COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 171 
stresses, but in this case there is very little experimental evidence available. 
It is certain that some materials do not fail in the same manner under all 
systems of stress, and Mallock’s double limit is an attempt to meet this 
difficulty,”” by which the material is supposed to fail according to the 
limit which is first reached. 
It is probable that each of the principal theories contains a germ of 
truth if its application be properly limited. Difficulty has arisen because 
a theory has been assumed to apply under too wide a range of conditions. 
The experimental evidence will be shown later to indicate that a 
ductile body fails when the maximum stress difference (or shear stress) 
reaches a certain value. So far as is known, the intermediate principal 
stress 1s without effect on the failure. If the failure be limited to yield, 
this limit will apply for compressive as well as tensile stresses. 
A brittle material appears to fracture at a definite maximum principal 
stress when this is a tension. Under compressive stress failure seems to be 
by shearing modified by friction on the plane of the shear. 
Theories (a) and (d) are so far justified under definite conditions by the 
experimental evidence, and (c) is a particular case of (d), and not very 
different from it, which applies to ductile steels because the coefficient 
of internal friction is zero.8° 
It will be noticed that Mallock’s double limits cover all the above if the 
shear theory is modified by friction, and maximum stress is used to replace 
the volume extension. The double limits apply to brittle materials, and the 
relation between them should be determined. It is difficult to conceive a 
volume extension limit, because in two-dimensional stress a material 
under tension in one direction would be strengthened by a compression 
perpendicular to the tension. Further, the very different strengths of 
most brittle substances in simple tension and simple compression are 
accompanied by very different strains at fracture, and a maximum strain 
theory could not always apply. 
It is clear that tests will be incomplete unless they employ most of the 
possible combinations of the principal stresses. 
5. The Rate of Loading, and Repeated Loading. 
The most common and simplest method of applying the stresses is by a 
slow rate of increase so that the material fails under sensibly static con- 
ditions. In engineering practice combined stresses are also applied under 
rhythmically repeated and shock conditions. The former is of special 
interest because it is one of the most common cases of combined stresses, 
and causes fractures which resemble those of brittle materials under 
similar, but static loading. It is probable that the practical cases which 
involve shock will require special treatment, but some attention might be 
given to the matter in our investigations. 
6. The Mechanism of Failure. 
Consideration of this subject has been largely dissociated from that 
of combined stresses, to the detriment of both. That there is the closest 
connection is evident, and it is possible that a study of the mechanism of 
failure might be of assistance in the case of a material to which the more 
usual methods cannot be applied. The matter is noted here as a reminder 
rather than for present discussion. 
