Inelastic deformation and dislocation structure of a nickel alloy: Effects of deformation and thermal histories
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I.
INTRODUCTION
DURINGrecent years there has
been substantial interest in developing elastic-viscoplastic constitutive equations ['-71 which are suitable for arbitrary loading histories over a broad temperature range. These constitutive equations have often been referred to as "unified" models because all aspects of inelastic deformation, including plasticity under monotonic and cyclic loading, creep, and stress relaxation, are represented by a single inelastic strain rate term.J8] One of the essential features of this class of constitutive theories is the use of hardening variables with appropriate evolution equations for representing material resistance to inelastic flow. The general framework of the evolution equations of the hardening variables is based on the well-known BaileyOrowan t h e o r y , [9'1~ w h i c h theorizes inelastic deformation to occur under a hardening process proceeding with deformation and a static thermal recovery process proceeding with time. The evolution rate, X, of a hardening variable, X, is then the difference between the hardening rate and the thermal recovery rate as given by [8]
= F(X, T) = h(X, T)~,I - r(X, T)
[1]
where h and r are the hardening and the thermal recovery functions, respectively, and h and r are functions of X, and temperature, T. The hardening measure, M, is either the inelastic strain rate or the plastic work rate, depending on the constitutive model. Most of the models use two hardening variables, one for representing isotropic hardening and the other for directional (kinematic) hardening. The hardening function usually contains a linear hardening term and a dynamic recovery term which leads to a limiting value for X and a saturation stress for low temperature deformation. For high temperature deformation, the presence of a static thermal recovery term leads to steady-state conditions whose values of hardening variables are lower than their respective limiting values.
K. S. CHAN, Principal Engineer, and R. A. PAGE, Staff Scientist, are with Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78284. Manuscript submitted November 6, 1987. METALLURGICAL TRANSACTIONS A
One of the advantages of the unified constitutive equations approach is that it is applicable for arbitrary deformation and thermal histories. In general, material constants in a unified model are determined using isothermal test data, such as those obtained from tensile and creep tests. [lq The unified model can then be used for structural analysis applications under complex deformation and thermal loading histories, e.g., a finite-element analysis of an airfoil over a flight cycle. [5'12'13]In such applications, Eq. [1] is assumed to remain essentially valid for arbitrary thermomechanical loading histories, provided that corrections are made for the change of material constants with temperature.[5'14'ls] For thermomechanical loading under a temperature rate T,
= h(X, T)~4 - r(X, T) + O(X, T)J"
[2]
with
O(X, T) = OF(X,T)/OT
[31
and F(X, T) defined in Eq. [l]. Recent experim
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