# Gradient damage models applied to dynamic fragmentation of brittle materials

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## Abstract

This paper is devoted to the use of gradient damage models in a dynamical context. After the setting of the general dynamical problem using a variational approach, one focuses on its application to the fragmentation of a brittle ring under expansion. Although the 1D problem admits a solution where the damage field remains uniform in space, numerical simulations show that the damage field localizes in space at a certain time and then a fragmentation of the ring rapidly occurs. To understand this phenomenon from a theoretical point of view, one develops a stability analysis of the homogeneous response by studying the growth of small perturbations. A dimensional analysis shows that the problem essentially depends on two dimensionless parameters \(\tilde{\ell }\) and \(\tilde{\dot{\varepsilon }}_0\), \(\tilde{\ell }\) being related to the characteristic length present in the damage model and \(\tilde{\dot{\varepsilon }}_0\) to the applied expansion rate. Then, since the product \(\tilde{\ell }\tilde{\dot{\varepsilon }}_0\) is small in practice, the problem of stability is solved in a closed form by using asymptotic expansions. The comparison with the numerical results allows us to conclude that the time at which the damage localizes and the number of fragments are really governed by the growth of the imperfections. To conclude, a numerical simulation of the fragmentation of a 2D ring is presented.

## Keywords

Damage Variational approach Dynamical fracture Fragmentation Stability analysis Asymptotic expansion method## Notes

## References

- Alessi R, Marigo J-J, Vidoli S (2014) Gradient damage models coupled with plasticity and nucleation of cohesive cracks. Arch Rat Mech Anal 214(2):575–615CrossRefGoogle Scholar
- Alessi R, Marigo J-J, Vidoli S (2015) Gradient damage models coupled with plasticity: variational formulation and main properties. Mech Mater 80(B):351–367CrossRefGoogle Scholar
- Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55(5):1017–1040CrossRefGoogle Scholar
- Benallal A, Marigo J-J (2007) Bifurcation and stability issues in gradient theories with softening. Model Simul Mater Sci Eng 15(1):283–295CrossRefGoogle Scholar
- Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95CrossRefGoogle Scholar
- Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826CrossRefGoogle Scholar
- Bourdin B, Francfort GA, Marigo J-J (2008) The variational approach to fracture. J Elast 91(1–3):5–148CrossRefGoogle Scholar
- Bourdin B, Larsen CJ, Richardson CL (2011) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168(2):133–143CrossRefGoogle Scholar
- Bourdin B, Marigo J-J, Maurini C, Sicsic P (2014) Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys Rev Lett 112(1):014301CrossRefGoogle Scholar
- Braides A (2002) \(\Gamma \)-Convergence for beginners volume 22 of Oxford lecture series in mathematics and its applications. Oxford University Press, OxfordGoogle Scholar
- Comi C (2001) A non-local model with tension and compression damage mechanisms. Eur J Mech A Solid 20(1):1–22CrossRefGoogle Scholar
- Dal-Maso G, Toader R (2001) A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch Rat Mech Anal 162(2):101–135CrossRefGoogle Scholar
- de Borst R, Brekelmans WAM, Peerlings RHJ, de Vree JHP (1996) Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39(19):3391CrossRefGoogle Scholar
- Drugan WJ (2001) Dynamic fragmentation of brittle materials: analytical mechanics-based models. J Mech Phys Solids 49:1181–1208CrossRefGoogle Scholar
- Francfort GA, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342CrossRefGoogle Scholar
- Glenn LA, Chudnovsky A (1986) Strain-energy effects on dynamic fragmentation. J Appl Phys 59:1379–1380CrossRefGoogle Scholar
- Grady DE (1982) Local inertial effects in dynamic fragmentation. J Appl Phys 53:322–325CrossRefGoogle Scholar
- Griffith AA (1921) The phenomena of rupture and flows in solids. Philos Trans R Soc Lond A221:163–197CrossRefGoogle Scholar
- Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57(2):342–368CrossRefGoogle Scholar
- Larsen CJ (2010) Models for dynamic fracture based on Griffith’s criterion. In Hackl K (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials. Springer, Dordrecht, pp 131–140Google Scholar
- Levy S, Molinari J-F (2010) Dynamic fragmentation of ceramics, signature of defects and scaling of fragment sizes. J Mech Phys Solids 58:12–26CrossRefGoogle Scholar
- Li T, Marigo J-J (2017) Crack tip equation of motion in dynamic gradient damage models. J Elast 127(1):25–57CrossRefGoogle Scholar
- Li T, Marigo J-J, Guilbaud D, Potapov S (2016) Gradient damage modeling of brittle fracture in an explicit dynamics context. Int J Numer Methods Eng 108(11):1381–1405CrossRefGoogle Scholar
- Lorentz E, Benallal A (2005) Gradient constitutive relations: numerical aspects and application to gradient damage. Int J Numer Methods Eng 194(50–52):5191–5220Google Scholar
- Mardal K-A, Wells GN, Logg A (2012) Automated solution of differential equations by the finite element method—the FeniCS book. Springer, BerlinGoogle Scholar
- Marigo J-J (1989) Constitutive relations in plasticity, damage and fracture mechanics based on a work property. Nucl Eng Des 114:249–272CrossRefGoogle Scholar
- Mercier S, Molinari A (2003) Predictions of bifurcation and instabilities during dynamic extension. Int J Solids Struct 40(8):1995–2016CrossRefGoogle Scholar
- Miehe C, Hofacker M, Schänzel L-M, Aldakheel F (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic–plastic solids. Comput Methods Appl Mech Eng 294:486–522CrossRefGoogle Scholar
- Miller O, Freund LB, Needleman A (1999) Modeling and simulation of dynamic fragmentation in brittle materials. Int J Fract 96:101–125CrossRefGoogle Scholar
- Molinari J-F, Gazonas G, Raghupathy R, Rusinek A, Zhou F (2007) The cohesive element approach to dynamic fragmentation: the question of energy convergence. Int J Numer Methods Eng 69:484–503CrossRefGoogle Scholar
- Moré JJ, Toraldo G (1991) On the solution of large quadratic programming problems with bound constraints. SIAM J Optim 1:93–113CrossRefGoogle Scholar
- Pandolfi A, Krysl P, Ortiz M (1999) Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture. Int J Fract 95:279–297CrossRefGoogle Scholar
- Pham K, Marigo J-J (2010a) The variational approach to damage: I. The foundations. Comptes Rendus Mécanique 338(4):191–198CrossRefGoogle Scholar
- Pham K, Marigo J-J (2010b) The variational approach to damage: II. The gradient damage models. Comptes Rendus Mécanique 338(4):199–206CrossRefGoogle Scholar
- Pham K, Marigo J-J (2011) From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Continuum Mech Thermodyn 25:147–171CrossRefGoogle Scholar
- Pham K, Amor H, Marigo J-J, Maurini C (2011a) Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech 20(4, SI):618–652CrossRefGoogle Scholar
- Pham K, Marigo J-J, Maurini C (2011b) The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. J Mech Phys Solids 59(6):1163–1190CrossRefGoogle Scholar
- Ravi-Chandar K (1998) Dynamic fracture of nominally brittle materials. Int J Fract 90(1):83–102CrossRefGoogle Scholar
- Ravi-Chandar K, Triantafyllidis N (2015) Dynamic stability of a bar under high loading rate: response to local perturbations. Int J Solids Struct 58:301–308CrossRefGoogle Scholar
- Rodríguez-Martinez JA, Vadíllo G, Fernández-Sáez J, Molinari A (2013) Identification of the critical wavelength responsible for the fragmentation of ductile rings expanding at very high strain rates. J Mech Phys Solids 61(6):1357–1376CrossRefGoogle Scholar
- Sicsic P, Marigo J-J, Maurini C (2014) Initiation of a periodic array of cracks in the thermal shock problem: a gradient damage modeling. J Mech Phys Solids 63:256–284CrossRefGoogle Scholar
- Simo JC, Hughes TJR (1998) Computational inelasticity. Interdisciplinary applied mathematics. Springer, BerlinGoogle Scholar
- Vaz-Romero A, Rodríguez-Martinez JA, Mercier S, Molinari A (2017) Multiple necking pattern in nonlinear elastic bars subjected to dynamic stretching: the role of defects and inertia. Int J Solids Struct 125:232–243CrossRefGoogle Scholar
- Zhou FH, Wang YG (2009) Dynamic tensile fragmentations of Al(2)O(3) rings under radial expansion loading. In: DYMAT 2009: 9th international conference on the mechanical and physical behaviour of materials under dynamic loading, vol 1, pp 325–330Google Scholar
- Zinszner J, Erzar B, Forquin P, Buzaud E (2015) Dynamic fragmentation of an alumina ceramic subjected to shockless spalling: an experimental and numerical study. J Mech Phys Solids 85:112–127CrossRefGoogle Scholar