Constitutive relations in 3-D for a wide range of strain rates and temperatures – Application to mild steels

Abstract

An original phenomenological thermo-visco-plastic model is reported that encompasses strain hardening, strain rate and temperature sensitivity. The model is based to some extent on the concept of physical modeling proposed earlier by Klepaczko [Klepaczko, J.R. 1975. Thermally activated flow and strain rate history effects for some polycrystalline FCC metals. Mater. Sci. Engng. 18, 121–135] , and also by different authors, for example: [Becker, R. 1925. Uber die plastizitat amorpher und kristalliner fester korper. Z. Physik 26, 919–925; Seeger, A., 1957. Dislocations and Mechanical Properties of Crystals, Wiley, New York; Conrad, H., 1964. Thermally activated deformation of metals J. Metals 16, 582; Gilman, J.J. 1968. Dislocation dynamics and response of materials to impact Appl. Mech. Rev. 21, 767–783; Gibbs, G.B., 1969. Thermodynamic analysis of dislocation glide controlled by dispersed local obstacles. Mater. Sci. Engng. 4, 313–328; Kocks, U.F., Argon, A.S., Ashby, M.F., 1975. Thermodynamics and kinetics of slip. In: Progress in Materials Science, vol. 19. Pergamon Press, New York, p. 19; Kocks, U.F. 1976. Laws for work-hardening and low-temperature creep. J. Eng. Mater. Technol. 98, 76–85, and later by many others.

The thermo-visco-plastic formulation, called RK and applied in this paper, has been verified experimentally for strain rates of ${10}^{-4}{\text{s}}^{-1}⩽{\dot{\overline{\epsilon }}}_{\text{p}}⩽5\times {10}^3{\text{s}}^{-1}$ and temperatures 213 K ⩽ T ⩽ 393 K, it covers the range of dynamic loadings observed during crash tests and other impact problems. In order to implement the RK constitutive relation, a thermo-visco-plastic algorithm based on the J₂ theory of plasticity is constructed. The type of algorithm is a return mapping one that introduces the consistency condition $(f=\overline{\sigma }-{\sigma }_y\text{,}f=0)$, without the overstress state proposed by Perzyna [Perzyna, P. 1966. Fundamental problems in viscoplasticity. Advances in Applied Mechanics, vol. 9. Academic Press, New York, pp. 243–377]. The coupling of the RK constitutive relation with the integration scheme of the thermo-visco-plastic algorithm has demonstrated its efficiency for numerical analyses of different dynamic processes such as Taylor test [Zaera, R., Fernández-Sáez, J. 2006. An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations. Int. J. Solids Struct., 43, 1594–1612.], ring expansion [Rusinek, A., Zaera, R., 2007. Finite element simulation of steel ring fragmentation under radial expansion. Int. J. Impact Eng. 34, 799–822], dynamic tension test [Rusinek, A., Zaera, R., Klepaczko, J.R., Cherigueme, R., 2005. Analysis of inertia and scale effects on dynamic neck formation during tension of sheet steel. Acta Mater. 53, 5387–5400], perforation of metallic sheets [Rusinek, A., 2000, Modélisation thermoviscoplastique d’une nuance de tôle d’acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la traction et de la perforation, Ph.D. thesis, University of Metz, France], and other cases. All the equations are implemented via the user subroutine VUMAT in the ABAQUS/Explicit code for adiabatic conditions of plastic deformation.