156
CONTACT OF ELASTIC SOLIDS
We see that in the said plane § and ʼn are proportional to
the forces exerted by an infinitely long elliptic cylinder, which
stands on the surface of pressure and whose density increases
inwards, according to the law of increase of the pressure in the
surface of pressure. In general then, & and n are given by
complicated functions; but for points close to the axis they
can be easily calculated. Surrounding the axis we describe a
very thin cylindrical surface, similar to the whole cylinder;
this [small] cylinder we may treat as homogeneous, and since
the part outside it has no action at points inside it, the com-
ponents of the forces in question, and therefore also έ and ŋ,
must be equal to a constant multiplied respectively by x/a and
by y/b. Hence
૬ _ m
ду
.მუ
-b
= 0.
On the other hand we have
૬. શ્ t
3p
1
=σ-
Əz
дх ду
4K(1+20)π ab
From these equations we find for the three quantities
which we sought
૦૬
3p
1
=
Эх 4K(1+20)π a(a+b)'
>
θη
3p
1
ду
4K(1+20)π b(a + b)
*
3p
1
=
Əz 4K(1+20)π ab
The negative sign of these three quantities shows that the
element in question is compressed in all three directions.
The compressions vary as the cube root of the total pressure.
It is easy to determine from them the pressures at the origin.
These pressures are the most intense of all those occurring
throughout the bodies pressed together; we may therefore
say that the limit of elasticity will not be exceeded until
these pressures become of the order of magnitude required for
transgressing the elastic limit. In plastic bodies, e.g. in the