# American Institute of Mathematical Sciences

February  2013, 6(1): 131-146. doi: 10.3934/dcdss.2013.6.131

## Some remarks on the viscous approximation of crack growth

 1 Universität Würzburg, Institut für Mathematik, Emil-Fischer-Straße 40, 97074 Würzburg, Germany 2 Università degli Studi di Udine, DIMI, Via delle Scienze 206, 33100 Udine, Italy

Received  May 2011 Revised  September 2011 Published  October 2012

We describe an existence result for quasistatic evolutions of cracks in antiplane elasticity obtained in [16] by a vanishing viscosity approach, with free (but regular enough) crack path. We underline in particular the motivations for the choice of the class of admissible cracks and of the dissipation potential. Moreover, we extend the result to a model with applied forces depending on time.
Citation: Giuliano Lazzaroni, Rodica Toader. Some remarks on the viscous approximation of crack growth. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 131-146. doi: 10.3934/dcdss.2013.6.131
##### References:
 [1] G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Problems, 18 (2002), 1333-1353. doi: 10.1088/0266-5611/18/5/308. [2] B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148. [3] D. Bucur and N. Varchon, A duality approach for the boundary variation of Neumann problems, SIAM J. Math. Anal., 34 (2002), 460-477. doi: 10.1137/S0036141002389579. [4] D. Bucur and J. P. Zolésio, $N$-dimensional shape optimization under capacitary constraint, J. Differential Equations, 123 (1995), 504-522. doi: 10.1006/jdeq.1995.1171. [5] A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal., 167 (2003), 211-233. doi: 10.1007/s00205-002-0240-7. [6] G. Dal Maso, F. Ebobisse and M. Ponsiglione, A stability result for nonlinear Neumann problems under boundary variations, J. Math. Pures Appl. (9), 82 (2003), 503-532. [7] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187. [8] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799. doi: 10.1142/S0218202502002331. [9] P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l'intégrale de Rice en th\'eorie de la rupture fragile, Math. Methods Appl. Sci., 3 (1981), 70-87. [10] A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163-198. doi: 10.1098/rsta.1921.0006. [11] P. Grisvard, "Singularities in Boundary Value Problems,'' Research Notes in Applied Mathematics, 22, Masson, Paris, Springer-Verlag, Berlin, 1992. [12] 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. doi: 10.1142/S0218202508003121. [13] V. A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations, Z. Angew. Math. Phys., 54 (2003), 410-423. doi: 10.1007/s00033-003-0143-y. [14] C. Larsen, Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl. Math., 63 (2010), 630-654. [15] G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity, J. Math. Pures Appl. (9), 95 (2011), 565-584. [16] G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), 2019-2047. [17] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations'' (eds. C. M. Dafermos and E. Feireisl), Handbook of Differential Equations,Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [18] A. Mielke, R. Rossi and G. Savaré, $BV$ solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80. doi: 10.1051/cocv/2010054. [19] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585. [20] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925. doi: 10.1142/S0218202508003236. [21] U. Stefanelli, A variational characterization of rate-independent evolution, Math. Nachr., 282 (2009), 1492-1512. doi: 10.1002/mana.200810803. [22] V. Šverák, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551. [23] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital., (9), 2 (2009), 1-35.

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##### References:
 [1] G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Problems, 18 (2002), 1333-1353. doi: 10.1088/0266-5611/18/5/308. [2] B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148. [3] D. Bucur and N. Varchon, A duality approach for the boundary variation of Neumann problems, SIAM J. Math. Anal., 34 (2002), 460-477. doi: 10.1137/S0036141002389579. [4] D. Bucur and J. P. Zolésio, $N$-dimensional shape optimization under capacitary constraint, J. Differential Equations, 123 (1995), 504-522. doi: 10.1006/jdeq.1995.1171. [5] A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal., 167 (2003), 211-233. doi: 10.1007/s00205-002-0240-7. [6] G. Dal Maso, F. Ebobisse and M. Ponsiglione, A stability result for nonlinear Neumann problems under boundary variations, J. Math. Pures Appl. (9), 82 (2003), 503-532. [7] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187. [8] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799. doi: 10.1142/S0218202502002331. [9] P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l'intégrale de Rice en th\'eorie de la rupture fragile, Math. Methods Appl. Sci., 3 (1981), 70-87. [10] A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163-198. doi: 10.1098/rsta.1921.0006. [11] P. Grisvard, "Singularities in Boundary Value Problems,'' Research Notes in Applied Mathematics, 22, Masson, Paris, Springer-Verlag, Berlin, 1992. [12] 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. doi: 10.1142/S0218202508003121. [13] V. A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations, Z. Angew. Math. Phys., 54 (2003), 410-423. doi: 10.1007/s00033-003-0143-y. [14] C. Larsen, Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl. Math., 63 (2010), 630-654. [15] G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity, J. Math. Pures Appl. (9), 95 (2011), 565-584. [16] G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), 2019-2047. [17] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations'' (eds. C. M. Dafermos and E. Feireisl), Handbook of Differential Equations,Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [18] A. Mielke, R. Rossi and G. Savaré, $BV$ solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80. doi: 10.1051/cocv/2010054. [19] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585. [20] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925. doi: 10.1142/S0218202508003236. [21] U. Stefanelli, A variational characterization of rate-independent evolution, Math. Nachr., 282 (2009), 1492-1512. doi: 10.1002/mana.200810803. [22] V. Šverák, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551. [23] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital., (9), 2 (2009), 1-35.
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