
464 Ballistics: Theory and Design of Guns and Ammunition
If we write the force balance in the x-direction, we obtain
ρ
τ
fxyz yz
x
xy
z
xz
xxx
xx
xx
yx
()() ()
()
dddddddd dd
dd
++
∂
−
++
∂
σσ
σσ
ττ
τ
yx
yx x
y
yxzxzxyzu
∂
−=dddddddd()()
ρρ
(16.171)
After we cancel terms and divide by the volume dxdydz, we obtain
ρ
σ
ρ
f
xy
u
x
xx
yx
x
+
∂
∂
+
∂
=
(16.172)
Examined in three dimensions, the equation would be
ρ
σ
τ
ρρ
f
xyz
uf
x
u
x
xx
yx
zx
xi
ij
j
i
+
∂
∂
+
∂
+
∂
∂
=→+
∂
=
ρ
(16.173)
If we recall Hooke’s law in its one-dimensional form, we get
(16.174)
In three dimensions, it is written for a homogeneous material using two material con-
stants, called the Lamé constants, as
ij kk ij ...