Thermo-elastic-plastic transition

Abstract

The classical theory of elasticity and plasticity does not recognize explicitly the existence of a "transition zone" between elastic and plastic states, which instead, makes extensive use of ad-hoc, semi-empirical laws, such as yield conditions, at the "yield surface" to match both the extreme states. In the present investigation, it is shown that these ad-hoc, semi-empirical laws turn out to be totally unnecessary if one appreciates the existence of a "transition zone" introduced by Seth in recent years, which is quite realistic from the point of view of physical considerations. A transition phenomenon, being an asymptotic one, should be dealt with as a limiting process, and so the transition state should be obtainable from the basic system of equations characterizing the elastic state as a limiting case. The plastic state is similarly to be obtained from the transition state when a certain parameter is made to approach zero. In order to appreciate the existence of a transition state, it is basically important at the outset to identify the "transition points" from the differential system which characterizes the physical phenomenon. It is examined in this thesis how and in what manner transition is to be understood in the case of physical phenomena. It is found that there are three ways in which a transition could be identified analytically: at transition, the differential system characterizing the elastic state should attain some criticality, the complete breakdown of the macroscopic structure at transition should correspond to the degeneracy of the material or spatial strain ellipsoid, if we consider the plastic state as an image of the elastic state, then at transition the Jacobian of transformation is bound to behave singularly. The last condition turns out to be the most general one from which a general yield condition is deduced and it is found that most of the yield conditions present in current literature come out as special cases. Also, it has been seen that our results take into account Bauschinger's effect, while neither Tresca's yield condition nor von- Mises yield condition does. It has also been shown that transition fields naturally being non-linear in character, are sub-harmonic (super-harmonic) fields. Once one recognizes the "transition zone" as a separate state, the natural question of determining the constitutive equation also arises. In order to answer this question and to illustrate the procedure, four problems of practical interest are discussed in detail. The problems of elastic-plastic transition of shells and tubes subjected to external pressure are solved. Further, by considering the effect of a steady state temperature, the problems of thermo-elastic-plastic transition of shells and tubes are also solved. No yield conditions have been assumed. It is found that if they exist, they come out of the differential system as a consequence of the transition analysis. Some of the results have been compared with those of the classical theory. Possible scope of future work, where the transition concept may be profitably exploited, has also been discussed.