Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics

Robert Lipton1
1Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

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Tài liệu tham khảo

Ambrosio, L.: A compactness theorem for a special class of functions of bounded variation. Boll. Unione Mat. Ital. 3-B, 857–881 (1989)

Ambrosio, L., Brades, A.: Energies in SBV and variational models in fracture mechanics. In: Cioranescu, D., Damlamian, A., Donnato, P. (eds.) Homogenization and Applications to Materials Science 9, pp. 1–22. Gakuto/Gakkotosho, Tokyo (1997)

Ambrosio, L., De Giorgi, E.: Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 82, 199–210 (1989)

Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon, Oxford (2000)

Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2006)

Bobaru, F., Hu, W.: The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176, 215–222 (2012)

Borden, M., Verhoosel, C., Scott, M., Hughes, T., Landis, C.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217–220, 77–95 (2012)

Bourdin, B., Francfort, G., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)

Bourdin, B., Larsen, C., Richardson, C.: A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 168, 133–143 (2011)

Braides, A.: Discrete approximation of functionals with jumps and creases. In: Homogenization, Naples, 2001. Gakuto Internat. Ser. Math. Sci. Appl., vol. 18, pp. 147–153. Gakkotosho, Tokyo (2003)

Dal Maso, G., Larsen, C.J.: Existence for wave equations on domains with arbitrary growing cracks. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 22, 387–408 (2011)

Driver, B.: Analysis Tools with Applications E-Book. Springer, Berlin (2003)

Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23, 493–540 (2013)

Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K.: J. Elast. 113(2), 193–217 (2013)

Dyal, K., Bhattacharya, K.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)

Emmrich, E., Weckner, O.: On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007)

Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (2010)

Francfort, G., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

Freund, L.B.: Dynamic Fracture Mechanics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1998)

Gerstle, W., Sau, N., Silling, S.: Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)

Gobbino, M.: Finite difference approximation of the Mumford–Shah functional. Commun. Pure Appl. Math. 51, 197–228 (1998)

Gobbino, M.: Gradient flow for the one-dimensional Mumford–Shah functional. Ann. Sc. Norm. Super. Pisa, Cl. Sci. XXVII(4), 145–193 (1998)

Gobbino, M., Mora, M.G.: Finite difference approximation of free discontinuity problems. Proc. R. Soc. Edinb. A 131, 567–595 (2001)

Hu, W., Ha, Y.D., Bobaru, F.: Modeling dynamic fracture and damage in a fiber-reinforced composite lamina with peridynamics. Int. J. Multiscale Comput. Eng. 9, 707–726 (2011)

Larsen, C.J.: Models for dynamic fracture based on Griffith’s criterion. In: Hackl, K. (ed.) IUTAM Sympo-sium on Variational Concepts with Applications to the Mechanics of Materials, pp. 131–140. Springer, Berlin (2010)

Larsen, C., Ortner, C., Suli, E.: Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20, 1021–1048 (2010)

Mengesha, T., Du, Q.: Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. (2013). doi: 10.1007/s10659-013-9456-z

Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 17, 577–685 (1989)

Schmidt, B., Fraternali, F., Ortiz, M.: Eigenfracture: an eigendeformation approach to variational fracture. Multiscale Model. Simul. 7, 1237–1266 (2009)

Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)

Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)

Silling, S.A., Bobaru, F.: Peridynamic modeling of membranes and fibers. Int. J. Non-Linear Mech. 40, 395–409 (2005)

Silling, S., Lehoucq, R.: Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)

Silling, S., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010)

Weckner, O., Abeyaratne, R.: The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)