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COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 359 
in an extension of the theory by which, in 1849, he had already calculated the depression 
of the freezing point by pressure, Thomson states: ‘ Stresses (of any kind) tending to 
change the form of any crystals in the saturated solutions from which they have 
crystallised must give them a tendency to dissolve away, and to generate, in sub- 
stitution for themselves, other crystals free from the applied stresses.’ It does not 
appear that Thomson applied this principle for metals; nor was the subject then 
ripe for such an application. Thomson states: ‘I have not... any clear conception 
of continuous crystalline structure admitting of what may be called ductile or malle- 
able bending. . . . What may be the nature of the molecular arrangement induced 
by bending them I cannot say ; but I suppose that... their crystalline structure is 
materially altered, and rendered discontinuous where, before, it was continuous.’ 
Beilby’s memorable work identified the change in question as the physical change 
from the crystalline to the vitreous state. Ewing, Rosenhain, and Humphrey demon- 
strated microscopically that the change occurs on gliding surfaces within the grains ; 
and showed that ductile strain is due to numerous small displacements on surfaces 
spaced at finite distances that vary in different circumstances. It is understood that 
the ‘ vitreous’ theory of plastic strain is now widely accepted, although, since it 
asserts that the phenomena of strain depend upon a change of physical state, its 
acceptance involves a variety of thermodynamic consequences which hitherto have 
not been fully investigated or verified. 
While the vitreous theory of non-elastic strain affords an explanation of plastic 
flow, on the assumption that the changed metal behaves as a viscous liquid, and offers 
an incomplete explanation of the hardening action of cold-work, on the assumption 
that the same changed metal behaves as a rigid vitreous substance, it remains to be 
shown that the two requirements are not mutually incompatible. As a deduction 
from the first law of thermodynamics, the author will show that the changed metal 
must be formed at high temperature, such that it may well be viscous during the 
short period of time that elapses before it cools to the temperature of the surrounding 
metal. 
It will further be shown that the phenomena of hysteresis are attributable to the 
action of a dual process of decrystallisation and recrystallisation occurring in a cyclic 
manner that is always irreversible in the thermodynamic sense, but may be mechani- 
cally reversible or irreversible according as the range of applied stress lies within or 
beyond the fatigue limit of the metal; and that the incidence of fatigue may be ex- 
plained by the formation of small cavities in the metal, in consequence of the imperfect 
mechanical reversibility of the action when it occurs with undue energy and on too 
large a scale. 
Thermodynamic Elasticity. 
In the usual definition of elasticity, reliance is placed on a phrase ‘ no permanent 
strain after removal of stress’ ; and attention is focussed on the comparison of measure- 
ments taken before and after loading. Since the behaviour of the material during 
loading and unloading is not usually specified, the phenomena of ‘ elastic hysteresis ’ 
often escape attention, or are regarded as of minor importance and to be ascribed to 
‘molecular friction.’ For theoretical purposes it is desirable to define a more ideal 
type of elasticity, as exhibited by cold vitreous substances such as glass, free from 
hysteresis. 
Thermodynamic elasticity may be defined—without reference to Hooke’s approxi- 
mate law of proportionality—by regarding as an ideally elastic change an action that 
is thermodynamically reversible : the work done on the metal, in straining it, shall be 
completely recovered when the stress is released. On this basis, a definition may be 
given as follows :—A test-piece is thermodynamically elastic if, in an experiment in 
which the applied stress is varied within limits and under specified control of tem- 
perature and other external conditions, the strain observed at any stage of loading or 
unloading is found to depend solely on the stress applied at that stage, being inde- 
pendent of whether that stress is rising or falling. 
Thermodynamic elasticity does not necessarily entail fulfilment of Hooke’s law : 
thus the compression of water is thermodynamically elastic although the bulk modulus 
varies with the applied pressure. In the absence of hysteresis, however, metals would 
doubtless fulfil Hooke’s law even more approximately than they do in actual 
experiment. 
