Quasistatic crack growth in 2d-linearized elasticity

Friedrich, Manuel; Solombrino, Francesco

doi:10.1016/j.anihpc.2017.03.002

Friedrich, Manuel ; Solombrino, Francesco

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 27-64.

Résumé

In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation ( $G S B D$ ). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the $G S B D$ setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.

MR Zbl | 3 citations dans Numdam

DOI : 10.1016/j.anihpc.2017.03.002

Classification : 74R10, 49J45, 70G75
Mots-clés : Brittle materials, Variational fracture, Free discontinuity problems, Quasistatic evolution, Crack propagation

@article{AIHPC_2018__35_1_27_0,
     author = {Friedrich, Manuel and Solombrino, Francesco},
     title = {Quasistatic crack growth in 2d-linearized elasticity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {27--64},
     publisher = {Elsevier},
     volume = {35},
     number = {1},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.03.002},
     mrnumber = {3739927},
     zbl = {1386.74124},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.002/}
}

TY  - JOUR
AU  - Friedrich, Manuel
AU  - Solombrino, Francesco
TI  - Quasistatic crack growth in 2d-linearized elasticity
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 27
EP  - 64
VL  - 35
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.002/
DO  - 10.1016/j.anihpc.2017.03.002
LA  - en
ID  - AIHPC_2018__35_1_27_0
ER  -

%0 Journal Article
%A Friedrich, Manuel
%A Solombrino, Francesco
%T Quasistatic crack growth in 2d-linearized elasticity
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 27-64
%V 35
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.002/
%R 10.1016/j.anihpc.2017.03.002
%G en
%F AIHPC_2018__35_1_27_0

Friedrich, Manuel; Solombrino, Francesco. Quasistatic crack growth in 2d-linearized elasticity. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 27-64. doi : 10.1016/j.anihpc.2017.03.002. http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.002/

Bibliographie
Cité par

[1] Ambrosio, L. A compactness theorem for a new class of functions of bounded variation, Boll. Unione Mat. Ital., B (7), Volume 3 (1989), pp. 857–881 | MR | Zbl

[2] Ambrosio, L. Existence theory for a new class of variational problems, Arch. Ration. Mech. Anal., Volume 111 (1990), pp. 291–322 | DOI | MR | Zbl

[3] Ambrosio, L.; Coscia, A.; Dal Maso, G. Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal., Volume 139 (1997), pp. 201–238 | DOI | MR | Zbl

[4] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000 | DOI | MR | Zbl

[5] Baldi, A.; Montefalcone, F. A note on the extension of BV functions in metric measure spaces, J. Math. Anal. Appl., Volume 340 (2008), pp. 197–208 | DOI | MR | Zbl

[6] Bellettini, G.; Coscia, A.; Dal Maso, G. Compactness and lower semicontinuity properties in $SBD (Ω)$ , Math. Z., Volume 228 (1998), pp. 337–351 | DOI | MR | Zbl

[7] Bucur, D.; Varchon, N. Boundary variation for a Neumann problem, Ann. Sc. Norm. Super., Cl. Sci., Volume 29 (2000) no. 4, pp. 807–821 | Numdam | MR | Zbl

[8] Burago, Yu.D.; Maz'ja, V.G. Potential Theory and Function Theory for Irregular Regions, Semin. Mat., vol. 3, V.A. Steklov Math. Ins., Leningrad, 1969 (translated from Russian) | MR | Zbl

[9] Chambolle, A. A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal., Volume 167 (2003), pp. 211–233 | DOI | MR | Zbl

[10] Chambolle, A. An approximation result for special functions with bounded deformation, J. Math. Pures Appl., Volume 83 (2004), pp. 929–954 | DOI | MR | Zbl

[11] Chambolle, A.; Doveri, F. Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Commun. Partial Differ. Equ., Volume 22 (1997), pp. 811–840 | DOI | MR | Zbl

[12] Conti, S.; Focardi, M.; Iurlano, F. Integral representation for functionals defined on $S B D^{p}$ in dimension two, Arch. Ration. Mech. Anal., Volume 223 (2017), pp. 1337–1374 | DOI | MR | Zbl

[13] Dal Maso, G. Generalised functions of bounded deformation, J. Eur. Math. Soc. (JEMS), Volume 15 (2013), pp. 1943–1997 | DOI | MR | Zbl

[14] Dal Maso, G.; Francfort, G.A.; Toader, R. Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 176 (2005), pp. 165–225 | MR | Zbl

[15] Dal Maso, G.; Toader, R. A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal., Volume 162 (2002), pp. 101–135 | DOI | MR | Zbl

[16] Giorgi, E. De; Dautray, R. Free-discontinuity problems in calculus of variations, Frontiers in Pure and Applied Mathematics, a Collection of Papers Dedicated to J.-L. Lions on the Occasion of His 60th Birthday, North-Holland, 1991, pp. 55–62

[17] De Giorgi, E.; Carriero, M.; Leaci, A. Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., Volume 108 (1989), pp. 195–218 | DOI | MR | Zbl

[18] Fonseca, I.; Leoni, G. Modern Methods in the Calculus of Variations: $L^{p}$ Spaces, Springer, New York, 2007 | MR | Zbl

[19] Francfort, G.A.; Larsen, C.J. Existence and convergence for quasi-static evolution in brittle fracture, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1465–1500 | DOI | MR | Zbl

[20] Friedrich, M. A derivation of linearized Griffith energies from nonlinear models, 2015 (preprint) | arXiv | MR

[21] Friedrich, M. A Korn-type inequality in SBD for functions with small jump sets, 2015 (preprint) | arXiv | MR

[22] Friedrich, M. A piecewise Korn inequality in SBD and applications to embedding and density results, 2016 (preprint) | arXiv | MR

[23] Francfort, G.A.; Marigo, J.-J. Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, Volume 46 (1998), pp. 1319–1342 | DOI | MR | Zbl

[24] Giacomini, A.; Ponsiglione, M. Discontinuous finite element approximation of quasi-static growth of brittle fractures, Numer. Funct. Anal. Optim., Volume 24 (2003), pp. 813–850 | DOI | MR | Zbl

[25] Gołab, S. Sur quelques points de la théorie de la longueur, Ann. Soc. Pol. Math., Volume 7 (1928), pp. 227–241 | JFM

[26] Griffith, A. The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond., Volume 221-A (1920), pp. 163–198

[27] Iurlano, F. A density result for GSBD and its application to the approximation of brittle fracture energies, Calc. Var. Partial Differ. Equ., Volume 51 (2014), pp. 315–342 | DOI | MR | Zbl

[28] Maggi, F. Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Camb. Stud. Adv. Math., vol. 135, Cambridge University Press, Cambridge, 2012 | MR | Zbl

[29] Mumford, D.; Shah, J. Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., Volume 42 (1989), pp. 577–684 | DOI | MR | Zbl

[30] Murat, F. Nonlinear Variational Problems, Res. Notes Math., Volume vol. 127 (1985) (Isola d'Elba, 1983) | MR | Zbl

[31] Nitsche, J.A. On Korn's second inequality, RAIRO. Anal. Numér., Volume 15 (1981), pp. 237–248 | DOI | Numdam | MR | Zbl

[32] Temam, R. Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985 | MR

Cité par Sources :