Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics
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An Asymptotic Method of Boundary-Value
Problems Solution of Elasticity Theory for Thin
Bodies
Lenser A. Aghalovyan
Institute of Mechanics of National Academy of Sciences of Armenia, Armenia
aghal@mechins.sci.am
Abstract. By an asymptotic method the solution of boundary value problems of
elasticity theory for isotropic, anisotropic, layered beams, plates and shells is built.
The first, second and the mixed boundary problems for one-layered and multy-
layered beams, plates and shells are solved. The asymptotic method permits us to
solve effectively dynamic problems for thin bodies. Free and forced vibrations are
considered. General asymptotic solutions are obtained. The conditions of reso-
nance rise are established.
1 Introduction
Thin bodies, met in various spheres of technics are characteristic so that one of
their dimensions sharply differs from the two others. For beams and bars the length
is much more than the dimensions of the cross-section, and for plates and shells the
thickness is much less than their tangential dimensions. Therefore, passing to di-
mensionless coordinates in the equations of elasticity theory and using dimen-
sionless displacements, these equations will contain small geometrical parameter
ε
. It has become clear, that the equations system of elasticity theory for thin bod-
ies is singularly perturbed by small parameter [1]. The solution of such systems is
combined from the solution of the inner problem (basic solution) and the solution
for the boundary layer [1-4]:
int
b
II I=+
(1)
In the paper by an asymptotic method the solution of the inner problem for iso-
tropic, anisotropic, layered beams and bars, for plates and shells is built.
The solution for the boundary layer of the strip-rectangle, plates and shells is
built. By the asymptotic method the second and the mixed boundary problems for
thin bodies are solved [1,5,6]. The method permits us to solve effectively dynamic
problems for thin bodies [6,7].
10 L.A. Aghalovyan
2 The Asymptotic Solution of the First Boundary Value
Problem of Elasticity Theory for the Orthotropic
Thermoelastic Strip: The Connection with the Classical
Theories of Beams and Bars
Consider a plane, correspondingly simple problem, on the example of which the
clear and sequential presentation of the essence of the asymptotic method and de-
termination of possibilities of its application for solving more complicated prob-
lems is possible.
It is required to find the solution of the plane problem of thermoelasticity for
orthotropic strip-rectangle
},,0:),{( hhyhxyxD >>≤≤−≤≤= AA un-
der the boundary conditions of the first boundary-value problem of elasticity the-
ory at
hy ±=
)(),(),(),( xYhxxXhx
yyxy
±±
±=±±=±
σσ
(2)
and under the conditions at
A,0=x (conditions of fastening), which are for now
considered to be arbitrary.
In the equations of thermoelasticity by Duhamel-Neuman model, passing
to dimensionless coordinates
A/ξ x= , hy /=
ζ
and displacements
AA v/,/ == VuU , we get a singularly perturbed by small parametre
A/ε h= system
0),(
1
=+
∂
∂
+
∂
∂
−
ζξ
ζ
σ
ε
ξ
σ
x
xy
xx
FA
0),(
1
=+
∂
∂
+
∂
∂
−
ζξ
ζ
σ
ε
ξ
σ
y
yyxy
FA
),(
),(
),(
1266
1
222212
1
111211
ζξθασ
ξζ
ε
ζξθασσ
ζ
ε
ζξθασσ
ξ
+=
∂
∂
+
∂
∂
++=
∂
∂
++=
∂
∂
−
−
xy
yyxx
yyxx
a
VU
aa
V
aa
U
(3)
An Asymptotic Method of Boundary-Value Problems Solution of Elasticity Theory 11
where
ij
σ
are the stresses tensor components,
ij
a are the elasticity constants,
),(),(
0
yxTyxT −=
θ
is the change of the temperature field,
ij
α
are the
coefficients of the temperature extension,
)(),( xYxX
±±
are the surface loading,
yx
FF , are the volume forces, for example, the weight ( gFF
yx
ρ
−== ,0 ,
ρ
is
density) or the reduced seismic loading by Mononobe model
(
xysx
FFkF 75,0, ≈=
βγ
,
s
k is the coefficient of seismicness,
β
is the coeffi-
cient of dynamicness,
γ
is the specific weight). The solution of system (3) has the
form of (1),
int
I is sought in the form of
int ( )
,0,
Q
qs
s
IQsN
ε
+
==
(4)
The notation means
Ns ,0= summing by the umbral index from 0=s up to
the number of the approximations
Ns = , Q is any of the sought stresses and
displacements, the values
Q
q are integers, characterizing the intensity (the as-
ymptotic order) of the given value. They should be established so, that after the
substitution (4) into the equations (3) the noncontradictory system for the succes-
sive determination of the coefficients
)(s
Q could be obtained. The obtained
solution should satisfy the conditions on the facial surfaces of the thin body. The
values
Q
q sensitively reacts on the type of the boundary conditions on the facial
surfaces. These values are determined by the only way. In case of the first bound-
ary plane problem (2) for strip (isotropic and anisotropic).
v3,2
0,1,2
forquforq
forqforqforq
yyxyxx
−=−=
=−=−=
σ
σ
σ
(5)
Substituting (4), (5) into system (3) and equalizing in each equation the coeffi-
cients at the same degrees of
ε
, we obtain a recurrent system for determining the
coefficients
)(s
Q . With this the contribution of the volume forces and the tem-
perature field will be the contribution order of the surface forces, if
0,0,,,
,,
)()()(2)0()0(2)0(
)(2)(1)(2
≠======
===
+−+−+−
sFFFFFF
FFFF
ss
y
s
xyyxx
sss
y
s
y
s
x
s
x
θθεθεε
θεθεε
AA
AA
(6)
12 L.A. Aghalovyan
Having solved the recurrent system, we get
),(
1v1
),()(
v
),(v)(v
)(
*
)(
11
2
)(2
11
)(
)(
*
)(
)(
)(
)(
*
)()(
ζξσ
ξ
ζ
ξ
σ
ζξξζ
ξ
ζξξ
s
xx
ss
s
xx
ss
s
s
sss
d
du
a
d
d
a
uu
d
d
U
V
++−=
++−=
+=
),()(
1
2
v1
)(
*
)(
0
2
)(2
11
2
3
)(3
11
)(
ζξσξσζ
ξ
ζ
ξ
σ
s
xy
s
xy
ss
s
xy
d
ud
a
d
d
a
++−=
),()(
)(
2
1
6
v1
)(
*
)(
0
)(
0
2
3
)(3
11
3
4
)(4
11
)(
ζξσξσζ
ξ
ξσ
ζ
ξ
ζ
ξ
σ
s
yy
s
yy
s
xy
ss
s
yy
d
d
d
ud
a
d
d
a
++−+−=
(7)
where
0,0,0
][,][
][
1
]
v
[
][v
)(
*
)(
0
)(
*
)()(
*
0
)(
*
)()(
*
)(
11
)2(
12
)(
*
11
)(
*
0
)(
*
)1(
12
)2(
66
)(
*
0
)2(
22
)4(
22
)2(
12
)(
*
<≡≡
∂
∂
+−=
∂
∂
+−=
−+
∂
∂
=
∂
∂
−+=
++=
∫∫
∫
∫
−
−−
−−−
mwhenQQ
dFdF
a
u
a
dau
daa
mm
s
xy
s
y
s
yy
s
xx
s
x
s
xy
ss
yy
s
s
xx
s
ss
xy
s
ss
yy
s
xx
s
ζ
ξ
σ
σζ
ξ
σ
σ
θασ
ξ
σ
ζ
ξ
θασ
ζθασσ
ζζ
ζ
ζ
(8)
In solution (4), (7) the functions
() () () ()
00
(),v (), ,
ss so
xy yy
u
ξξ
σσ
are not
known yet. They should be determined from the boundary conditions.
Having satisfied boundary conditions (2),
)(
0
)(
0
,
o
yy
s
xy
σσ
are expressed by
)()(
v,
ss
u which are determined from the equations
)1,()1,(][
2
)(
*
)(
*
)()()(
)(
2
)(2
11
−−++−=
=
−+
ξσξσ
ξ
s
xy
s
xy
sss
x
s
x
s
XXq
q
ud
a
(9)
An Asymptotic Method of Boundary-Value Problems Solution of Elasticity Theory 13
)]1,()1,([
)1,()1,(
v
3
2
)(
*
)(
*
)()(
)(
*
)(
*
)()()(
)(
4
)(4
11
−−−−+
+−+−+=
=
−+
−+
ξσξσ
ξ
ξσξσ
ξ
s
xy
s
xy
ss
s
yy
s
yy
sss
s
s
XX
d
d
YYq
q
d
a
(10)
0,0,,
)()()0()0(
≠====
±±±±±±
sYXYYXX
ss
ε
therefore
)(
6
)(
5
2
)(
4
3
)(
3
0
)(
000
)(
11
0
)(
2
0
)(
1
)()(
11
2!3
v
3
2
2
ssssss
sss
x
s
CCCCdqddd
a
CCdqdu
a
++++=
++=
∫∫∫∫
∫∫
ξ
ξξ
ξξξξ
ξξξ
ξξξξ
ξξ
(11)
According (7) the stresses values depend on the constants
)(
4
)(
3
)(
1
,,
sss
CCC , and
the constants
)(
6
)(
5
)(
2
,,
sss
CCC will characterize the rigid displacement, which can
be eliminated, fastening one point, for example
)0,0(O :
0,0)0,0(,0)0,0(
0
0
)()(
)()(
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
==
=
=
ζ
ξ
ξζ
ss
ss
VU
VU
If on the edges
A,0=x the stresses or displacements values are given, for
example
)(),0(v),(),0( b)
)(),0(),(),0()a
21
21
ζψξζψζ
ζ
ϕ
ξ
σ
ζ
ϕ
ζ
σ
==
==
u
xyxx
(12)
then it is obvious that it is impossible to satisfy conditions (12) by three constants
)(s
i
C in each point of the end-wall 0=x or A=x , which once again confirms
that the initial boundary-value problem is singularly perturbed. For satisfying con-
ditions (12) it is necessary to have principally a new solution too, the solution of a
boundary layer type is like that.
In order to build the solution of the boundary layer type near the end-wall
0=x , in equations (3) without accounting the volume forces and the temperature
field (they are taken into account in the solution of the inner problem) we intro-
duce a new substitution of the variable
ε
ξ
γ
/= , we ascribe index “b” to all the
unknown and the solution of the new system will be sought in the form of the
functions of the boundary layer type:
14 L.A. Aghalovyan
)exp())(v),((),(
,0),exp()(),(
)()(
)(1
λγζζε
λγζσεζγσ
−=
=−=
+−
s
b
s
b
s
bb
s
ijb
s
ijb
uVU
Ns
(13)
where
λ
is for now an unknown number which characterizes the velocity of the
values diminution when removing from the boundary. The sought values may be
expressed through
)(s
yyb
σ
by formulae
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
+−=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+−=
==
ζ
σ
λζ
σ
λ
σ
λζ
σ
λ
ζ
σ
λ
σ
ζ
σ
λ
σ
d
d
aa
d
d
aa
d
d
a
u
d
d
d
d
s
yyb
s
yyb
s
b
s
yyb
s
yyb
s
b
s
yyb
s
xyb
s
yyb
s
xxb
)(
2
6612
3
)(3
4
11
)()(
12
2
)(2
3
11
)(
)(
)(
2
)(2
2
)(
v,
1
,
1
(14)
where
)(s
yyb
σ
is determined from the equation
0)2(
)(
22
4
2
)(2
1266
2
4
)(4
11
=+++
s
yyb
s
yyb
s
yyb
a
d
d
aa
d
d
a
σλ
ζ
σ
λ
ζ
σ
(15)
The solution of the boundary layer should satisfy the boundary conditions
0)1(,0)1(
)()(
=±==±=
ζσζσ
s
xyb
s
yyb
(16)
From the correlations (14) and conditions (16) a very important property follows –
the stresses
xybxxb
σσ
, in the arbitrary cross-section
k
γγ
= are self-balanced
kxybxxbxxb
ddd
γγζζγσζζγζσζζγσ
=∀≠==
∫∫∫
+
−
+
−
+
−
,0),(,0),(,0),(
1
1
1
1
1
1
(17)
the displacements do not have this character, i.e.
kbbb
ddudu
γγζζγζζγζζζγ
=∀≠=≠
∫∫∫
+
−
+
−
+
−
,0),(v,0),(,0),(
1
1
1
1
1
1
(18)
The solution of equation (15) depends on the sign of the discriminant
2211
2
1266
4)2( aaaa −+=Δ . Determining
)(s
yyb
σ
and satisfying conditions (16),
we shall have
)(
)()(
ζσ
n
s
n
s
yyb
FA=
(19)
In the skew-symmetric problem (bend)
a)
ζβλβλζζβλβλζ
nnnnn
F cossinsincos)(,0 −==Δ
(20)
An Asymptotic Method of Boundary-Value Problems Solution of Elasticity Theory 15
n
λ
is the root of the equation
4
2
1
,022sin
E
E
nn
==−
ββλβλ
b)
ζλβλβζλβλβζ
nnnnn
F
2112
sinsinsinsin)(,0 −=>Δ
(21)
21
12
21
,)(,0sinsin
ββ
β
β
ωλββωω
+
−
=+==−
nnnn
zzz
c)
ζαλζβλαλβλζαλζβλαλβλζ
nnnnnnnnn
F shcoschsinchsinshcos)(,0 −=<Δ
(22)
10,,2,0shsin <<===−
ω
β
α
ωβλωω
nnnn
zzz
Parameters
βαββ
,,,
21
are expressed through the constants of the elasticity by
the formulae, brought in [1]. In the analogous way the solution in the symmetric
(tension-compression) problem is written [1]. It is considered, that in (19) summa-
tion in correspondence with all the roots
n
λ
with 0Re >
n
λ
of the transcenden-
tal equations takes place.
Transcendental equations (20)-(22) have denumerable set of the complex-
conjugate roots. Considering this, we write any of the stresses and displacements
b
Q in the form of
)exp()(),(
~
),,(
~
)()(
γλζγζγζ
nbnbnbn
s
n
s
bn
QQQAQ −==
(23)
then representing
nnn
s
n
s
n
s
n
iyxiAAA +=−=
λ
),(
2
1
)(
2
)(
1
)(
we shall have
knQAQAQ
bn
s
nbn
s
n
s
bn
,1,
~
Im
~
Re
)(
2
)(
1
)(
=+=
(24)
Virtue of homogeneity of the boundary layer boundary-value problem the
solution will be determined with the exactness of the arbitrary constant.
Solution (24) for every
s is exact. The solution corresponding to the end-wall
A=x can be obtained from solution (13) by the formal change
γ
into
hx /)(
1
1
−=−= A
γ
ε
γ
.
The boundary layer values when removing from the end-wall
0=x diminish
as )Reexp(
γλ
n
− . Usually when satisfying conditions (12) is neglected by the
influence of the boundary layer at
A=x , which is equivalent to the condition
16 L.A. Aghalovyan
1Reexp1
1
≈
⎟
⎠
⎞
⎜
⎝
⎛
−+
h
A
λ
which is always fulfilled for beams from real materials and real lengths
(
h10≥A ). For example, for the beams made of glassplastic STET
n
λ
is deter-
mined from equation (21), we have 398.3Re,297.2Re
21
≈≈
λλ
,
622.5Re
3
≈
λ
i.e. this condition is practically always fulfilled. As the bound-
ary-value problem of the boundary layer is homogeneous, its solution is deter-
mined with the exactness of the constant multiplier, which is often represented as
χ
ε
, the value
χ
depends on the type of the conditions at A,0=x .
Such approach permits us to produce the conjugation of the inner problem and
the boundary layer solutions.
3 Conjugation of the Inner Problem and the Boundary Layer
Solutions: Proof of Validity of Saint-Venant Principle
Consider the question of satisfaction of conditions (12). According to (1), (4), (5),
(13), (23)
),(
~
),(
),(
~
),(
)(1)(1
)(1)(2
ζγσεζξσεσ
ζγσεζξσεσ
χ
χ
s
xyb
ss
xy
s
xy
s
xxb
ss
xx
s
xx
+−+−
+−+−
+=
+=
(25)
After the substitution (25) into (12) we obtain noncontradictory conditions only at
1−=
χ
:
0,0),()(),0(
),()(),0(
)()2(
2
)()1(
)0()2(
1
)()(
≠==+
==+
−−
−
m
m
i
ss
xyb
s
xy
ii
ss
xxb
s
xx
ϕζϕζσζσ
ϕϕζϕζσζσ
(26)
Using the self-balance property of the boundary layer stresses, from (17), (26) the
conditions
ζϕζζσ
ζζϕζζζσζϕζζσ
dd
dddd
ss
xy
ss
xx
ss
xx
∫∫
∫∫∫∫
+
−
−
+
−
+
−
−
+
−
+
−
−
+
−
=
==
1
1
)1(
2
1
1
)(
1
1
)2(
1
1
1
)(
1
1
)2(
1
1
1
)(
),0(
),0(,),0(
(27)
will follow.
From three conditions (27) three constants
)(
4
)(
3
)(
1
,,
sss
CCC are determined in
the solution of the inner problem, therefore the solution of the inner problem itself.
An Asymptotic Method of Boundary-Value Problems Solution of Elasticity Theory 17
Returning to conditions (26), for determining the constants
)(
2
)(
1
,
s
n
s
n
AA in the solu-
tion of the boundary layer we get the conditions:
),0()(
),0()(
)1()2(
2
)(
)()2(
1
)(
ζσζϕσ
ζσζϕσ
−−
−
−=
−=
s
xy
ss
xyb
s
xx
ss
xxb
(28)
or
),0()(),0(
~
Im),0(
~
Re
),0()(),0(
~
Im),0(
~
Re
)1()2(
2
)(
2
)(
1
)()2(
1
)(
2
)(
1
ζσζϕζσζσ
ζσζϕζσζσ
−−
−
−=+
−=+
s
xy
s
xybn
s
nxybn
s
n
s
xx
s
xxbn
s
nxxbn
s
n
AA
AA
(29)
kn ,1= , k is a number of selected
n
λ
.
From conditions (29)
)(
2
)(
1
,
s
n
s
n
AA may be determined by Fourier, collocations,
least squares methods. The right parts of conditions (29) are self-balanced, there-
fore, the boundary layer in the first boundary-value problem of elasticity theory
takes the self-balanced part of the end-wall loading on itself, it did not influence
on the solution of the inner problem. This purely mathematical result proves the
validity of Saint-Venant principle in the first boundary-value problem of elasticity
theory.
In the case of conditions (12,b) we have
A
A
/),0(v),0(
/),0(),0(
2
)()(3
1
)()(2
ψζεζε
ψζεζε
χ
χ
=+
=+
++−
++−
s
b
sss
s
b
sss
V
uU
(30)
Conditions (30) will be noncontradictory at
3−=
χ
, therefore
0,0,),0(v),0(
/,),0(),0(
)()3(
2
)()(
)0()3(
1
)()1(
≠==+
==+
−
−−
mV
uU
s
i
ss
b
s
ii
ss
b
s
ψψζζ
ψψψζζ
A
(31)
According to (18) the displacements of the boundary layer are not self-balanced,
that is why we can’t use the previous method as Saint-Venant principle for the
displacements is not valid.
For the cojugation of the solutions it is possible to use the method of least
squares, i.e. minimization of the error integral
I
ζψζζψζζ
dVuUI
ss
b
sss
b
s
∫
+
−
−−−
−++−+=
1
1
2)3(
2
)()(2)3(
1
)()1(
])),0v),0(()),0(),0([(
18 L.A. Aghalovyan
From the system of algebraic equations
kn
A
J
A
J
i
C
J
s
n
s
n
s
i
,...,1,0,0,0
4,3,10
)(
2
)(
1
)(
==
∂
∂
=
∂
∂
==
∂
∂
(32)
Considering that
)v,(),,0(
~
Im),0(
~
Re),0(
)(
2
)(
1
)(
uuAuAu
b
s
nb
s
n
s
b
ζζζ
+= the
constants
,
)(
1
s
C
)(
4
)(
3
,
ss
CC of the inner problem and
)(
2
)(
1
,
s
n
s
n
AA of the boundary
layer are determined.
The asymptotics (4), (5) and the above described method of the solution are ap-
plicable for layered beams and bars, too. The structure of solution (1) remains in
force when solving space problems of plates and shells.
For orthotropic plates
},0,0:),,{( hzhbyaxzyxD ≤≤−≤≤≤≤ after
the change of the variables
),min(,,, bahzyx ==== AAA
ζ
η
ξ
and pass-
ing to dimensionless displacements
AAA /,/v,/ wwVuU === we again
come to the singularly perturbed system. In the inner problem we have asymptotics
2−=q for 1,,, −=q
yyxyxx
σσσ
for
yzxz
σσ
, , q=0 for
zz
σ
2−=q for u,v; 3−=q for w
(33)
The solution of the inner problem is reduced to the solution of the system
)(
4
)(4
22
22
)(4
6612
4
)(4
11
)(
2
)(
22
)(
12
)(
1
)(
12
)(
11
)2(2
v,v
s
sss
ssssss
q
w
B
w
BB
w
B
pupu
=
∂
∂
+
∂∂
∂
++
∂
∂
=+=+
ηηξξ
AAAA
(34)
where
2
1222116666121211222211
2
661212
2
2
66
2
2
1111
,/1,/,/,/
)(
),;2,1(,
aaaaBaBaBaB
BB
BB
−=Δ=Δ−=Δ=Δ=
∂∂
∂
+=
∂
∂
+
∂
∂
=
ηξ
ηξ
ηξ
A
A
(35)
At
0=s the first two equations (34), written in dimensional coordinates coincide
with the equations of the generalized plane problem, and the last equation coin-
cides with the equation of the bend according to Kirchhoff-Love classical theory