COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 363 
“mathematical terms. The ultimate tensile strength is the maximum value of the 
nominal stress given by the term (f.a/A), where A is the original cross-section of 
the test-piece, and f is the stress on the reduced cross-section a. It follows that 
the ultimate strength occurs when the differential of (f.«) is zero; t.e. when 
(f.da) + (a. df) =0; 
where df is the increase of yield stress produced by a further elongation, dl, on the 
already extended length J. The ultimate tensile strength is therefore the particular 
“nominal ’ stress that corresponds to a definite stage of elongation at which 
th rgb 
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that is, at which 2 per cent. increase of yield stress (reckoned on the reduced area, as 
in wire-drawing) is produced by a further 1 per cent. elongation (reckoned on the 
already extended length). 
The necking that occurs in a ductile metal strained beyond this stage produces 
lateral pull stresses in the core of the bar: the exterior layers, subject to tension, are 
held in their curved positions by these lateral tensions. When the curvature is con- 
siderable, it may happen that the core is subject to an almost symmetrical ‘ triple- 
tensile’ stress. It is of interest to consider, therefore, the probable action of this 
unusual type of stress, regarding which we have but little direct experimental 
evidence. According tothe ‘ strain-energy’ theory, symmetrical triple-tensile stress of 
sufficient intensity promotes change of state, from crystalline to vitreous, but causes 
no tendency to glide. The metal becomes ‘ pseudo-rigid,’ ceasing to glide although 
suffering the change of state. If such an action occurs in the core of a tensile test-piece, 
we may expect that that core will take up an undue share of the applied load, will 
suffer increasingly rapid change without the relief that might be derived from gliding, 
and will rupture in the manner characteristic of a brittle metal that breaks without 
plastic strain. 
The above description of the action of triple-tensile stress, in promoting fracture 
in a ductile metal, is supported by evidence from wire-drawing operations. In the 
process of wire-drawing, the occurrence of triple-tensile stress is prevented by the 
lateral pressure exerted by the dies; and, as a consequence, more shear strain can 
be imposed without fracture, so that the metal can be hardened much further than by 
simple tensile elongation. The profile and lubrication of the dies are of importance, 
not only on account of their dragging action but also because they affect the mag- 
nitude and location of the lateral pressure that neutralises the destructive triple- 
tension which, in the absence of the dies, would promote rupture. In ‘ cupped’ 
wire, internal cracks occur in a manner suggestive of a deficiency of this action. 
The view that rupture can be caused by change of physical state resulting in the 
formation of cavities, with or without gliding motion, is supported by several con- 
siderations. When the densities of crystalline and vitreous metals are compared, 
using dilatometer methods at temperatures such that the change occurs without 
stress, the vitreous metal is found to occupy the smaller volume ; and, on account of 
the different thermal contractions, it is probable that the difference of volume is still 
greater at ordinary temperatures. Moreover, the slight but definite expansions that 
accompany distortion by tension, shear, and even compression, are regarded as evidence 
that the change of state results in the formation of cavities, even when it is followed 
by gliding movements which tend to seal together the gliding faces. The author 
holds the view that gliding is to be regarded not as the cause of rupture. but rather 
as an imperfect protection against rupture caused by the opening of cavities and the 
spreading of a crack, much in the same manner as occurs evidently in a ‘ notched-bar’ 
test for brittleness. 
When fatigue occurs under a pulsating pull that attains only a moderate crest 
value, not necessarily exceeding the yield-point and much lower than the ultimate 
tensile stress, the fatigue fracture assumes a characteristic form intermediate between 
the typical tensile fracture and the typical fatigue fracture under alternating 
stress. Fig. (6 illustrates the comparison, and shows how the pulsating stress 
