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.