1908-9.] Internal Friction in Cases of Compound Stress. 427 
XXVII. — On the Effect of Internal Friction in Cases of Compound 
Stress. By G. H. Gulliver, B.Sc., A.M.I.Mech.E., Lecturer in 
Engineering in the University of Edinburgh. 
(MS. received January 19, 1909. Head July 12, 1909.) 
In a previous paper * certain results were deduced for bars supporting 
simple tensile or compressive loads, on the assumption that materials exhibit 
an internal resistance to deformation of the nature of a frictional resistance 
to the sliding of the particles along the cleavage surfaces. These results 
are easily extended to any system of stress in a solid body. 
Let p v p 2 , an d Ps he the principal stresses, acting normally to the three 
pairs of faces of a small cubic element, p x being the maximum tension or 
minimum compression, and p 3 the minimum tension or maximum compres- 
sion. If tensions are taken as positive, Pi>p 2 >p s • Then it is a well- 
known result that the plane of maximum shear is parallel to the direction 
of p. v and inclined at 45° to both p x and p 3 , and that the value of the 
maximum shearing stress is i(p 1 —p B )- 
In the same manner, but taking into consideration the effect of internal 
friction, as already explained, the plane of sliding is parallel to the direction 
of p 2 , and is inclined at a = (45° + (p/2) to p v and at /3 = (45 u — 0/2) to p B . 
The minimum resistance to sliding is, 
ju,K ~Pi sin a cos a + pp 1 sin 2 a-p 3 sin /3 cos /3 + pp 3 sin 2 {3 
= sin a COS a(p 1 —p 3 ) + p(p x sill 2 a +p 3 COS 2 a) 
~ 2 {( cos ^ M s in <f>){p { -p 3 ) + p( p l +p 3 )\ 
= h{Pi( J l +p 2 + p) ~P 3 ( Jl+p 2 - p)} 
where K is the cohesion of the material as defined before, and p ( = tan 0) 
is the coefficient of internal friction. 
Few investigations of the effect of compound loading upon metal bars 
have been carried out. The most complete series of experiments are those 
of Guest ; some interesting results have been obtained also by Hancock, by 
Scoble, and by Goodman. It occurred to the writer that these exjDeriments 
might afford some evidence for or against the influence of internal friction. 
Guest’s experiments f were made with ductile materials — steel, copper, 
and brass — in the form of thin tubes. These tubes were subjected to 
tension, to torsion, and to internal pressure ; and also to tension combined 
with torsion, to tension combined with internal pressure, and to torsion 
combined with internal pressure. The loads were so regulated as to obtain 
* Proceedings , xxviii., 1908, 374. t Phil. Mag., 1., 1900, 69. 
