Damage Models for Cyclic Plasticity

Abstract

The process of void growth in ductile materials under alternating plastification is studied by means of unit-cell calculations. Constitutive models for damage evolution in ductile materials as the micromechanical model by Gurson, Tvergaard and Needleman and the continuum damage model by Rousselier are systematically investigated with respect to their ability of describing this process. Nonlinear kinematic hardening effects are taken into account and a respective extension of the GTN model is presented. The models are finally applied for simulating the mechanical behaviour of tensile specimens under cyclic loading, and the results are compared with experimental results. Introduction Fracture toughness and resistance to ductile crack extension under monotonous loading can be predicted by means of constitutive models for damage evolution in ductile materials [1, 2, 3]. Components, however, are also subject to alternating plastic deformations and may fail after some load cycles though their ultimate load under monotonous loading has not been exceeded. Failure mechanisms under cyclic deformation are not completely understood. Whereas empirical models for fatigue like Paris' law are more or less successfully applied in engineering practice, approaches for describing damage under cyclic plasticity are comparably new. Micromechanically based damage models as that by Gurson [4], Tvergaard and Needleman [5], addressed as GTN model in the following, and the continuum damage model by Rousselier [6] (ROU model) are generally suited for describing phenomena of cyclic deformations, although they have been developed and succesfully applied for monotonous loading conditions. The underlying micromechanical process of void growth can be studied on unit cells as under monotonous loading [7, 8]. These unit cells represent a periodic microstructure of spherical cavities in a ductile matrix. Generally, cyclic deformations can not be simulated realistically by isotropic hardening, however, which is assumed in the original GTN and the Rousselier model. Kinematic hardening effects have to be taken into account. In addition, failure by alternating plastification can only be simulated by models which show an increase of the void-volume fraction, so called ratcheting, with the number of load cycles. The damage models are systematically studied with respect to this ability. The GTN-model has been extended to nonlinear kinematic hardening based on the model proposed by Leblond et al. [9] (LPD model). Key Engineering Materials Vols. 251-252 (2003) pp 389-398 Online available since 2003/Oct/15 at www.scientific.net © (2003) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.251-252.389 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 188.193.42.12-20/08/09,11:27:48) 2 Title of Publication (to be inserted by the publisher) Inelastic Deformation under Cyclic Loading Constitutive Equations. Inelastic deformations are described in the framework of the classical theory of rate-independent plasticity. The yield function has the general form Φ q, p,h ( )≤ 0 ; q = 2 ′ s ij ′ s ij , p = − 3 skk , (1) where the dependence on p vanishes for incompressible plastic deformations, but appears in pressure-dependent plasticity. The effective stress, sij, is defined as the difference between Cauchy stesses, σij, and back stresses, aij, for kinematic hardening sij = σ ij − aij . (2) The scalar quatities h denote internal variables for isotropic hardening. Strain rates are decomposed in an elastic and a plastic contribution, Ý e ij = Ý e ij el + Ý e ij pl , (3) which is equivalent to the multiplicative decomposition of the respective displacement gradients for small elastic strains. Hooke's law is assumed for elastic deformation rates and Jaumann stress rates according to the Hughes-Winget approach [10] σ ∇ ij = Cijkl Ý e kl el = Cijkl Ý e ij − Ý e ij pl ( ). (4) An associated flow rule with a plastic potential which equals the yield function of eq. (1), Ý e ij pl = Ý λ ∂Φ ∂σ ij = Ý λ ∂Φ ∂q nij − ∂Φ ∂p δ ij = 3 2q Ý e q ′ s ij + 1 3 Ý e pδij , (5) following the notation of Aravas [11], is postulated for the plastic deformation rates. The consistency condition Ý Φ = 0 requires that Ý e p ∂Φ ∂q + Ý e q ∂Φ ∂p = 0 . (6) Finally, evolution equations for the internal variables of isotropic and kinematic hardening, Ý h (α ) = gα e ij pl ,sij , h (α) ( ) a ∇ ij = Fij e ij , sij ,h (α ) ( ) , (7) complete the constitutive relations. Strain hardening materials contain a single scalar hardening variable, the accumulated equivalent plastic strain, e = 3 Ý e ij pl Ý e ij pl dτ