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Merida, Yucatan, Mexico
Higher order theories of Multiscale and
Multifield Models of Beams, Rods, Plates
and Shells Using COMSOL Multiphysics
Simulation Software
By Professor Volodymyr Zozulya
COMSOL Conference 2019, Boston, US
02-04 October, 2018
Centro de Investigación Científica de Yucatán, A.C.
Main Publications
Thermoelastic contact of plates and shells at macro scale.
Kantor B.Ya., Zozulya V.V. Connected problem on contact plate with rigid body though the heat-conducting layer, Docl. Akad. Nauk Ukr.SSR,
1988, 4, pp. 31-33. (in Russian)
Zozulya V.V. Contact cylindrical shell with a rigid body though the heat-conducting layer, Docl. Akad. Nauk Ukr.SSR, 1989, 10, pp.48-51. (in
Russian)
Zozulya V.V. The combines problem of thermoelastic contact between two plates though a heat conducting layer, Journal of Applied Mathematics
and Mechanics, 1989, 53(5), pp.791-797.*
Zozulya V.V. Bending of a plate in temperature field under restrictions, Izvestiya vuzov. Engineering, 1990, 1, pp. 24-27. (in Russian)
Zozulya V.V. Contact cylindrical shell with a rigid body though the heat-conducting layer in transitional temperature field, Mechanics of Solids,
1991, 2, pp.160-165. *
Zozulya V.V., Borodenko Yu.N. Thermoplastic contact of rigidly fixed shell with a rigid body though the heat-conducting layer, Docl. Akad. Nauk
Ukr.SSR, 1991, 7, pp. 47-53. (in Russian)
Zozulya V.V., Borodenko Yu.N. Thermoelastic condition of cylindrical shell, which interaction with a rigid body though the heat-conducting
layer, Izvestiay vuzov. Engineering, 1990, 8, pp. 47-52. (in Russian)
Zozulya V.V., Borodenko Yu.N. Connecting problem on contact of cylindrical shells with a rigid body in temperature though the heat-conducting
layer, Docl. Akad. Nauk Ukraine, 1992, 4, pp.35-41. (in Russian)
Romanenko L.G., Zozulya V.V. Stability of non ideal thermoelastic contact of plates, Docl. Akad. Nauk Ukraine, 1999, N 5, P. 73-77. (in
Russian).
Zozulya V.V., Aguilar M. Thermo-elastic contact and heat transfer between plates through the heat-conducting layer, in: Heat transfer 2000 , Eds.
Sunden B., Brebbia C.A. 2000, Computational Mechanics Publications, Southampton, UK and Boston, USA, 123-132.
Zozulya V.V. Contact of a shell and rigid body through the heat-conducting layer temperature field. International Journal of Mathematics and
Computers in Simulation, Iss.2, Vol. 1, (2007) pp. 138-145
Zozulya V.V. Contact of the thin-walled structures and rigid body through the heatconducting layer. Theoretical and Experimental Aspects of Heat
and Mass Transfer, 2008, pp. 145-150
Theory of high order shells, plates and rods and application in nuclear engineering
Zozulya V. V. A high order theory for linear thermoelastic shells: comparison with classical theories, Journal of Engineering. 2013, Article ID
590480, 19 pages*
Zozulya V.V. A higher order theory for shells, plates and rods. International Journal of Mechanical Sciences, 2015, 103, 40-54. *
Zozulya V.V. Mathematical Modeling of Pencil-Thin Nuclear Fuel Rods. In: Structural Mechanics in Reactor Technology (Ed. A.Gupta), SMIRT,
Toronto, 2007, pp. C04-C12. *
Main Publications
Laminate composites
Zozulya V.V. and Herrera-Franco P.J. New model of laminated composites with considering unilateral contact and friction between laminas.
Proceedings of the 14th ASCE Engineering Mechanics Division Conference, The University of Texas at Austin, Austin, Texas, USA, May
21-24, 2000. CD-ROM Proceedings (J.L. Tassoulas, ed.)
Zozulya V.V. Nonperfect contact of laminated shells with considering debonding between laminas in temperature field. Theoretical and Applied
mechanics, No. 42 (2006), 92-97.
Zozulya V.V. Laminated shells with debonding between laminas in temperature field. International Applied mechanics, Vol. 42, No. 7, 2006,
135-141. *
Functionally graded beams and shells
Zozulya, V. V. A high order Theory for Functionally Graded Shell, World Academy of Science, Engineering and Technology, Vol. 59, 2011. pp.
779-784.
Zozulya V.V. New high order theory for functionally graded shells, Theoretical and Applied Mechanics, 2012, 4(50), pp. 175-183.
Zozulya V. V. A high-order theory for functionally graded axially symmetric cylindrical shells, Archive of Applied Mechanics, 2013, 83(3), 331–
343. *
Zozulya V. V., Zhang Ch. A high order theory for functionally graded axisymmetric cylindrical shells, International Journal of Mechanical
Sciences, 2012, 60(1), 12-22. *
Zozulya V.V. A higher order theory for functionally graded beams based on Legendre’s polynomial expansion. Mechanics of Advanced Materials
and Structures, 2016, 24(9), 745-760.*
Application to the MEMS/NEMS analysis and simulations at micro and nano scale
Zozulya V.V., Saez A. High-order theory for arched structures and its application for the study of the electrostatically actuated MEMS devices,
Archive of Applied Mechanics, 2014, 84(7), 1037-1055*
Zozulya V.V., Saez A. A high order theory of a thermo elastic beams and its application to the MEMS/NEMS analysis and simulations. Archive of
Applied Mechanics, 2015, 84, 1037–1055*
Zozulya V.V. Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures, 2017, 4, 104–
118*
Zozulya V.V. Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures,
2017, 4, 119–132*
Zozulya V.V. Nonlocal theory of curved beams. 2-D, high order, Timoshenko’s and Euler-Bernoulli models, Curved and Layered Structures,
2017, 4, 221–236.
Main Publications
2018-2019
Zozulya VV. Higher order theory of micropolar plates and shells. The Journal of Applied Mathematics and Mechanics
(ZAMM). 2018, 98(6), 886–918.
Zozulya V.V. Higher order couple stress theory of plates and shells. The Journal of Applied Mathematics and Mechanics
(ZAMM). 2018, 98(10), 1834–1863 .
Czekanski A., Zozulya V.V. A higher order theory for functionally graded shells, Mechanics of Advanced Materials and
Structures, 2018, 1-19 p.
Zozulya V.V. Nonlocal Theory of a Thermoelastic Beams and Its Application to the MEMS/NEMS Analysis and Simulations.
In: Altenbach H., Öchsner A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg, 2018, 1-12,
Zozulya V.V. Higher Order Theory of Micropolar Curved Rods. In: Altenbach H., Öchsner A. (eds) Encyclopedia of
Continuum Mechanics. Springer, Berlin, Heidelberg, 2018, 1-11,
Zozulya V.V. Higher Order Theory of Functionally Graded Shells In: Altenbach H., Öchsner A. (eds) Encyclopedia of
Continuum Mechanics. Springer, Berlin, Heidelberg, 2018, 1-16.
Czekanski A., Zozulya V.V. A higher order theory of beams and its application to the MEMS/NEMS analysis and
simulations, The Canadian Society for Mechanical Engineering International Congress 2018, May 27-30, 2018,
Toronto, ON, Canada. http://csme2018.lassonde.yorku.ca
Czekanski A., Zozulya V.V. Nonlocal theory of plates, shells and beams. Higher order, Timoshenko’s and Euler-Bernoulli
models. ICMAMS 2018. First International Conference on Mechanics of Advanced Materials and Structures, Turin,
Italy, 17-20 June 2018. www.icmams.com
Zozulya V.V. Exploration of the high order theory for functionally graded beams based on Legendre's polynomial expansion,
Composites. Part B., 2019, 158, 373–383.
Zozulya V.V. Carrera unified formulation (CUF) for the micropolar beams: Analytical solutions. Mechanics of Advanced
Materials and Structures, 2019, 1-25.
Zozulya V.V. Higher Order Theory of Electro-Magneto-Elastic Plates and Shells, In: H. Altenbach et al. (eds.), Recent
Developments in the Theory of Shells, Springer Nature Switzerland AG 2019, 727-769.
Content• Thermoelastic contact problems for plates and shells
• Modeling of laminated composites with unilateral contact and
friction between laminas
• A high order theory of an elastic beams and its application to the
MEMS/NEMS analysis and simulations
• Micropolar theory of curved beams. 2-D, high order, Timoshenko’s
and Euler-Bernoulli models
• Nonlocal theory of curved beams. 2-D, high order, Timoshenko’s
and Euler-Bernoulli models
• A High Order Theory for Functionally Graded Beams, Plates and
Shells
• Application of the boundary integral equation method to the
arbitrary geometry shells
Merida, Yucatan, Mexico
THERMOELASTIC CONTACT
PROBLEMS FOR PLATES AND
SHELLS
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
Contact of shells and plates in the temperature fields
3-D statement of the problem
2 1 ; ( )
2
; ( 3 )
( )
j ij j t i ij i j j i
ij ijkl kl ij ij ij
ijkl ij kl ik jl il kj
b u u u
c
c
+ = = +
= + = +
= + +
Equations of thermoelasticity
)
2
2
,
( ) ,
, t 0,
ij j i t i ij j
ij ij k k i j
ij ijkl k l
A u b u
A V
A c T
+ = +
= + +
= =
x
0 0
0
, ; u
; ; ,
i ij j i p i i u
i i t i i
p n V V
u u u v V t t
= = =
= = =
x x
x
( )1 2
0 0
1 2 1 2
1
, 0 , 0
0 ;
, , ,
n n n n n n
f n f n
e e e e
u u u h q u h q
q k q u q k q u p
p p q x V V V V n n n
= − − =
→ = = → = −
= − = = = = = −
Equations of motion
Boundary and initial conditions
One - side restrictions Equations of heat conductivity
ij i j t ij t ijc− = + 0
Boundary and initial conditions
( ) 0 0
, , ,
0 ; , ,
b b q
ij n ij b
V V
V t t
= =
+ − = = =
x q q x
x
Heat conductivity through the heat conducting layer
* * * * * *
* * *
- , ,
, ,
ij i j t
ij n ij n e
c x V t
x V t
=
= =
3-D integral equations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
−+
+−+−=
V
ijij
V
ii
V
jijjiji
dVpUppUpb
dSpUpqpZpc
dSpWpupUpppu
,,,,,,
,,,,,,,,,,,,, 0
xyyxyy
xyyxyyxyyxyyx
( )
( )
iiiijijikikjikij
iiiiikikkii
iijijikkiikjkij
UTTnUPWnUPW
UTcTUc
TUpUAUpUA
0
00nin00
000000
0
2
0
2
, Z, =F, ˆ ; ˆ
; 0
0 ;
==−=−=
−=+−=+−
=−−−=−−
yx
yx
3-D fundamental solutions
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
−+
+−+−=
V
ii
V
iiii
dVpUpbpTp
dSpFppTpqc
pUpppWpup
,,,,,,
]},,,,,,[,,,,,,{,
0
00
0
xyyxyy
xyyxyyxyyxyyx
Reciprocal Betty-Rayleigh theorem for thermoelasticity
( ) ( ) ( )
( ) ( ) ( )
' ' ' ' ' '
0
' ' ' ' ' '
0 - - - -
0 - - - -
i i i i i ii i
V V V
n n
V V
u b u b dV u p u p dS dV
dV dS dV
= +
= +
3-D integral representations for thermoelasticity
2-D statement of the problem
Expansion into Legendre’s polynomial series
( ) ( ) ( ) ( ) ( ) ( ) 3
0 -
2 1 ,
2
h
k k
ij ij k ij ij k
k h
kx x P x x P dx
h
=
+= =
2-D equations of thermoelasticity in displacements
( ) ( ) 2
11 1 12 2 13 3 1 2 13 13 2 1 0 1 2 1 1 2 1
0 0
2 11
2
kkl l k k kl l k k k k
t
l l
kL u L u L u A A A A A b A A u
h
+ −
= =
+ + + + − − − − + =
( ) ( ) k
t
kkkk
l
lklkk
l
lkl uAAbAAAh
kAAuLuLuL 2
2
21221021232321
0
323121
0
222 12
12 =+−−−−
++++ −+
=
=
( ) ( ) ( )
( ) k
t
kkkk
kkkk
l
l
l
lkl
l
lkl
uAAh
kAA
kkAAbAAh
kAAuuLuL
3
2
21
3
0
31
0
1
21
02121321333321
0
3
0
232
0
131
...2
122
12
12
=−+−+
+
+−+++−−+
+++
−−−−
−+
=
=
=
( ) ( )
( ) ( ) ( ) ( )
++++
+++
+=+++++
−−−+
+
+
++
−−−+
kkkkk
k
t
kkkkkkk
ukkuuh
kuAuA
AA
aQkkQQ
h
kQQ
h
k
A
A
A
A
AA
321
3
3
1
3212121
21
0
0
321
3
3
1
3332
2
1
21
1
2
1
21
...121
1...
121
2
121
2-D equations of heat conductivity
2-D statement of the problem
( )
=
−−
=
++ =−=
====
00
kk
i
, 1 ; ,
, ; , , , u ; ,
k
i
k
i
k
k
i
k
i
q
k
b
k
b
k
u
k
ip
k
i
k
i
pppp
tp
xx
xqqxxx
u u u vi
ki
k
i
ki
k k= = = 0 0
0 , , , x V , t = tt
k
0
( ) ( ) ( )
( ) ( ) ( ) ( ) ; 1 ; 1; 1 ; 1
11q ; ; 1
0
)(3
0
)2(3
00
)2(
0
)(3
0
)1(2
00
)1(
0
)2(
0
)1(i
0
)2(
0
)1(
=
=
=
=
=
=
=
=
=
=
−
=
+
=
−=−==−=−
−−=−===−
k
kk
k
k
k
kk
k
k
k
k
k
kk
k
k
k
kk
k
k
i
k
k
k
i
k
n
k
k
nn
k
k
n
k
QQQQ
ppuuuu
2-D contact conditions
2-D boundary conditions
2-D initial conditions
2-D integral equations
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
=
=
=
−+
−+−=
0
0
0
0
,,,,,,,,,-
,,,,,,,,,,
n
kn
i
nkn
ji
n
j
kn
i
n
n
kn
i
n
n
kn
ji
n
j
kn
ji
n
j
k
i
dpUppUpbdSpUpq
pZpc
dSpWpupUpppu
xyyxyyxyy
xyyxyyxyyx
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
=
=
=
−+−
−+−=
0 0
00
0
0
0
,,,,,,,,,
,,,,,,,,,,
n
kn
i
n
i
knnknn
n
knn
n
kn
i
n
i
kn
i
n
i
k
dpUpbpTpdSpFp
pTpqc
dSpUpppWpup
xyyxyyxyy
xyyxyyxyyx
Nonlinear integral equations of thermoelasticity
( ) ( )
( ) ( )
( ) ( )kk
kk
nnnn
Th
Qh
Th
Th
Qh
Th
QkkQQh
n
+=+−=+
−=++=−
+++−−+
=
++−+
++−+
−+
2
3 ;
2
5
2
3
2
1 ;
2
3
4
3
12
12
1
3
1
33
0
3
0
33
0
32133
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( ) 210
2
3
1
30
2
0
1
1
0
120
1
2
0
210
2
3
1
30
2
02
210
2
3
1
30
2
0
1
2
0
210
1
1
0
120
2
3
1
30
2
01
99
106310639
99
106310639
hhuuh
hhuuhT
hhuuh
hhuuhT
k
k
++−−
−++−++−−=
++−−
−++−++−−=
+
−
−
+
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )tdtdtTt
tdtdtVtqtv
kn
n
nk
ijiji
,,,,,,
,,,,,,
1
0
1
xxyyx
xxyyx
•+−=
•+−=
=
Integral equations in the mechanical contact area
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
ququq
xxyyx
xxyy
−=→==→
=−
•+−=
•+−=
=
tftf
n
i
e n
knnk
i
e
jii
kk
hhu
tddtTt
tddtVqh
nn
n0nn0
0
0
0
q , 0q
;0qu , 0q ,
,,,,,,
,,,,,
Integral equations in the area free of mechanical contact
Zozulya V.V. The combines problem of thermoelastic contact between two plates though a heat conducting layer // Journal of
Applied Mathematics and Mechanics. 1989,53(5), P. 622-627.
Zozulya V.V. Contact cylindrical shell with a rigid body though the heat-conducting layer in transitional temperature field //
Mechanics of Solids, 1991, 2, P. 160-165.
Approximate 2-D equationsVekua’s shells
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
0 1 0 1 30 1
0 1 0 1 30 1
;
;
ij ij ij i i i
ij ij ij
xx x P x P u x u x u x
h
xx x P x P x x x
h
= + = +
= + = +
Differential equations in displacements
( )
( )
00 0 01 1 0 0 0 0 2 0
0
10 0 11 1 1 1 1 1 2 1
0
-
-
ij j ij j i i t i
ij j ij j i i t i
L u L u L b u
L u L u L b u
+ + + =
+ + + =
Differential equations of heat conductivity
( ) ( )
( ) ( )
00 - 0 0 0 0
0 3 3 1 2 3 0
0
11 - 1 1 1 1
0 3 3 1 2 3 0
0
1 1-
2
3 1-
2
t t
t t
Q Q k k Qh a
Q Q k k Qh a
+
+
+ + + + = +
+ + + + = +
Temperature prescription on + and -
( ) ( ) ( ) ( )
( )( )( ) huh
huhT
hT
hhT
hQQT
hhT
hQQ
k
kkkk
−
+
++−+++−+
+−
+−+−=
−+=−−=−−=++=−
300
10
300
11
3
1
33
0
3
0
33
9
1063
2
3
2
3Q ,
2
5
2
3,
2
1Q ,
2
3
4
3
Approximate 2-D equations
Timoshenko’s shells
( )( ) ( )
( )( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
xxxxx
xxxxxx
xx
xxx
x
333
33333
33
3
33
3
;
;0 ; ;
;0 ; 2
;3
2
vuxvu
exe
h
n
h
xm
h
n
=+=
==+=
==+=
Differential equations in displacements
( )
( )
0 0 0 2
0
1 1 1 2
0
-
-
u u
ij j i i i t i
u
ij j i t
L v L L b v
L v L L m
+ + + =
+ + + =
Kirchhoff-Lore’s shells
1 1
3 3 3 30;j j u v = = =
3 0 =
( )( )
( ) ( ) ( )3
1-x v x x v x
A x
= + ( )1
2
0
0
-k k k
ij j i i t i
k
L v L b v =
+ + =
Comparative study of shells
Zozulya V. V. A high order theory for linear thermoelastic shells: comparison with classical theories, Journal of
Engineering. 2013, Article ID 590480, 19 pages
0-D problem
Temperature distribution and displacements.
00
02
0 =− F 01
12
1 =− F
04 0
0
4 =− w
Equations for
Temperature and displacements
2120
15 ,
3
hh==
,
( )22
24
4
13
rh
−=
,
,
,
( ) ( )−− −+= kk Thr
TF2
1+5.0 00
,
( ) ( ) 1
11
3-
2
3+5.0
hrT
hrTF kk
−− +−=
( )
rh 20
13
−=,
Data for calculations:Geometry:
Cm
V
Cm
V
CE
ooo10 , 20,
1105.2 , 0.25= , GPa 100 1
5 ====
− Material:
mr 01.0= m.h= 00140 mh 0007.00 =
Simple example
( )
212
12
12
12
0
02
02
02
202
12
1
0
0
4
4
4
15 ,0,0
,3
,1
4
hF
dx
dF
dx
d
hqp
Ddx
dw
dx
wd
==+−=+−
=−=−−+
( ) ( ) ( ) ( ) ( ) ( ) 1
, ; , 30 wdyyFyqypD
yxWdyyFyxGl
i
l
ii =
−−=
( ) ( ) ( ) ( ) ( ) ( ) ( ) 300 −−=ll
dyyFβypx,yW=xF,xD+Fhdyyqx,yW
Differential equations in displacements
Integral equations of Hammerstein’s type
Fredgolm’s first-kind integral equation in contact region
Fundamental solutions
( ) ( ) ( ) ( ) ( ) ( ) yxyxyxD
yxW,, i=yxyxG iii −+−−−=−−=
sincosexp8
1,;10 2/exp,
3
Contact conditions
( ) ( ) ( ) ( )
( ) ( )( )
( ) ( )h
αν, β
rh
αν , β
ν
Eh Dθ
hr-θT
hr+θTε.F
rh
ν βθ-βθεFβF θT
hr+Tθε.F
ττkk
kk
+=
−=
−=+−=
−=+=−+=
−−
−−
113
13
2,
3
2
350
4
13,,
2
150
1202
31
11
22
240
0
12
111300
( )( ) hwh
hwhTθ, Tq; w=hqhw kk
−
+
−
+−
−−==→=→
01
01
00 ,00
Simple example
Data for calculations:
Geometry: m.m ; l=.m ; h.m ; h=.r 50005001070 0 ==
Cm
V
Cm
V
CE
ooo10 , 20,
1105.2 , 0.25= , GPa 100 1
5 ====
− Material:
Simple exampleData for calculations:
Geometry:
Cm
V
Cm
V
CE
ooo10 , 20,
1105.2 , 0.25= , GPa 100 1
5 ====
− Material:
mr 01.0=
,
,
,
m.h= 00140
,
mh 0007.00 =
Contact of a cylindrical shell with
foundation
The nuclear fuel rod
Zozulya V.V. Mathematical Modeling of Pencil-Thin Nuclear Fuel Rods. In: Structural Mechanics in Reactor
Technology (Ed. A.Gupta), SMIRT, Toronto, 2007, pp. C04-C12.
Merida, Yucatan, Mexico
MODELING OF LAMINATED
COMPOSITES WITH UNILATERAL
CONTACT AND FRICTION BETWEEN
LAMINAS
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
Laminate composite shells
For the laminated shell we have
−+
====
====−== )1()(
)(
1
)(
1
)(
11
)( ,,,,],[, SSSVVVSSSShhSShh Ql
q
q
l
Q
ql
q
l
Q
qll
Q
q
q
We introduce the following notations: is the middle surface of the
layer, is its boundary, and are the up and down sides,
. is a lateral side, is a volume
occupied by the layer and is its boundary
)(qS
)(qS +
)(qS −
)(qS
],[ )()(
)()(
qql
q
l
q hhSS −= ],[ )()()()( qqqq hhV −=−+ = )()()(
)(
l
q
q SSSV
Some notations
Zozulya V.V. and Herrera-Franco P.J. New model of laminated composites with considering unilateral contact and friction between laminas. Proceedings
of the 14th ASCE Engineering Methanics Division Conference, The University of Texas at Austin, Austin, Texas, USA, May 21-24, 2000. CD-ROM
Proceedings (J.L. Tassoulas, ed.)
Zozulya V.V. Nonperfect contact of laminated shells with considering debonding between laminas in temperature field. Theoretical and Applied
mechanics, No. 42 (2006), 92-97.
Zozulya V.V. Laminated shells with debonding between laminas in temperature field. International Applied mechanics, Vol. 42, No. 7, 2006, 135-141.
3-D formulation
)()()( q
k
k
ij
q
ji
q
ji uuu −=
Relations of 3-D elasticity take place:
QqVbuuc qi
q
ij
qj
q
ij
q
ji
q
ij
q
kl
ijkl
q
ij
q ,...,1,,0,)(2
1, )(
)()(
)()()()(
)()( ==++== x
are covariant derivativesk
ij
The differential equations in displacements
QqVCAbuA q
lk
ijkl
q
ij
q
i
q
q
j
ij
q ,...,1,,,0)()( )(
)()()(
)(
)( ===+ xxx
are the Christoffel`s symbols
On the outer surfaces:
QqSppSpp ii
Q
ii
Q ,...,1,,)()(;,)()( )1()()1()()()( === −
−
+
+ xxxxxx
On the lateral side: QqSuSp u
l
q
i
q
i
p
l
i
q
i
q ,...,1,,)()(;,)()( )()(
)()( === xxxxxx
3-D boundary conditions
)()()( q
k
k
ij
q
ji
q
ji uuu −=
Contact conditions between laminas
1. Boundary conditions in areas of complete adhesion:QqSSSppuu qq
a
q
i
q
i
q
q
i
q
i ,...,1,),()(,)()( 1)1()(
)1()( === −+
−−
− xxxxx
2. Boundary conditions in areas of weak adhesion:QqSSuu qq
a
qa
n
q
n ,...,1,,,)()( 1
)()1()( === −+
−
−xpxx
Conditions in the tip of the sliding mode (II). cracks
−=C
ij
ij dsunn ])([ u
][)( )1(
)1(
)(
)(21 −
−+= q
ij
ij
q
q
ij
ij
q u)1(
)1(
)(
)(
−
− += q
ij
ij
q
q
ij
ij
qij
ij ununun where
The criterion for crack initiation in this case takes the formc=
3. Boundary conditions in areas of debonding:
c
q
q
t
q
t
q
n
t
q
n
q
c
q
q
n
q
n
q
n
q
n
qkqk
Qqququ
−=→==→
==
xququq
x
,,0
,...,1,,0,0,0
)()()()()()()(
)()()()(
Reduction to 2-D problem
Expansion into Legendre's polynomial series:
−
=
+==
)(
)(
33)(
)(
)(
0
)()( )(),(2
12)(,)()()(
q
q
h
h
n
q
iq
nq
i
n
n
nq
i
q
i dxPxuh
nuPuu xxxx
−
=
+==
)(
)(
33
)()()(
0
)()( )(),(2
12)(,)()()(
q
q
h
h
n
ij
ij
nq
n
n
ij
nq
ij
q dxPxh
nP xxxx
−
=
+==
)(
)(
33)(
)(
)(
0
)()( )(),(2
12)(,)()()(
q
q
h
h
n
q
ijq
nq
ij
n
n
nq
ij
q
ij dxPxh
nP xxxx
−
=
+==
)(
)(
33
)()()(
0
)()( )(),(2
12)(,)()()(
q
q
h
h
n
i
i
nq
n
n
i
nq
i
q dxPxph
npPpp xxxx
−
=
+==
)(
)(
33
)()()(
0
)()( )(),(2
12)(,)()()(
q
q
h
h
n
i
i
nq
n
n
i
nq
i
q dxPxbh
nbPbb xxxx
2-D equations for coefficients in the Legendre's polynomial series
QqmnSbuA
buuc
q
i
nq
mq
j
ij
nmq
i
nq
ij
nqj
nq
ij
nq
ji
nq
ij
nq
kl
ijkl
q
ij
nq
,...,1,,...,1,0,,,0)()(
0,)(2
1,
)()(
)(
)(
)()(
)()()()(
)()(
===+
=++==
xxx
2-D boundary conditionsOn the outer surfaces:
−
−
+
+ == )1()()1()()()( ,)())((;,)()( SppSpp i
n
i
Q
i
n
i
nQ xxxxxxx
On the lateral side:u
l
nq
i
nq
i
p
l
i
nq
i
nq SuSp == xxxxxx ,)()(;,)()( )()(
)()(
Boundary conditions in areas of complete adhesion:−+
−−
a
q
i
nq
i
nq
nq
i
nq
i SSSppuu 1)1()(
)1()( ),()(,)()( xxxxx
Boundary conditions in areas of weak adhesion:−+
−
=
a
qa
n
n
nqnq
n
nq
n SSPuu 1
0
)()1()( ,)()(,)()( xxqxx
Boundary conditions in areas of debonding:
c
q
n
n
nq
n
n
n
nq
n
n
n
nq
n
n
n
nq
n PqPuPqPu =
=
=
=
=
x,0)()(,0)(,0)(0
)(
0
)(
0
)(
0
)(
0)()()(0
)(
0
)(
0
)( =→
=
=
= n
n
nq
t
n
n
nq
n
n
n
nq PPqkP uq
=
=
=
=
−=→=0
)(
0
)(
0
)(
0
)( )()()()(n
n
nq
t
n
n
nq
t
n
n
nq
n
n
n
nq PPPqkP quq
First-approximation theory
The stress-strain state parameters have the form:
)()()()()(,)()()()()(
)()()()()(,)()()()()(
1
1)(
0
0)()(
1
1)(
0
0)()(
1
1)(
0
0)()(
11)(00)()(
PpPppPP
PuPuuPP
q
i
q
i
q
i
q
ij
q
ij
q
ij
q
i
q
i
q
i
ij
q
ij
q
ij
q
xxxxxx
xxxxxx
+=+=
+=+=
The differential equations in displacements
)(1)(
1)(
11)(
0)(
10)(
0)(
1)(
01)(
0)(
00)(
,0)()()(
0)()()(
q
i
q
q
j
ij
q
q
j
ij
q
i
q
q
j
ij
q
q
j
ij
q
SbuAuA
buAuA
=+
=++
xxxx
xxx
Timoshenko’s theory
033
)( =q 0)(
33 =qHypothesis:
Components of the stress tensor: 0)(,2
)()(,
)(2
2
)()( 33
)(
)(3
)(3
3
)()(
)( ==+= xx
xxx
x q
q
q
qh
n
h
xm
h
n
Components of the strain tensor: 0)(,)()(,)()()( )(
33
)(
3
)(
3
3)()()( ==+= xxxxxx qqqqqq exe
Components of the displacement vector: )()(,)()()( )(
3
)(
3
3)()()(
xxxxxqqqqq vuxvu =+=
0)()()(
0)()()(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
=++
=++
xxx
xxx
q
q
q
q
j
j
vq
i
q
qi
vq
q
j
ij
vq
hAvA
fAvA
The differential equations in displacements
Merida, Yucatan, Mexico
A high order theory of an elastic beams
and its application to the MEMS/NEMS
analysis and simulations
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
Application to the MEMS/NEMS analysis and simulations
Microbeam settles above the rigid foundation in the thermal field
00 01 0 0 0 0
11 12 1 1 1 1
00 01 0 0 0 0
22 12 2 2 2 2
01 11 1 1 1 1
12 11 1 1 1 1
10 11 1 1 1 1
21 22 2 2 2 2
0 0
0 0, , , ,
0 0
0 0
u
L L L u f
L L L u f
L L L u f
L L L u f
= = = = =L L E u θ f
00 0
2
11 1
2
0, ,
0
QL
QL
= = =L Q χ
0u + + =L u L θ f
( ) ( ) ( ) ( )
( ) ( )
1 1
0 0
1
0 0
( ) , ( ) ,
( ) , ( ) ( ) ( ) .
k k
ij ij k ij ij k
k k
k
i i
k
k
k
k
k
x P x
u P P
P
u x
= =
=
=
= =
==
x x
x x
x
( ) ( ) ( ) ( )
( ) ( )
1 1 1 1
1
2 2 2 2
2 2 21 2
, ,
2 1, ( ) ( ,
2 1 2 1, ( ) , ( )
2 2
2 1) ( ) ., ( )
2 2
k k
ij ij k ij i
h h
hh
j k
h h
h
i
h
k
k
k
i k
k kx x x P dx x x x P dx
h h
ku x u x x P dx
kx P dx
hh
− −
− −
+ +
+= =
=
+
=
x x
Differential equations of thermoelasticity
The Legendre polynomials series expansions Coefficients of the expansions
Zozulya V.V., Saez A., A high order theory of a thermo elastic beams and its application to the MEMS/NEMS
analysis and simulations. Archive of Applied Mechanics, 2015, 84, 1037–1055
The temperature set 0 0550 , 0C C + −= = The temperature set 0 030 , 170C C + −= =
Study displacements, stress and temperature distribution
The Legendre polynomials
coefficients
The Legendre polynomials
coefficients
Displacements, stress and temperature Displacements, stress and temperature
Application to modeling electrostatically
actuated microbeam
Zozulya V.V., Saez A.P., High Order Theory for Arched Structures and its Application for Study of the
Electrostatically Actuated MEMS Devices. Archive of Applied Mechanics, 2014, 84(7), 1037-1055.
Electrode
Beam
. Schematic diagram of an electrostatically actuated beam
0u + =L u f 00 00 0 0 0 0
11 12 11 12 1 1
00 00 0 0 0
21 22 21 22 2 2
0 0
11 12 11 12 1 1
0 0
21 22 21 22 2 2
, ,
n n
n nn
u
n n nn nn n n
n n nn nn n n
L L L L u f
L L L L u f
L L L L u f
L L L L u f
= = =L u f
Legendre’s polynomials series expansion
( ) ( ) ( ) ( )
( ) ( )
1 1
0 0
1
0 0
( ) , ( ) ,
( ) ( ) ( ) .( ) ,
k k
ij ij k ij ij k
k k
k
i
k
ki k
k k
p P
x P x P
u u x P p
= =
=
=
= =
==
x x
x x
x
( ) ( ) ( ) ( )
( ) ( )
1 1 2 1 1
1 1
2 2 2
2 2 1 1 2 2
, ,
2 1, ( ) ( , ) ( )
2 1 2 1, ( ) , ( )
2 2
2 1, ( )
2.
2
k k
ij ij k ij ij k
h h
k
i i k
h
h h
hh
k
k
h
k kx x x P dx x x x P dx
h h
ku x u x x P dx
h
kp x p x x P dx
h
− −
−−
+ += =
+ +==
Differential equations of elasticity in displasements
2 2
0
2 3 3 2
2 0 2
3,
ˆ ˆ(1 ) 4 (1 )e
V lF
u bh h u
= =
− −
Electrostatic force
Second order model of micro beam00 00 01 01 0 0
11 12 11 12 1 1
00 00 01 01 0 0
21 22 21 22 2 2
10 10 11 11 12 1 1
11 12 11 12 12 1 1
10 10 11 11 12 1 1
21 22 21 22 22 2 2
21 21 22 22 2 2
11 12 11 12 1 1
21 12 22 22 2 2
21 21 21 22 2 2
0 0
0 0
0, ,
0
0 0
0 0
u
L L L L u f
L L L L u f
L L L L L u f
L L L L L u f
L L L L u f
L L L L u f
= = = L u f
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
0 1 2
1 0 1 1 1 2
0 1 2
1 0 1 1 1 2
0 1 2
1 0 1 1 1 2
,
,
.
ij ij ij ij
ij ij ij ij
i i i i
x P x P x P
x P x P x P
u u x P u x P u x P
= + +
= + +
= + +
x
x
x
Euler-Bernoulli model of beam and pull-in instability 1ˆ ˆ( )n nw T w+ = 1 2
0
ˆ( ) ( , )ˆ(1 ( ))
ln
nT w G x d
w
=
+
2 21 1 1
13
2 21 1
3
1
1
( ) (3 2 ) for
6
( ) (3 2 ) for
( , )
6
x l l lx xx
l
l x l lx
xG
xl
=
− − −
− − −
Fixed-fixed beam Simply supported beam2 2
1 11
2 21 1 1
1
1
( )( 2 ) for
6
( )( 2 ) for
6
( , )
x l x lx
x l x x lx
G x
− + −
− + −
=
Stress-strain parameters
Fixed-fixed micro beam Simply supported micro beam
Displacements, stress and temperature
Merida, Yucatan, Mexico
Micropolar theory of curved beams. 2-d,
high order, Timoshenko’s and Euler-
Bernoulli models
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
3-D formulation of the problem for shells Curvilinear orthogonal coordinates
1, ,ki
i ij k i
i i j i i i
HH x x x x x
= = = =
eR R R Re e
1 1
2
j k i jk i i kij ik jk ik ij
i j i k i j k
H H H HH H H
H x H H x x x
= − + + −
1 1 1,
2
1
j kiij ij k ijk k
j j i i
j k
ij ij k
i i
uuu
H x H x
H x
= + + +
= −
Kinematic relations Equations of motion2
2
2
2
1,
1,
ij ij k jik iik ki i
j j k i
ij ij k jik iik ki ijk jk i
j j k i
ub
H x H H t
m jH x H H t
+ + + =
+ + + + =
Constitutive relations
, .2 2 2 2 2 2
,ij ij ji ij ii jj ij ij ji ij ij ij
ij i
j
j
ii j
E EE
+ − + −= + + + +
= =
+
( ) ( ) , ( ) ( )ij rr ij ij ji ij rr ij ij ji = + + + − = + + + −
( ) ( ) 2
( ) ( ) 4 2
i j j j j i ijk j k i i
i j j j j i i ijk i j i i
u u b u
u m
+ − + + + + =
+ − + + − + + =
Equations of motion in displacements
3-D formulation in special coordinates
31 1 2 2 22 1 3
1 1 1 2 2 1 1 1 2 1 1 1
31 1 1 2 21 2 3
2 2 1 2 2 2 2 2 1 1 2 2
31 21 1 2 2
3 3 3
, , ( )
1 1 1 1
1 1 1 1
i j
uu A u u Au k u
A x A A x A x A A x A x
uu u A u Au k u
A x A A x A x A A x A x
uu uk u k u
x x x
= − = =
+ + −
= − + +
− −
ε u ω χ ω ω ω e e
u
Kinematic relations Equations of motion
32 1 1 21 2 3
1 2 1 2 3
32 1 1 23
3 1 2
3
1 2 3
( ) ( ) ( ) 2
( )
( ) ( )1 1( ) ( )
( ) ( )1( )
4 2
uA u A uk k u
A x A A x x x
uA u A u
x A A x x x
j
+ + + − + +
= + + + +
=
+ − + +
=
=
+ + +
u u ω b
ω
u
u
ω u m
u 1 2 3
2
2 11 2 2
1 2 1 1 1 2 2 2 3 3
( )
1( )
k k u
A Ak k
A A x A x x A x x x
+
= + + + +
u u u uu
Equations of motion in displacements
3 3
33
3
(1 ) for 1,2 and 1
(1 ), , 0i
H A k x H
H A H Hk x k A
x x x x
= + = =
= + = =
2 1 1 2 31 2 3 1 2
1 2 3 1
2 31 1 32 1 2
31 2
( ) 0, ( ( ) ( ) ) 0,
0, 0,
( ) ( )
( ) ( ) (
ij i j
V V
ij i j
dV dV
AA A AA A A A k
x x x x x
A A A A
x x
+ = + + + + =
+ = + + = =
= + + + + −
= + +
σ b r σ P i i μ m
σ b μ σ m σ e e
σ
σ 3311 1 2 1 22 1 2 2
3
)A A k A A k
x
− −
3
1 2 3 22 122 11 11 1 1 22 2 2
1 1 2 2
1 1 2 21 221 31 1 1 12 32 2 2
2 1 1 1
1 1, , , ,
1 1, ,
( ) ( ) ( )
,
A Ak A k A
A x A x
A Ak A k A
A x
x
A x
= − = − = − = −
= = =
= +
=
R x r x n x
2-D formulationExpansion into Legendre’s polynomial series
Derivatives with respect to
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )( )
3 3
3
3
1
3 3 3
3
3 2
3 3 3
3 1
3
3 3 3 3 3 3
3
,2 1 2 11
2
,2 1 2 11
2
,
,
2 1 2 1,
k
i
k
i
k
hki
k i i
h
hki
k i i
h
i i i
k k k k k
i i i
xk kP dx
h x h
x
k k
k kP dx
h
h h
x h
+ −
−
+
−
−
− − −
−
+ + = − − −
+ +
+ +
= − − −
= + + = + +
xx x x
xx x x
x x x x x x
3x
( ) ( ) , ( ) ( )k k k k k k k k
ij rr ij ij ij ij rr ij ij ij = + + + − = + + + −
2-D constitutive relations
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3
0 0
0 0
0 0
3
3 3
3 3
( ), ( ),, ,
, ,( ), ( ),
(, , ( ,,) )
i i i i
ij ij ij ij
ij ij
k k
k k
k k
k k
k k
k k
k k
k k
k
j
k
ij i
u u P P
P P
P P
x x
x x
x x
= =
= =
= =
= =
= =
= =
x x x x
x x x x
x x x x
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 3 3 3
3 3 3 3
3 3 3 3
, , , ,
, , , ,
2 1 2 1( ) ( )
2 2
2 1 2 1( ) ( )
2 2
2 1 2 1( ) ( )
2, , ,
2,
h h
i i i i
h h
ij ij ij ij
h h
ij ij ij ij
k k
k k
h h
k k
k k
h h
k k
k k
h h
k ku u P dx P dx
h h
k kP dx P dx
h h
k k
x x
x x
x xP dx P dxh h
− −
− −
− −
+ += =
+ += =
+ += =
x x x x
x x x x
x x x x
( ) ( )( ) ( ) ( ) ( ) ( )( )
( ) ( )( ) ( ) ( ) ( ) ( )( )
3
3
3 3
3
3
1
3
1
3
3
32 1, ,
2 1
,2 1
2
,2 1
2, ,
k k k
k h
i i
k i i
h
k h
i i
k i i
h
k
i i
k k k k
i i
ku u u u
u u xkP dx
x h x
xkP dx
x h x
h
k
h
+ +
+ +
−
−
+= =
+= =
+= + +
+= + +
x xx x x x
x xx x x x
2-D formulation2-D equations of motion
u + =L u b u
2-D kinematic relations
2-D equations of motion in displacements
2-D constitutive relations
( ) ( )
( ) ( )22 2
2 1 1 2
2 3 1 2 3
1 2
2 31 1 32
11 1 2 1 1 2 33
1 23
0, 0k k k k k
k
k k
k k k k
k k
k k k
k
k k
A A AAA A k
x x x x
A AA A k A A k
x x
+ = + + =
+ + + − −
+ − − −
=
=
σ b μ σ m
σ
σ
1 32
1 1 1 2 1 1 1
31
2 2 1 2 2 2
1 2 22 1 3
1 1 1 2 2
1 1 2 21 2 3
2 2 1 2
1 1
2 1
2 221
, ,
1 1
1 1
1 1
1 1
k k kk k
k k
k k k k k k
k
kk k
k k k
kk k
uu u A
A x A A
u Au k u
A x A A x
u A
x A x
uu u A
A x A A x A xu
u k
k uA x
u k u
A
u
A x
= − =
−
= −
−
+ +
+ +
−
ε u ω χ ω
u
3
k ku
( ) ( ) , ( ) ( )ij rr ij ij ji ij rr ij ij j
k k k k k k k k
i = + + + − = + + + −
2-D formulation
u + =L u b u
2-D equations of motion in displacements
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 0
0 0
0 0
3 3
3 3
3 3
( ), ( ),
( ), ( ),
(
, ,
, ,
, ,), ( ) ,
N N
i i i i
N N
ij ij ij ij
N N
ij ij i
k k
k k
k k
k k
k k
k k
k k
k k
k k
j ij
u u P P
P P
x x
x x
x xP P
= =
= =
= =
= =
= =
= =
x x x x
x x x x
x x x x
N-order approximation equations
000 00 0 0 0111 16 11 16 1
000 00 0 0 0361 66 16 66 3 2
2
0 0
11 16 11 16 1 1
0 0
61 66 16 66 1 3
0 0 0
0 0 0
, , ,
0 0
n n
n n
u u
n n nn nn n n
n n nn nn n n
bL L L L u
m jL L L L
tL L L L u b
L L L L m
= = = =
L u b M
0
0 0 0 j
0, 0, 0, for 0 and f ork k k k k nu = = =
2-D equations of micropolar elasticity
Kinematic relations 33
3 3 3 3 3
1 1, ,
uu
H x H x
= − + = −
Equations of motion
11 12 13 11 12 13 1
21 22 23 21 22 23 2
31 32 33 3
0 0 0 0 0 0 0
0 , 0 0 , 0 , 0 0 , , 0
0 0 0 0 0 0 0 0 0 0
u
u
= = = = = =σ μ ε χ u ω
Curvilinear orthogonal coordinates1
, ,kii ij k i
i i j i i i
HH x x x x x
= = = =
eR R R Re e
1 1
2
j k i jk i i kij ik jk ik ij
i j i k i j k
H H H HH H H
H x H H x x x
= − + + −
2 23 3
3 32 2
1 1,
ub m
H x H H t H x t
+ + + = + + =
Constitutive relations3 3( ) ( ) , ( ) = + + + − = +
3 3
3 3 3 3 3
( ) ( ) ( ) 2
( ) 4 2 ( ) M j
+ + + − + + =
+ − + + =
u u e b u
u
Equations of motion in displacements
Zozulya V.V. Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and
Layered Structures, 2017, 4, 104–118.
2-D formulation in special coordinates
Kinematic relations 1 2 2
11 1 2 22 12 1 1 3
1 1 2 1 1
3 3121 1 2 3 13 23
2 1 1 2
1 1, , ,
1, ,
u u uk u k u
A x x A x
uk u
x A x x
= + = = − −
= − + = =
( ) ( )
( )
2 2
11 21 1 12 22 21 21 12 1 1 22 11 22 2
1 1 2 1 1 2
2
13 23 3
2
1 1
12 21 3
2
1 1,
1
u uk b k b
A x x t A x x
m
t
jA x x t
+ + + + = + + − + =
+ − + =
+
Equations of motion
1 2 1( ) ( ) ( )x x x= +R x r n 1 11 1 2 1 1 1 2 2 1 2
1 1
1 2 11 1 1 2 1 1 1 1
2 2 2
( , ) ( )(1 ), 1, (1 ),
1, 0, 1 1 , ,
H AH x x A x k x H k x
x x
H H Hk A k x H A k A
x x H x
= + = = +
= = + → =
2-D equations of motion2 2
31 1 1 1 2 11 1 1 2 1 12 2 2 2
1 1 2 2 1 1 1 1 2 2
2
32 2 2 1 21 1 1 2 12 2 2
1 1 2 2 2 1 1 2 1
1 1 1( ) ( ) 2
1 1 1( ) ( ) 2
u u u u u uk k A u A b
A x x x A x A x x x t
u u u u uk k A u A
A x x x x A x x A
+ + + + + − + + + + =
+ + + + + − + + −
2
22 2
1
2 2
3 3 3 32 13 1 1 1 3 32 2 2 2
1 1 2 2 1 1 2
1 1( ) 2 4
ub
x t
u uk k u m j
A x x x A x x t
+ =
+ + + + − − − + =
1 2 1 2
22 11 1 1 21 1 1 120, , , 0k A k A = = − = =
1-D formulation
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 1 3 1 2 3 1
1 2 1 3 1 2 3 1
1 2 1 3 1 2
0 0
0
3 1
0
0 0
( ), ( ), ,
, ,
, ,
,
( ), ( ),
( ), ( ) ,
k k
k k
k k
k k
k k
k k
k k
k k
k k
x x x x x x
x x x x x x
x x x
u u P P
P
x x
P
P Px
= =
= =
= =
= =
= =
= =
Expansion into Legendre’s polynomial series
( )( )
( ) ( ) ( ) ( ) ( )( )
( )( ) ( ) ( ) ( ) ( )( )
1 2
1 2 1 1 1 1
2 2
3 1 2
2
1 3
1 3
1 1 3 13 3 1
2
3
, 2 1, ,
2 1
2
,2 1
2
2 1, ,
h
k k kk
k
h
h
k
k
k k k k
h
u x xu kx P dx x x x x
ku u u u
h
k
x h x
x xkP dx x x x
h x hx
−
+
+ +
−
++= + +
+
+= =
+= =
+ +
Derivatives with respect to
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
2 1 2
2 2 1 2 1 1
2
23 1 2
2
2 3 1 3 1 1
2
23
,,2 1 2 1
12
,2 1 2 11
2
hk
k
h
hk
k
h
k
k
x xk kP dx x x x
h x h
x xk kP dx x x x
h x h
+ −
−
+ −
−
+ + = − − −
+ + = − − −
2x
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 1 2 2 3 1 3 1 2 2
1 1 2 2 3 1 3 1 2 2
1 1 2 2 3 1 3 1 2 2
, , , ,
, , , ,
, , , ,
2 1 2 1( ) ( )
2 2
2 1 2 1( ) ( )
2 2
2 1 2 1( ) ( )
2 2
k k
k k
h h
k k
k k
h h
k
h h
h h
k
k k
h h
h h
k ku u P dx P dxx x x x x x
x x x x x x
x x x x
h h
k kP dx P dx
h h
k kP dx P dx
h hx x
− −
− −
− −
+ += =
+ += =
+ += =
( ) ( ) ( )( ) ( ) ( ) ( )( )1 3 1 3
2 2 2 23 23 231 1 1 1 1 1
2 1 2 1,k k k k k kk k
h hx x x x x x − − − −+ +
= + + = + +
1-D formulation
( ) ( ) ( )13
22 2
31 222 11 12 2 2
11 1221 12 1 21 1 22 2 12 21 23 3
1 1 1 11 1
1 1 1, ,
kk kk k k k k k k
kkk k k k
kku u
jt t
k b k b mA x A x A x t
+ − + = + − − + = + − − +
=
+
1-D equations of motion
u u + = L u f M u
1-D kinematic relations
1-D equations of motion in displacements
33 21 1 1 2 3 2
1 211 1 2 12 1 1 22 13 23 3
1 11 1 1 1
1, , , , ,
1 1k k
k k k k kk
k k k k k k kku uk u k u
Au k u u
Ax A x x
− =
= + = −
=
− ==
+
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 1 3 1 2 3 1
1 2 1 3 1 2 3 1
1 2 1
0 0
0 0
0
1 1
0
3 2 3
, ,
,
( ), ( ),
( ), ( ),
( ),
,
( ) ,, ,
N N
N
k k
k k
k k
k k
k k
k k
k k
k k
k
N
N N
k
x x x x x x
x x x x x x
x x x x
u u P P
P
x x
P
P P
= =
= =
= =
= =
= =
= =
N-order approximation equations 00 00 00 0 0 0
11 12 13 11 12 13
00 00 00 0 0 0
21 22 23 21 22 23
00 00 00 0 0 0
31 32 33 31 32 33
0 0 0
11 12 13 11 12 13
0 0 0
21 22 23 21 22 23
0 0 0
31 32 33 31 32 33
,
n n n
n n n
n n n
u
n n n nn nn nn
n n n nn nn nn
n n n nn nn nn
L L L L L L u
L L L L L L
L L L L L L
L L L L L L
L L L L L L
L L L L L L
= =L u
0011
0022
0033
1 1
22
33
,n n
nn
nn
b
bu
m
u b
u b
m
=b
0, 0, 0, for 0 and f ork k k k k nu = = =
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )1 1 1 1 3 1 3 1 23 12 2 2 13
2 1 2 1( 1) , ( 1)k k k k k kb x x
k kb m
h hx x m x x x x + − + −+ +
= + − − = + − −
1-D approximation equations First order approximation equations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
0 1 0 1
1 2 1 0 1 1 3 1 2 3 1 0 3 1 1
0 1 0 1
1 2 1 0 1 1 3 1 2 3 1 0 3 1 1
0 1 0 1
1 2 1 0 1 1 3 1 2 3 1 0 3 1 1
, , ,
, , ,
, , ,
u x x u x P u x P x x x P x P
x x x P x P x x x P x P
x x x P x P x x x P x P
= + = +
= + = +
= + = +
000 00 00 01 01 0111 12 13 11 12 1
000 00 01 01 01 0221 22 23 21 22 2
000 00 00 01 0331 32 33 31 3
10 10 10 11 11 11 1 111 12 13 11 12 13 1 2
10 10 11 11 11 12
21 22 21 22 23 21
11 11 11 11
31 32 33 33
0
0
0 0, ,
0
0 0 0
u
bL L L L L u
bL L L L L u
mL L L L
L L L L L L u b
L L L L L u b
L L L m
= = =L u b2
2
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0,
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
u
j
t
j
=
M
Equations of motion for straight rod
( ) ( )
( )( ) ( )
( )( )
02 0 2 0
31 21 22 2
1 1 1
0 2 1 00
2 0 2 00 01 2
2 2
2 0 2 0 2 10 0 13 3 13 1 32 2
12 1 23 12
1 1
2 2
1 1
2 1
2
2 2
1
1
1
3
2 , 2
3 36( )
3
2 4 , 2
22
u u
x x x
u u um u
x x h x
u
x x h
u ub b
t t
ub
x t h h t
+ +
+ −
+ = − + =
+ = − + + =
− −
+ + − + −
++
2 1 00 11 11 2
2 3
1 1
2 12 11 13 3 32
2 32 2 2
1 1
3 32
( )1, ( ) 4
u uu m
h x x h
ub
t x x t
+− + −
+ = =
+ − +
u u + = L u f M u
Micropolar Timoshenko’s theory
0 1 0 0 1
22 22 3 3 11 11 11 2 12 12 21 21 11 11
11 11 2 12 12 2 21 21 2 11 11 2 2 13 13 2
0, 0, , , , , , , , ,
, , , , ,
h h h h h
h h h h h
u u u h e x e e e
n dx n dx n dx m x dx dx
− − − − −
= = = + = =
= = = = =
Static and kinematic assumptions
Kinematic relations
Equations of motion in displacements
31 2 111 1 2 22 12 1 1 3 11 21 1 1 2 3 13 23
1 1 1 1 1 1 1 1
1 1 1 1, 0, , , , , 0
u ue k u e e k u e k u
A x A x A x A x
= + = = − − = = − + = =
Equations of motion
( )
( )
2 2
1 21 1 22 2
2
11 1221 12 1 11 1
1 1 1 1
13 1112 21 1
2
3 1
2 21 3
1 1 1
2
1
1 1
1,,
, ,
1
u ub F b F
t t
n nn n k n k A
A x A x
jF Jm
n n m n mA At xx t
+ + + = − + =
+ − + = − + =
Constitutive relations0
11 11 11 11 13 13 12 12 21 21 21 12, , ( ) , ( ) ( ) , ( ) ( )n EFe m EJ F n Fe Fe n Fe Fe = = = + = + + − = + + −
1 111 12 11 11
2221 22 21 21 2
2
131 32 31 31 1
341 42 41 41 3
0 0 0
0 0 0, , ,
0 0 0
0 0 0
u u
u u
u uu u
u u
u b FL L L L
u FL L L L b
JtL L L L m
jFL L L L m
= = = =
L u b M
Micropolar Euler-Bernoulli theory
21 1
0 1 0 1
22 22 11 11 11 2 12 12 21 21 11 11
11 11 2 12 12 2 11 111
1
2 2
1
0, 0, , , , , , , ,
1
, , , ,
h h h
h h h
u u u h e x e e e
n dx n dx mu
xk duA
xx
− − −
= = = + = =
= =− + =
=
Static and kinematic assumptions
Kinematic relations
Equations of motion in displacements
2
31 2 2 1 211 1 2 22 12 1 1 3 11 21 1 1 1 2 3 13 232 2
1 1 1 1 1 1 1
1
1 1 1 1 1
1 1 1 1 1, 0, , , , 0,
u u u k ue k u e e k u e k u k u
A x A x A x
u
xA x A x A
= + = = − − = − + = − − + =
=
Equations of motion
( )
( )
2 2 2
3
1112
31 11 2 21 1 22 2 2 2
1 1 1
1121 12 1 11
1 1
12 21
1 1 1 1
2
13 3
2
1 1
1 3
1 1
1 1 ( ) 2
1 ,
1
, ,mu m u uF
b F b FA t A x A x A x
mn
t
FA x
nn n k n k
x x
nA
n mt
mx
−− −
+ + + = − + =
+ −
−
= −+ =
Constitutive relations
( )2
1 2 1 211 1 2 11 12 1 1 1 2 32 2
1 1 1 1 1 1
3221 1 1 1 2 3
1
1
1
3
1 1 1 1
1
1 1 1, , 2 2
1 12 ( ) 2 , ( ) ,
u u k un EF k u m EJ n F k u Fk u
A x A x A x
un F k u Fk u F
u
A x
A x A x
= + = − = − − − −
= − + + + = +
111 12 12 1 2
21 22 22 2 2 2
31 32 31 3 3
0 0
, , , 0 0
0 0
u u
u u
u u
u u
bL L L u F
L L L u b Jt
L L L m jF
= = = =
L u f M
Micpolar equations for straight rod
( ) ( )
( )( ) ( )
02 0 2 0
31 21 22 2
1 1 1
0 2 1 00 12 1 23 12
1 1 1
2 0 2 00 01 2
2 2
2 0 2 0 2 10 0 13 3 13 1 32 2 2 2
1
2 , 2
3 36( ) 2 4 , 2
u ub b
u u
t t
u
x x x
u u um u
x x hb
x t h h tx
+ +
+ −
+ = − + =
+ = − + + =
+ + − + −
( )( )1 2 1 0 2 12 1
1 13 3 3 322 3
2 1 0 11 12 1 2
2 32 2
1 1 1
2 2 2
1 11
3 2 ( )3 32
12 , ( ) 4
ub
t x x
u u uu m
x x h h x x h t
− − + = + − +
+ ++ − + −
=
First order approximation
2 2 2 2
31 1 2 1 21 22 2 2 2
1 1
2 22 21 31 2 1 2
1 3 1 32 2
1
1 1
0
1 2
1 1 1 1
3
2 , , ( ) ( )
( ) ( ) 2 2 2 ( ) 4,
u u u uEF b F F F b
x t x t
u uEJ J J J b
F Ax x
mF F F Fx x t x x
+ = + − + =
−
+ −
− − − + − + + −
+ = +
= 3
2t
Timoshenko’s equations
2 4 2
3 31 1 2 21 22 2 4 2
1 1 1
2 2
3 33 1 2
1
2
2
1
, 2
( ) 4
mu u u uEF b F EJ F b
x x
m
Ft x x t
x t
+ = + =
+ =
− −
+ −
Euler-Bernoulli equations
2-D equations of couple stress elasticity
Kinematic relations 3
3 3
3
1 1 1 , , ,
2
uuu
H x H H x
= − = = + =
ε u ω ω u
Equations of motion
11 12 13 11 12 13 1
21 22 23 21 22 23 2
31 32 33 3
0 0 0 0 0 0 0
0 , 0 0 , 0 , 0 0 , , 0
0 0 0 0 0 0 0 0 0 0
u
u
= = = = = =σ μ ε χ u ω
Curvilinear orthogonal coordinates1
, ,kii ij k i
i i j i i i
HH x x x x x
= = = =
eR R R Re e
1 1
2
j k i jk i i kij ik jk ik ij
i j i k i j k
H H H HH H H
H x H H x x x
= − + + −
( ), 0,
1
H x H H
+ = + = = =
= + +
σ b u μ σ σ e e e e
σ e
Constitutive relations
( )
1( , ) 8 , 2 , 8
2kk kk ij ij i iE e e e e e e = + + = + = −e κ
( )2( 2 ) u u u b u + − − + =
Equations of motion in displacements
Zozulya V.V. Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models.
Curved and Layered Structures, 2017, 4, 119–132.
2-D formulation in special coordinates
Kinematic relations 1 2 2 1 2 1
11 1 2 22 12 1 1 3 21 1 1
1 1 2 1 1 2 1 1 2
1 1 1 1 1, , ,
2 2
u u u u u uk u k u k u
A x x A x x A x x
= + = = + − = = − −
( ) ( )
( )
2 2
1 21 1 21 12 1 2 1 22 11 22 2
3 12 21
,
0
u uk b k b
t t
+ + + = + − + =
− + =
Equations of motion
1 2 1( ) ( ) ( )x x x= +R x r n 1 11 1 2 1 1 1 2 2 1 2
1 1
1 2 11 1 1 2 1 1 1 1
2 2 2
( , ) ( )(1 ), 1, (1 ),
1, 0, 1 1 , ,
H AH x x A x k x H k x
x x
H H Hk A k x H A k A
x x H x
= + = = +
= = + → =
2-D equations of motion2 2
1 2 1 2 1 11 2 1 2 12
1 1 1 1 2 1 1 1 1 1 1 2 2 2
2 2 22 2 1 1 2 1 1
1 1 1 1 12 2
1 1 2 2 2 1 1 2
1 1
1 1 1 1 12
1 12
1 1
u u u u u uk u k u k
A x A x x A x A x A x x x x
u u u u u uk k k u b
A x x x x A x x t
A x A
+ + + + + + − +
+ − − + + − + =
22 1 2 1 1 1 21 1 1 1 22
1 1 2 1 1 1 2 1 2 1 1 2
2 2
2 2 1 21 1 2 22 2
2 2 1 1
1 1
12 2
u u u u u u uk u k k u
x x A x x x x x A x x
u u u uk k u b
x x A x t
+ − − − − + + + +
+ + − − + =
1 2 1 2
22 11 1 1 21 1 1 120, , , 0k A k A = = − = =
Constitutive relations
( ) ( ) 2 2
11 11 22 11 22 11 22 22 12 12 3 21 21 32 , 2 , 2 2 2 2 = + + = + + = − = +
1-D formulation
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 1 3 1 2 3 1
1 2 1 3 1 2 3 1
1 2 1 3 1 2
0 0
0
3 1
0
0 0
( ), ( ), ,
, ,
, ,
,
( ), ( ),
( ), ( ) ,
k k
k k
k k
k k
k k
k k
k k
k k
k k
x x x x x x
x x x x x x
x x x
u u P P
P
x x
P
P Px
= =
= =
= =
= =
= =
= =
Expansion into Legendre’s polynomial series
( )( )
( ) ( ) ( ) ( ) ( )( )
( )( ) ( ) ( ) ( ) ( )( )
( )( ) ( ) ( ) ( ) ( )
1 2
1 2 1 1 1 1
2 2
3 1 2
2 1 1 3 1 3 1
2
2
3 1 2
2
1 3
1 1 1 1
1 3
3 3
1 3
32 3
2
3 3
,2 1
2
,2 1
2
,2 1 2 1.
2
2 1, ,
2 1, ,
,
k k k k
k k k k
k k k k
hk
k
h
h
k
h
h
k
h
u x xu kx P dx x x x x
x h x
x xkP dx x x x x
h x
ku u u u
h
x xk kP dx x x x x
h x
k
h
h
+ +
+ +
+
−
+
−
−
+= =
+=
+= + +
+=
+ +
+ += + +
=
( )..
Derivatives with respect to
( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
2 1 2
2 2 1 2 1 12
1 3
2 21 1 2
2
1
,
2 1
,2 1 2 11
2
hk
k
k
k k
h
k
x xk kP dx x x x
h h
x xk
x
xh
+ −
− −
−
+ + = −
+= +
−
−
+
2x
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 1 2 2 3 1 3 1 2 2
1 1 2 2 3 1 3 1 2 2
1 1 2 2 3 1 3 1 2 2
, , , ,
, , , ,
, , , ,
2 1 2 1( ) ( )
2 2
2 1 2 1( ) ( )
2 2
2 1 2 1( ) ( )
2 2
k k
k k
h h
k k
k k
h h
k
h h
h h
k
k k
h h
h h
k ku u P dx P dxx x x x x x
x x x x x x
x x x x
h h
k kP dx P dx
h h
k kP dx P dx
h hx x
− −
− −
− −
+ += =
+ += =
+ += =
1-D formulation
( ) ( )11 121 2
2 2
1 211 12 21 1 22 2
1 1 1 1
22 112 2
1 1, ,
k k kkk k k k k k k ku u
k b k bxt tA x A
+ − + = + − − + =
+
1-D equations of motion
2
2u ut
+ =
L u f I u
1-D kinematic relations
1-D equations of motion in displacements
1 2 211 1 2 22 2 12 1 1 1 1
1 1 1 1 1
3 1 1
1
1, , 2 ,
2
1 1 1kk k k
k k kk k k k kku u uk u uu k uk u u
A x A x A x
= + = = + − −
= −
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 1 3 1 2 3 1
1 2 1 3 1 2 3 1
1 2 1
0 0
0 0
0
1 1
0
3 2 3
, ,
,
( ), ( ),
( ), ( ),
( ),
,
( ) ,, ,
N N
N
k k
k k
k k
k k
k k
k k
k k
k k
k
N
N N
k
x x x x x x
x x x x x x
x x x x
u u P P
P
x x
P
P P
= =
= =
= =
= =
= =
= =
N-order approximation equations
000 00 0 0 0111 12 11 12 1
000 00 0 0 0221 22 21 22 2
0 0
11 12 11 12 1 1
0 00
21 22 21 22 22
, ,
n n
n n
u
n n nn nn n n
n nn nn nn
bL L L L u
bL L L L u
L L L L u b
L L L L u b
= = =L u b
0, 0, 0, for 0 and f ork k k k k nu = = =
( ) ( ) ( ) ( )( )1 12 21 1
2 1( 1) ,k k kk
bh
b x x x x + −+= + − −
1-D approximation equations First order approximation equations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
0 1 0 1
1 2 1 0 1 1 3 1 2 3 1 0 3 1 1
0 1 0 1
1 2 1 0 1 1 3 1 2 3 1 0 3 1 1
0 1 0 1
1 2 1 0 1 1 3 1 2 3 1 0 3 1 1
, , ,
, , ,
, , ,
u x x u x P u x P x x x P x P
x x x P x P x x x P x P
x x x P x P x x x P x P
= + = +
= + = +
= + = +
000 00 01 01 0111 12 11 12 1
000 00 01 01 0221 22 21 22 2
10 01 11 11 1 111 12 11 12 1 1
10 10 11 11 11
21 22 21 22 22
, ,u
bL L L L u
bL L L L u
L L L L u b
L L L L u b
= = =L u b
Equations of motion for straight rod2 0 1 2 0 1 0 1
1 2 2 2 11 22 2
1 1 1 1
0 0 2 1
2 0 4 3 2 00 01 1
112 2 1
12
1 1
0 2 1 4 1
1 2 2
2
2 4 3 2
1 1
3 2 2 111 1
13 2 2 2
2
1
2
1 1
1
2
3 3 3 32
, ,
,
3
u u u u u ub b
x x x h x h
u u
h t x
u u u uu b
x h
x t
u
h x h h x t
h
x
u u u
x x
+ + = + − + =
− − + =
+
+ +
− + + +
− −
12 1
1
4
22 222
1
,3( 2 )
uh
bx
u
t
− + =
+
2
2u ut
+ =
L u f I u
1-D equations of motion in displacements
Timoshenko’s couple stress theory
0 1 0 0 1
22 22 3 3 11 11 11 2 12 12 21 21 11 11
11 11 2 12 12 2 21 21 2 11 11 2 2 13 13 2
0, 0, , , , , , , , ,
, , , , ,
h h h h h
h h h h h
u u u h e x e e e
n dx n dx n dx m x dx dx
− − − − −
= = = + = =
= = = = =
Static and kinematic assumptions
Kinematic relations
Equations of motion in displacements
1 2 1 211 1 2 22 12 1 1 1 11 3 21 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1, 0, ,
2 2
u u ue k u e e k u k u
A x A x A x A x
= + = = − − = = = − +
Equations of motion
( )11 12 1121 12 1 1
2
1 21 3
1 1 1 1
2 2
1 2 11 1 22 2
1 1
2
1 1 1, , ,
u ub F b
n n mF Jn n k n k n m
A x A x A xt t t
+ + + =
− + = −
+ =
Constitutive relations
2 2
1
2 2
3 311 11 11 11 12 12 21 122 2
1 11
, , 2 2 , 2 2F F
n EFe m EJ n Fe n Fex xA A
= = = − = +
111 12 11 1
21 22 21 1 2
31 32 31 1 1
, , ,
u u
u u
u
u u
bL L L u
L L L u b
L L L m
= = =L u b
Euler-Bernoulli couple stress theory
21 1
0 1 0 1
22 22 11 11 11 2 12 12 21 21 11 11
11 11 2 12 12 2 11 111
1
2 2
1
0, 0, , , , , , , ,
1
, , , ,
h h h
h h h
u u u h e x e e e
n dx n dx mu
xk duA
xx
− − −
= = = + = =
= =− + =
=
Static and kinematic assumptions
Kinematic relations
Equations of motion in displacements
2
1 2 1 211 1 2 22 12 11 3
1
1
21 1 12 2
1 1 1 1 1 1 1
1 1 1, 0, 0, ,
u u k ue
uk u e e k u
A x A x x A xA
= + = = = − = = −
Equations of motion
( )2 2
1 21 1 22 2
11
11 1221 12 1 11
1 1 1 1
1121 12 3
1 1
3
1 1
1 1, ,
1,0
1 m
un nn
ub F b F
t tn k n k
A x A x
m
A xn
Am n m
x
+ + + = − + =
− +
=
= −
Constitutive relations2
1 2 111 1 2 11 2 2
1 1 1 1
3 2 3 2
2 1 1 2 1 112 213 2 3 2
1 1
1
1 1
3 2 3 2
1 1 1111
1 1, ,
2 22 2, 2
u u kn EF k u m EJ
A x A x
u Fk u u Fk uF Fn n
x x
u
x x
A x
A A A A
= + = −
= − + = −
1 111 12
221 22 2
, , ,u u
u u u
u bL L
uL L b= = =L u b
Couple stress equations for straight rod
2 0 1 2 0 1 0 1
1 2 2 2 11 22 2
1 1 1 1
0 0 2 1 112 2 1
12
1
2 0 4 3 2 00 01 1 2
2 4 3 2
1 1
3 2 2 111 1
13 2 2 2 2
1 11
2 ,
3 3 3 3 2
,
,
u u u u u ub b
x x x h x h
u u u uu b
x
u u
h t x x t
u
h x h hx th x
+ + = + − + =
+ +
− + +− − =
+
+
0 2 1 4 111 2 2
2
2 11 2
2
1
22 4
1 1
2
3 3( 2 ) ,
u u u u
h hu b
x x tx
+ − − +
=
+
−
First order approximation
2 4 3 2
1 1 2
2 2
2 2
1 1
2 1 1 21 22 4 3 2
1 1
3 2 2
2 2 1 1 11 3 2 2
1 1
1
2
2
1 1
3
,u u u u u
EF b F F F F bt x x t
u uF F F EJ F
x x x
mxx x x t
−
+ + +
+ = − − + =
− + +
=
Timoshenko’s equations
2
112 2
1
4 2
2 22 2
1
1
4
2
( 2 )
,
,
uEF b
t
u uEJ F b
x t
u
x
−
+
=
=
+
Euler-Bernoulli equations
Merida, Yucatan, Mexico
Nonlocal theory of curved beams. 2-D,
high order, Timoshenko’s and Euler-
Bernoulli models
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
3-D formulation of the problem for shells
( ) 0( ) , ( ) , / ,c
ij ij e
V
H d e a l V = − = x x x x x x
Curvilinear orthogonal coordinates1
, ,kii ij k i
i i j i i i
HH x x x x x
= = = =
eR R R Re e
1 1
2
j k i jk i i kij ik jk ik ij
i j i k i j k
H H H HH H H
H x H H x x x
= − + + −
1 1 1
2
j kiij ij k
j j i i
uuu
H x H x
= + +
Cauchy relations
Nonlocal constitutive relations
2 2 2
0 0 0[ ( , )] ( ), [ ] , [ ] (1 )[ ]c
eL H L L l − = − = → = − x x x x σ σ
Equations of motion1 ij ij k jik
ik ki i i
j j k i
b uH x H H
+ + + =
2c
ij rr ij ije e = +
( ) ( ) 02 3/2 2 2 2
1 1, exp , , ,
( ) 2e e e e
H H Kl l l l
= − =
x x x xx x
2 2 2(1 ) 2e ij kk ij ijl e e − = +
3-D formulation in special coordinates
Kinematic relations 1 1 2 2
11 2 1 3 22 1 2 3
1 1 1 2 2 2 2 2 1 1
3 1 2 2 133 12 2 1
3 2 2 1 1 1 1 2 2
3 31 213 1 1 23 2 2
3 1 1 3 2 2
1 1 1 1, ,
1 1 1 1,
1 1,
u A u Au k u u k u
A x A A x A x A A x
u u A u Au u
x A x A x A x A x
u uu uk u k u
x A x x A x
= + + = + +
= = − + −
= − + = − +
132 11 1 12 1 21 2 12 13 1 2 1 22 1 2 1 1 2 1
1 2 3 2 1
232 21 1 22 2 11 2 21 23 1 2 2 11 1 2 2 1 2 2
1 2 3 1 2
2 31 1 32 1 2 33
1 2
( ) ( ),
( ) ( ),
( ) ( ) ( )
A A A AA A A A k A A b A A u
x x x x x
A A A AA A A A k A A b A A u
x x x x x
A A A A
x x
+ + + + − + =
+ + + + − + =
+ +
11 1 2 1 22 1 2 2 1 2 3 1 2 3
3
.A A k A A k A A b A A ux
− − + =
Equations of motion
3( ) ( ) ( )x = +R x r x n x
2 2 2 2 2 2 2( ) ( ) , (1 )e el l + + + = − = − u u b u u b b
3-D equations of motion
3 3
33
3
(1 ) for 1,2 and 1
(1 ), , 0i
H A k x H
H A H Hk x k A
x x x x
= + = =
= + = =
Kinematic relations
3( ) ( ) ( )x = +R x r x n x 3 3
33
3
(1 ) for 1,2 and 1
(1 ), , 0i
H A k x H
H A H Hk x k A
x x x x
= + = =
= + = =
2-D formulation
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3
0
3 3
0
3 3
0
2 1( ) , , ( )
2
2 1( ) , , ( )
2
2 1( ) , , (
,
2
,
.)
k k
i i k i i k
k h
k k
ij ij k ij ij k
k h
k k
ij ij k ij ij k
k h
h
h
h
ku u P u u x P dx
h
kP x P dx
h
kP x P dx
h
= −
= −
= −
+= =
+= =
+= =
x x x x
x x x x
x x x x
Expansion into Legendre’s polynomial series
22 2 2 2 2 1
1 22
1 2 1 1 1 2 2 2 3 3
12 , ( )k k k k
ij e ij rr ij ij
A Al k k
A A x A x x A x x x
− = + = + + + +
( ) ( )( ) ( ) ( ) ( ) ( )( )3
3
3 3
1 3,2 2 1
2, ,
1k h
i i
k i i
k k
i
h
k k
i
u u xkP dx
x h x
ku u u u
h
−
+ + +
= =
+= + +
x xx x x x
3-D constitutive relations in special coordinates
( ) ( )( )
( ) ( )( )
( ) ( )( )
3
3
3
3
2 2
3
32 2
,2 1,
2
,2 1,
2
,2 1,
2
k h
k
h
k h
k
h
k h
k
h
u u xkP dx
x h x
xkP dx
x h x
xkP dx
x h x
−
−
−
+=
+=
+=
x x
x x
x x
Derivatives with respect to
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )( )
3
03 3
2
1
3
203 3
3
0
1 3
0
( ) 2 1( ), ,
( )( ),
( , )
( , ) 2 1... ,
k k k k kkij k ij
k
k k k k kkij ij k
ij
ij ij ij
k
i
ij
j
k
ij ij
k
x
x x
x k
x x h
P kP
h
PP
=
=
+ +
=
+ +
=
+=
= =
+= =
+ +
= + +
xx x x x x
xx x x x x
3x
2-D formulation
( ) ( )
( ) ( )
( ) ( )
2 11 1 12 1 212 13 1 2 1 22 13 1 2 1
1 2 2 1
2 21 1 22 212 23 1 2 2 11 23 1 2 2
1 2 1
2 31
2
1 2 1
211 2 2
2
2
1 32
11 1 2 1 1
1 2
2
,
,
k k
k k k k k
k k
k k k k k
k k
t
k
t
k k
kA A A A
A A k A A fx x x x
A A A AA A k A A f
x x x x
A AA A k A A
x
A A u
A A u
x
+ + + − − + =
+ + + − − + =
+ − −
2
2 3 1 22 33 2 31 .k k k
tA Ak A uA f − + =
2-D equations of motion
2 22 2 2
1( (2 , ( () ) ) )k k k k k k
ij e ij rr ij ij ij
k k
ij ijijl k − = + = + +x x x x
( )2 2 2
1u el k + = − + +L u b u u u u
2-D kinematic relations
2-D equations of motion in displacements
2-D constitutive relations
1 1 2 211 2 1 3 22 1 2 3
1 1 1 2 2 2 2 1 2 1
1 2 2 112 2 1
2 2 1 1 1 1 2 2
3 313 1 1 23 2 31 2 2 3
21 1 2
3
1 1 1 1
1 1 1 1,
1 1.
, ,
, ,
k kk k k k k k
k kk k k
k k kk k
k k k k
u A u Ad u k u u k u
A x A A x A x A A x
u A u Au u
A x A x A x A x
u uk u k u
A x Au
xu u
+
= + + = + +
= − + −
= − = − =
+
2-D approximation equations
( ) ( )
( ) ( )
( ) ( )
3
3
0
3
0
0
( ),
( ),
( )
,
,
,
,
n
i i
n
i
k
k
k
k
k
k
k
j
k
k
ij
n
ij ij
u u
P
x P
x
x
P
=
=
=
=
=
=
x x
x x
x x
N-order approximation equations
First order approximation equations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 0 1 0 1
0 1 0 1 0 1, ,ij ij ij i i i ij ij ijP P u u P u P P P = + = + = +x x x x x x x x x
00 00 00 01 01 01 0
11 12 13 11 12 13 1
00 00 00 01 01 01 0
21 22 23 21 22 23 2
00 00 00 01 01 01 0
31 32 33 13 32 33 3
10 10 10 11 11 11 1
11 12 13 11 12 33 1
10 10 10 11 11 11 1
21 22 23 21 22 23 2
10 10 10 11 11 11
31 32 33 31 32 33
,
L L L L L L u
L L L L L L u
L L L L L L u
L L L L L L u
L L L L L L u
L L L L L L
= =L u
01
11
0 12 2
0 13 3
1
1
1
21
133
/
/
/, ,
0
0
0
b u h
b u h
b u h
b
b
u b
= =b u
00000 00 0 0 0 11111 12 12 13 1
00000 00 0 0 022 221 22 22 23 2
0 0
21 22 22 23 2 2 2 20 00
31 32 32 33 33 3
3
, , , ,
n n
n n
u
n n nn nn n n nn
n nn nn n nn n
uubL L L L u
ub uL L L L u
L L L L u b u u
L L L L u b u u
= = = = =L u b u u
( )2 2 2
1u el k + = − +L u b u u u
Zozulya V. V. A high order theory for linear thermoelastic shells: comparison with classical theories, Journal of
Engineering. 2013, Article ID 590480, 19 pages
Zozulya V.V. A higher order theory for shells, plates and rods. International Journal of Mechanical Sciences,
2015, 103, 40-54.
2-D formulation of the problem for rods Curvilinear orthogonal coordinates1
, , , 1,2, HH x x x x x
= = = = =
eR R R Re e
1 2 1 22 1 1 222 11 21 12
1 1 2 2 2 2 1 1
1 1 1 1, , ,
H H H H
H x H x H x H x
= − = − = =
( )1 1
( ) ,Tuu u
H x H x H
= + = = +
eε u u u e e e
Cauchy relations
Nonlocal constitutive relations
Equations of motion
1,
H x H H
+ = = + +
σ b u σ e
2 2 2(1 ) ( ) 2e ijl tr − = +σ ε I ε
Zozulya V.V. Nonlocal theory of curved beams. 2-D, high order, Timoshenko’s and Euler-Bernoulli models,
Curved and Layered Structures, 2017, 4, 221–236.
2-D formulation in special coordinates
Kinematic relations 1 2 2
11 1 2 22 12 1 1
1 1 2 1 1
1 1, ,
u u uk u k u
A x x A x
= + = = −
1 2
1 1 21 12 2 22 11
1 1 2
1, , ,k
A x x
+ = = + + = + = −
σ b u σ e
Equations of motion
1 2 1( ) ( ) ( )x x x= +R x r n 1 11 1 2 1 1 1 2 2 1 2
1 1
1 2 11 1 1 2 1 1 1 1
2 2 2
( , ) ( )(1 ), 1, (1 ),
1, 0, 1 1 , ,
H AH x x A x k x H k x
x x
H H Hk A k x H A k A
x x H x
= + = = +
= = + → =
Nonlocal constitutive relations2
2 2 2 2
12
1 1 1 1 2 2
1 12 ,el k
A x A x x x
− = + = + +
2-D equations of motion2 2 2
u el + = − L u b u u
2
1 1 1 12 2 2
1 1 2 2 1 1 1 1 2
2
1 1 1 12 2 2
2 1 1 1 1 2 2 2 2
1 1 1( )
1 1( ) ( )
u
k k A AA x x x A x A x x
k k A Ax A x A x x x x x
+ + + + +
=
+ + + + + +
L
1 2 1 2
22 11 1 1 21 1 1 120, , , 0k A k A = = − = =
1-D formulation
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 1 1 1 2 2
1 2
0
0
0
1 1 1 2 2
1 2 1 1 1 2 2
2 1( ) , ( )
2
2 1( ) , ( )
2
2 1( ) ,
, , ,
, , ,
, ( ,),2
k k
k k
k h
k
h
h
h
k
k k
k h
k k
k k
k h
ku u P u u P dx
h
kP P dx
h
kP P dx
h
x x x x x x
x x x x x x
x x x x x x
= −
= −
= −
+= =
+= =
+= =
Expansion into Legendre’s polynomial series
( )( )
( ) ( ) ( ) ( ) ( )( )1 2
1 2 1 1 1 1
2
3
2
1,2 1
2
2 1, ,k
k k k
hk
h
ku x xu k
x P dx x x x xk
u u u ux hh x
−
+ + + +
= += =
+
( ) ( )( )
( ) ( )( )
( ) ( )( )
1 1 2
2
1 1
1 1 2
2
1 1
2 2
1 1 2
22 2
1 1
,2 1,
2
,2 1,
2
,2 1,
2
k h
k
h
k h
k
h
k h
k
h
u x u x xkP dx
x h x
x x xkP dx
x h x
x x xkP dx
x h x
−
−
−
+=
+=
+=
Derivatives with respect to
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )( )
1 2
1 1 1 1 1
02 3
2
1 2
1 1 1 1 12
1 3
0
2
3
0
1
03
( ) 2 1( ), ,
( )
( , )
( , ) 2 1.( ), ,..
k k k k kkk
k
k k k k kkk
k
k
k
x xx x x x x
x x
x x kx x x x x
x x h
P kP
h
PP
+ +
=
+
+
=
= =
= =
+
+= + +
= = + +
=
2x
2 22 2 2
1 1 1 1 12 , ) ()( ( ) )(k k k k k k k k
el x x k x x − = + = ++
2-D constitutive relations
1-D formulation
( ) ( ) ( ) ( )( )
1
1
1 1
1 1
2
1 3
12 2 2 2 2
2 1 2 1( 1) ,
1 , ,
k
k k k k
k k k k
k k
k kk kb
h
kA x
b x x xh
+ − − −
+ = = + −
+ += + − −
= + +
σ b u σ e
1-D equations of motion
( )2 2 2
1u el k + = − + +L u b u u u u
1-D kinematic relations
1-D equations of motion in displacements
( ) ( ) ( )( )1 31 211 1 2 22 12 1 1 1
1 1 1
2 1 1
1
1, , 2 ,1 1 2 1
k k
k k k k k k k k kku u kk u u k u u u u
A x A x hu x x x + + +
= + = = + − = + +
( ) ( )
( ) ( )
( ) ( )
1 2 1
1 2
0
0
0
1
1 2 1
( ),
( ),
( )
,
, ,
,
k
k
k
k
k
k
n
k
k
nk
n
u u P
P
x x x
x
x P
x x
x x
=
=
=
=
=
=
N-order approximation equations 0
0000 00 0 0 0 11111 12 11 12 100000 00 0 0 022221 22 21 22 2
0 0
11 12 11 12 1 1 1 10 00
21 22 21 22 22 2
2
, , , ,
n n
n n
u
n n nn nn n n nn
n nn nn n nn n
uubL L L L u
ub uL L L L u
L L L L u b u u
L L L L u b u u
= = = = =L u b u u
0, 0, 0, for 0 and f ork k k k k nu = = =
1-D approximation equations
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 1
1 2 1 0 1 1
0 1
1 2 1 0 1 1
0 1
1 2 1 0 1 1
, ,
, ,
, ,
x x x P x P
x x x P x P
u x x u x P u x P
= +
= +
= +
First order approximation equations
Second order approximation equations
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
0 1 2
1 0 1 1 1 2
0 1 2
1 0 1 1 1 2
0 1 2
1 0 1 1 1 2
,
,
.
x P x P x P
x P x P x P
u u x P u x P u x P
= + +
= + +
= + +
x
x
x
000 00 01 01 0 1
111 12 11 12 1 1
000 00 01 01 0221 22 21 22 2
110 10 11 11 12 1111 12 11 12 12 1
10 10 11 11 12 1 121 22 21 22 22 2 2
21 21 22 22 22
11 12 11 12 11
21 12 22 22 2221 21 21 22 22
0 0
0 0
0, , ,
0
0 0
0 0
u
bL L L L u u
bL L L L u
bL L L L L u
L L L L L u b
L L L L u b
L L L L u b
= = = =L u b u
2 2
1
1 2 2
2 2
2
1
2
2
/ 3 /
/ 3 /
3 / 0,
3 / 0
0 0
0 0
u
u
u
u
h h
u h h
h
h=u
000 00 01 01 0 1111 12 11 12 1 1
000 00 01 01 0 1221 22 21 22 2 2
10 01 11 11 1 111 12 11 12 1 1
10 10 11 11 11
21 22 21 22 22
/
/, , ,
0
0
u
bL L L L u u h
bL L L L u u h
L L L L u b
L L L L u b
= = = =L u b u
( )2 2 2
1u el k + = − +L u b u u u
( )2 2 2
1u el k + = − + +L u b u u u u
Nonlocal Timoshenko’s theory
0 1 0 1
22 22 11 11 11 2 12 12 21 21 11 11
11 11 2 12 12 2 11 11 2 2
0, 0, , , , , , , ,
, , ,
h h h
h h h
u u u h e x e e e
n dx n dx m x dx
− − −
= = = + = =
= = =
Static and kinematic assumptions
2 2 2
u el + = − L u b u u
Kinematic relations
Equations of motion in displacements
1 2 11 1 2 1 1 2 1 1 2 1 2 2 11 1 2 22 12 1 1 1 11
1 1 1 1 1 1
1 1 1 1( , ) ( ) ( ), ( , ) ( ), , 0, ,
2
u uu x x u x x x u x x u x e k u e e k u
A x A x A x
= − = = + = = − − =
Equations of motion11 12 11
2
12 1 11 12 3
1 1 1 1 1 1
2 2
1 2 11 1 22 2 2
1 1 1, , ,
n n mn k n k
u un mb F b F J
t t tA x A x A x
+ + = − + = − + =
Constitutive relations2 2 2 2 2 2 2 2 21 1 2
11 11 1 2 11 11 12 12 1 1
1 1 1 1 1 1
1 1 1, ,
u un l n EF k u m l m EJ n l n F k u
A x A x A x
− = + − = − = −
111 12 13 1
21 22 23 2 2
31 32 33 1 1
, , ,
u u
u u
u
u u
bL L L u
L L L u b
L L L m
= = =L u b
Nonlocal Euler-Bernoulli theory
21 1
0 1 0 1
22 22 11 11 11 2 12 12 21 21 11 11
11 11 2 12 12 2 11 111
1
2 2
1
0, 0, , , , , , , ,
1
, , , ,
h h h
h h h
u u u h e x e e e
n dx n dx mu
xk duA
xx
− − −
= = = + = =
= =− + =
=
Static and kinematic assumptions
2 2 2
u el + = − L u b u u
Kinematic relations
Equations of motion in displacements
2
1 2 2 11 1 2 1 1 2 1 1 2 1 2 2 11 1 2 22 12 1 1 11 2 2
1 1 1 1 1 11
1
1
1 1 1( , ) ( ) ( ), ( , ) ( ), , 0,
u u u ku x x u x x x u x x u x e k u e e k u
A x A x
u
xx AA
= − = = +
= = − = − +
Equations of motion2 2 2
311 1 11 21 3 1 22 2 2 2
1 1 1
11 11112 3
1 1 1
1
1
1 1
1 11
1 1 1,
1,
mm u m ub m
n kk kF b F
A An
x
mn m
A x t A x A xx t
+ + = − + =
− −
=
−
Constitutive relations2
2 2 2 2 2 2 2 2 21 2 1 211 11 1 2 11 11 1
1
1
2 12 1 12 2
1 1 11 1 1 1
1 1 1, , 2
u u k u
A
un l n EF k u m l m EJ n l n F k u
A x A x A xx
− = + − = − − = −
1 111 12
221 22 2
, , ,u u
u u u
u bL L
uL L b= = =L u b
22 2 2 2 2 2 3
1 1 2 2 2
1 1
1(1 ) , (1 )e e
mb l b b l b
A x
= − = − −
Nonlocal equations for straight rod
( )
( )
2 0 1 2 1 2 0 1 2 1
1 2 1 2
2 0 2 0 2 0 2 02 2 2 0 2 2 2 2 2 0 2 21 1 1 2 2 1
22 2 2 2 2 2
2 1 22 2 2 11 1
1 212 2
1 1 1 1
2 1 011 21 2 12 2
1 1
, ,
3
2
23
e e e e
e
u u u k u u kl l
u u u u ub b
x x x x
u
l lh t t h t h t t h t
u ul
t
uu b
x h x h
+ + − −
+ −
+ + = + + + =
− + + =
( )1 2 1 2 12 2 2 12 2
22 2 2
2 1 012 122
1 1
2
3 13, ,
2e
u uu b
x h x h
u ul
t h t t
+ + + =
+ −
First order approximation
( )
( )
2 0 2 0 2 0 2 02 2 2 0 2 2 2 2 2 0 2 21 1 1 1
2 0 1 2 1 2 2 2 0 1 2 1 2 2
1 2 1 1 1 1 2 21 22 2 2 2
1 1 1 1
2 1 2
1 2
2
1
2 1
2 2 2 2 2 2 2
1
22
32
3 3, ,e e e e
u u u uu u k uu u u ub
u kl l l l
h t t h t t t t h t tb
x x h x x
x h
h h
u u
x
+ + = + +
+ + − −
+
+ + = +
+
( )
( )
0 2 212 11 12
1
2 1 0 2 212 1 22
2 1 22 2 2 1 2 21 1 1
2 2 2
2 1 22 2 2 1 2 22 2 1
2 2 2
2
22 2
1 1
2 2 12
22 2 2 1
2
1 212 2
1 1
3 3,
33,
5
2
15
3
3
2
k
e e
k
e e
e
u uu b
h x h
u u uu
x h
u u kl l
t t h t
u u k
x h
u uu b
x h x
l b lt t h
h
t
ul
t
+ + =
+ + =
+
− − −
+ − − −
+
+ − −
( )2 2 122 1
2 2 2 2 2 22 2 2 2 21 2 2
1 2 2 22
1
2
1
2 2
155 2, ,e
u uu b
x h x
u u ul
t t th
+ − −
= + + =
Second order approximation
2 2 22 2 2 2 2 2 2 2 21 2 1 2 2 1
1 2 1 1
2 2 2
1 1
2 2 2
1 1
2 2 2
11 1
(1 ) , (1 ) , (1 ) , e e e
u uEF b l F b l EJ F m l
t x t t
u u u
x x x x
+ − + − − + −
= − = −
=
Timoshenko’s equations
2 4 22 2 2 2 2 21 1 2 2
1 12
2
2 4 2
1 1
(1 ) , (1 ) e e
u u u uEF b l EJ b l
t tx x
− = −
+ =
+
Euler-Bernoulli equations
Merida, Yucatan, Mexico
A High Order Theory for Functionally
Graded Beams, Plates and Shells
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
3-D formulation
( ) ( ) 0 0 0( ) ( ) ( ) 0 , ,ij i ij ij ij ij
j kl k lL u E L Lb V L c+ = = = x x x x x x
Cauchy relations
1 1( ) ( )
2 2
k
ij i i j i i i j i ij ku u u u u = + = + −
0, ij i ij ij i kj j ik
j j j jk jkb + = = + +
Equations of equilibrium
( ) ( )
( ) ( )
0
00 0 0 0 0
( ) ( ) ( ), ,
1 22 , ,
2 1 1 2
ij ij ij ij
kl kl kl kl
ij ij kl il jk
kl
c c c
c
E
g g g g
==
= + = =+ −
x x x x x
Generalized Hook’s law
Boundary-value problem
( ) ( ) , ,
( ) ( ) ( ) [ ( )] ( ) ,
i i u
i ij ij i
j j p
u V
p n P u V
=
= = =
x x x
x x x x x x
3-D formulation in special coordinates
Cauchy relations
1 1 2 211 2 1 3 22 1 2 3
1 1 1 2 2 2 2 2 1 1
3 1 2 2 133 12 2 1
3 2 2 1 1 1 1 2 2
3 31 213 1 1 23 2 2
3 1 1 3 2 2
1 1 1 1, ,
1 1 1 1,
1 1,
u A u Au k u u k u
A x A A x A x A A x
u u A u Au u
x A x A x A x A x
u uu uk u k u
x A x x A x
= + + = + +
= = − + −
= − + = − +
132 11 1 12 1 21 2 12 13 1 2 1 22 1 2 1
1 2 3 2 1
232 21 1 22 2 11 2 21 23 1 2 2 11 1 2 2
1 2 3 1 2
2 31 1 32 1 2 3311 1 2 1 2
1 2 3
( ) ( )0,
( ) ( )0,
( ) ( ) ( )
A A A AA A A A k A A b
x x x x x
A A A AA A A A k A A b
x x x x x
A A A AA A k
x x x
+ + + + − + =
+ + + + − + =
+ + − −
2 1 2 2 1 2 3 0.A A k A A b+ =
Equations of equilibrium
3( ) ( ) ( )x = +R x r x n x
2-D formulation
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3
0
3 3
0
3 3
0
2 1( ) , , ( )
2
2 1( ) , , ( )
2
2 1( ) , , (
,
2
,
.)
k k
i i k i i k
k h
k k
ij ij k ij ij k
k h
k k
ij ij k ij ij k
k h
h
h
h
ku u P u u x P dx
h
kP x P dx
h
kP x P dx
h
= −
= −
= −
+= =
+= =
+= =
x x x x
x x x x
x x x x
Expansion into Legendre’s polynomial series
( ) ( ) ( )0
3
1
, ( ) ( ) ( )n nrm m nrm
ij ijkl kl m
r m h
h
r
n rc E P P P dx
= −
== x x x
( ) =E L u f
2-D Hook’s law
3( ) ( ) nm
m
h
n
h
P P dx −
=
Equations of equilibrium in displacements
00 01 00 01 0 0
10 11 10 11 1 1, , ,
ij ij ij ij j j
ij ij ij ij j j
E E L L u f
E E L L u f= = = =E L u f
2-D approximation equations
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 1
0 1
0 1
0 1
0 1
0 1
,
,
,
ij ij ij
ij ij ij
i i i
P P
P P
u u P u P
= +
= +
= +
x x x
x x x
x x x
First approximation equations
Second approximation equations
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
0 1 2
0 1 2
0 1 2
0 1 2
0 1 2
0 1 2
,
,
,
ij ij ij ij
ij ij ij ij
i i i i
P P P
P P P
u u P u P u P
= + +
= + +
= + +
x x x x
x x x x
x x x x
00 00 01 0 0
11 13 13 1 1
00 00 01 01 0 0
00 01 0231 33 31 33 3 3
10 11 11 12 1 1
10 1111 11 13 13 1 1
10 10 11 11 12 1 1
31 33 31 33 33 3 3
21 22 22 2 2
13 11 13 1 1
21 21 22 22 2 2
31 33 31 33 3 3
0 0 0
0 0
0 0, , ,
0
0 0 0
0 0
ij ij ij
ij ij
L L L u f
L L L L u fE E E
L L L L u fE E E
L L L L L u f
L L L u f
L L L L u f
= = = =L u f12
02 12 22
ij
ij ij ij
E
E E E
00 00 01 0 0
11 13 13 1 1
00 0100 00 01 01 0 0
31 33 31 33 3 3
10 1110 11 11 1 1
13 11 13 1 1
10 10 11 11 1 1
31 33 31 33 3 3
0
, , ,0
ij ij
ij ij
L L L u f
E EL L L L u fE
E EL L L u f
L L L L u f
= = = =L u f
Zozulya, V. V. A high order Theory for Functionally Graded Shell, World Academy of Science,
Engineering and Technology, 2011, Vol. 59, 779-784.
Finite element formulation
Lagrangean type functional
( ) 0 ( ) ( ) ( ) ( ) ( )1
( ) ( )2
p
lj ki i i i i i
V V V
jklJ dV u dS Vuc dbE
= − − u xx x x x x x
Approximation of the displacements and strain
1 1
1
( ) ( )( )
1 , ,( ) ( )
( )( ( ) ( ) ( )
2
1( ) )
2( )
e
e e e
Q Qq qq q q
i i q ij i j
q q i j
Nq q T q q q q q
e V V V
E d
N Nu u N u u
x x
J V dVdS
= =
=
+
= − −
Nu D u C D u ψ u f N u
x xx x x
x x x x x
1
1 2
1 2
1 1 1 1
1 2 1 2
1
1 1
1
2 2
( ) ( ) ( )
( ) ( ) ,
,
00 0,
0 0 0
, ,
e
e e
bN N
e
NT
e V
q q
V Ve
Q
Q
T Tq Q Q q Q Q
dS
N N N
NN N
u u u u f
E dV
d
f f
V
f
= =
=
=
=
=
=
+
=
N N
K D C D
f ψ f
N
u f
x x x
x x
Some notations in 2-D case
Finite elements equations( )
0,q
q
i
J
u
= → =
uK u f
1 2
1 1 1
1 2
2 2 2
1 1 2 2
2 1 2 1 2 1
( )( ) ( )0 0 0
( )( ) ( )( ) 0 0 0
( ) ( )( ) ( ) ( ) ( )
Q
Q Q
NN N
x x x
NN N
x x x
N NN N N N
x x x x x x
=D
xx x
xx xx
x xx x x x
Constutuve relations for functionally graded materials
( ) 33
3 2 1 1 3 1( 0), ( ) , ( )2
n
xx hV n E x E E V E E x E e
h
+ = = − + =
Volume fraction power and exponential law distribution
( ) ( ) ( ) 2
2 1 2 1 2 12 3 2
1
2
15 ( 1)
, , ,1 2 3 (1 )(2
1log
)(3 ) 2
E E n E E nh E E n nhE E E
n n n n n n
E
h E
+ − − −−
+ + +
= = = =
+ + +
Volume fraction distribution versus non-dimensional thickness for various exponents
Legendre polynomials coefficients for the effective Young’s modulus
( ) ( )2 2
1 2 3
2 3
5 (3 )sinh( ) 3 cosh( )3 cosh( ) sinh( )sinh( ), , .
h h h hh h hhE E E
h hh
+ −−= = =
Power law
Exponential law
Axisymmetric cylindrical shell
Axisymmetric equations of equlibrium and Cauchy relations
13111
1 3
31 33 22
1 3
3
0
0
+ + =
+ − + =
bx x
bx x R
3111 22
1
3 133 13
3 3
3
1
, ,
,
= =
= = +
uu
x R
u uu
x x x
31 2
1113 1
1
33 3
1
2
0,
0,
− + =
− − + =
kk k
k kk k
fx
fx R
1-D equations for cylindrical shell
( ) ( ) ( )( )
3 3111 22 33 13
1 1
1
1 1
3
3 1
1
,,
2 1.
, ,
+ +
= = = =
++ +
+
=
k kk
kk k k k
k k k
i
k
i i
uu uu
x R x
ku u
h
u
x xux
( ) ( ) ( ) ( )( )
( ) ( ) ( )( )
1 1 1 1
1 1 1
3 3
1 3
3 3 3
2 1( 1) ,
2 1.
+ −
− −
+= + + −
+= + +
k k k
i i i i
k k k
i i i
kf b
h
k
h
x x x x
x x x
V.V. Zozulya , Ch. Zhang, A High Order Theory For Functionally Graded Axisymmetric Cylindrical
Shells, International Journal of Mechanical Sciences, 2012, , 60(1), 12-22.
Zozulya V. V. A high-order theory for functionally graded axially symmetric cylindrical shells, Archive of
Applied Mechanics, 2013, 83(3), 331–343.
First order approximation
Second order approximation
0 1
00 00 01 0 0
11 13 13 1 10 100 00 01 01 0 0
31 33 31 33 3 3
10 11 11 1 11 0 13 11 13 1 1
10 10 11 11 1 1
31 33 31 33 3 3
1 0
22 0 0
302
0 2 03
, , , .2 2 0
0 03 3
2 20 0
3 3
E E
L L L u fE E
L L L L u f
L L L u fE E
L L L L u f
E E
= = = =E L u f
0 1 2
0 1 2
1 0 2 1
1 0 2 1
2 1 0 2
2 1 0 2
2 22 0 0 0
3 5
2 20 2 0 0
3 5
2 2 4 40 0 0
3 3 15 15,
2 2 4 40 0 0
3 3 15 15
2 4 2 40 0 0
5 15 5 35
2 4 2 40 0 0
5 15 5 35
E E E
E E E
E E E E
E E E E
E E E E
E E E E
+
=
+
+
+
E
00 00 01 0 0
11 13 13 1 1
00 00 01 01 0 0
31 33 31 33 3 3
10 11 11 12 1 1
11 11 13 13 1 1
10 10 11 11 12 1 1
31 33 31 33 33 3 3
21 22 22 2 2
13 11 13 1 1
21 21 22 22 2 2
31 33 31 33 3 3
0 0 0
0 0
0 0, , .
0
0 0 0
0 0
L L L u f
L L L L u f
L L L L u f
L L L L L u f
L L L u f
L L L L u f
= = =L u f
Results and discussion
31
1 3 1 2 1Dimetionless coordinates and ,Young’s modulus 1 Pa and / 2,
Paramiters of shell 0.125 and 0.1 .
xxE E E
L h
R L h R
= = = =
= =
Axisymmetrical cylindrical shell
Displacements and stresses versus normalized length and thickness.
First approximation Second approximation
Displacements and stresses versus Τℎ 𝑅 for various exponents
A higher order functionally graded beams
2-D equations of equlibrium and Cauchy relations in
Cartecian coordinates
11 121
1 2
21 222
1 2
0
0
bx x
bx x
+ + =
+ + =
111
1
112
222
2
2
2 1
, ,
1.
2
u u
x x
u u
x x
= =
= +
1112 1
1
2122 2
1
0 ,
0 .
kk k
kk k
fx
fx
− + =
− + =
1-D equations for straight beam
( ) ( ) ( )( )
1 211 12 22
1 1
1 3
1 2
1 1 1
.
2 1
, ,
.
,k k
k k k
k k k
i i
k
i
ku u
x x
u u
x x
ku xu u
h
+ +
= = =
+= +
+
+
( ) ( ) ( ) ( )( )
( ) ( ) ( )( )
1 1 1 1
1 1 1
2 2
1 3
2 2 2
2 1( 1) ,
2 1.
k k k
i i i i
k k k
i i i
x x x x
x x x
kf b
h
k
h
+ −
− −
+= + + −
+= + +
Zozulya V.V. A higher order theory for functionally graded beams based on Legendre’s polynomial expansion.
Mechanics of Advanced Materials and Structures, 2016, 24(9), 745-760.
Finite element formulation
Lagrangean type functional
11 1 1 1 1
1
1 1 11 1 1 1 1 1 1
1 1 1
( )( ) ( ) ( ) (( )
)
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
l
l
l
l
l
l
dJ
d
d d d
d
xx x x x dx
x
x x xx x x x x x
d ddx
x x x
−
−
−
−
+
=
− −
− +
Cu
u u b u
u u uu uC A uB
Approximation of the displacements, ,
1 11
111
1
1
( )( )( ( ,) )
kQ Qqk k q k qi
i i q i
q q
dNduu u N u
dx d
xxx x
x= =
1 1 11 1 1
1 1
1
1
1
1
1 1 1
,
( ) ( ) ( )( ) ( ) ( )
( ) ( )
T TlN
e l
ll
le
q q
l
N
x x xx x x
x x d
d d ddx
dx dx
x
dx
−
=
−=
−
= +
+
+ =
N N N
K C B N
N
N A N
f ψ b
Some notations in 2-D case
Finite elements equations( )
0,q
q
i
J
u
= → =
uK u f
First order approximation
Second order approximation
00 00 01 0 00 1
11 12 12 1 1
00 00 01 01 0 00 1
21 22 21 22 2 2
10 11 11 1 11 0
12 11 12 1 1
10 10 11 11 1 11 0
21 22 21 22 2 2
00 0
0 0, , , ..
00 0
0 0
L L L u fE E
L L L L u fE E
L L L u fE E
L L L L u fE E
= = = =E L u f
0 1 2
0 1 2
1 0 2 1
1 0 2 1
2 1 0 2
2 1 0 2
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
E E E
E E E
E E E E
E E E E
E E E E
E E E E
+=
+
+
+
E
00 00 01 0 0
11 12 12 1 1
00 00 01 01 0 0
21 22 21 22 2 2
10 11 11 12 1 1
11 11 12 12 1 1
10 10 11 11 12 1 1
21 22 21 22 22 2 2
21 22 22 2 2
12 11 12 1 1
21 21 22 22 2 2
21 22 21 22 2 2
0 0 0
0 0
0 0, , .
0
0 0 0
0 0
L L L u f
L L L L u f
L L L L u f
L L L L L u f
L L L u f
L L L L u f
= = =L u f
1-D equations in displacements
00 01 00 01 0 0
10 11 10 11 1 1 0, , , , .
0
ij ij ij ij j j nm
nm
ij ij ij ij j j ij nm
E E L L u fE
E E L L u f EE
= = = = =E L u f
( ) , =E L u f
High order theory
Results and discussion31
1 3 1 2 1Dimetionless coordinates and ,Young’s modulus 1 Pa and / 2,
Paramiters of beam: widness 1, and 0.1 .
xxE E E
L h
b h L
= = = =
= =
Displacements and stresses versus normalized length and thickness.
First approximation Second approximation
Free supported Free supportedClamped-clamped Clamped-clamped
Free supported
Clamped-clamped
Free supported
Clamped-clamped
Merida, Yucatan, Mexico
APPLICATION OF THE BOUNDARY
INTEGRAL EQUATION METHOD TO
THE ARBITRARY GEOMETRY
SHELLS
By Professor Volodymyr Zozulya
Centro de Investigacion Cientifica de Yucatan, A.C.
BIE for shells of arbitrary geometry
,,])([
,0)()(
k
k
ijjijilk
jkilklijij
i
j
ij
uuuggggA
VbuA
−=++=
=+
xxx
3-D differential equations
3-D fundamental solutions in Cartesian coordinates
2
2
)1(8
))21(())(21())(),((
))()())(()((,)1(16
)43())()((
R
RRRRnRnW
zzzzRR
RRU
njiijjiij
ij
iiii
jiij
ij
−
+−−−−=
−−=−
+−=−
yzxz
yxyxyzxz
3-D fundamental solutions in curvilinear coordinates
0,0and,0,,
))(),(()()(),()),()(()()()(
===
=−=−
i
j
i
j
k
j
i
k
jii
j
jii
j
k
l
l
j
i
k
i
jkl
l
j
k
iij
bdetadetaaxzbzxa
WabWUaaU yzxzyxyxyzxzyxyx
Reciprocal theorem of Betty and Raleigh
),,(),,( ii
i
ii
i bpuLbpuL =
+=V
i
i
V
i
iii
i dSupdVubbpuL )()(2
1)()(
2
1),,( xxxx
Somigliano’s integral representation in curvilinear coordinates
−+−−−= V
ji
j
V
j
ijji
j
i dVUpdSWuUpu )()())()()()(()( yxxyxxyxxy
2-D BIE for shells of arbitrary geometry
−
=
+==
h
h
ni
n
i
n
n
n
ii dxPxuh
nuPuu 33
0
)(),(2
12)(,)()()( xxxx
Expansion into Legendre’s polynomial series
2-D reciprocal theorem of Betty and Raleigh
),,(),,(ni
ni
i
nn
i
n
ii
n
bpuLbpuL−−−
=
=
=
+= dlupdufbpuLn
n
i
ni
n
n
i
ni
ni
ni
i
n
00
)()(2
1)()(
2
1),,( xxxx
2-D Somigliano’s integral representation for shells of arbitrary geometry
)0,1,2,..., (m,,)(),()(),()(),()(00
=+
−=
=
=
yxyxxyxxyxy dSfUdluWpUun
ni
nm
ij
n
i
nmi
j
n
ni
nm
ij
m
i
2-D fundamental solutions
−−
−−
+
+=
−+
+
=
h
h
mn
j
i
h
h
nmi
j
h
h
mnji
h
h
nm
ij
dydxPPWh
m
h
nW
dydxPPUh
m
h
nU
33
33
)()()),(2
12
2
12),(
)()()(2
12
2
12),(
yxyx
yxyx
2-D BIE for shells of arbitrary geometry
)0,1,2,..., (,,)(),()(),()(),()(2
1
00
=+
−=
=
=
mdSfUdluWpUun
ni
nm
ij
n
i
nmi
j
n
ni
nm
ij
m
i yxyxxyxxyxy
BIE for 2-D elastostatics
−+−−= V
ijj
V
ijjijji dVUbdSWuUpu )()()),()()()(()( yxxyxxyxxy
Fundamental solutions
),)ln()43(()1(8
1)( rrrU jiijij −−
−
−=−
yx
( )rrrrnrnr
W njiijjiijij +−−−−−
= )2)21(())(21()1(4
1),(
yx
Expansion into Legendre’s polynomial series
−
=
+==
h
h
nj
n
jn
n
n
jj dxPxxuh
nxuPxuu 2211
01 )(),(
2
12)(,)()()( x
−
=
+==
h
h
nj
n
jn
n
n
jj dxPxxph
nxpPxpp 2211
01 )(),(
2
12)(,)()()( x
−
=
+==
h
h
nj
n
jn
n
n
jj dxPxxbh
nxbPxbb 2211
01 )(),(
2
12)(,)()()( x
For 0-approximation
−
==h
h
jjjj dxxxuh
xuxuu 2211
0
1
0
1),(2
1)(,1)()(x
−
==h
h
jjjj dxxxph
xpxpp 2211
0
1
0
1),(2
1)(,1)()(x
−
==h
h
jjjj dxxxbh
xbxbb 2211
0
1
0
1),(2
1)(,1)()(x
Transformation to the BIE
For 0-approximation
=
−+
=
=
=
=
=
−−+−+
+
−−−=
],[ 0
111
0
1
0
1
0
1
)()1()()()()(
)()()()()()()()(
1
1
ba n
j
n
jn
n
jij
bx
ax
ij
n
n
n
jij
n
n
n
j
n
n
n
i
dVxpxpPxbU
WPxuUPxpPyu
yx
yxyx
+
−==
=
= V
jj
bx
ax
jjjj dSyxUxfyxWxuyxUxpyui ),()(),()(),()()( 1 11
00
11
0
11
00
11
0
11
00
11
0
1
0
1
1
1
(14)
+
−==
=
= V
jj
bx
ax
jjjj dSyxUxfyxWxuyxUxpyui ),()(),()(),()()( 2 11
00
21
0
11
00
21
0
11
00
21
0
1
0
2
1
1
Fundamental solutions
=
=
=−
0 0
2211 )()(),()(
n mmnij
nm
ijh
yP
h
xPyxUU yx
=
=
=
0 0
2211 )()(),(),(
n mmnij
nm
ijh
yP
h
xPyxWW yx
− −
−+
+
=h
h
h
h
mnij
nm
ij dydxh
yP
h
xPU
h
m
h
nyxU 22
2211 )()())(
2
12
2
12),( yx
(11)
− −
+
+=
h
h
h
h
mnij
nm
ij dydxh
yP
h
xPW
h
m
h
nyxW 22
2211 )()()),(
2
12
2
12),( yx
, (*)
− −− −
==−==−
h
h
h
h
ijijijij
h
h
h
h
ijijijij dydxWhh
yxWyxWWdydxUhh
yxUyxUU 2211
00
11
00
2211
00
11
00
)),(2
1
2
1),(,),(),(,))(
2
1
2
1),(,),()( yxyxyxyx
For 0-approximation
Fundamental solutions
( )( ) − − − −
−−−
−
+
+
−
−=
h
h
h
h
mnjjii
h
h
h
h
mnij
nm
ij dydxh
yP
h
xPyxyx
rdydx
h
yP
h
xPr
h
m
h
nyxU 22
22
22222
11
1)ln()43(
2
12
2
12
)1(8
1),(
− −
−
−−+−−
−−
−−
+
+
−=
h
h
h
h
mn
jjiiij
jj
iii
j
nm
ij dydxh
yP
h
xPn
r
yx
r
yx
r
yx
r
yxn
r
yxn
rh
m
h
nyxW 22
221
1111 )()(2)21()21(
1
2
12
2
12
)1(4
1),(
Integrals to be calculated
− −
=
h
h
h
h
mnmn dydxh
yP
h
xPrJ 22
22, )ln(
,
( )22
22
2
22,,,2 dydx
h
yP
h
xP
r
yxJ mn
h
h
h
h
k
mnk
−=
− −
,
( )22
22
4
22,,,4 dydx
h
yP
h
xP
r
yxJ mn
h
h
h
h
k
mnk
−=
− −
,
The BE equations for 0-approximation
FzA =
),(),(),(),(),(),(),(),(
),(),(),(),(),(),(),(),(
),(),(),(),(),(),(),(),(
),(),(),(),(),(),(),(),(
22
00
21
00
22
00
21
00
22
00
21
00
22
00
21
00
12
00
11
00
12
00
11
00
12
00
11
00
12
00
11
00
22
00
21
00
22
00
21
00
22
00
21
00
22
00
21
00
12
00
11
00
12
00
11
00
12
00
11
00
12
00
11
00
bbUbbUaaUbaUbbWbbWaaWbaW
bbUbbUaaUbaUbbWbbWaaWbaW
abUabUaaUaaUabWabWaaWaaW
abUabUaaUaaUabWabWaaWaaW
−−−−−
−−−−−
−−−−−
−−−−−
=
A
)(
)(
)(
)(
)(
)(
)(
)(
2
01
02
01
0
2
0
1
0
2
0
1
0
bp
bp
ap
ap
bu
bu
au
au
=z
1
],[
1
00
221
0
21
00
211
0
1
1
],[
1
00
121
0
21
00
111
0
1
1
],[
1
00
221
0
21
00
211
0
1
1
],[
1
00
121
0
21
00
111
0
1
),()(),()(
),()(),()(
),()(),()(
),()(),()(
dxbxUxfbxUxf
dxbxUxfbxUxf
dxaxUxfaxUxf
dxaxUxfaxUxf
ba
ba
ba
ba
+−
+−
+−
+−
=F
Very simple example
,
),(2/1),(
),(),(
11
00
11
00
11
00
11
00
baUbbW
aaUbbWA
−−−
−−=
)(
)(
1
0
1
0
ap
buz =
),(
),(
200
11
00
11
bbU
abU
h
Q−=F
The BE equations
QdxPxQh
Qh
h
=−= −
202
0
)()0(2
1
The fundamental solutions
)4ln(3)1(16
)43(),(),( 2
11
00
11
00
haaUbbU −−
−==
( ) − − − −
−−−−
−==
h
h
h
h
h
h
h
h
dydxr
badydxrhh
abUbaU 222
2
22
00
11
00
11
1)ln()43(
2
1
2
1
)1(8
1),(),(
( )( )
− −− −
−+−−
−−=−=
h
h
h
h
h
h
h
h
dydxr
badydxrhh
baabWbaW 224
2
222
00
11
00
11
12
1)21(
2
1
2
1
)1(4),(),(
)1(4),(),(
00
11
00
11
−
−=−=
hbbWaaW
Zozulya V.V. Somigliana identity and fundamental solutions for arbitrary geometry shells, Docl. Akad. Nauk Ukraine, 1997, N 6, P. 60-65. (in
Russian).
Zozulya V.V. Integral boundary equations for shells of arbitrary geometry, International Applied Mechanics, 1998, V.34, N 5, P. 454-463.
Zozulya V.V. Boundary Integral Equations for Arbitrary Geometry Shells. In: Integral Methods in Science and Engineering. Computational and
Analytic Aspects. (Eds. C.Constanda and P.J. Harris), Birkhäuser, 2011, pp. 430-441.
Thank you very much for your
attention