166
VI
ON HARDNESS
Here K, denote the coefficients of elasticity in Kirchhoff's
notation. Young's modulus of elasticity is expressed in terms
of these coefficients by the equation
E = 2K
1+30
1+20
The ratio between lateral contraction and longitudinal exten-
sion is
με
1+20
For bodies like glass or steel, this ratio is nearly, or
nearly 1, and K is nearly 3 E. For slightly compressible
bodies the ratio is nearly; here then = ∞, K=E. As
K = }E.
a matter of fact a particular combination of K and will
play the principal part in our formulæ, for which we shall
therefore introduce a special symbol.
We put
Ꮽ .
=
2(1+0)
K(1+20)
In bodies like glass, 9 = 4/3K 32/9E; in all bodies lies
· = 9
between 3/E and 4/E, since lies between 0 and ∞. In
regard to the II's we must note that calculated by the above
formulæ they have infinite values; but their differential co-
efficients, which alone concern us, are finite. It would only
be necessary to add to the II's infinite constants of suitable
magnitude to make them finite. By a simple differentiation,
remembering that 2P0, we find
V²II,
=
-
2 JP
K, Əz'
V²II,
-
2 ap
K, Əz
We now assume the following expressions for the displace-
ments in the two bodies:-
=
ап,
ax
1
=
әп,
dy
=
ап,
+29₁P,
dz
€2
==
whence follow
=
01-
૬.
Əx
+
ar₂
Xxx
+
ay
n2
ası
Əz
=
=
all2
dy
52
=
ap
²I₁ +291 z
=
-
ar 2
dz
-
2.9.P,
2
ap
Əz K₁(1+20) Əz'