휎휃(0)=−
푎^3 푞
2푅^33
−
푎^3 푞
2푅^34
, (18)
푢(0)푟 =
푎^3 푞
4퐺
(
푡+푟
푅^33
+
푡+푟−푚
푅^34
), (19)
푢(0)푧 =
푎^3 푞
4퐺
(
푧−ℎ
푅^33
+
푧−ℎ+푛
푅^34
), (20)
where the cavity depth below the ground surface is denoted
byℎ,thedistancefromthesource표 1 to the calculation point
푝is푅 3 ,andtheimage표 2 to the point푝is푅 4. According to the
relationship of geometry shown inFigure 4, the expressions
for푅 3 and푅 4 are
푅 3 =√(푡+푟)^2 +(푧−ℎ)^2 ,
푅 4 =√(푚−푡−푟)^2 +(푛+푧−ℎ)^2.
(21)
With the increasing of depth of the spherical cav-
ity, the image cavity gradually moves from above ground
(Figure 4(a))tothegroundbelow(Figure 4(c)). Accordingly,
the expressions of푅 3 and푅 4 will be changed with the depth
of cavities. Specifically, when the virtual image cavity is just
on the ground surface as shown inFigure 4(b),the푅 4 can be
simplified to be
푅 4 =√(푚−푡−푟)^2 +푧^2 , (22)
푚=2(ℎtan훽+푡)cos^2 훽, (23)푛=(ℎtan훽+푡)sin2훽, (24)where푚and푛are defined as the horizontal (푟direction)
and vertical (푧direction) distances between the source and
its image, respectively.
3.3. The Correction of the Stresses on the Horizontal Ground.
According to the models shown inFigure 4,theactual
expansion cavity and its image do produce not only nonzero
normal stress but also shear stress on the horizontal ground
surface푧=0,whichcanbeshowntobeasfollows:
휎(0)푧
儨儨
儨儨儨푧=0=
푎^3 푞
2
[
3ℎ^2
푅耠5 3
−
1
푅耠3 3
+
3 (푛−ℎ)^2
푅耠5 4
−
1
푅耠3 4
], (25)
휏(0)푟푧
儨儨
儨儨儨푧=0=
−3푎^3 푞ℎ(푡+푟)
2푅耠5 3
−
3푎^3 푞(ℎ−푛)(푡+푟−푚)
2푅耠5 4
, (26)
where 푅耠 3 = √(푡 + 푟)^2 +ℎ^2 and 푅耠 4 =
√(푚−푡−푟)^2 +(푛−ℎ)^2.
In order to satisfy the condition of the free horizontal
boundary (푧=0), the different correction functions are
introduced to deal with the normal stress (see ( 25 )) and
shear stress (see ( 26 ))ontheboundary.Basedonthetheory
of superposition, the stresses on the ground surface can be
divided into two parts:
(i) only normal stress on the horizontal ground surface:
휎(1)푧|푧=0=−휎(0)푧|푧=0,휏푟푧(1)|푧=0=0;
(ii) only shear stress on the horizontal ground surface:
휎(2)푧|푧=0=0,휏푟푧(2)|푧=0=−휏푟푧(0)|푧=0.Usingtheaxiallysymmetricstressfunction푓(푟,푧)of
Kassir and Sih [ 21 ] in the first part (i), the corresponding
stress and displacement solutions are written as follows:휎푟(1)=2퐺[(1−2])
휕^2 푓
휕푟^2
−2]
휕^2 푓
휕푧^2
+푧
휕^3 푓
휕푟^2 휕푧
], (27)
휎휃(1)=2퐺[
1
푟
휕푓
휕푟
+2]
휕^2 푓
휕푟^2
+
푧
푟
휕^2 푓
휕푟휕푧
], (28)
휎푧(1)=2퐺[−
휕^2 푓
휕푧^2
+푧
휕^3 푓
휕푧^3
], (29)
휏푟푧(1)=2퐺푧
휕^3 푓
휕푟휕푧^2
, (30)
푢(1)푟 =(1−2])
휕푓
휕푟
+푧
휕^2 푓
휕푟휕푧
, (31)
푢(1)푧 =−2(1−])
휕푓
휕푟
+푧
휕^2 푓
휕푧^2
, (32)
in which푓=퐴(
1
푅3푡
+
1
푅4푡
), (33)
where퐴is a constant. The horizontal ground surface is
treatedastheplaneofsymmetry.푅3푡denotes the distance
between the point푝and the symmetrical position of the
actual cavity푂 1 .Likewise,푅4푡is the distance from the point
푝to the symmetrical position of the image cavity푂 2 .Asa
result, their expressions can be written as follows:푅3푡=√(푡+푟)^2 +(푧+ℎ)^2 ,
푅4푡=√[푚−(푡+푟)]^2 +[푧−(푛−ℎ)]^2 ;
(34)
fis a harmonic function, that is,∇^2 푓=0. The corresponding
stresses on the free surface are:휎(1)푧
儨儨儨
儨儨푧=0=2퐺퐴[
1
푅耠33푡
−
3ℎ^2
푅耠53푡
+
1
푅耠34푡
−
3 (푛−ℎ)^2
푅耠54푡
], (35)
휏푟푧(1)
儨儨儨
儨儨푧=0=0, (36)
where푅耠3푡=푅耠 3 ,and푅耠4푡=푅耠 4.
Thus,thenormalstressonthehorizontalgroundsurface
푧=0could be eliminated by the stress function푓with accu-
rate value of퐴. According to the following equation: