ELASTOSTATICS 121
δσ
η
θσ
12
δσ
ί3
ϋ—
+
^—
+
~ä—
+
bl =ü
tfXi 0Χ
2
ν
χ
3
^ + ^ + ^ +
ί>
2
= 0 (5.1)
öx
x
dx
2
cx
3
θσ
3ί
δσ
32
θσ
33
+
&
3
= 0
cbcj δχ
2
<3χ
where σ
21
=
σ
12
,σ
31
=
σ
139
σ
32
= σ
23
.
Equations (5.1) can be written in indicial notation, simply as
pl + b
t
= 0 i= 1,2,3 )=l,2,3 (5.2)
OXj
The prescribed surface force intensities or tractions on the surface
Γ
2
of the body are denoted by
p
h
p
2
,p
3
(Figure 5.1). Equilibrium on
the boundary requires the satisfaction of the following boundary
conditions:
Pi =σ
11
Μ
1
+<7
12
Μ
2
+ σ
13
η
3
= ρ
1
Pi = σ
21
η
1
+σ
22
η
2
+ σ
23
η
3
= Ρι (5.3)
P3 = σ
31
«
1
+σ
32
η
2
+ σ
33
η
3
= p
3
where n
l
, n
2
, n
3
are the direction cosines of the normal with respect to
the x
1?
x
2
,x
3
axes, i.e.
n
x
= cos(n,x,) (5.4)
Equations (5.3) can be written in indicial form as,
Pi = Gijitj = ^ i = 1,2,3 7 = 1,2,3 (5.5)
The displacement field is defined by the three displacement
components u
l9
u
2
,u
3
. Let Γ
ί
denote the portion of
the
boundary on
which displacements are prescribed. The displacement constraints
there are,
W
= Mi, U
2
=M
2
, U
3
= Ü
3 /c
,.
or
γ ι 2 ό ό
(5.6)
Ui = U
t
.
The
üi
are the prescribed values. Note that the total surface Γ of the
body is equal to Γ
ι
+Γ
2
.