THE PHENOMENA OF RUPTURE AND FLOW IN SOLIDS. 
193 
It is a fundamental assumption of the mathematical theory that it is legitimate to 
replace summation of the molecular forces by integration. In general this can only 
be true if the smallest material dimension, involved in the calculations, is large compared 
with the unit of structure of the substance. In crystalline metals the crystals appear, 
from the foregoing investigation, to be anisotropic and they must therefore be regarded 
as the units of structure. Hence the theory of isotropic homogeneous solids may 
break down if applied to metals in cases where the smallest linear dimension involved 
is not many times the length of a crystal. 
Similar considerations apply to solids such as glass, save that here the units of 
structure are probably curved. 
The most important practical case of failure is that of a re-entrant angle or groove. 
Here the theory may break down if the radius of curvature of the re-entering corner 
is but a small number of crystals long. An extreme instance is that of a surface 
scratch, where the radius of curvature may be but a fraction of the length of the 
crystals. 
In the case of brittle materials the general nature of the effect of scratches on strength 
may be inferred from the theoretical criterion of rupture enunciated in section 2 above. 
Whether the material be isotropic or anisotropic, homogeneous or heterogeneous, it is 
necessary on dimensional grounds that the strain energy shall depend on a higher power 
of the depth of the scratch than the surface energy. It follows that small scratches 
must reduce the strength less than large ones of the same shape. Hence, where the 
tenacity of the material, under “ uniform ” stress, is determined by the presence of 
“ flaws,” it must always be possible to find a certain depth of scratch whose breaking 
stress is equal to that of the flaws. Evidently such a scratch can have no influence 
on the strength of the piece. Deeper scratches must have some weakening effect, 
which must increase with the depth, until in the limit the strength of very large grooves 
may be found by means of the elastic theory and the appropriate empirical hypothesis 
of rupture. 
In the case of ductile metals, the effect of scratches is important only under alternating 
or repeated stresses. On the theory advanced in the preceding section, fatigue failure 
under such stresses is determined by phenomena which occur at the intercrystalline 
boundaries. Hence the strength of a scratched piece is fixed, not by the maximum 
stress range in the corner of the scratch, but by the stress range at a certain distance 
below the surface. This distance cannot be less than the width of one crystal, and it 
may be greater. Elastic theory suggests that the stress due to a scratch falls off very 
rapidly with increasing distance from the re-entrant corner, so that the relatively small 
effect of scratches in fatigue tests is readily explained. 
Possibly many published results bearing on this matter depend more on initial skin 
stresses than on sharp corner effects. 
