008), 1529-1569. [19] D. Knees, C. Zanini and A. Mielke, Crack growth in polyconvex materials, Physica D, 239 (2010), 1470-1484. [20] D. Knees and A. Schröder, "Global Spatial Regularity for an Elasticity Model with Cracks and Contact," Math. Methods Appl. Sci., accepted (2012). [21] V. A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 10 (1967), 227-313. [22] G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), 2019–2047. [23] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," Vol. I. Translated from the French by P. Kenneth, Grundlehren der Mathematischen Wissenschaften, Band 181, Springer-Verlag, 1972. [24] V. G. Maz'ya, S. A. Nazarov and B. A. Plamenevsky, "Asymptotische Theorie elliptischer Randwertaufgaben in Singulär Gestörten Gebieten I,II," Akademie-Verlag, Berlin, 1991. [25] A. Mielke, Evolution of rate-independent systems, in "Evolutionary equations.'' (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [26] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst. Ser. A, 25 (2009), 585-615. [27] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differ. Equ., 31 (2008), 387-416. [28] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925. [29] S. Nicaise and A. M. Sändig, Transmission problems for the Laplace and elasticity operators: Regularity and boundary integral formulation, Math. Models Methods Appl. Sci., 9 (1999), 855-898. doi: 10.1142/S0218202599000403. [30] S. Nicaise and A.-M. Sändig, Dynamic crack propagation in a 2D elastic body: The out-of-plane case, J. Math. Anal. Appl., 329 (2007), 1-30. [31] J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comput., 28 (1974), 937-958. doi: 10.1090/S0025-5718-1974-0373325-9. [32] W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill Book Company, 1976. [33] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boun dary conditions, Math. Comput., 54 (1996), 483-493. doi: 10.1090/S0025-5718-1990-1011446-7. [34] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser, Basel, 1983. [35] H. Triebel, "Theory of Function Spaces," Reprint of the 1983, original Basel: Birkhäuser, 2010.

show all references

##### References:
 [1] M. Costabel and M. Dauge, Crack singularities for general elliptic systems, Math. Nachrichten, 235 (2002), 29-49. doi: 10.1002/1522-2616(200202)235:1<29::AID-MANA29>3.0.CO;2-6. [2] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. [3] P. Destuynder and M. Djaoua, Sur une interpretation mathématique de l'intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci., 3 (1981), 70-87. doi: 10.1002/mma.1670030106. [4] P. Destuynder, M. Djaoua and S. Lescure, On a numerical method for fracture mechanics, Singularities and constructive methods for their treatment, Proc. Conf., Oberwolfach 1983, Lect. Notes Math., 1121 (1985), 69-84. [5] C. Ebmeyer and J. Frehse, Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains, Math. Nachrichten, 203 (1999), 47-74. [6] Messoud A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167. [7] Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput., 28 (1974), 963-971. doi: 10.1090/S0025-5718-1974-0391502-8. [8] G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. [9] A. Giacomini and M. Ponsiglione, Discontinuous finite element approximation of quasistatic crack growth in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 77-118. [10] M. Giaquinta and S. Hildebrandt, "Calculus of Variations I," Springer-Verlag, Berlin, Heidelberg, 1996. [11] P. Grisvard, Singularité en élasticité, Arch. Ration. Mech. Anal., 107 (1989), 157-180. doi: 10.1007/BF00286498. [12] D. Gross, "Bruchmechanik," Springer Verlag, Berlin, Heidelberg, 1996. [13] A. M. Khludnev and J. Sokolowski, Griffith formulae for elasticity systems with unilateral conditions in domains with cracks, Eur. J. Mech., A/Solids, 19 (2000), 105-119. doi: 10.1016/S0997-7538(00)00138-8. [14] A. M. Khludnev, V, A. Kovtunenko and A. Tani, Evolution of a crack with kink and non-penetration, J. Math. Soc. Japan, 60 (2008), 1219-1253. [15] N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," Philadelphia, PA: SIAM, 1988. doi: 10.1137/1.9781611970845. [16] D. Knees, "A Short Survey on Energy Release Rates," Oberwolfach Report, Mini-Workshop Mini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Organizers: G. Dal Maso, C. J. Larsen, C. Ortner, 2011. [17] D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci., 31 (2008), 501-528. [18] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci., 18 (2008), 1529-1569. [19] D. Knees, C. Zanini and A. Mielke, Crack growth in polyconvex materials, Physica D, 239 (2010), 1470-1484. [20] D. Knees and A. Schröder, "Global Spatial Regularity for an Elasticity Model with Cracks and Contact," Math. Methods Appl. Sci., accepted (2012). [21] V. A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 10 (1967), 227-313. [22] G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), 2019–2047. [23] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," Vol. I. Translated from the French by P. Kenneth, Grundlehren der Mathematischen Wissenschaften, Band 181, Springer-Verlag, 1972. [24] V. G. Maz'ya, S. A. Nazarov and B. A. Plamenevsky, "Asymptotische Theorie elliptischer Randwertaufgaben in Singulär Gestörten Gebieten I,II," Akademie-Verlag, Berlin, 1991. [25] A. Mielke, Evolution of rate-independent systems, in "Evolutionary equations.'' (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [26] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst. Ser. A, 25 (2009), 585-615. [27] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differ. Equ., 31 (2008), 387-416. [28] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925. [29] S. Nicaise and A. M. Sändig, Transmission problems for the Laplace and elasticity operators: Regularity and boundary integral formulation, Math. Models Methods Appl. Sci., 9 (1999), 855-898. doi: 10.1142/S0218202599000403. [30] S. Nicaise and A.-M. Sändig, Dynamic crack propagation in a 2D elastic body: The out-of-plane case, J. Math. Anal. Appl., 329 (2007), 1-30. [31] J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comput., 28 (1974), 937-958. doi: 10.1090/S0025-5718-1974-0373325-9. [32] W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill Book Company, 1976. [33] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boun dary conditions, Math. Comput., 54 (1996), 483-493. doi: 10.1090/S0025-5718-1990-1011446-7. [34] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser, Basel, 1983. [35] H. Triebel, "Theory of Function Spaces," Reprint of the 1983, original Basel: Birkhäuser, 2010.
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