COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 169 
oints out that the stress difference theory of Tresca leads to a limit which 
is little different from the shear strain hypothesis enunciated by Coulomb. 
The Stress Difference Theory was applied by G. H. Darwin and C. Chree.™ 
It is probable that they were influenced by Tresca, and also by the 
knowledge that a brittle material fractures by shearing when loaded in 
compression. 
The Maximum Strain Theory is usually named after St. Venant,® but he 
attributed it to Mariotte,? who wrote: ‘ que c’est le degré d’extension qui 
fait rompre les corps.’ St. Venant adapted this theory to the elastic 
breakdown of a material by assuming that after the limit of mathematical 
elasticity is reached, the body will ultimately be ruptured if it has to sus- 
tain the same load.* He also rejected Coulomb’s theory when applied to 
rupture in compression, and followed Poncelet,®? who ascribed rupture by 
compression to the transverse stretch which accompanies a longitudinal 
squeeze. 
Lamé?‘ assumed that the greatest tension had a limiting value to ensure 
safety. This view was adopted by Rankine, who was followed by British 
and American engineers, but when known as the Maximum Stress Theory 
it is usually assumed to apply in compression as well as in tension. 
A modification of the Stress Difference Theory was suggested by Navier, 
and is based on the assumption that the shear-stress at failure is modified 
by the internal friction of the material to an extent proportional to the 
stress normal to the plane of the shear. Perryf has been the principal 
exponent of this modified shear stress theory in this country. He notices 
that cast iron, stone, brick, and cement fracture at angles greater than 45 
degrees with the cross section. For cast iron the angle is 542 degrees, 
which corresponds to a coefficient of internal friction equal to 0:35. 
Perry also suggested that there is no internal friction in wrought iron and 
mild steel, in which case the modification is eliminated, and the simple law 
holds. Mohr™* has proposed a further development of the shear theory to 
take account of the kind of stress which is developed within the body. 
Poisson’s Theory indicated that the ratio which bears his name should 
be 4. The early determinations by Wertheim? have been noted here 
because they disproved the theory by giving different values, and thus 
had a great influence on the rariconstancy and multiconstancy controversy 
in the Theory of Elasticity. 
1. Theories. 
We desire to define the conditions which determine the failure of a 
material when subjected to any system of stress. Theories have been 
advanced which suggest as a criterion of strength: (a) the maximum 
stress; (b) the maximum strain; (c) the greatest stress difference, or 
shear stress or strain ; (d) the maximum value of the shear stress modified 
by a friction term proportional to the stress perpendicular to the plane of 
the shear. 
The shearing stress and the stress normal to the plane of the shear have 
received increasing attention, not always on the lines indicated by theories 
(c) and (d), but these have to a great extent eclipsed the other suggestions. 
In a recent paper, Mallock *” ** considers the limit of shear and the limit 
of volume extension as the fundamental limits of a material, and failure 
is assumed to occur according to which is first reached. 
* Todhunter and Pearson, History of the Theory of Elasticity, vol. ii., pt. i., p. 107, 
{ Perry, Applied Mechanics, 1898, pp. 345 to 348, 356, 
