GREEN’S FUNCTION FOR AN ELASTIC LAYER
LOADED HARMONICALLY ON ITS SURFACE
TOMAŽ PLIBERŠEK, ANDREJ ŠTRUKELJ and ANDREJ UMEK
About the authors
Tomaž Pliberšek
University of Maribor,
Faculty of Civil Engineering
Smetanova ulica 17, 2000 Maribor, Slovenia
E-mail: tomaz.plibersek@uni-mb.si
Andrej Štrukelj
University of Maribor,
Faculty of Civil Engineering
Smetanova ulica 17, 2000 Maribor, Slovenia
E-mail: andrej.strukelj@uni-mb.si
Andrej Umek
University of Maribor,
Faculty of Civil Engineering
Smetanova ulica 17, 2000 Maribor, Slovenia
E-mail: umek@uni-mb.si
Abstract
The Green’s function in surface displacement plays an
important role in soil structure interaction. In evaluating
the Green’s function, several difficulties occur because
it is formulated in the infinite integral form. This paper
outlines a method of analyzing the steady-state dynamic
response of an elastic layer subjected to general point
load excitation. It is assumed that the load is applied at
the surface. The application Hankel integral transform,
to the governing differential equations and boundary
conditions yields the response displacements at the surface
in integral representation. It will be shown that these semiinfinite integrals can be reduced to the integral with the
finite range of integration, which can be efficiently taken
numerically. The numerical results are presented, which
show the efficiency of the developed procedure.
Keywords
Soil-Structure Interaction, Layered Half-Space,
Elastodynamics, Green’s Function
1 INTRODUCTION
The fundamental dynamic solutions for homogeneous
half-space as well as for a layered one are studied in
depth and known in the literature. They are given in two
fundamentally different mathematical forms. On one
hand, there are approximate solutions, e.g. the ingenuous thin layer method introduced by Kausel [1], and on
the other hand analytical methods leading to the solutions in form of integrals with infinite or semi-infinite
path of integration, e.g. Vostroukhov [2], Jin and Liu [3],
and others. The sufficiently accurate evaluation of these
integrals, as needed in practical engineering problems
primarily in dynamic soil-structure interaction, is time
consuming if not tedious. Kobayashi [4] made a step
forward showing that in the case of a homogeneous
half-space the integrals of semi-infinite extension
representing displacements could be transformed to
the integrals with the finite path of integration. The
numerical evaluation of these integrals is then easy and
straightforward. The drawback of these techniques is
that it applies to homogeneous half-space only. In order
to extend the Kobayashi [4] method also to the layered
half-spaces, the authors first show that the Kobayashi [4]
method can be applied to a homogeneous layer, where it
is understood that a homogeneous half-space is a special
example of such a layer only, i.e. a layer of semi-infinite
depth. This is the topic of this paper. In the forthcoming
papers, the solutions for the layers will be combined
leading to a solution for a layered half-space expressed
in the form of integrals with the finite integration path.
2 METHOD OF ANALYSIS
Let us consider an elastic layer subjected to general
point load excitation, which can be represented by two
components, the vertical and the horizontal one (Fig.1).
The model is analyzed under the following assumptions:
• The load varies in time harmonically.
• A general point load is applied at the surface of the
medium.
• The material constants of an elastic layer are the
shear modulus µ , Poisson’s ratio ν , the mass
⌢
density ρ and the damping coefficient µ .
• Material damping in the elastic layer is introduced in
accordance with Voigt's rheological model.
ACTA GEOTECHNICA SLOVENICA, 2005/1
5.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
With these assumptions in mind the problem at hand
has a steady-state solution, which varies in time in the
same manner as the load, namely as e i⋅ω⋅t .
The equation of motion [5]:
So, to formulate the problem mathematically we employ
the displacement potentials
by means of which the
displacement vector U of the homogeneous half-space
may be decomposed as [6]:
U = ∇⋅ ϕ + ∇× ψ .
..
µ ⋅∇2U + (λ + µ) ⋅∇⋅ (∇ • U ) + ρ ⋅ F − ρ ⋅U = 0 ,
(1)
which is well known as Navier equation of motion,
serves as the starting point.
The system of equations (1) presents a disadvantageous
feature as it couples three displacement components.
Of course, we can uncouple this system of equations
by eliminating two of three displacement components
through two of three equations, but this results in partial
differential equations of the sixth order. A far more
convenient approach is to express the components of the
displacement vector in terms of derivatives of potentials.
These potentials satisfy uncoupled wave equations.
The above equation has in the cylindrical coordinate
system r, ϑ and z the following form:
∂ϕr 1 ∂ψz ∂ψϑ
+ ⋅
−
∂r
r ∂ϑ
∂z
ϕ
∂
ψ
∂
ψ
∂
1
uϑ = ⋅ ϑ + r − z
r ∂ϑ
∂z
∂r
ur =
uz =
ACTA GEOTECHNICA SLOVENICA, 2005/1
(3)
.
(4)
.
∂ϕz 1 ∂ (ψϑ ⋅ r ) 1 ∂ψr
,
− ⋅
+ ⋅
r
r ∂ϑ
∂r
∂z
(5)
with ϕ and ψ = (ψr , ψϑ , ψz ) , i.e. the scalar and the
vector Helmholtz potentials that satisfy the following
wave equations in the absence of body forces:
Figure 1. An elastic layer subjected to the surface with a general harmonic point load.
6.
(2)
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
∇2 ϕ =
1 ∂ 2ϕ
⋅
,
cL2 ∂t 2
Generally the components of ψ are taken to be related
in some way. Usually, but not always, the relation:
(6)
ψ
2 ∂ψ
1 ∂2 ψ
∇ ψr − 2r − 2 ⋅ θ = 2 ⋅ 2r
r
r ∂θ
cT ∂t
(7)
ψθ
2 ∂ψ
1 ∂2 ψ
+ 2 ⋅ r = 2 ⋅ 2θ
2
r
r ∂θ
cT ∂t
(8)
2
∇2 ψθ −
1 ∂ ψz
⋅
,
cT2 ∂t 2
∇⋅ ψ = 0
is taken as an additional constraint condition. This
relation has the advantage that it is consistent with the
Helmholtz decomposition of a vector.
In the case of the vertical point load, the corresponding
elastodynamic problem is axi-symmetrical. Displacement components are independent of coordinate ϑ and
are given by:
⎧⎪ ∂ϕ ∂ψϑ
⎫⎪
r
⎪⎪
⎪⎪
−
⎪⎪
⎪⎪
r
z
∂
∂
⎪⎪
⎪⎧⎪ur (r , t )⎪⎫⎪ ⎪⎪
⎪⎪
⎪⎪ ⎪⎪
⎪⎪
u =⎨ 0 ⎬=⎨
0
⎬ . (14)
⎪⎪
⎪⎪ ⎪⎪
⎪⎪
⎪⎪⎩uz (r , t )⎪⎪⎭ ⎪⎪
⎪⎪
∂
⋅
r
ψ
(
)
ϕ
∂
⎪⎪ r 1
ϑ ⎪
⎪
+
⋅
⎪⎪ ∂z
∂r ⎪⎪⎪⎭
r
⎪⎩
2
∇2 ψz =
(9)
where the Laplacian ∇2 is defined as
∇2 =
1 ∂2
∂2 1 ∂
∂2
+ ⋅ + 2⋅ 2+ 2.
2
r ∂r r ∂θ
∂r
∂z
(10)
and:
cL =
ω
λ +2⋅µ
=
kL
ρ
cT =
ω
µ
=
kT
ρ
(11)
The two scalar wave potentials have to satisfy the partial
differential equations (6) and (8), the boundary conditions at the top of the layer:
(12)
P ⋅ H (t ) ⋅ δ (r )
(15)
2⋅π ⋅r
τ zr = 0 , (16)
σz = −
are the velocity of the dilatational (P-waves) and the
velocity of the shear waves (S-waves).
It should be noted that Eq. (2) relates the three components of the displacement vector to four other functions,
i.e. the scalar potential and the three components
of the
vector potential. This indicates that ϕ and the ψ should
be subjected to an additional constraint condition.
w (r , 0, ω) = −
and any axi-symmetric conditions on z = h .
After application of the Hankel transform r → ξ to the
above stated problem and the solution of the resulting
ordinary differential equation we obtain:
∞
ξ ⋅ ξ 2 − kL2
P kT2
⋅ ⋅∫
⋅ J 0 (ξr ) ⋅ dξ
2⋅π µ 0
D(ξ )
∞
− 2 ⋅ kT2 ⋅ ∫
ξ ⋅ (ξ 2 − kT2 ) ⋅ ξ 2 − kL2
D(ξ )
0
∞
− 4 ⋅ kT2 ⋅ ∫
⋅ C1 (ξ ) ⋅ J 0 (ξr ) ⋅ dξ
ξ 2 ⋅ ξ 2 − kL2 ⋅ ξ 2 − kT2
D(ξ )
0
u (r , 0, ω) = −
(13)
(17a)
⋅ C3 (ξ ) ⋅ J 0 (ξr ) ⋅ dξ
(
)
∞ ξ 2 ⋅ 2 ⋅ ξ 2 − k2 − 2 ⋅ ξ 2 − k2 ⋅ ξ 2 − k2
T
L
T
P
⋅∫
⋅ J1 (ξr ) ⋅ dξ
2⋅π ⋅µ 0
D(ξ )
∞
− 4 ⋅ kT2 ⋅ ∫
0
∞
−2⋅k ⋅ ∫
2
T
0
ξ 2 ⋅ ξ 2 − kL2 ⋅ ξ 2 − kT2
D(ξ )
ξ ⋅ (ξ 2 − kT2 ) ⋅ ξ 2 − kL2
D(ξ )
⋅ C1 (ξ ) ⋅ J1 (ξr ) ⋅ dξ
(18a)
⋅ C3 (ξ ) ⋅ J1 (ξr ) ⋅ dξ
ACTA GEOTECHNICA SLOVENICA, 2005/1
7.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
where D (η ) is defined as:
D(ξ ) = (2 ⋅ ξ 2 − k
)
2 2
T
− 4 ⋅ ξ 2 ξ 2 − kL2 ⋅ ξ 2 − kT2 . (19)
For a half-space, which is understood as a layer of semiinfinite depth, the undetermined constants C1 and C3
in (17a) and (18a) can be set to be equal zero. This gives:
w (r , 0, ω) = −
∞
ξ ⋅ ξ 2 − kL2
P kT2
⋅ ⋅∫
⋅ J 0 (ξr ) ⋅ dξ
2⋅π µ 0
D(ξ )
∞
u (r , 0, ω) = −
P
⋅
2 ⋅ π ⋅ µ ∫0
(
ξ 2 ⋅ 2 ⋅ ξ 2 − kT2 − 2 ⋅ ξ 2 − kL2 ⋅ ξ 2 − kT2
(17)
⎡ 1 ∂u
∂u ⎤
σ ϑz = µ ⋅ ⎢ ⋅ z + ϑ ⎥
⎢⎣ r ∂θ
∂z ⎥⎦
⎡ ∂u
∂u ⎤
σ zr = µ ⋅ ⎢ r + z ⎥
⎢⎣ ∂z
∂r ⎥⎦
∇2 ϕ = −
) ⋅ J (ξr )⋅ dξ
1
D(ξ )
(18)
The above solutions are analogous to these given by
Achenbach [5] for the Laplace domain.
The integrals of semi-infinite integration range in equations (17) and (18), which are to be reduced to the finite
integration range, can be clearly identified.
Now we turn our attention to a layer loaded on its
upper surface with a concentrated horizontal point load
Without loss of generality, we can assume that it acts on
positive x-axis direction.
The relevant stress-displacement relations for an elastic
layer may be written as:
⎡ ∂u
∂u
∂u ⎤
u
1 ∂u
σz = λ ⋅ ⎢ r + r + ⋅ ϑ + z ⎥ + 2 ⋅ µ ⋅ z
⎢⎣ ∂r
r r ∂θ
∂z
∂z ⎥⎦
⎡ 1 ∂u
∂u ⎤
σ ϑz = µ ⋅ ⎢ ⋅ z + ϑ ⎥
⎢⎣ r ∂θ
∂z ⎥⎦
⎡ ∂u
∂u ⎤
∂u
u
1 ∂u
σz = λ ⋅ ⎢ r + r + ⋅ ϑ + z ⎥ + 2 ⋅ µ ⋅ z
⎢⎣ ∂r
r r ∂θ
∂z
∂z ⎥⎦
(31)
∇2 ψθ −
ψθ
ω2
2 ∂ψr
=
−
⋅ ψθ
+
⋅
r 2 r 2 ∂θ
cT2
(32)
∇2 ψz = −
∂ (ψr ⋅ r )
∂r
+
(34)
The loading conditions in Cartesian coordinate system
are given by:
Fx =
Q (ω )
⋅ δ (r )
2⋅π ⋅r
Fz = 0 .
(35)
(24)
σϑz (r , ϑ, 0, ω) =
(25)
(37)
(38)
Q (ω) ⋅ δ (r )
⋅ cos (ϑ)
2⋅π ⋅r
Q (ω) ⋅ δ (r )
⋅ sin(ϑ)
2⋅π ⋅r
σ zz (r, ϑ, 0, ω) = 0
(26)
(36)
Using the above results, the boundary conditions on the
surface z=0 of an elastic layer can be written as:
σrz (r , ϑ, 0, ω) = −
ACTA GEOTECHNICA SLOVENICA, 2005/1
(33)
∂ψθ
∂ψ
+r ⋅ z = 0 .
∂θ
∂z
thus we obtain:
8.
ω2
⋅ ψz ,
cT2
Fϑ = −Fx ⋅ sin(ϑ)
(23)
∂ϕ
1 ∂ (ψϑ ⋅ r ) 1 ∂ψr
,
uz = z + ⋅
− ⋅
r
r ∂ϑ
∂r
∂z
(30)
ψr 2 ∂ψθ
ω2
−
⋅
=
−
⋅ ψr
r 2 r 2 ∂θ
cT2
Fr = Fx ⋅ cos (ϑ)
For the problem at hand we again assume the harmonic
time dependence:
∂ψ ∂ψ
1 ∂ϕ
uϑ = ⋅ ϑ + r − z
r ∂ϑ
∂z
∂r
(29)
Their transformation to the cylindrical coordinates
yields:
⎡ ∂u
∂u ⎤
σ zr = µ ⋅ ⎢ r + z ⎥ . (22)
⎢⎣ ∂z
∂r ⎥⎦
∂ϕ
1 ∂ψ ∂ψ
ur = r + ⋅ z − ϑ
∂r
r ∂ϑ
∂z
(28)
∇2 ψr −
(20)
(21)
a (r , ϑ, z , t ) = a (r , ϑ, z , ω) ⋅ e i⋅ω⋅t
ω2
⋅ϕ ,
cL2
(27)
(39)
(40)
(41)
and they complete the statement of the problem.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
The ϑ -dependence of the loading as given by equations
(36) to (38) and equations (39) to (41) respectively and
the geometry, which is axi-symmetric permit us to seek
the solution to the problem in the following form
ϕ (r , ϑ, z , ω) = ϕ (r , z , ω) ⋅ cos (ϑ)
ψr (r , ϑ, z , ω) = ψr (r , z , ω) ⋅ sin(ϑ)
ω
(57)
kL
ω
cT =
. (58)
kT
cL =
(42)
Eqs. (53) and (54) are coupled. To decouple these equations, it is customary to introduce the new potentials χ
and κ :
(43)
ψϑ (r , ϑ, z , ω) = ψϑ (r , z , ω) ⋅ cos (ϑ)
(44)
ψz (r , ϑ, z , ω) = ψz (r , z , ω) ⋅ sin(ϑ) ,
(45)
which correspond for the following ϑ -dependencies of
the displacements and stresses:
ur (r , ϑ, z , ω) = ur (r , z , ω) ⋅ cos (ϑ)
(46)
uϑ (r , ϑ, z , ω) = uϑ (r , z , ω) ⋅ sin(ϑ)
(47)
uz (r , ϑ, z , ω) = uz (r , z , ω) ⋅ cos (ϑ)
(48)
σrz (r , ϑ, z , ω) = σrz (r , z , ω) ⋅ cos (ϑ)
(49)
σϑz (r , ϑ, z , ω) = σϑz (r , z , ω) ⋅ sin(ϑ)
(50)
σ zz (r , ϑ, z , ω) = σ zz (r , z , ω) ⋅ cos (ϑ) .
(51)
⎡ ∂ ϕ 1 ∂ϕ 1
∂ ϕ⎤
⎢ 2 + ⋅
− 2 ⋅ ϕ + 2 ⎥ + kL2 ⋅ ϕ = 0
⎢⎣ ∂r
r ∂r r
∂z ⎥⎦
(52)
⎡ ∂2 1 ∂
2
2
∂2 ⎤
⎢ 2 + ⋅ − 2 + 2 ⎥ ⋅ ψr + 2 ⋅ ψϑ + kT2 ⋅ ψr = 0 (53)
⎢⎣ ∂r
⎥
r ∂r r
r
∂z ⎦
⎡ ∂2 1 ∂
2
2
∂2 ⎤
⎢ 2 + ⋅ − 2 + 2 ⎥ ⋅ ψθ + 2 ⋅ ψr + kT2 ⋅ ψθ = 0 (54)
⎢⎣ ∂r
r ∂r r
r
∂z ⎥⎦
⎡ ∂ψ
∂ψ ⎤
ψr − ψθ + r ⋅ ⎢⎢ r + z ⎥⎥ = 0 , (56)
∂z ⎥
⎢⎣ ∂r
⎦
κ = ψr − ψϑ
(59)
1
1
ψϑ = ⋅ (χ − κ ) . (60)
ψr = ⋅ (χ + κ )
2
2
In terms of these newly introduced potentials wave
equations (52)-(55), Eq. (56) and displacements, Eqs.
(24)-(26), can be rewritten as:
⎡ ∂2 1 ∂
∂2 ⎤
⎢ 2 + ⋅ + 2 ⎥ ⋅ χ + kT2 ⋅ χ = 0
⎢⎣ ∂r
r ∂r ∂z ⎥⎦
(61)
(62)
⎡ ∂2 1 ∂
4
∂2 ⎤
⎢ 2 + ⋅ − 2 + 2 ⎥ ⋅ κ + kT2 ⋅ κ = 0
⎢⎣ ∂r
r ∂r r
∂z ⎥⎦
(63)
⎡ ∂2 1 ∂
1
∂2 ⎤
⎢ 2 + ⋅ − 2 + 2 ⎥ ⋅ ψz + kT2 ⋅ ψz = 0
⎢⎣ ∂r
r ∂r r
∂z ⎥⎦
2
⎡ ∂2 1 ∂
1
∂2 ⎤
⎢ 2 + ⋅ − 2 + 2 ⎥ ⋅ ψz + kT2 ⋅ ψz = 0
⎢⎣ ∂r
r ∂r r
∂z ⎥⎦
χ = ψr + ψϑ
⎡ ∂2 1 ∂
1
∂2 ⎤
⎢ 2 + ⋅ − 2 + 2 ⎥ ⋅ ϕ + kL2 ⋅ ϕ = 0
⎢⎣ ∂r
r ∂r r
∂z ⎥⎦
The substitution of Eqs. (42)-(45) into wave Eqs.
(30)-(33) and Eq. (34) yields:
2
where:
(55)
⎡ 1 ∂ ( χ + κ ) ∂ψ ⎤
κ + r ⋅ ⎢⎢ ⋅
+ z ⎥⎥ = 0 .
2
∂
∂z ⎥
r
⎢⎣
⎦
ur =
1 ∂ (χ − κ )
∂ϕ 1
+ ⋅ ψz − ⋅
2
∂z
∂r r
1
1 ∂ (χ + κ ) ∂ψz
uϑ = − ⋅ ϕ + ⋅
−
r
∂z
2
∂r
uz =
1 ∂ (χ − κ )
∂ϕ 1
.
− ⋅κ + ⋅
2
∂r
∂z r
(64)
(65)
(66)
(67)
(68)
The general solutions of wave equations (61)-(64) is
found by applying the Hankel transform method with
respect to the radial coordinate. This transform is
defined as in [8]:
ACTA GEOTECHNICA SLOVENICA, 2005/1
9.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
∞
f Hn (ξ ) = ∫ r ⋅ f (r ) ⋅ J n (ξr ) ⋅ dr
(69)
κ˘ H2 = C5 ⋅ e β ⋅z + C6 ⋅ e−β ⋅z
(70)
˘
ψzH1 = C7 ⋅ e β ⋅z + C8 ⋅ e−β ⋅z . (80)
(79)
0
∞
f (r ) = ∫ ξ ⋅ f Hn (ξ ) ⋅ J n (ξr ) ⋅ dξ ,
0
where n is the order of the transform and J n (ξr ) is the
ordinary Bessel function of order n.
For Eqs. (61) and (64) the order of the transform should
be equal to 1, for Eq. (62) to 0 and for Eq. (63) to 2.
Applying these transform to Eqs. (61)-(64) one obtains:
d ϕ˘ H1 (ξ )
− α2 ⋅ ϕ˘ H1 (ξ ) = 0
∂z 2
We continue the analysis by expressing the relationship
given Eq. (65) through the transformed potentials. They
are defined of as the Hankel transforms, expression for
of ϕ , χ , κ and ψz and are given as:
∞
ϕ (r , z , ω) = ∫ ξ ⋅ϕ˘ H1 (ξ , z , ω) ⋅ J1 (ξr ) ⋅ dξ
d 2 χ˘ H0 (ξ )
− β 2 ⋅ χ˘ H0 (ξ ) = 0
∂z 2
d 2κ˘ H2 (ξ )
− β 2 ⋅ κ˘ H2 (ξ ) = 0
∂z 2
(71)
∞
χ (r , z , ω) = ∫ ξ ⋅χ˘ H0 (ξ , z , ω) ⋅ J 0 (ξr ) ⋅ dξ
(82)
0
(72)
∞
(73)
˘
˘
d ψz H1 (ξ )
− β 2 ⋅ ψz H1 (ξ ) = 0 , (74)
2
∂z
2
with:
κ (r , z , ω) = ∫ ξ ⋅κ˘ H2 (ξ , z , ω) ⋅ J 2 (ξr ) ⋅ dξ
(83)
0
∞
˘
ψz (r , z , ω) = ∫ ξ ⋅ψz H1 (ξ , z , ω) ⋅ J1 (ξr ) ⋅ dξ
(84)
0
The substitution of these expressions into Eq. (65) yields:
2
α = ξ2 −
ω
= ξ 2 − kL2
cL2
(75)
β = ξ2 −
ω2
= ξ 2 − kT2 .
cT2
(76)
The general solutions of Eqs. (71)-(74) can be found
easily as:
ϕ˘ H1 = C1 ⋅ e α⋅z + C2 ⋅ e−α⋅z
(77)
χ˘ H0 = C3 ⋅ e β ⋅z + C4 ⋅ e−β ⋅z
(78)
˘
∂ψz H1 (ξ , z , ω)
ξ ⋅ κ˘ H2 (ξ , z , ω) − ξ ⋅ χ˘ H0 (ξ , z , ω) + 2 ⋅
= 0 (85)
∂z
which establishes r-independent relationship between
the transformed components of the vector-potential.
Having taken into account relationship (85) and using
recurrent relation between the Bessel functions [8], the
substitution of expressions (81)-(84) into transformed
displacements (66)-(68) yields:
∞
˘
⎪⎧
1
ur (r , z , ω) = ⋅ ∫ ξ ⋅ ⎪⎨⎡⎣ ξ ⋅ϕ˘ H1 (ξ , z , ω) + ξ ⋅ ψz H1 (ξ , z , ω) +
2 0 ⎪⎪⎩
⎤
˘H
2
∂κ˘ H2 (ξ , z , ω) 2 ∂ ψz 1 (ξ , z , ω) ⎥
⎥ ⋅ J 0 (ξr )
−
− ⋅
⎥
∂z
∂z 2
ξ
⎥⎦
⎫⎪
⎡
˘
∂κ˘ H2 (ξ , z , ω) ⎤⎥
⎪⎪
+ ⎢⎢−ξ ⋅ ϕ˘ H1 (ξ , z , ω) + ξ ⋅ ψz H1 (ξ , z , ω) +
ξ
⋅
J
r
(
)
⎬ ⋅dξ
⎥ 2
⎪⎪
∂
z
⎢⎣
⎥⎦
⎪⎭
10. ACTA
(81)
0
2
GEOTECHNICA SLOVENICA, 2005/1
(86)
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
˘H
∞
2
⎪⎧⎪ ⎡
∂κ˘ H2 (ξ , z , ω) 2 ∂ ψz 1 (ξ , z , ω)
1
˘
⎢
H1
⎪
uϑ (r , z , ω) = ⋅ ∫ ξ ⋅⎨− ⎢ ξ ⋅ ϕ (ξ , z , ω) −
− ⋅
+
∂z
∂z 2
2 0 ⎪⎪ ⎢
ξ
⎪⎩ ⎣
⎤
˘
+ ξ ⋅ ψz H1 (ξ , z , ω)⎥ ⋅ J 0 (ξr ) +
⎦⎥
(87)
⎡
⎤
⎪⎫⎪
˘
∂κ˘ H2 (ξ , z , ω)
+ ⎢⎢−ξ ⋅ϕ˘ H1 (ξ , z , ω) +
+ ξ ⋅ ψz H1 (ξ , z , ω)⎥⎥ ⋅ J 2 (ξr )⎪⎬ ⋅dξ
⎪⎪
∂z
⎢⎣
⎥⎦
⎪⎭
∞
uz (r , z , ω) = ∫
0
⎡ ˘H
⎤
˘
⎢ ∂ϕ 1 (ξ , z , ω) ∂ψz H1 (ξ , z , ω)
⎥
˘
H2
−
− ξ ⋅ κ (ξ , z , ω)⎥ ⋅ J1 (ξr ) ⋅ dξ
ξ ⋅⎢
⎢
⎥
∂z
∂z
⎢⎣
⎥⎦
(88)
where we can see, that the displacements in Eqs. (86)-(88) are written only with three of four potentials. Substituting the
expressions for transformed displacements into expressions for stresses (27)-(29) yields:
σrz (r , z , ω) =
∞
µ ⎪⎧⎪
⋅⎨ ∫ ξ ⋅
2 ⎪⎪⎩ 0
{
˘H
3
∂ϕ˘ H1
2 ∂ ψz 1 (ξ , z , ω)
⎡
⋅
ξ
⋅
ξ
ω
−
⋅
,
z
,
+
(
)
⎢2
⎢⎣
∂z
ξ
∂z 3
⎤
∂ 2κ˘ H2 (ξ , z , ω)
⎥ ⋅ J (ξr ) +
2 ˘ H2
,
,
ξ
κ
ξ
ω
−
⋅
z
(
)
⎥ 0
2
∂z
⎥⎦
˘
⎡
∂ψz H1 (ξ , z , ω)
∂ϕ˘ H1 (ξ , z , ω)
⎢
+ ⎢−2 ⋅ ξ ⋅
+
+2⋅ξ ⋅
∂z
∂z
⎢⎣
⎤
⎪⎫ ⎪⎫
∂ 2κ˘ H2 (ξ , z,, ω)
⎥ ⋅ J (ξr )⎪⎪ ⋅ dξ ⎪⎪
2 ˘ H2
ξ
κ
ξ
,
,
ω
+
+
⋅
z
(
)
⎬ ⎬
⎥
2
⎪⎪ ⎪⎪
∂z 2
⎥⎦
⎪⎭ ⎪⎭
−
⎧⎪ ⎡
∞
∂ϕ˘ H1 (ξ , z , ω)
⎪
µ ⎪⎧⎪
⋅ ⎨ ∫ ξ ⋅⎪⎨ ⎢⎢−2 ⋅ ξ ⋅
+ ξ 2 ⋅ κ˘ H2 (ξ , z , ω) +
∂z
2 ⎪⎪⎩ 0 ⎪⎪ ⎢
⎪⎩ ⎣
⎤
˘
H
3
˘
2 ∂ ψz 1 (ξ , z , ω) ∂ 2κ H2 (ξ , z , ω) ⎥
⎥ ⋅ J 0 ( ξr ) +
+ ⋅
+
⎥
ξ
∂z 3
∂z 2
⎦⎥
⎡
∂ 2κ˘ H2 (ξ , z , ω)
∂
+ ⎢−2 ⋅ ξ ⋅ ϕ˘ H1 (ξ , z , ω) +
+
⎢⎣
∂z 2
∂z
⎤
˘
⎪⎫⎪ ⎪⎫⎪
⎥
∂ψz H1 (ξ , z , ω)
⎪ ⎪
2 ˘ H2
⎥
+ ξ ⋅ κ (ξ , z , ω) ⋅ J 2 (ξr )⎪⎬ ⋅ dξ ⎪⎬
+2⋅ξ ⋅
⎥
⎪⎪ ⎪⎪
∂z
⎥⎦
⎪⎪⎭ ⎪⎪⎭
(89)
σ z ϑ (r , z , ω) =
(90)
ACTA GEOTECHNICA SLOVENICA, 2005/1
11.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
∞
⎤
⎧⎪⎛ λ + 2 ⋅ µ ⎞ ⎡⎢ ∂ 2ϕ˘ H1 (ξ , z , ω)
⎥
2 ˘ H1
⎟⎟ ⋅
σ zz (r , z , ω) = µ ⋅ ∫ ξ ⋅⎪⎨⎜⎜⎜
ξ
ϕ
ξ
ω
,
,
−
⋅
z
(
)
⎥
⎟⎟ ⎢
2
⎪
∂
z
µ
⎝
⎠
⎢⎣
⎥⎦
⎪⎩
0
∂
+ 2 ⋅ ξ 2 ⋅ ϕ˘ H1 (ξ , z , ω) − 2 ⋅ ξ ⋅ κ˘ H2 (ξ , z , ω) +
∂z
˘ H1
⎪⎫⎪
2
∂ ψz (ξ , z , ω)⎪
−2⋅
⎬ ⋅ J 1 ( ξr ) ⋅ d ξ
⎪⎪
∂z 2
⎪⎪⎭
As an example of a homogeneous layer a homogeneous
half space will be considered from here onwards. To
satisfy the radiation conditions at h → ∞ we introduce
branch cuts in such a way that α and β in Eqs. (77)(80) are positive for all values of ξ , which further
implies that constant C1 , C3 , C5 , and C7 must be equal
to zero:
ϕ˘ H1 = C2 ⋅ e−α⋅z (92)
χ˘ H0 = C4 ⋅ e−β ⋅z
(93)
κ˘ H2 = C6 ⋅ e−β ⋅z
(94)
˘
ψzH1 = C8 ⋅ e−β ⋅z
(95)
(91)
Boundary conditions on the surface of the half-space
can be stated as:
σrz (r , z , ω)
z =0
σϑz (r , z , ω)
Q (ω) ⋅ δ (r )
2⋅π ⋅r
(96)
Q (ω) ⋅ δ (r )
2⋅π ⋅r
(97)
=−
z =0
=
σ zz (r , z , ω)
z =0
=0
(98)
Introducing expressions (92)-(95) into the boundary
conditions (96)-(98) and taking into account Eqs.
(89)-(91) yields:
˘H
∞
3
∂ϕ˘ H1 (ξ , 0, ω) 2 ∂ ψz 1 (ξ , 0, ω)
µ ⎪⎧⎪
⎡
σrz (r , 0, ω) = ⋅ ⎨ ∫ ξ ⋅ − ⎢−2 ⋅ ξ ⋅
+ ⋅
+
⎢⎣
2 ⎪⎪⎩ 0
∂z
∂z 3
ξ
{
⎤
∂ 2κ˘ H2 (ξ , 0, ω)
⎥ ⋅ J (ξr ) +
2 ˘ H2
0
,
,
ξ
κ
ξ
ω
+
⋅
(
)
⎥ 0
∂z 2
⎥⎦
˘
⎡
∂ψz H1 (ξ , 0, ω)
∂ϕ˘ H1 (ξ , 0, ω)
⎢
+
+2⋅ξ ⋅
+ ⎢−2 ⋅ ξ ⋅
∂z
∂z
⎢⎣
⎫⎪ ⎫⎪
⎤
∂ 2κ˘ H2 (ξ,, 0, ω)
⎪ ⎪
Q (ω) ⋅ δ (r )
⎥
2 ˘ H2
+
+ ξ ⋅ κ (ξ , 0, ω)⎥ ⋅ J 2 (ξr )⎪⎬ ⋅ dξ ⎪⎬ = −
2
⎪
⎪
∂z
2⋅π ⋅r
⎥⎦
⎪⎪⎭ ⎪⎪
⎭
+
∞
⎪⎧⎪ ⎡
∂ϕ˘ H1 (ξ , 0, ω)
µ ⎪⎧⎪
⋅ ⎨ ∫ ξ ⋅⎪⎨ ⎢⎢−2 ⋅ ξ ⋅
+ ξ 2 ⋅ κ˘ H2 (ξ , 0, ω) +
2 ⎪⎪⎩ 0 ⎪⎪ ⎢
∂z
⎩⎪ ⎣
⎤
˘
H
3
˘
2 ∂ ψz 1 (ξ , 0, ω) ∂ 2κ H2 (ξ , 0, ω) ⎥
⎥
+ ⋅
⋅ J (ξr ) +
+
⎥ 0
ξ
∂z 3
∂z 2
⎥⎦
⎡
∂ 2κ˘ H2 (ξ , 0, ω)
∂
+ ⎢−2 ⋅ ξ ⋅ ϕ˘ H1 (ξ , 0, ω) +
+
⎢⎣
∂z 2
∂z
⎤
˘
⎪⎫⎪ ⎫⎪⎪
⎥
∂ψz H1 (ξ , 0, ω)
⎪ ⎪ Q (ω) ⋅ δ (r )
2 ˘ H2
+ ξ ⋅ κ (ξ , 0, ω)⎥ ⋅ J 2 (ξr )⎪⎬ ⋅ dξ ⎪⎬ =
+2⋅ξ ⋅
⎥
⎪⎪ ⎪⎪
∂z
2⋅π ⋅r
⎥⎦
⎪⎪⎭ ⎪⎪⎭
(99)
σ z ϑ (r , 0, ω) =
12. ACTA
GEOTECHNICA SLOVENICA, 2005/1
(100)
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
∞
⎤
⎪⎧⎛ λ + 2 ⋅ µ ⎞⎟ ⎡⎢ ∂ 2ϕ˘ H1 (ξ , 0, ω)
⎥
2 ˘ H1
⎟⎟ ⋅ ⎢
σ zz (r, 0, ω) = µ ⋅ ∫ ξ ⋅⎪⎨⎜⎜
−
⋅
0
ξ
ϕ
ξ
ω
,
,
(
)
⎥
2
⎟
⎜
⎪
∂
z
µ
⎝
⎠
⎢
⎥⎦
⎪⎩
0
⎣
∂
+ 2 ⋅ ξ 2 ⋅ ϕ˘ H1 (ξ , 0, ω) − 2 ⋅ ξ ⋅ κ˘ H2 (ξ , 0, ω) +
∂z
⎫⎪
˘
∂ 2 ψz H1 (ξ , 0, ω)⎪⎪
−2⋅
⎬ ⋅ J 1 ( ξr ) ⋅ d ξ = 0
⎪⎪
∂z 2
⎪⎪⎭
(101)
˘H
It ca be shown that Eqs. (99)-(101) can be written in the following form using just three of four potentials ϕ 1 , κ˘ H2
˘
and ψz H1 :
˘H
3
∂ϕ˘ H1 (ξ , 0, ω) 2 ∂ ψz 1 (ξ , 0, ω) ∂ 2κ˘ H2 (ξ , 0, ω)
Q (ω )
−2 ⋅ ξ ⋅
+ ⋅
+
+ ξ 2 ⋅ κ˘ H2 (ξ , 0, ω) =
3
2
∂z
∂z
∂z
ξ
π ⋅µ
˘
∂ψz H1 (ξ , 0, ω) ∂ 2κ˘ H2 (ξ , 0, ω)
∂ϕ˘ H1 (ξ , 0, ω)
−2 ⋅ ξ ⋅
+2⋅ξ ⋅
+
+ ξ 2 ⋅ κ˘ H2 (ξ , 0, ω) = 0
∂z
∂z
∂z 2
⎡ ˘
⎤
1 ⎛⎜ λ + 2 ⋅ µ ⎞⎟ ⎢ ∂ 2ϕ H1 (ξ , 0, ω)
⎥+
2 ˘ H1
⎟⎟ ⋅ ⎢
0
⋅ ⎜⎜
,
,
ξ
ϕ
ξ
ω
−
⋅
(
)
⎥
2 ⎝ µ ⎟⎠ ⎢
∂z 2
⎥⎦
⎣
˘
∂ 2 ψz H1 (ξ , 0, ω)
∂
+ ξ 2 ⋅ ϕ˘ H1 (ξ , 0, ω) −
− ξ ⋅ κ˘ H2 (ξ , 0, ω) = 0
∂z 2
∂z
(102)
(103)
(104)
The substitution of general solutions for wave potentials Eqs. (92)-(95) evaluated at z=0 into Eqs. (102)-(104) yields:
⎡2
⎤ Q (ω )
C2 ⋅[2 ⋅ ξ ⋅ α ] + C6 ⋅ ⎡⎣ ξ 2 + β 2 ⎤⎦ − C8 ⋅ ⎢ ⋅ β 3 ⎥ =
⎢⎣ ξ
⎥⎦ π ⋅ µ
C2 ⋅[2 ⋅ ξ ⋅ α ] + C6 ⋅ ⎡⎣ ξ 2 + β 2 ⎤⎦ − C8 ⋅[2 ⋅ ξ ⋅ β ] = 0
⎡ α2
C2 ⋅ ⎢⎢
⎣2
⎛ λ + 2 ⋅ µ ⎞⎟ ξ 2
⎟−
⋅ ⎜⎜
⎜⎝ µ ⎟⎟⎠ 2
(105)
(106)
⎤
⎛ λ + 2 ⋅ µ ⎞⎟
⎟ + ξ 2 ⎥ + C6 ⋅ ⎡⎣ β ⋅ ξ ⎤⎦ − C8 ⋅ ⎡⎣ β 2 ⎤⎦ = 0
⋅ ⎜⎜
⎥
⎜⎝ µ ⎟⎟⎠
⎦
(107)
The above equations for constants C2 , C6 and C8 can be presented in a matrix form as:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢ α2
⎢
⎢ 2
⎣
2 ⋅ ξ ⋅α
ξ2 + β2
2 ⋅ ξ ⋅α
ξ2 + β2
⎛ λ + 2 ⋅ µ ⎞⎟ ξ 2
⎟−
⋅ ⎜⎜
⎜⎝ µ ⎟⎟⎠ 2
⎛ λ + 2 ⋅ µ ⎞⎟
⎟+ ξ2
⋅ ⎜⎜
⎜⎝ µ ⎟⎟⎠
β ⋅ξ
⎪⎧⎪ Q (ω) ⎪⎫⎪
⎤
⎪
⎪
⎥
⎥ ⎪⎧ C ⎪⎫ ⎪⎪⎪ π ⋅ µ ⎪⎪⎪
⎥ ⎪ 2⎪ ⎪
⎪⎪
⎥ ⎪ ⎪ ⎪
−2 ⋅ ξ ⋅ β ⎥ ⋅ ⎪⎨ C6 ⎪⎬ = ⎪⎨ 0 ⎪⎬
⎪⎪
⎥ ⎪⎪ ⎪⎪ ⎪⎪
⎪⎪
⎥ ⎪⎪⎩ C8 ⎪⎪⎭ ⎪⎪
2
⎪⎪
⎪
−β ⎥
⎥
⎪⎪ 0 ⎪⎪⎪
⎦
⎩⎪
⎭⎪
2
− ⋅ β3
ξ
(108)
ACTA GEOTECHNICA SLOVENICA, 2005/1
13.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
Upon the solution of system (108) we can write the final form of solution for Eqs. (71)-(74):
2
Q (ω) η ⋅ η −1 −α⋅z
⋅e
ϕ˘ H1 = −
⋅ 2
π ⋅ µ kT ⋅ F (η )
χ˘ H0 = 0 ⋅ e−β ⋅z = 0
(110)
Q (ω ) E (η )
⋅ e −β ⋅ z
κ˘ H2 =
⋅
π ⋅ µ kT2 ⋅ F (η )
where:
(109)
(111)
˘
Q (ω )
η
⋅ e−β ⋅z , (112)
ψzH1 =
⋅
π ⋅ µ 2 ⋅ k 2 ⋅ (η 2 −1)
T
F (η ) = (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2
2
(113)
E (η ) = η 2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥ .
⎣
⎦
(114)
Displacements on the surface can be derived from Eqs. (86)-(88) in the following form:
ur (r , 0, ω) = ur1 (r , 0, ω) + ur 2 (r , 0, ω)
(115)
uϑ (r , 0, ω) = uϑ1 (r , 0, ω) + uϑ 2 (r , 0, ω)
(116)
⎡⎛
⎞⎟
∞
⎢⎜⎜
η ⋅ η 2 −1
⎟
Q (ω )
uz (r , 0, ω) =
⋅ ∫ ξ ⋅ ⎢⎜⎜
⋅ α⎟⎟⎟ +
2
⎢
2
2
2
2
2
2
π ⋅ µ 0 ⎢⎜⎜ k ⋅ ⎡ (2 ⋅ η −1) − 4 ⋅ η ⋅ η −1 ⋅ η − γ ⎤ ⎟⎟
⎜ T ⎢⎣
⎥⎦ ⎠
⎣⎝
⎞⎟ ⎡ ⎡
⎛
⎜⎜
η
⎟
+ ⎜⎜
⋅ β ⎟⎟ + ⎢ ⎢ η 2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥
⎣
⎦
⎜⎜ 2 ⋅ kT2 ⋅ (η 2 −1) ⎟⎟ ⎢⎣ ⎢⎣
⎠
⎝
where:
(117)
⎫
2
⎤ ⎤ ⎪
kT2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥ ⎥ ⋅ ξ ⎥ ⎬ J1 (ξr ) ⋅ dξ
⎣
⎦ ⎦ ⎥⎦ ⎪
⎪
⎭
⎛
⎞⎟
⎪
∞
⎪
⎜⎜
⎟⎟
⎪
η ⋅ η 2 −1
1 Q (ω )
⎜
⎪
⋅ ξ ⎟⎟ +
ur1 (r , 0,ω) = ⋅
⋅ ∫ ξ ⋅⎨⎜⎜−
2
⎟
⎡
⎤
⎪
2
2
2
2
2
2
2 π ⋅ µ 0 ⎪⎜⎜ k ⋅ (2 ⋅ η −1) − 4 ⋅ η ⋅ η −1 ⋅ η − γ
⎥ ⎟⎟⎟⎠
⎪
⎜⎝ T ⎢⎣
⎦
⎪
⎩
⎞⎟ ⎛
⎞⎟
⎛
⎜
η
η
2
⎟ ⎜
⎟
⋅ ξ ⎟⎟ − ⎜⎜⎜
⋅ ⋅ β 2 ⎟⎟ +
+ ⎜⎜⎜
⎟⎟
⎜⎜ 2 ⋅ kT2 ⋅ (η 2 −1) ⎟⎟ ⎜⎜ 2 ⋅ kT2 ⋅ (η 2 −1) ξ
⎠ ⎝
⎠
⎝
⎡⎡
+ ⎢⎢ ⎢ η 2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥
⎢
⎣
⎦
⎣⎣
⎫
2
⎤ ⎤ ⎪
kT2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥ ⎥ ⋅ β ⎥ ⎬ J 0 (ξ ) ⋅ dξ
⎣
⎦ ⎦ ⎥⎦ ⎪
⎪
⎭
14. ACTA
GEOTECHNICA SLOVENICA, 2005/1
(118)
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
⎡⎛
⎞⎟
∞
⎢⎜⎜
η ⋅ η 2 −1
⎟
1 Q (ω )
⋅ ξ ⎟⎟⎟ +
ur 2 (r , 0,ω) = ⋅
⋅ ∫ ξ ⋅ ⎢⎜⎜
2
⎢
2 π ⋅ µ 0 ⎢⎜⎜ k 2 ⋅ ⎡ (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤ ⎟⎟
⎜ T ⎣⎢
⎦⎥ ⎠
⎣⎝
⎞⎟ ⎡ ⎡
⎛
⎜
η
⎟
+ ⎜⎜⎜
⋅ ξ ⎟⎟ + ⎢⎢ ⎢ η 2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥
⎣
⎦
⎜⎜ 2 ⋅ kT2 ⋅ (η 2 −1) ⎟⎟ ⎣ ⎢⎣
⎠
⎝
(119)
2
⎤ ⎤ ⎪⎫
kT2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥ ⎥ ⋅ β ⎥ ⎬ J 2 (ξr ) ⋅ dξ
⎣
⎦ ⎦ ⎥⎦ ⎪⎭⎪
⎞⎟
⎪⎧⎪ ⎛⎜
∞
⎟⎟
⎪⎪ ⎜⎜
η ⋅ η 2 −1
1 Q (ω )
⎟⎟ +
⋅
ξ
uϑ1 (r , 0, ω) = ⋅
⋅ ∫ ξ ⋅⎨−⎜⎜−
2 π ⋅ µ 0 ⎪⎪ ⎜⎜ k 2 ⋅ ⎡ (2 ⋅ η 2 −1)2 − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤ ⎟⎟⎟
⎥ ⎟⎠
⎪⎪ ⎜⎝ T ⎢⎣
⎦
⎩
⎞⎟ ⎛
⎞
⎛
⎜⎜
2 2 ⎟⎟⎟
η
η
⎟⎟ ⎜⎜
⋅ ξ ⎟ −⎜
⋅ ⋅β ⎟+
+ ⎜⎜
⎟⎟
⎜⎜ 2 ⋅ kT2 ⋅ (η 2 −1) ⎟⎟ ⎜⎜⎜ 2 ⋅ kT2 ⋅ (η 2 −1) ξ
⎠ ⎝
⎠
⎝
(120)
⎡⎡
+ ⎢⎢ ⎢ η 2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥
⎢
⎣
⎦
⎣⎣
2
⎤ ⎤
kT2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥ ⎥ ⋅ β ⎥
⎣
⎦ ⎦ ⎥⎦
⎪⎫
⎬ J 0 (ξ ) ⋅ dξ
⎪⎪⎭
⎡⎛
∞
⎢⎜⎜
η ⋅ η 2 −1
1 Q (ω )
uϑ 2 (r , 0, ω) = ⋅
⋅ ∫ ξ ⋅ ⎢⎜⎜
2 π ⋅ µ 0 ⎢⎢⎜⎜ k 2 ⋅ ⎡ (2 ⋅ η 2 −1)2 − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2
⎜ T ⎢⎣
⎣⎝
⎞⎟
⎟
⋅ ξ ⎟⎟⎟ +
⎤ ⎟
⎥⎦ ⎟⎠
⎞⎟ ⎡ ⎡
⎛
⎜⎜
η
⎟
+ ⎜⎜
⋅ ξ ⎟⎟ + ⎢⎢ ⎢ η 2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥
⎟ ⎢
2
2
⎣
⎦
⎜
⎝⎜ 2 ⋅ kT ⋅ (η −1) ⎟⎠ ⎣ ⎣
(121)
2
⎤ ⎤ ⎪⎫
kT2 ⋅ ⎡⎢ (2 ⋅ η 2 −1) − 4 ⋅ η 2 ⋅ η 2 −1 ⋅ η 2 − γ 2 ⎤⎥ ⎥ ⋅ β ⎥ ⎬ J 2 (ξr ) ⋅ dξ
⎣
⎦ ⎦ ⎥⎦ ⎪⎭⎪
Transformation of our results from cylindrical to Cartesian coordinates reconfirms Kobayashi’s [4] findings.
Finally, evaluating the integrals in Eqs. (115)-(117) one
can obtain the solution surface Green’s function in the
frequency domain. The strait forward evaluation of In
evaluating the inverse Hankel transform integrals in the
above mentioned equations can be very problematic.
Due to their complex structure they cannot be evaluated
analytically. Their numeric evaluation through a FFT like
scheme, as proposed for a similar problem by Vostrukov
[2], is some how problematic. At a numerical evaluation
one has to reduce the semi-infinite integration range to
a finite one, therefore neglecting the contributions to
the integral value coming at large values of integration
variable and one is faced with the problems coming from
integration through or nearby a singularity of integrand.
The closer inspection of Eqs. (115)-(117) shows that
the inversion integrals in these equations have the same
basic mathematical structure as the semi-infinite integrals Kobayashi [4] succeeded to transform to the ones
with the finite range of integration. For the evaluation of
ACTA GEOTECHNICA SLOVENICA, 2005/1
15.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
integrals in Eqs. (115)-(117), despite the differences in
detail, we adopt the concept, which has been forwarded
by Kobayashi showing hereby that his approach to deal
with homogeneous half-space can be generalized and
extended to a homogeneous layer.
3 NUMERICAL EXAMPLE
3.1 INTEGRAND
The surfaces shown in Figs. (3)-(10) represent integrands
drawn for ν =1 3 . In Fig. (3)-(6) the discontinuities
at η = γ can be clearly seen. Borders between meshed
and unmeshed areas show curves for a=10. As shapes of
integrands are very smooth, a numerical integration can
be performed without difficulties.
A numerical example is given for homogeneous halfspace as a special example of a layer, subjected to a
general point load.
Figure 3. Real part of the integrand of the horizontal
fundamental function of axi-symmetric componet
Figure 4. Imaginary part of the integrand of the horizontal
fundamental function of axi-symmetric component
16. ACTA
GEOTECHNICA SLOVENICA, 2005/1
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
Figure 5. Real part of the integrand of the horizontal
fundamental function of anti-symmetric component
Figure 6. Imaginary part of the integrand of
the horizontalfundamental function
of anti-symmetric component
Figure 7. Real part of the integrand of
the horizontal fundamental function
of axi-symmetric component
ACTA GEOTECHNICA SLOVENICA, 2005/1
17.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
Figure 8. Imaginary part of the integrand of
the horizontal fundamental function
of axi-symmetric component
Figure 9. Real part of the integrand of
the horizontal fundamental function
of anti-symmetric component
Figure 10. Imaginary part of the integrand of
the horizontal fundamental function
of anti-symmetric componen
18. ACTA
GEOTECHNICA SLOVENICA, 2005/1
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
3.2 FUNDAMENTAL FUNCTION
The horizontal Green’s function is represented as:
ux (r , 0, ω) =
Q (ω ) 1
⋅ ⋅ { f H1 0 + i ⋅ f H20 + ⎡⎢⎣ f H32 + i ⋅ f H42 ⎤⎥⎦ ⋅ cos (2ϑ)
4⋅π ⋅µ r
}
(129)
1
2
where, f H0 , f H0 are the horizontal fundamental function
of the axi-symmetric component, and f H32 , f H42 are the
horizontal fundamental function of the anti-symmetric
component. These fundamental functions are defined by
two variables, i.e. Poisson’s ratio ν and non-dimensional
frequency a. The blue curve in Figs. (11)-(12) shows the
real part and the red one the imaginary part of the horizontal fundamental function of axi and anti-symmetric
component.
Figure 11. Horizontal fundamental function of axi-symmetric component
Figure 12. Horizontal fundamental function of anti-symmetric component
ACTA GEOTECHNICA SLOVENICA, 2005/1
19.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
The numerical solution for vertical Green’s function
is obtained through the similar process and is shown
in Fig. (13). The blue curve in Fig. (13) shows the real
part and the red one the imaginary part of the vertical
Green’s function.
Figure 13. The vertical Green’s function
4 CONCLUSIONS
The integral representations of the surface displacements
for an elastic layer due to a harmonic point load are
presented. The form of these expressions permits us to
consider the homogeneous, elastic half-space as a layer
of infinite depth. They are expressed by integrals of
semi-infinite range. These integrals are usually evaluated
by numerical integration. It has been demonstrated in
this paper that they can be reduced to the integrals of
finite range along the branch cuts of their integrands
and the residues at the poles of the integrands. This
method yields accurate expressions for singularities of
the Green’s functions and considerably better results
for distances where contributions at large values of
integration variable are significant. Numerical results for
a homogeneous half-space are presented, which are in
agreement with the findings of other researchers derived
by a different approach.
Our results for a homogeneous layer, which are believed
to be original, allows us to extend our method through
20. ACTA
GEOTECHNICA SLOVENICA, 2005/1
superposition of layers to the case of a layered halfspace. Our findings in dealing with this problem will be
reported in the forthcoming papers.
REFERENCES
[1]
[2]
[3]
Kausel, E. (1981). An Explicite Solution for the
Green’s Function for Dynamic Loads in Layered
Media, Research Report R81-13, Massachusetts
Institute of Tehnology.
Vostroukhov, A.V., Verichev, S.N., Kok, A.W.M.
and Esveld, C. (2004). Steady-state response of
stratified half-space subjected to a horizontal
arbitrary buried uniform load applied at a circular
area. Soil Dynamics and Earthquake Engineering
24, 449-459.
Jin, B. and Liu H. (1999). Exact solution for
horizontal displacement at center of the surface of
an elastic half space under horizontal impulsive
punch loadinga. Soil Dynamics and Earthquake
Engineering 18, 495-498.
T. PLIBERŠEK ET AL.: GREEN'S FUNCTION FOR AN ELASTIC LAYER LOADED HARMONICALLY ON ITS SURFACE
[4]
Kobayashi, T. and Sasaki, F. (1991). Evaluation of
Green’s function on semi-infinite elastic medium.
KICT Report, No. 86
[5] Achenbach, J.D. (1973). Wave propagation in elastic
solids. North-Holland, Amsterdam, London.
[6] Miklowitz, J. (1980). The theory of elastic waves
and waveguides. North-Holland Series in Applied
Mathematics and Mechanics, New York.
[7] Sneddon, I.H. (1972). The use of integral transforms. McGraw-Hill Book Company, New York.
[8] Abramovitz, M. and Stegum, I.A. (1972). Handbook of mathematical functions. Dover, New York.
[9] Arfken, G.B. and Weber, H.J. (1995). Mathematical
methods for physicists. Academic Press Inc, San
Diego.
[10] Kobayashi, T. (1981). Evaluation of response
to point load excitation on semi-infinite elastic
medium. Tran. Archi. Institute Japan, Vol. 302,
UDC : 624.042.7:620.1.
[11] Premrov, M. and Špacapan, I. (2004). Solving
exterior problems of wave propagation based on an
iterative variation of local DtN operators. Appliedmathematical modelling, 28, 291-304.
ACTA GEOTECHNICA SLOVENICA, 2005/1
21.