Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 6, 2022

Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case

  • Gianni Dal Maso ORCID logo and Rodica Toader ORCID logo EMAIL logo

Abstract

We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.

MSC 2010: 35R35; 74R10; 74C05

Communicated by Irene Fonseca


Funding statement: This paper is based on work supported by the National Research Project (PRIN 2017) “Variational Methods for Stationary and Evolution Problems with Singularities and Interfaces”, funded by the Italian Ministry of University and Research. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References

[1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. B (7) 3 (1989), no. 4, 857–881. Search in Google Scholar

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar

[3] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4th ed., Springer Monogr. Math., Springer, Dordrecht, 2012. 10.1007/978-94-007-2247-7Search in Google Scholar

[4] B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity 91 (2008), no. 1–3, 5–148. 10.1007/s10659-007-9107-3Search in Google Scholar

[5] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973. Search in Google Scholar

[6] M. Brokate and P. Krejčí, Wellposedness of kinematic hardening models in elastoplasticity, RAIRO Modél. Math. Anal. Numér. 32 (1998), no. 2, 177–209. 10.1051/m2an/1998320201771Search in Google Scholar

[7] H. Bruno, G. Barros, I. F. M. Menezes and L. F. Martha, Return-mapping algorithms for associative isotropic hardening plasticity using conic optimization, Appl. Math. Model. 78 (2020), 724–748. 10.1016/j.apm.2019.10.006Search in Google Scholar

[8] A. Chambolle and V. Crismale, Equilibrium configurations for nonhomogeneous linearly elastic materials with surface discontinuities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2022), 10.2422/2036-2145.202006_002. 10.2422/2036-2145.202006_002Search in Google Scholar

[9] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), no. 2, 165–225. 10.1007/s00205-004-0351-4Search in Google Scholar

[10] 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), no. 2, 101–135. 10.1007/s002050100187Search in Google Scholar

[11] G. Dal Maso and R. Toader, Quasistatic crack growth in elasto-plastic materials: The two-dimensional case, Arch. Ration. Mech. Anal. 196 (2010), no. 3, 867–906. 10.1007/s00205-009-0258-1Search in Google Scholar

[12] J. Desai, G. Allaire, F. Jouve and C. Mang, Topology optimization in quasi-static plasticity with hardening using a level-set method, Struct. Multidiscip. Optim. 64 (2021), no. 5, 3163–3191. 10.1007/s00158-021-03034-7Search in Google Scholar

[13] D. C. Drucker, Stress-strain relations for strain hardening materials: Discussion and proposed experiments, Proceedings of Symposia in Applied Mathematics Vol. I, American Mathematical Society, Providence (1949), 181–187. 10.1090/psapm/001/0030425Search in Google Scholar

[14] D. C. Drucker, Some implications of work hardening and ideal plasticity, Quart. Appl. Math. 7 (1950), 411–418. 10.1090/qam/34210Search in Google Scholar

[15] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. Search in Google Scholar

[16] G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), no. 10, 1465–1500. 10.1002/cpa.3039Search in Google Scholar

[17] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), no. 8, 1319–1342. 10.1016/S0022-5096(98)00034-9Search in Google Scholar

[18] G. A. Francfort and U. Stefanelli, Quasi-static evolution for the Armstrong–Frederick hardening-plasticity model, Appl. Math. Res. Express. AMRX 2 (2013), 297–344. 10.1093/amrx/abt001Search in Google Scholar

[19] J. Frehse and D. Löbach, Regularity results for three-dimensional isotropic and kinematic hardening including boundary differentiability, Math. Models Methods Appl. Sci. 19 (2009), no. 12, 2231–2262. 10.1142/S0218202509004108Search in Google Scholar

[20] M. Friedrich, A compactness result in GSBV p and applications to Γ-convergence for free discontinuity problems, Calc. Var. Partial Differential Equations 58 (2019), no. 3, Paper No. 86. 10.1007/s00526-019-1530-3Search in Google Scholar

[21] A. Giacomini and M. Ponsiglione, A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications, Arch. Ration. Mech. Anal. 180 (2006), no. 3, 399–447. 10.1007/s00205-005-0392-3Search in Google Scholar

[22] M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University, Cambridge, 2010. 10.1017/CBO9780511762956Search in Google Scholar

[23] W. Han and B. D. Reddy, Plasticity. Mathematical Theory and Numerical Analysis, Interdiscip. Appl. Math. 9, Springer, New York, 1999. Search in Google Scholar

[24] C. Johnson, On plasticity with hardening, J. Math. Anal. Appl. 62 (1978), no. 2, 325–336. 10.1016/0022-247X(78)90129-4Search in Google Scholar

[25] J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University, Cambridge, 1990. 10.1017/CBO9781139167970Search in Google Scholar

[26] G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Texts Appl. Math., Cambridge University, Cambridge, 1992. 10.1017/CBO9781139172400Search in Google Scholar

[27] A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Appl. Math. Sci. 193, Springer, New York, 2015. 10.1007/978-1-4939-2706-7Search in Google Scholar

[28] W. Prager, Strain hardening under combined stresses, J. Appl. Phys. 16 (1945), 837–840. 10.1063/1.1707548Search in Google Scholar

[29] W. Prager, Recent developments in the mathematical theory of plasticity, J. Appl. Phys. 20 (1949), 235–241. 10.1063/1.1698348Search in Google Scholar

[30] U. Stefanelli, A variational principle for hardening elastoplasticity, SIAM J. Math. Anal. 40 (2008), no. 2, 623–652. 10.1137/070692571Search in Google Scholar

Received: 2022-04-01
Revised: 2022-08-31
Accepted: 2022-10-11
Published Online: 2022-12-06
Published in Print: 2024-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 21.5.2024 from https://www.degruyter.com/document/doi/10.1515/acv-2022-0025/html
Scroll to top button