Asymptotics near crack tips in hereditarily-elastic anisotropic aging two dimensional body M. C OSTABEL ∗ , M. DAUGE † , S.A. NAZAROV ‡ , J. S OKOLOWSKI § Abstract A model involving a Volterra kernel is considered for a hereditarily-elastic anisotropic ag- ing two-dimensional body with a straight crack. The asymptotics of the time dependent solution near the crack tips is investigated. We prove that, depending on the regularity of the material law and the Volterra kernel, these asymptotics are simple homogeneous func- tions of degree 1 2 or have a more complicated dependence on the distance variable r to the crack tips. In the latter situation, asymptotics involve a function of ln r growing in time, which requires a modification of usual fracture criteria. Contents 1 Introduction 2 1.A Crack in hereditarily-elastic aging body ..................... 3 1.B Mathematical formulation of the problem .................... 4 1.C Main results and structure of the paper ...................... 7 2 Existence of solutions and exponential estimates 8 2.A General method for Volterra equations ...................... 9 2.B Energy solutions ................................. 10 * Institut de Recherche Math´ ematique de Rennes, Universit´ e de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, Martin.Costabel@univ-rennes1.fr † Institut de Recherche Math´ ematique de Rennes, Universit´ e de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, Monique.Dauge@univ-rennes1.fr ‡ Institute of Mechanical Engineering Problems, Laboratory of Mathematical Methods, Russian Academy of Sciences, V.O. Bol’shoi 61, 199178 St. Petersburg, Russia, serna@snark.ipme.ru § Institut Elie Cartan, Laboratoire de Math´ ematiques, Universit´ e Henri Poincar´ e Nancy I, B.P. 239, 54506 Van- doeuvre l` es Nancy Cedex, France, Jan.Sokolowski@iecn.u-nancy.fr 1