Eulerian elastoplasticity: Basic issues and recent results

Abstract

Traditional formulations of elastoplasticity in the presence of finite strain and large rotation are Eulerian type and widely used; they are based upon, among other things, the additive decomposition of the stretching or the Eulerian strain-rate into elastic and plastic parts. In such formulations, yield functions and objective rate constitutive equations are expressed in terms of objective Eulerian tensor quantities, including the stretching, the Kirchhoff stress, internal state variables, etc. Each of these quantities transforms in a corotational manner under a change of the observing frame. According to the principle of material frame-indifference or objectivity, each constitutive function should be invariant, whenever the observing frame is changed to another one by any given time-dependent rotation. In this work the general form of constitutive equations is discussed. Several frequently used objective rates are analyzed with respect to their serviceability to develop a self-consistent formulation, i.e. to be integrable to deliver an elastic in particular hyperelastic relation for vanishing plastic deformation. This would be of great importance, e.g., for so-called spring back calculations in metal forming.Tradicionalne formulacije elastoplasticnosti u prisustvu konacne deformacije i velikog obrtanja su Ojlerovskog tipa i nasiroko se koriste; zasnovane su, pored ostalog, na aditivnoj dekompoziciji brzine deformacije (ili njene Ojlerovskog predstavnika) u elasticne i plasticne delove. Pri takvim formulacijama, funkcije tecenja i konstitutivne jednacine, u kojima se pojavljuju objektive brzine, su izrazene preko objektivnih Ojlerovskih tenzorskih velicina ukljujcujuci brzinu deformacije, Kirhofov napon, unutrasnje promenljive stanja itd. Svaka od ovih velicina se tranformisen na korotacioni nacin pri promeni posmatrackog sistema. Saglasno principu objektivnosti, svaka konstitutivna funkcija treba da bude invarijantna, kadgod se posmatracki sistem menja bilo kojim obrtanjem promenljivim u vremenu. U ovom radu se diskutuje opsti oblik konstitutivnih jednacina. Nekoliko cesto koriscenih objektivnih brzina se analizira u odnosu na njihovu usluznost razvoju samousaglasavajuce formulacije, tj. integrabilnosti koja daje elasticnu i, posebno, hiperelasticnu, relaciju pri iscezavajucoj plasticnoj deformaciji. Ovo je od velike vaznosti, naprimer, za takozvana odskocna izracunavanja u slucaju obrade deformacijom.