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 & 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. doi: 10.1088/0266-5611/18/5/308. Google Scholar

[2]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, J. Elasticity, 91 (2008), 5. Google Scholar

[3]

D. Bucur and N. Varchon, A duality approach for the boundary variation of Neumann problems,, SIAM J. Math. Anal., 34 (2002), 460. doi: 10.1137/S0036141002389579. Google Scholar

[4]

D. Bucur and J. P. Zolésio, $N$-dimensional shape optimization under capacitary constraint,, J. Differential Equations, 123 (1995), 504. doi: 10.1006/jdeq.1995.1171. Google Scholar

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications,, Arch. Ration. Mech. Anal., 167 (2003), 211. doi: 10.1007/s00205-002-0240-7. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s002050100187. Google Scholar

[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. doi: 10.1142/S0218202502002331. Google Scholar

[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. Google Scholar

[10]

A. A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163. doi: 10.1098/rsta.1921.0006. Google Scholar

[11]

P. Grisvard, "Singularities in Boundary Value Problems,'', Research Notes in Applied Mathematics, (1992). Google Scholar

[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. doi: 10.1142/S0218202508003121. Google Scholar

[13]

V. A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations,, Z. Angew. Math. Phys., 54 (2003), 410. doi: 10.1007/s00033-003-0143-y. Google Scholar

[14]

C. Larsen, Epsilon-stable quasi-static brittle fracture evolution,, Comm. Pure Appl. Math., 63 (2010), 630. Google Scholar

[15]

G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity,, J. Math. Pures Appl. (9), 95 (2011), 565. Google Scholar

[16]

G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation,, Math. Models Methods Appl. Sci., 21 (2011), 2019. Google Scholar

[17]

A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461. Google Scholar

[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. doi: 10.1051/cocv/2010054. Google Scholar

[19]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Adv. Math., 3 (1969), 510. Google Scholar

[20]

M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion,, Math. Models Methods Appl. Sci., 18 (2008), 1895. doi: 10.1142/S0218202508003236. Google Scholar

[21]

U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492. doi: 10.1002/mana.200810803. Google Scholar

[22]

V. Šverák, On optimal shape design,, J. Math. Pures Appl. (9), 72 (1993), 537. Google Scholar

[23]

R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth,, Boll. Unione Mat. Ital., 2 (2009), 1. Google Scholar

show all references

References:
[1]

G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements,, Inverse Problems, 18 (2002), 1333. doi: 10.1088/0266-5611/18/5/308. Google Scholar

[2]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, J. Elasticity, 91 (2008), 5. Google Scholar

[3]

D. Bucur and N. Varchon, A duality approach for the boundary variation of Neumann problems,, SIAM J. Math. Anal., 34 (2002), 460. doi: 10.1137/S0036141002389579. Google Scholar

[4]

D. Bucur and J. P. Zolésio, $N$-dimensional shape optimization under capacitary constraint,, J. Differential Equations, 123 (1995), 504. doi: 10.1006/jdeq.1995.1171. Google Scholar

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications,, Arch. Ration. Mech. Anal., 167 (2003), 211. doi: 10.1007/s00205-002-0240-7. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s002050100187. Google Scholar

[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. doi: 10.1142/S0218202502002331. Google Scholar

[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. Google Scholar

[10]

A. A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163. doi: 10.1098/rsta.1921.0006. Google Scholar

[11]

P. Grisvard, "Singularities in Boundary Value Problems,'', Research Notes in Applied Mathematics, (1992). Google Scholar

[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. doi: 10.1142/S0218202508003121. Google Scholar

[13]

V. A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations,, Z. Angew. Math. Phys., 54 (2003), 410. doi: 10.1007/s00033-003-0143-y. Google Scholar

[14]

C. Larsen, Epsilon-stable quasi-static brittle fracture evolution,, Comm. Pure Appl. Math., 63 (2010), 630. Google Scholar

[15]

G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity,, J. Math. Pures Appl. (9), 95 (2011), 565. Google Scholar

[16]

G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation,, Math. Models Methods Appl. Sci., 21 (2011), 2019. Google Scholar

[17]

A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461. Google Scholar

[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. doi: 10.1051/cocv/2010054. Google Scholar

[19]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Adv. Math., 3 (1969), 510. Google Scholar

[20]

M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion,, Math. Models Methods Appl. Sci., 18 (2008), 1895. doi: 10.1142/S0218202508003236. Google Scholar

[21]

U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492. doi: 10.1002/mana.200810803. Google Scholar

[22]

V. Šverák, On optimal shape design,, J. Math. Pures Appl. (9), 72 (1993), 537. Google Scholar

[23]

R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth,, Boll. Unione Mat. Ital., 2 (2009), 1. Google Scholar

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