Materials Science and Engineering, 46 (1980) 69 - 74 69 © Elsevier Sequoia, S.A., Lausanne -- Printed in the Netherlands On the Average Internal Stresses in Each Constituent Phase in Plastically Deformed Two-ductile-phase Alloys Y. TOMOTA, S. NAKAMURA and K. KUROKI Department of Mechanical Engineering, Ibaraki University, Nahanarusawa-machi, Hitachi 316 (Japan) I. TAMURA Department of Metal Science and Technology, Kyoto University, Sakyo-ku, Kyoto 606 (Japan) (Received December 19, 1979; in revised form March 10, 1980) SUMMARY The average internal stresses in each consti- tuent phase of plastically deformed ~-7 two- ductile-phase Fe-Cr-Ni steels are studied Firstly, the average internal stresses produced by the misfit strain between two phases are calculated from a self-consistent continuum model (an unrelaxed model). The effects of the 0.2% proof stress ratio C* (where C* /s the 0.2% proof stress of the ~ phase divided by the 0.2% proof stress of the 7 phase) and the volume fraction of the ~ phase on these stresses are calculated. Secondly, in order to evaluate the influence of plastic relaxation caused by the high local internal stress, these average internal stresses are calculated by an elastic-plastic analysis using the finite element method. The average internal stress is shown to be reduced by the inhomogeneous distribution of plastic strain in each constituent phase After the applied load has been removed, most of the average internal stresses are con- sidered to remain in the specimen. These re- maining stresses are called phase stresses or residual stresses. Finally, in order to check the effects of C* and of the ~ phase volume frac- tion on the average internal stresses predicted from our calculations, the residual stresses in the ~ phase were measured using an X-ray technique on the surface of plastically de- formed specimens. 1. INTRODUCTION It has been recognized that two<luctile- phase alloys show inhomogeneous plastic deformation between the two constituent phases. The average stress condition in each constituent phase is described by six stress components which are composed of the ap- plied stresses and the average internal stresses yielded by the inhomogeneous distribution of dislocations. The term "average internal stress" used in this paper does not mean the stress obtained by the double strain rate cycling test or the stress relaxation test. The relation be- tween these stresses is not well established. When the applied load is removed, the value of the average internal stress will be reduced because of a small change in dislocation struc- ture due to backward movement of disloca- tions and recovery. However, most of the stresses will remain in the specimen. These remaining stresses are called phase stresses or residual stresses. We have previously calculated [1] the flow stresses of ferrite (~)-austenite (~/) Fe-Cr-Ni alloys using a continuum model where the average internal stress produced by an inhomo- geneous distribution of plastic strain was taken into account. In our work, the discrepancy between the calculations and the experimental data was relatively large for an alloy with a low ~ phase volume fraction and a large C* value defined as the 0.2% proof stress O0.e ratio (i.e. the 0.2% proof stress of the hard phase divided by the 0.2% proof stress of the soft phase) [2]. The effect of plastic relax- ation associated with the local internal stress near the interface was suggested as one of the reasons for this disagreement. Araki et al. [3] have calculated the flow stresses of dual-phase steels (~ and martensite (~')) using basically the same model. They ob- tained good agreement with experimental flow