Control of the strain field near the crack in elastic layer
In the present paper we consider the strain analysis near the crack parallel to boundary surfaces of a linear isotropic elastic layer of constant thickness. We pose the problem to minimize opening of the crack’s faces, when a normal force is applied to a boundary of the layer. The control secondar...
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irk-123456789-1780252021-02-18T01:28:07Z Control of the strain field near the crack in elastic layer Ciarletta, M. Iovane, G. Sumbatyan, M. A. In the present paper we consider the strain analysis near the crack parallel to boundary surfaces of a linear isotropic elastic layer of constant thickness. We pose the problem to minimize opening of the crack’s faces, when a normal force is applied to a boundary of the layer. The control secondary loading is a normal stress uniformly distributed over the boundary surfaces as to oppose the opening caused by the external applied force. We show that the most dangerous are tangential deformations, and the normal ones are always by many times less dangerous Проаналiзовано пружнiсть бiля трiщини, яка паралельна межi поверхнi лiнiйного iзотропного пружного шару сталої товщини. Ставиться проблема мiнiмiзацiї ширини трiщини у випадку, коли на межi шару дiє нормальна сила. Вторинним керуючим навантаженням є нормальне напруження, рiвномiрно розподiлене по межi поверхонь, яке протидiє розширенню трiщини, що виникає пiд дiєю зовнiшньо прикладеної сили. Показано, що найбiльш загрозливими є тангенцiальнi деформацiї, а нормальнi деформацiї завжди в багато разiв менш загрозливi. 2005 Article Control of the strain field near the crack in elastic layer / M. Ciarletta, G. Iovane, M.A. Sumbatyan // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 265-277. — Бібліогр.: 5 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178025 517.9 en Нелінійні коливання Інститут математики НАН України |
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In the present paper we consider the strain analysis near the crack parallel to boundary surfaces of a linear
isotropic elastic layer of constant thickness. We pose the problem to minimize opening of the crack’s faces,
when a normal force is applied to a boundary of the layer. The control secondary loading is a normal stress
uniformly distributed over the boundary surfaces as to oppose the opening caused by the external applied
force. We show that the most dangerous are tangential deformations, and the normal ones are always by
many times less dangerous |
| format |
Article |
| author |
Ciarletta, M. Iovane, G. Sumbatyan, M. A. |
| spellingShingle |
Ciarletta, M. Iovane, G. Sumbatyan, M. A. Control of the strain field near the crack in elastic layer Нелінійні коливання |
| author_facet |
Ciarletta, M. Iovane, G. Sumbatyan, M. A. |
| author_sort |
Ciarletta, M. |
| title |
Control of the strain field near the crack in elastic layer |
| title_short |
Control of the strain field near the crack in elastic layer |
| title_full |
Control of the strain field near the crack in elastic layer |
| title_fullStr |
Control of the strain field near the crack in elastic layer |
| title_full_unstemmed |
Control of the strain field near the crack in elastic layer |
| title_sort |
control of the strain field near the crack in elastic layer |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/178025 |
| citation_txt |
Control of the strain field near the crack in elastic layer / M. Ciarletta, G. Iovane, M.A. Sumbatyan // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 265-277. — Бібліогр.: 5 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT ciarlettam controlofthestrainfieldnearthecrackinelasticlayer AT iovaneg controlofthestrainfieldnearthecrackinelasticlayer AT sumbatyanma controlofthestrainfieldnearthecrackinelasticlayer |
| first_indexed |
2025-07-15T16:23:35Z |
| last_indexed |
2025-07-15T16:23:35Z |
| _version_ |
1837730750006296576 |
| fulltext |
UDC 517 . 9
CONTROL OF THE STRAIN FIELD NEAR
THE CRACK IN ELASTIC LAYER
КЕРУВАННЯ ПОЛЕМ НАПРУГИ БIЛЯ ТРIЩИНИ
В ПРУЖНОМУ ШАРI
M. Ciarletta, G. Iovane
Univ. Salerno
Via Ponte don Melillo, 84084 Fisciano (SA), Italy
e-mail: iovane@diima.unisa.it
M. A. Sumbatyan
Rostov State Univ.
Zorge Str., 5, Rostov-on-Don 344090, Russia
e-mail: sumbat@math.rsu.ru
In the present paper we consider the strain analysis near the crack parallel to boundary surfaces of a linear
isotropic elastic layer of constant thickness. We pose the problem to minimize opening of the crack’s faces,
when a normal force is applied to a boundary of the layer. The control secondary loading is a normal stress
uniformly distributed over the boundary surfaces as to oppose the opening caused by the external applied
force. We show that the most dangerous are tangential deformations, and the normal ones are always by
many times less dangerous.
Проаналiзовано пружнiсть бiля трiщини, яка паралельна межi поверхнi лiнiйного iзотропно-
го пружного шару сталої товщини. Ставиться проблема мiнiмiзацiї ширини трiщини у випад-
ку, коли на межi шару дiє нормальна сила. Вторинним керуючим навантаженням є нормальне
напруження, рiвномiрно розподiлене по межi поверхонь, яке протидiє розширенню трiщини, що
виникає пiд дiєю зовнiшньо прикладеної сили. Показано, що найбiльш загрозливими є тангенцi-
альнi деформацiї, а нормальнi деформацiї завжди в багато разiв менш загрозливi.
1. Introduction. Strain-stress control of cracked elastic materials is very important, both in
theoretical and practical aspects, problems in civil engineering, soil mechanics, mechanics of
composite and other inhomogeneous materials, in fracture mechanics, medicine, and some
other fields of science and technology [1]. Typically, the researchers’ main goal is to control
stress concentration near the edges of the cracks arising in damaged materials, but to control
the crash process and its consequence, including the control of the internal crack faces behavior
is also very important in medicine and some other application [2].
From the mathematical point of view, crack analysis can be usually reduced to some integral
equations, which admit explicit-form solutions only for simplest geometries [3]. If the elastic
medium occupies an infinite layer of a constant thickness, then the problem admits an explicit
analytical solution in the form of some Fourier integral quadratures. The most direct method
reduces the crack problems in elastic media to hypersingular integral equations that require a
nonstandard theoretical or numerical treatment.
In our recent work [4] we have developed and justified a numerical method, which allows
one to directly treat hypersingular integral equations without any additional transformations (in
c© M. Ciarletta, G. Iovane, and M. A. Sumbatyan, 2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 265
266 M. CIARLETTA, G. IOVANE, AND M. A. SUMBATYAN
Fig 1. A crack dislocated in the elastic layer of a constant thickness h = h+ + h−,
under the applied external load P0 and the uniform controlling stress σ0.
contrast with a standard approach reducing the problem to an infinite system of linear algebraic
equations). In that work we demonstrate an application of the proposed technique to crack
problems in unbounded classical elastic medium. Then we further develop these results for the
crack problem in a porous elastic medium. In this work we consider the problem of cracks
for the linear isotropic elastic layer, which is subjected to a process. As it can be seen below,
in the case of a classical linear isotropic elastic layer, the problem is reduced to a system of
hypersingular integral equations. So, the main purpose of the present work is to show that our
approach is efficient not only for a single hypersingular equation but also for the system of
such equations. Besides, we aim at a practical application in control of behavior of the free
faces of the crack, under various load (i.e., the character of boundary conditions) applied to
boundaries of the layer. We stress, when the applied load causes higher opening of the crack and
a higher relative displacement of its faces, that is not desired in applications such as treatment
and restoration of cracked bones, gluing together different materials during a technological
process under the loading.
2. Mathematical formulation of the problem. Let us consider the crack problem in the layer
of thickness h = h+ + h− (see Fig. 1). Then the basic partial differential equations for the
displacement vector components ū = {ux(x, y), uy(x, y), 0} in the (in-plane) 2-dimensional
problem are (c2 = µ/(λ+ 2µ) = c2s/c
2
p , λ and µ-elastic moduli)1
∂2ux
∂x2
+ c2
∂2ux
∂y2
+ (1− c2) ∂
2uy
∂x∂y
= 0,
(2.1)
∂2uy
∂y2
+ c2
∂2uy
∂x2
+ (1− c2) ∂
2ux
∂x∂y
= 0.
1 Note that the constant c in these formulas is a ratio of transverse and longitudinal wave speeds.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
CONTROL OF THE STRAIN FIELD NEAR THE CRACK IN ELASTIC LAYER 267
As soon as the system (2.1) is solved, the components of the stress tensor (σ) can be
calculated in terms of displacements, as follows:
σxx
λ+ 2µ
=
∂ux
∂x
+ (1− 2c2)
∂uy
∂y
,
σyy
λ+ 2µ
= (1− 2c2)
∂ux
∂x
+
∂uy
∂y
, (2.2)
τxy
µ
=
∂ux
∂y
+
∂uy
∂x
,
where only representation for the most physically important components is given.
The system (2.1) is to be solved in the domain of the x-variable with x ∈ [−∞,∞]. So, we
can apply the Fourier transform along this variable, so that any function f(x, y) transforms to
its Fourier image (f =⇒ F ) as follows:
F (s, y) =
∞∫
−∞
f(x, y)eisxdx,
(2.3)
f(x, y) =
1
2π
∞∫
−∞
F (s, y)e−isxds,
with the following evident and well known properties: if f =⇒ F, then
∂f(x, y)
∂x
=⇒ (−is)F (s, y),
∂2f(x, y)
∂x2
=⇒ −s2F (s, y),
(2.4)
∂f(x, y)
∂y
=⇒ F ′(s, y),
∂2f(x, y)
∂y2
=⇒ F ′′(s, y).
Here primes denote respective derivatives applied to the variable y (in fact, the variable s can
be considered as a parameter when solving the system with respect to y).
According to the Eqs. (2.3), the system (2.1) of partial differential equations in the Fourier
transformed space becomes a system of ordinary differential equations with constant coeffici-
ents (all capital letters denote Fourier images for relevant original functions),
c2U ′′x − s2Ux + (1− c2)(−is)U ′y = 0,
(2.5)
(1− c2)(−is)U ′x + U ′′y − c2s2 Uy = 0,
where all derivatives here are again applied with respect to the variable y.
It is well known that exact analytical solution of such a system can be constructed in terms
of a characteristic polynomial. Let us consider separately the upper (+) layer 0 < y < h+ and
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
268 M. CIARLETTA, G. IOVANE, AND M. A. SUMBATYAN
the lower (−) one −h < y < 0. Then it can be verified directly that a general solution to the
system (2.5) is
U±x = A±i ch(sy) +B±i sh(sy)− C±i
[
1 + c2
s(1− c2)
ch(sy) + y sh(sy)
]
−
−D±i
[
1 + c2
s(1− c2)
sh(sy) + y ch(sy)
]
,
(2.6)
U±y = −A± sh(sy)−B± ch(sy) + C±y ch(sy) +D±y sh(sy),
where sh and ch are hyperbolic sine and cosine, respectively.
Eight unknown constants A±, B±, C±, D± arising in Eqs. (2.6) should be defined by sati-
sfying the boundary conditions that we are now beginning to formulate. Let us assume that the
both sides of the layer are free of tangential stress. Taking into account (2.2)3 and (2.4), the first
two boundary conditions are: τxy = 0, y = h+, y = −h− : U ′x − isUy = 0. Consequently by
using (2.6) we obtain
A+s sh(sh+) +B+s ch(sh+)− C+
[
sh(sh+)
1− c2
+ (sh+) ch(sh+)
]
−
−D+
[
ch(sh+)
1− c2
+ (sh+) sh(sh+)
]
= 0,
(2.7)
−A−s sh(sh−) +B−s ch(sh−) + C−
[
sh(sh−)
1− c2
+ (sh−) ch(sh−)
]
−
−D−
[
ch(sh−)
1− c2
+ (sh−) sh(sh−)
]
= 0.
Further, we assume that the normal load applied to the upper boundary surface y = h+ of
the layer is σ+(x), and the one applied to the lower boundary y = −h− is σ−(x).By considering
(2.2)2 and (2.4) the second pair of the boundary conditions is
σyy = σ±(x), y = h+, y = −h− : (1− 2c2)(−is)Ux + U ′y =
σ̃±(s)
λ+ 2µ
. (2.8)
By using (2.6) we get
−A+s ch(sh+)−B+s sh(sh+) + C+
[
c2 ch(sh+)
1− c2
+ (sh+) sh(sh+)
]
+
+D+
[
c2 sh(sh+)
1− c2
+ (sh+) ch(sh+)
]
=
σ̃+(s)
2µ
,
(2.9)
−A−s ch(sh−) +B−s sh(sh−) + C−
[
c2 ch(sh−)
1− c2
+ (sh−) sh(sh−)
]
−
−D−
[
c2 sh(sh−)
1− c2
+ (sh−) ch(sh−)
]
=
σ̃−(s)
2µ
,
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
CONTROL OF THE STRAIN FIELD NEAR THE CRACK IN ELASTIC LAYER 269
where σ̃±(s) denote the Fourier transform of the functions σ±(s).
We thus have already formulated four boundary conditions (2.7), (2.9) for eight constants.
The remaining four conditions should be put over the junction line y = 0, where we obviously
have the mixed boundary conditions. If the faces of the crack are free of load, then at y = 0,
τ+
xy = τ−xy, |x| > b,
(2.10)
τ+
xy = τ−xy = 0, |x| ≤ b,
σ+
yy = σ−yy, |x| > b,
(2.11)
σ+
yy = σ−yy = 0, |x| ≤ b,
where the first relations in the Eqs. (2.10), (2.11) mean continuity of the stress.
The last pair of the junction boundary conditions should take into account condition of the
displacements,
u+
x = u−x , u+
y = u−y , |x| > b, y = 0. (2.12)
It should be noted that (2.10), (2.11) imply continuity of both stresses τxy and σyy along the
total line y = 0,
τ+
xy = τ−xy, σ+
yy = σ−yy, |x| < ∞, (2.13)
which with the use of the Eqs. (2.2) gives the (third) pair of relations for the eight unknown
coefficients,
B+s−D+ 1
1− c2
= B−s−D− 1
1− c2
,
(2.14)
−A+s+ C+ c2
1− c2
= −A−s+ C−
c2
1− c2
.
In order to construct the 7-th and the 8-th conditions as to finally complete the 8 x 8 system, let
us introduce two auxiliary unknown functions, in accordance with Eq. (2.12),
u+
y − u−y =
{
gy(x), |x| ≤ b,
0, |x| > b,
y = 0, (2.15)
u+
x − u−x =
{
gx(x), |x| ≤ b,
0, |x| > b,
y = 0. (2.16)
Obviously, the physical meaning of these functions is the relative displacement (vertical and
horizontal, respectively) of the upper and the lower faces of the crack. If one considers the
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
270 M. CIARLETTA, G. IOVANE, AND M. A. SUMBATYAN
functions gy(x) and gx(x) which are defined over the total interval |x| ≤ b, then (2.15) and
(2.16) become the last pair of the required conditions in Fourier’s transformed space,
U+
y − U−y = Gy(s), U+
x − U−x = Gx(s), y = 0. (2.17)
Consequently by using (2.6) and (2.14) we obtain
−B+ +B− = Gy(s),
(2.18)[
A+ − C+ 1 + c2
s(1− c2
]
−
[
A− − C− 1 + c2
s(1− c2)
]
= −iGx(s).
So, we have finally arrived at the 8 x 8 system for the uknown coefficients A±, B±, C±, D±
that is given by Eqs. (2.7), (2.9), (2.14), (2.18).
The principal determinant of the considered 8 x 8 system is
∆ =
s2
[
2 ch(sh−) ch(sh+) ch(sh)− ch2(sh+)− ch2(sh−)− (sh)2
]
(1− c2)2
. (2.19)
Other determinants are such that application of the Kramer’s rule gives for the most important
stress tensor components the following expressions (|x| ≤ b):
τ̃+
xy(s, 0)
2µ
=
σ̃+(s)
2µ
L+
τ (s) +
σ̃−(s)
2µ
L−τ (s)−Gy(s)Ly
τ (s) + iGx(s)Lx
τ (s),
(2.20)
σ̃+
yy(s, 0)
2µ
=
σ̃+(s)
2µ
L+
σ (s) +
σ̃−(s)
2µ
L−σ (s) +Gy(s)Ly
σ(s) + iGx(s)Lx
σ(s),
where
L+
τ = − s2
∆(1− c2)2
{(sh+) sh(sh−) sh(sh) + (sh−)(sh) sh(sh+)} ,
L−τ =
s2
∆(1− c2)2
{(sh−) sh(sh+) sh(sh) + (sh+)(sh) sh(sh−)} ,
Ly
τ =
s3
∆(1− c2)
{
(sh+)2 sh2(sh−)− (sh−)2 sh2(sh+)
}
, Lx
σ = Ly
τ ,
Lx
τ = − s3
∆(1− c2)
{[
1 + (sh+)2
]
ch(sh−) sh(sh−) +
[
1 + (sh−)2
]
ch(sh+) sh(sh+)−
− ch(sh+) ch(sh−) sh(sh) −(sh−) sh2(sh+)− (sh+) sh2(sh−) + (sh+)(sh−)(sh)
}
, (2.21)
L+
σ = − s2
∆(1− c2)2
{[1 + (sh−)(sh)] ch(sh+)− (sh+) ch(sh−) sh(sh)−
− ch(sh−) ch(sh) + (sh) sh(sh+)} ,
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
CONTROL OF THE STRAIN FIELD NEAR THE CRACK IN ELASTIC LAYER 271
L−σ = − s2
∆(1− c2)2
{[1 + (sh+)(sh)] ch(sh−)− (sh−) ch(sh+) sh(sh)−
− ch(sh+) ch(sh) + (sh) sh(sh−)} ,
Ly
σ = − s3
∆(1− c2)
{
[1 + (sh+)(sh−)] (sh)−
[
1 + (sh−)2
]
ch(sh+) sh(sh+)−
− [ch(sh−) + (sh+) sh(sh−)] (sh+) ch(sh−)+
+ sh(sh+) ch(sh−) ch(sh)− (sh−) ch2(sh+)
}
.
3. Reduction to a system of hypersingular equations and its properties. In this section we
consider the above problem in the context of the applied mechanics, engineering, and biomedi-
cal science. Let us now come back to engineering aspects of the considered problem, as it is
discussed in Introduction. Recall that we plan to arrange an efficient control of the strain field
near the crack. More precisely, we are interested in a minimization of relative displacements of
the upper and lower crack’s faces, when some external force P0δ(x− x0) is applied in the point
x0 of the (for instance) upper boundary of the layer. Then we try to suppress the introduced
relative displacements, i.e., our functions gy(x) and gx(x), |x| ≤ b, by arranging a couple of
symmetric uniformly distributed normal loads σyy ≡ σ0, y = h, y = −h−. The main aspect
of the present study is to establish relations between σ0, P0, max |gx(x)|, max |gy(x)|, and the
variation of other physical and geometrical parameters. Strictly speaking, the problem posed in
such away is a classical inverse problem. However we do not aim at its full investigation, and
restrict our goal to a statement of some helpful relations between all these quantities.
First of all, let us note that for the chosen type of the applied load we have
σ+ =
{
−σ0, |x| ≤ a
0 , |x| > a
}
− P0 δ(x− x0), (3.1)
σ− =
{
−σ0 , |x| ≤ a
0, |x| > a
}
, (3.2)
so the Fourier transforms σ̃±, which are present in the system (2.20), now can be calculated
explicitly as follows:
σ+(x) =⇒ σ̃+(s) = −2σ0
sin(as)
s
− P0 e
i s x0 ,
(3.3)
σ−(x) =⇒ σ̃−(s) = −2σ0
sin(as)
s
.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
272 M. CIARLETTA, G. IOVANE, AND M. A. SUMBATYAN
Then application of the inverse Fourier transform to Eqs. (2.20) reduces the problem to the
following system of two integral equations with respect to the functions gy(x), gx(x), |x| ≤ b :
b∫
−b
K11(x− ξ)gy(ξ)dξ +
b∫
−b
K12(x− ξ)gx(ξ)dξ =
σ0
µ
fσ
1 (x) +
P0
2µ
fP
1 (x),
(3.4)
b∫
−b
K21(x− ξ)gy(ξ)dξ +
b∫
−b
K22(x− ξ)gx(ξ)dξ =
σ0
µ
fσ
2 (x) +
P0
2µ
fP
2 (x),
where
K11(x) =
∞∫
0
Ly
σ(s) cos(sx)ds, K12(x) = K21(x) =
∞∫
0
Lx
σ(s) sin(sx)ds,
K22(x) =
∞∫
0
Lx
τ (s) cos(sx)ds, fP
1 (x) =
∞∫
0
L+
σ (s) cos[s(x− x0)]ds,
fσ
1 (x) =
∞∫
0
[
L+
σ (s) + L−σ (s)
] sin(as)
s
cos(sx)ds, (3.5)
fP
2 (x) =
∞∫
0
L+
τ (s) sin[s(x− x0)]ds,
fσ
2 (x) =
∞∫
0
[
L+
τ (s) + L−τ (s)
] sin(as)
s
sin(sx)ds.
Here we have used the evident properties that functions Ly
σ, L+
σ , L
−
σ , L
x
τ are even, and Lx
σ, L
y
τ ,
L+
τ , L
−
τ are odd.
In order to evaluate qualitative properties of this system, let us note that all Fourier images
of the kernels (2.21) in the Eqs. (3.5) are continuos over 0 ≤ s < ∞. So,we only need to
estimate their behavior at infinity. It can be directly estimated that at s → +∞,
Ly
σ(s) = −1− c2
2
s+ L0
11(s), Lx
τ (s) =
1− c2
2
s+ L0
22(s),
(3.6)
L0
11(s), L
0
22(s), L
x
σ(s), Ly
τ (s), L
+
σ (s), L−σ (s), L+
τ (s), L−τ (s) = O
(
e− εs
)
,
where ε is some positive constant. So, it becomes clear that all off-diagonal kernels in (3.3), (3.4)
are regular, and such are all right-hand sides. To give an estimate of the qualitative properties
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CONTROL OF THE STRAIN FIELD NEAR THE CRACK IN ELASTIC LAYER 273
of the diagonal kernels, we should use the theory of generalized functions [5], from which it
follows that these kernels represent hypersingular integral operators (see also [4]), since
∞∫
0
s cos(sx)ds = lim
δ→+0
∞∫
0
e−δss cos(sx)ds = lim
δ→+0
[
− x2
(δ2 + x2)2
]
= − 1
x2
. (3.7)
Therefore, our system (3.4) is a system of two hypersingular integral equations. In our previ-
ous work [4] we developed a theory and proposed a numerical method to investigate a single
hypersingular equation. Thus, we have shown that a single characteristic equation (i.e., the one
with the kernel (3.7)) admits an explicit-form exact analytical solution. Then we constructed
also a collocation technique to solve the full hypersingular equation numerically when the
kernel is a sum of the characteristic part (3.7) and arbitrary regular function.
It can be proved that the same method is applicable to the system of hypersingular equati-
ons. Here we demonstrate this by an example which admits an exact analytical solution.
A model example. Let us study the 2 x 2 system
1∫
−1
K11(x− ξ)ϕ(ξ)dξ +
1∫
−1
K12(x− ξ)ψ(ξ)dξ = f1(x)),
(3.8)
1∫
−1
K21(x− ξ)ϕ(ξ)dξ +
1∫
−1
K22(x− ξ)ψ(ξ)dξ = f2(x),
with
K11 = K22 =
1
(x− ξ)2
, K12 = K21 = xξ,
(3.9)
f1(x) = −π +
πx
16
, f2(x) = −2πx.
It can be directly verified that a solution to this system is
ϕ(x) =
√
1− x2, ψ(x) = x
√
1− x2. (3.10)
Our collocation method consists of the following steps.
Let us first consider a single full hypersigular equation of the following type:
b∫
a
[
1
(x− ξ)2
+K0(x− ξ)
]
g(ξ)dξ = f(x), x ∈ (a, b), (3.11)
and arrange a mesh with two sets of nodes xi = −b + (i− 1/2) ∆x, i = 1, . . . , n, and tj =
= −b+ j∆x, j = 0, 1, . . . , n, taken with the constant step x = (b− a)/n. Then we prove that if
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274 M. CIARLETTA, G. IOVANE, AND M. A. SUMBATYAN
Fig 2. Comparison between the explicit-form exact solution and the numerical
collocation method for the model system (3.8), (3.9): the solid line is an
exact solution; the dashed line is an numerical solution, n = 40.
K0(x) ∈ C1(a− b, b− a), then for any x ∈ (a, b) the difference between a solution g(x) of the
linear algebraic system
n∑
j=1
[(
1
xi − tj−1
− 1
xi − tj
)
+K0(xi − tj)∆x
]
g(tj) = f(xi),
(3.12)
i = 1, 2, . . . , n,
and the solution of the Eq. (3.11) vanishes with ∆x → 0 (i.e., n → ∞).
It should be noted that the (second) regular term is a simple quadrature formula for numeri-
cal integration of regular functions. It can be also shown that any solution of the Eq. (3.11)
bounded at the both edges x = ±b, vanishes near these edges. This property is confirmed by a
direct numerical solution of the linear algebraic system (3.12).
Now let us come back to the system (3.8), (3.9). Our collocation method reduces the Eqs. (3.8),
(3.9) to a 2n× 2n system of linear algebraic equations
n∑
j=1
(
1
xi − tj−1
− 1
xi − tj
)
ϕ(tj) + ∆x
n∑
j=1
xitjψ(tj) = f1(xi),
(3.13)
∆x
n∑
j=1
xitjϕ(tj) +
n∑
j=1
(
1
xi − tj−1
− 1
xi − tj
)
ψ(tj) = f2(xi), i = 1, 2, . . . , n.
The Fig. 2 demonstrates comparison between analytical (Eq. (3.10), solid line) and numerical
solutions (from the system (3.13), dashed line) when the number of nodes is n = 40.
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CONTROL OF THE STRAIN FIELD NEAR THE CRACK IN ELASTIC LAYER 275
The same numerical method has been applied to the main system:
n∑
j=1
[(
1
xi − tj−1
− 1
xi − tj
)
+ ∆xK0
11(xi − ξj)
]
gy(tj)+
+ ∆x
n∑
j=1
K12(xi − tj)gx(tj) =
σ0
µ
fσ
1 (xi) +
P0
2µ
fP
1 (xi),
∆x
n∑
j=1
K21(xi − tj)gy(tj) +
n∑
j=1
[
−
(
1
xi − tj−1
− 1
xi − tj
)
+
+ ∆xK0
22(xi − ξj)
]
gx(tj) =
σ0
µ
fσ
2 (xi) +
P0
2µ
fP
2 (xi),
i = 1, 2, . . . , n,
where all kernels K0
11, K12 = K21, K
0
22 are regular (more precisely, infinitely differentiable).
4. Results of numerical treatment and physical conclusions. First of all, let us evaluate the
number of independent dimensionless parameters in the considered problem: 1) b/h, 2) a/h,
3) h+/h, 4) x0/h, 5) σ0/P0.We thus operate with a five-parametric problem, where the calculati-
on for any fixed combination of these five parameters requires solving a system of hypersingular
singular equations.
Physically, to provide a perfect contact between the crack’s faces (for medical or technologi-
cal purposes), we need to arrange such an optimization when the solution of the main system of
hypersingular equations gives the nonpositive function gy(x), |x| ≤ b, and the function gx(x),
|x| ≤ b, is as small (in modulus) as possible.
A relatively unexpected result is that even a small applied normal load σ0 provides vertical
opening of the crack negative, i.e., closed in vertical direction. Numerous computations show
that this conclusion remains valid uniformly over changes of all five parameters of the problem.
A typical example is presented in Fig. 3. As it can be seen from this figure, in the case when
the controlling stress σ0 is absent the force P0 makes the vertical opening to be of alternating
sign, so that it is closed in the central part of the interval (−b, b) and open near the crack’s edges.
However, even a small controlling pressure provides the crack to be closed over its total length.
Fig. 4 shows that in the case of asymmetric disposition of the crack in the layer, this arrange-
ment of closed interior space between the crack’s faces requires a higher controlling pressure
σ0. We also outline that the strain near the crack faces in the asymmetric case is around 7 times
larger than the one in the symmetric case.
Among other results, we stress the interesting property that the relative displacement |gx(x)|
of the crack’s faces in the horizontal direction is always considerably larger than the one in the
vertical direction, gy(x), when the latter is positive (i.e., in the dangerous case of really positive
vertical opening). This feature of the strain-stress process, under an external loa-
ding, indicates that the real obstacle for a perfect conglutination (for medical or technological
aims) consists of the impossibility to provide a small relative tangential crack faces’ displacement
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276 M. CIARLETTA, G. IOVANE, AND M. A. SUMBATYAN
Fig 3. The vertical opening of the crack: c2 = 0, 3, P0/2µ = 1, h+/h =
= h−/h = 0, 5, b/h = 0, 5, a/h = 1, x0/h = 3, 5 . Line 1: σ0/P0 =
= 0, Line 2: σ0/P0 = 0, 01, Line 3: σ0/P0 = 0, 02 .
Fig 4. The vertical opening of the crack: c2 = 0, 3, P0/2µ = 1, h+/h =
= 0, 8, h−/h = 0, 2, b/h = 0, 5, a/h = 1, x0/h = 3, 5 . Line 1:
σ0/P0 = 0, Line 2: σ0/P0 = 0, 05, Line 3: σ0/P0 = 0, 1, Line 4:
σ0/P0 = 0, 15 .
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CONTROL OF THE STRAIN FIELD NEAR THE CRACK IN ELASTIC LAYER 277
Fig 5. The ratio of the horizontal relative displacement to the vertical
opening versus the length of the crack: c2 = 0, 3l, P0/2µ = 1,
σ0 = 0, a/h = 1, x0/h = 3, 5. Line 1: h+/h = 0, 6, Line 2:
h+/h = 0, 7, Line 3: h+/h = 0, 8, Line 4: h+/h = 0, 9 .
under the controlling normal pressure σ0. Some striking examples are given in Fig. 5, where
the ratio of the maximums max{gx(x)}/max{|gy(x)|} over the interval x ∈ (−b, b) is shown,
which always happens when the controlling pressure σ0 = 0 (cf. Figs. (3), (4)). Many other
calculated examples show the same qualitative behavior like in Fig. 5, namely: the tangential
function |gx(x)| is always greater by many times than the vertical opening gy(x), regardless the
values of other parameters, including the controlling stress σ0.
1. Weertman J. Dislocation based fracture mechanics. — New York: World Sci., 1996.
2. Nightingale R. W. et al. Comparative strengths and structural properties of the upper and lower cervical spine
in flexion and extension // J. Biomech. — 2002. — 35. — P. 725 – 732.
3. Sneddon I. N. Mixed boundary value problems in potential theory. — Amsterdam: North-Holland, 1966.
4. Iovane G., Lifanov I. K., and Sumbatyan M. A. On direct numerical treatment of hypersingular integral
equations arising in mechanics and acoustics // Acta Mech. — 2003. — 162. — P. 99 – 110.
5. Gel’fand I. M., Shilov G. E. Generalized functions. — New York; London: Academic, 1964. — Vol. 1.
Received 28.02.2005
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