Eur. J. Mech. ASolids 18 1999 433–442
Elsevier, Paris
Constitutive equations of creep under changing multiaxial stresses
Jakob M. Klebanov
Department of Mechanical Engineering, Samara Federal Technical University, P.O. Box 4038, Samara 443110 Russia Received 7 May 1996; revised and accepted 1 November 1998
Abstract – Creep constitutive equations are derived here to describe creep behaviour of metals at non-proportional loading by generalisation of non- linear viscoelaticity equations equipped with temporal analogy of time-stress type. To obtain the new equations, integral dependences for the case of
degenerative integral operators are first transformed into differential ones. It is shown that the latters imply the kinematic hardening law. In the proposed constitutive equations, mixed hardening is adopted as a composition of three hardening mechanisms connected with translation, size and shape of the
potential surfaces respectively. Weight factors of these mechanisms are defined by two material constants which are obtained from the creep data of non-proportional loading. Two measures of isotropic hardening are introduced: the first is the equivalent strain and the second is a non-decreasing
parameter. The constitutive equations introduced are evaluated on the base of experimental data on creep behaviour of D16T aluminium alloy and VT3-1 titanium alloy under non-proportional step loading. Comparison of the theoretical results with the experimental data indicates that the former
give good predictions of the material responses. Noticeable differences between the two isotropic measures on predicting the isotropic hardening effect appear when the preceding stress vector rotation is not less than 90
◦
. The non-decreasing parameters are better in rotations of the kind.
Elsevier, Paris
creep hardening stress-rotation model anisotropy
1. Introduction
Various types of creep constitutive equations for metals at non-radial loading have been proposed by numerous authors: e.g. Lagneborg, 1972; Hart, 1976; Gittus, 1976; Miller, 1976 within framework of physical
and metallurgical points of view, Sosnin, 1970; Malinin and Khadjinsky, 1972; Chaboche, 1977; Kadashevich and Novogilov, 1980; Murakami and Ohno, 1982; Ding and Findley, 1984; Mroz and Trampczynski, 1984;
Gokhfeld and Sadakov, 1984; Ohno et al., 1985; Kawai, 1995; Rubin and Bodner, 1995; Yang, 1997 along the line of continuum mechanics approaches. Most constitutive equations are obtained by the way of generalisation
of the simplest classical time- and strain-hardening laws.
The classical laws of creep are based on the assumption of isotropic hardening of materials. To describe anisotropic hardening, kinematic hardening and a combination of isotropic and kinematic hardening are
introduced as extensions to the case of creep deformation of hardening theories in plasticity Malinin and Khadjinsky, 1972; Miller, 1976; Chaboche, 1977; Krieg et al., 1978; Ohashi et al., 1982.
Physical and metallurgical analyses confirm that both isotropic and anisotropic effects due to the change of micro-structure in material appear in inelastic deformation of real materials Murghabi, 1975; Miller,
1976; Krieg et al., 1978; Quesnel and Tsow, 1981; Ohashi et al., 1982; Kadashevich and Chernyakov, 1992. Anisotropic effects are connected with such mechanisms as dislocation pileup, bowing and recoil, or
non-uniform distributions of inelastic micro-strain through adjacent grains. Isotropic effects are caused by dislocation tangles, sell-structure formation and so on.
There are numerous modifications of the classical laws mostly based on the introduction of mixed isotropic- kinematic hardening. However a simple mixed hardening law can not accurately describe anisotropic creep
behaviour under large stress-rotations Ohashi et al., 1982. More complicated hardening mechanisms are considered in Mroz and Trampczynski, 1984; Ohno et al., 1985; Rubin and Bodner, 1995; Kawai, 1995.
434 J.M. Klebanov
In distinction from the classical time- and strain-hardening laws the theory of viscoelasticity reflects transient non-collinearity between the stress and the creep strain rate tensors under large stress-rotations. The theory
can describe creep at essential deviations from radial loading Bugakov, 1965. Equations of non-linear viscoelasticity equipped with temporal analogy of time-stress type are able to describe partial creep recovery
at unloading and some other effects in solids Shapery, 1969; Urgumtsev, 1982; Bugakov and Tchepovetskii, 1984. These properties allow to avoid the introduction of a superposition of recoverable and non-recoverable
creep strains considered in Findley and Lay, 1978; Ohno et al., 1985 as well as the decomposition of hardening mechanism into two parts associated with a creep loading process and a creep reorientation process suggested
in Mroz and Trampczynski, 1984. The equations of non-linear viscoelasticity equipped with temporal analogy lead to boundary value problems which are well-posed Klebanov, 1996.
Nevertheless analysis of experimental data of non-proportional loading steps in multiaxial creep demon- strates that the integral dependences of viscoelasticity equations can predict the material behaviour under stress
vector rotations limited by 50
◦
–60
◦
only.
2. Multiaxial creep constitutive equations
