We show the existence of globally stable quasistatic evolutions for a rate-independent material model with elastoplasticity and incomplete damage, in small strain assumptions. The main feature of our model is that the scalar internal variable which describes the damage affects both the elastic tensor and the plastic yield surface. It is also possible to require that the history of plastic strain up to the current state influences the future evolution of damage.
DOI : 10.1051/cocv/2015037
Mots clés : Variational models, quasistatic evolution, energetic solutions, elastoplasticity, damage models, incomplete damage, softening
@article{COCV_2016__22_3_883_0,
author = {Crismale, Vito},
title = {Globally stable quasistatic evolution for a coupled elastoplastic{\textendash}damage model},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {883--912},
publisher = {EDP-Sciences},
volume = {22},
number = {3},
year = {2016},
doi = {10.1051/cocv/2015037},
zbl = {1342.74026},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2015037/}
}
TY - JOUR AU - Crismale, Vito TI - Globally stable quasistatic evolution for a coupled elastoplastic–damage model JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 883 EP - 912 VL - 22 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015037/ DO - 10.1051/cocv/2015037 LA - en ID - COCV_2016__22_3_883_0 ER -
%0 Journal Article %A Crismale, Vito %T Globally stable quasistatic evolution for a coupled elastoplastic–damage model %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 883-912 %V 22 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015037/ %R 10.1051/cocv/2015037 %G en %F COCV_2016__22_3_883_0
Crismale, Vito. Globally stable quasistatic evolution for a coupled elastoplastic–damage model. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 883-912. doi : 10.1051/cocv/2015037. http://archive.numdam.org/articles/10.1051/cocv/2015037/
, and , Gradient damage models coupled with plasticity and nucleation of cohesive cracks. Arch. Ration. Mech. Anal. 214 (2014) 575–615. | DOI | Zbl
, and , Gradient damage models coupled with plasticity: Variational formulation and main properties. Mech. Mater. 80 (2015) 351–367. | DOI
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. The Clarendon Press, Oxford University Press, New York (2000). | Zbl
, A. Mielke and T. Roubcíˇek, A complete-damage problem at small strains. Z. Angew. Math. Phys. 60 (2009) 205–236. | DOI | Zbl
H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam-London; American Elsevier, New York (1973). | Zbl
V. Crismale and G. Lazzaroni, Viscous approximation of evolutions for a coupled elastoplastic-damage model. Preprint SISSA 05/2015/MATE (2015).
and , A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101–135. | DOI | Zbl
and , Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 257–290. | DOI | Numdam | Zbl
and , Quasistatic crack growth in elasto-plastic materials: the two-dimensional case. Arch. Ration. Mech. Anal. 196 (2010) 867–906. | DOI | Zbl
, and , Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237–291. | DOI | Zbl
, , and , Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media 3 (2008) 567–614. | DOI | Zbl
, , and , A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189 (2008) 469–544. | DOI | Zbl
, and , Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differ. Eq. 40 (2011) 125–181. | DOI | Zbl
, and , Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution. Calc. Var. Partial Differ. Eq. 44 (2012) 495–541. | DOI | Zbl
and , A Helly theorem for functions with values in metric spaces. Tatra Mt. Math. Publ. 44 (2009) 159–168. | Zbl
, and , Young-measure quasi-static damage evolution. Arch. Ration. Mech. Anal. 203 (2012) 415–453. | DOI | Zbl
and , Small-strain heterogeneous elastoplasticity revisited. Commut. Pure Appl. Math. 65 (2012) 1185–1241. | DOI | Zbl
and , Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33 (1996) 1083–1103. | DOI | Zbl
and , Sublinear functions of measures and variational integrals. Duke Math. J. 31 (1964) 159–178. | DOI | Zbl
, Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Ark. Math. 3 (1954) 181–186. | DOI | Zbl
, and , A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23 (2013) 565–616. | DOI | Zbl
D. Knees, R. Rossi and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains. Preprint (2013).
H. Matthies, G. Strang and E. Christiansen, The saddle point of a differential program, in Energy Methods in Finite Element Analysis, edited by Z.O. R. Glowinski and E. Rodin. Wiley, New York (1979) 309–318.
, Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351–382. | DOI | Zbl
A. Mielke, Evolution of rate-independent systems, in Evolutionary equations. Vol. II, Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam (2005) 461–559. | Zbl
and , Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 177–209. | DOI | Zbl
, and , BV solutions and viscosity approximations of rate-independent systems. ESAIM: COCV 18 (2012) 36–80. | Numdam | Zbl
A. Mielke, R. Rossi and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. To appear on J. Eur. Math. Soc. (2015).
and , Approche variationnelle de l’endommagement: II. Les modéles gáradient. C. R. Mécanique 338 (2010) 199–206. | DOI | Zbl
and , From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Contin. Mech. Thermodyn. 25 (2013) 147–171. | DOI | Zbl
, Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968) 1039–1045. | DOI | Zbl
W. Rudin, Real and Complex Analysis. McGraw-Hill, New York (1966). | Zbl
M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage. Vol. 276 of Pure Appl. Math. Chapman and Hall/CRC, Boca Raton, FL (2006). | Zbl
, Quasistatic evolution problems for nonhomogeneous elastic plastic materials. J. Convex Anal. 16 (2009) 89–119. | Zbl
R. Temam, Mathematical problems in plasticity. Gauthier-Villars, Paris (1985). [Translation of Problèmes mathématiques en plasticity. Gauthier-Villars, Paris (1983).] | Zbl
and , Duality and relaxation in the variational problem of plasticity. J. Mécanique 19 (1980) 493–527. | Zbl
, Quasistatic damage evolution with spatial BV-regularization. Discrete Contin. Dyn. Syst. Ser. S 6 (2013) 235–255. | Zbl
and , Damage of nonlinearly elastic materials at small strain-existence and regularity results. ZAMM Z. Angew. Math. Mech. 90 (2010) 88–112. | DOI | Zbl
Cité par Sources :
