FACTA UNIVERSITATIS
Series: Architecture and Civil Engineering Vol. 3, N
o
2, 2005, pp. 145 - 153
ANALYTICAL APPROACH FOR RESOLVING STRESS STATES
AROUND ELLIPTICAL CAVITIES
UDC 624.191.8(045)=20
Dragan Lukić
1
, Petar Anagnosti
2
1
University of Novi Sad, Faculty of Civil Engineering Subotica, Serbia and Montenegro
2
University of Belgrade, Faculty of Civil Engineering, Belgrade, Serbia and Montenegro
Abstract. The determination of stress states around cavities in the stressed elastic
body, regardless of cavity shapes, that may be spherical, cylindrical, elliptical etc. in
its analytical approach has to be based on selection of a stress function that will satisfy
biharmonic equation ∇
2
∇
2
Ψ = 0, under given boundary conditions. This paper is
concerned with formulation and solution of the cited differential equation using
elliptical coordinates in conformity with the cavity shape of oblong ellipsoid [1]. It is
therefore considered that the formulation of the stress tensor will be done in conformity
to the cited coordinates.
The paper describes basic statements and definitions in connection to harmonic
functions used for determination of stress states around cavities formed in the stressed
homogeneous space. The particular attention has been paid to the use of Legendre`s
functions, with definitions and derivation of recurrent formulas, that have been used
for determination of stress states around an oblong ellipsoidal cavity, [1]. The paper
also includes the description of procedures used in forming series based on Legendre`s
functions of the first order.
Key words: coordinates, biharmonic differential equation, stress state,
stress function, harmonic functions, recurrent formulas.
1. BIHARMONIC DIFFERENTIAL EQUATION SOLUTION BY A STRESS FUNCTION
In resolving stress states around a cavity having the oblong ellipsoidal shape, the
starting point is the basic equilibrium equation in terms of displacements, derived for a
stressed isotropic elastic body (neglecting the gravity). Following this approach the equi-
librium equations can be formulated in a general manner by using a function of coordi-
nates for an arbitrary point in the body, known as "stress function". Selection of stress
functions that are satisfying equilibrium equations and imposed boundary conditions in
either isotropic or not isotropic bodies, presents the common task for research activities in
Received February 01, 2005
D. LUKIĆ, P. ANAGNOSTI 146
this field of applied mechanics. An example of such solutions is referred to Papkovich –
Neuber [1] set of stress functions, taken as the basis for determination of stress tensor
coordinates in the following form:
3210
Φ+Φ+Φ+Φ=Ψ zyx (1)
that in the case of axial symmetry is reduced to:
30
Φ+Φ=Ψ z (2)
The selected stress function used for determination of stress tensor coordinates shall
satisfy biharmonic differential equation:
0
22
=Ψ∇∇ (3)
In an analytical approach this equation shall be formulated in the coordinates selected
in conformity with the geometry of the cavity around which the stress states are under
investigation.
2.
BIHARMONIC DIFFERENTIAL EQUATION IN ELLIPTICAL COORDINATES
The shape of the cited differential equation (3) that the selected stress functions shall
satisfy, depends on the expression of Laplace's operator (
∇
2
) in the chosen curvilinear
coordinates. In the case of coordinates suited for oblong rotational ellipsoid, Laplace's
operator is given in the following form:
)
)cos1)(1ch(
1
cos
)cos(1
cos
cosch
1
ch
)1ch(
ch
cosch
1
(
2
2
22
2
22
2
22
2
θ∂
∂
ϕ−−
+
⎥
⎦
⎤
⎢
⎣
⎡
ϕ∂
∂
ϕ−
ϕ∂
∂
ϕ−
+
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
−
∂
∂
ϕ−
=∇
u
u
u
u
u
u
(4)
where
u, φ, θ are elliptical coordinates defined by expressions x = L sh u sinφ y = L sh u
sinφ sinθ
z = L ch u cosφ (L − is the focal distance along larger axis, 0 ≤ u ≤ ∞ 0 ≤ φ ≤ π
0≤θ≤2 π).
Under condition of axial symmetry state independent of the coordinate θ, the operator
has the following form [2]:
⎥
⎦
⎤
⎢
⎣
⎡
ϕ∂
∂
ϕ−
ϕ∂
∂
ϕ−
+
+
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
−
∂
∂
ϕ−
=∇
cos
)cos(1
coscosch
1
ch
)1ch(
ch
cosch
1
2
22
2
22
2
u
u
u
u
u
(5)
Analytical Approach for Resolving Stress States around Elliptical Cavities 147
After substitution of expression (5) in equation (3), one can obtain:
]
cos
)cos(1
coscosch
1
ch
)1ch(
ch
cosch
1
[
2
22
2
22
⎥
⎦
⎤
⎢
⎣
⎡
ϕ∂
∂
ϕ−
ϕ∂
∂
ϕ−
+
+
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
−
∂
∂
ϕ−
u
u
u
u
u
0 ]
cos
)cos(1
coscosch
1
ch
)1ch(
ch
cosch
1
[
2
22
2
22
=
⎥
⎦
⎤
⎢
⎣
⎡
ϕ∂
Ψ∂
ϕ−
ϕ∂
∂
ϕ−
+
⎥
⎦
⎤
⎢
⎣
⎡
∂
Ψ∂
−
∂
∂
ϕ−
u
u
u
u
u
(6)
Therefore the expression (6) has the form of differential equation 0
22
=Ψ∇∇ in elliptical
coordinates, for axial symmetry case.
3.
DIFFERENTIAL EQUATION SOLUTION IN ELLIPTICAL COORDINATES
Differential equation (6) can be commonly transformed in two equations [3] as follows:
0A
2
=∇
(7)
A
2
=Ψ∇ (8)
The solution of equation (7) by separation of unknowns is to be searched in the form:
)(cos)F(ch A ϕΦ=
u (9)
By application of common mathematical transformations with separation of unknowns,
one can obtain two equations with independent unknowns "
u" and "ϕ", respectively:
0)(cos]
cos1
)1([ )]'(cos')cos1[(
0)F(ch ]
ch1
)1([ )]'(ch ' F)ch1[(
2
2
2
2
2
2
=ϕΦ
ϕ−
−++ϕΦϕ−
=
−
−++−
m
nn
u
u
m
nnuu
(10)
General solutions of equations (10) are known in the form:
)(cosQD)(cosPC)(cos
)ch (QB)(ch PA)ch (F
ϕ+ϕ=ϕΦ
+=
m
nnm
m
nnmnm
m
nnm
m
nnmnm
uuu
(11)
After insertion of (11) in (9) one can obtain the expression ( 9 ) in the form:
D. LUKIĆ, P. ANAGNOSTI 148
)](cosQD )(cosPC)][(ch QB )(ch PA[ A
00
ϕ+ϕ+=
∑∑
∞
==
m
nnm
m
nnm
m
nnm
m
nnm
n
n
m
uu (12)
Where:
)ch (Q ),(cosP u
m
n
m
n
ϕ are Legendre's polynomials and functions, that
For m=0 reduce to the following form:
)ch (Q ),(cosP u
nn
ϕ ; (13)
In order to satisfy the continuity of expression (12), the condition D
nm
= 0 is to be ful-
filled, and therefore the solution of equation (7) has the following form:
)(cosPC)](ch QB)(ch PA[ A
00
ϕ×+=
∑∑
∞
==
m
nnm
m
nnm
m
nnm
n
n
m
uu (14)
After insertion of the boundary condition:
;)ch (
ch
lim ∞=
∞→
uP
u
m
n
0)ch (Q
ch
lim =
∞→
u
u
m
n
(15)
one can obtain the value of parameter A
nm
= 0, and the solution of equation (7) gets the
following form:
)]ch (Q )(cosPBC[ A
00
u
m
n
m
nnmnm
n
n
m
ϕ=
∑∑
∞
==
(16)
By substitution a
nm
= C
nm
B
nm
, one can obtain the solution of equation (7) in its final form:
)]ch (Q )(cosP[ A
00
ua
m
n
m
nnm
n
n
m
ϕ=
∑∑
∞
==
(17)
By inserting the expression (17) into equation (8) one can obtain:
)]ch (Q )(cosP[
00
2
ua
m
n
m
nnm
n
n
m
ϕ=Ψ∇
∑∑
∞
==
(18)
After the searching its solution in the following form:
ph
Ψ+Ψ=Ψ
(19)
it appears that there is a possibility to obtain two differential equations, namely:
− general solution: 0
2
=Ψ∇
h
(20)
and
− particular solution: A
2
=Ψ∇
p
(21)
The homogeneous equation (20) can be resolved in the same manner as equation (7)
and solution can be obtained in the following form:
Analytical Approach for Resolving Stress States around Elliptical Cavities 149
− general solution )]ch (Q )(cosP[
00
ub
m
n
m
nnm
n
n
m
h
ϕ=Ψ
∑∑
∞
==
(22)
and
− particular solution is to be searched in the form:
)]ch (Q )(cosP[
00
ue
m
n
m
nnm
n
n
m
p
ϕ=Ψ
∑∑
∞
==
(23)
Applying differential operator (5) on Ψ
p
in expression (23) and after equating the left
side of equation (21), and the right side of equation (17) one can obtain the relationship
between quotients
a
nm
and e
nm
.
The final solution of equation (3) is therefore obtained in the form:
+ϕ=Ψ
∑∑
∞
==
)]ch (Q )(cosP[
00
ub
m
n
m
nnm
n
n
m
)]ch (Q )(cosP[
00
ue
m
n
m
nnm
n
n
m
ϕ
∑∑
∞
==
(24)
or,
)]ch (Q )(cosP[
00
uf
m
n
m
nnm
n
n
m
ϕ=Ψ
∑∑
∞
==
(25)
where:
nmnmnm
ebf +=
m
n
P, and
m
n
Q , are Legendre's polynomials and functions [4].
The unknown parameters
f
nm
are to be determined from boundary conditions existing
on the cavity surface, that have to be given in stresses., and the expression ( 2 ) is to be
rewritten as:
)(ch Q)cos(PA
00
0
∑∑
∞
==
ϕ=Φ
n
n
m
m
n
m
nnm
u and
)(ch Q)cos(PC cosch
10
11
0
0
3
∑∑
∞
−==
++
=
=
ϕϕ=Φ
n
n
m
m
n
m
nnm
m
n
uuLz (26)
where unknown parameters A
nm
and C
nm
are to be determined from boundary conditions
existing on the cavity surface, that have to be given in stresses.
Starting from expression (26) one can obtain analytically defined stresses σ
u,
σ
φ,
σ
θ,
τ
uφ
by well known expressions consisting of the derivatives of functions Ψ and Φ
3
. On the
other hand, the stresses acting on the cavity surface are to be derived also from a given
stress function "
f (u,φ,θ)" expressed in infinite series basing on Legendre's functions, and
equated to the values resolved by stress functions Φo and Φ
3
, thus forming up the final set
of linear equations that relate unknown Anm and Cnm parameters to the known values of
coefficients in the infinite series approximation of the boundary stresses.
4.
LEGENDRE'S POLYNOMIALS AND FUNCTIONS
Analytical solutions of problems related to stress states determination in vicinity of el-
liptical, spherical and cylindrical cavities formed in the stressed homogeneous space,