Technical information

5-3
zr
baau
∂∂
∂
⋅−⋅−=
φ
2
1112
)1()( ......................................................................... (5.1.3e)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⋅+
∂
∂
−
∂
∂
⋅⋅−⋅⋅=
rr
r
a
z
daaaw
φφφ 1
)2(
2
2
44
2
2
3313
........................................ (5.1.3f)
in which the parameters
a , b , c and d are functions of the elastic constants:
2
133311
121113
)(
aaa
aaa
a
−⋅
−⋅
=
....................................................................................... (5.1.4a)
2
133311
3312441313
)(
aaa
aaaaa
b
−⋅
⋅
−+⋅
=
....................................................................... (5.1.4b)
2
133311
4411121113
)(
aaa
aaaaa
c
−⋅
⋅
+−⋅
=
....................................................................... (5.1.4c)
2
133311
2
12
2
11
aaa
aa
d
−⋅
−
= ......................................................................................... (5.1.4d)
and the stress function
),( zr
φ
satisfies the ‘compatibility’ equation
0
22
=∇∇ φ
βα
in
which
2221222
/// zrrr ∂∂⋅+∂∂⋅+∂∂=∇
−−
α
α
and
2
β
∇
is identical to
2
α
∇ except for
β
in place of
α
. The terms
α
and
β
are also derived from the material properties, as
follows:
d
dcaca
⋅
⋅−+±+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
2
4)(
2
2
2
β
α
...................................................................... (5.1.5)
The above algorithm imposes additional restrictions (named algorithm
restrictions or ARs) on the material properties: (AR1)
0
2
133311
≠−⋅ aaa
; (AR2)
04)(
2
>⋅−+ dca ; (AR3) 0
2
> α ; and (AR4) 0
2
> β . In addition,
α
and
β
must be
distinct or the following derivation becomes singular; for this reason the isotropic case,
in which
1==
β
α
, can only be approached but not directly computed.
Consider a semi-infinite medium made of
1
−
n parallel layers lying over a half-
space. Each layer is identified by a subscript i with material properties
iz
E )(
,
ix
E )(
,