Abstract

This research deals with the stability analysis of shallow segments of the toroidal shell made of saturated porous functionally graded (FG) material. The nonhomogeneous material properties of porous shell are assumed to be functionally graded as a function of the thickness and porosity parameters. The porous toroidal shell segments with positive and negative Gaussian curvatures and nonuniform distributed porosity are considered. The nonlinear equilibrium equations of the porous shell are derived via the total potential energy of the system. The governing equations are obtained on the basis of classical thin shell theory and the assumptions of Biot's poroelasticity theory. The equations are a set of the coupled partial differential equations. The analytical method including the Airy stress function is used to solve the stability equations of porous shell under mechanical loads in three cases. Porous toroidal shell segments subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. Closed-form solutions are expressed for the elastic buckling behavior of the convex and concave porous toroidal shell segments. The effects of porosity distribution and geometrical parameters of the shell on the critical buckling loads of porous toroidal shell segments are studied.

References

1.
Biot
,
M. A.
,
1964
, “
Theory of Buckling of Porous Slab and Its Thermoelastic Analogy
,”
ASME J. Appl. Mech.
,
31
(
2
), pp.
194
198
.10.1115/1.3629586
2.
Stein
,
M.
, and
McElman
,
J. A.
,
1965
, “
Buckling of Segments of Toroidal Shells
,”
J. AIAA
,
3
(
9
), pp.
1704
1709
.10.2514/3.55185
3.
Hutchinson
,
J. W.
,
1967
, “
Initial Postbuckling Behavior of Toroidal Shell Segments
,”
J. Solid Struct.
,
3
(
1
), pp.
97
115
.10.1016/0020-7683(67)90046-7
4.
Sobel
,
L. H.
, and
Flugge
,
W.
,
1967
, “
Stability of Toroidal Shells Under Uniform External Pressure
,”
J. AIAA
,
5
(
3
), pp.
425
431
.10.2514/3.3997
5.
Jordan
,
P. F.
,
1973
, “
Buckling of Toroidal Shells Under Hydrostatic Pressure
,”
J. AIAA
,
11
(
10
), pp.
1439
1441
.10.2514/3.6937
6.
Radhamohan
,
S. K.
, and
Prasad
,
B.
,
1974
, “
Asymmetric Buckling of Toroidal Shells Under Axial Tension
,”
J. AIAA
,
12
(
4
), pp.
511
515
.10.2514/3.49277
7.
Magnucki
,
K.
, and
Stasiewicz
,
P.
,
2004
, “
Elastic Buckling of a Porous Beam
,”
J. Theor. Appl. Mech.
,
42
(
4
), pp.
859
868
.https://www.researchgate.net/publication/268416947_Elastic_buckling_of_a_porous_beam
8.
Magnucka-Blandzi
,
E.
,
2008
, “
Axi-Symmetrical Deflection and Buckling of Circular Porous-Cellular Plate
,”
Thin-Walled Struct.
,
46
(
3
), pp.
333
337
.10.1016/j.tws.2007.06.006
9.
Chen
,
D.
,
Yang
,
J.
, and
Kitipornchai
,
S.
,
2015
, “
Elastic Buckling and Static Bending of Shear Deformable Functionally Graded Porous Beam
,”
Compos. Struct.
,
133
, pp.
54
61
.10.1016/j.compstruct.2015.07.052
10.
Chen
,
D.
,
Yang
,
J.
, and
Kitipornchai
,
S.
,
2016
, “
Free and Forced Vibrations of Shear Deformable Functionally Graded Porous Beams
,”
Int. J. Mech. Sci.
,
108-109
, pp.
14
22
.10.1016/j.ijmecsci.2016.01.025
11.
Chen
,
D.
,
Kitipornchai
,
S.
, and
Yang
,
J.
,
2016
, “
Nonlinear Free Vibration of Shear Deformable Sandwich Beam With a Functionally Graded Porous Core
,”
Thin-Walled Struct.
,
107
, pp.
39
48
.10.1016/j.tws.2016.05.025
12.
Babaei
,
H.
,
Eslami
,
M. R.
, and
Khorshidvand
,
A. R.
,
2020
, “
Thermal Buckling and Postbuckling Responses of Geometrically Imperfect FG Porous Beams Based on Physical Neutral Plane
,”
J. Therm. Stresses
,
43
(
1
), pp.
109
131
.10.1080/01495739.2019.1660600
13.
Bich
,
D. H.
,
Ninh
,
D. G.
, and
Thinh
,
T. I.
,
2015
, “
Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments Under Mechanical Loads
,”
J. Eng. Mech.
, 142(1), pp.
1
10
. https://ascelibrary.org/doi/10.1061/%28ASCE%29EM.1943-7889.0000964
14.
Ninh
,
D. G.
,
Bich
,
D. H.
, and
Kien
,
B. H.
,
2015
, “
Torsional Buckling and Postbuckling Behavior of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments Surrounded by an Elastic Medium
,”
Acta Mech.
,
226
(
10
), pp.
3501
3519
.10.1007/s00707-015-1391-6
15.
Bich
,
D. H.
, and
Ninh
,
D. G.
,
2016
, “
Post-Buckling of Sigmoid-Functionally Graded Material Toroidal Shell Segment Surrounded by an Elastic Foundation Under Thermo-Mechanical Loads
,”
Compos. Struct.
,
138
, pp.
253
263
.10.1016/j.compstruct.2015.11.044
16.
Ninh
,
D. G.
, and
Bich
,
D. H.
,
2016
, “
Nonlinear Buckling of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments Under Torsional Load Surrounded by an Elastic Foundation in Thermal Environment
,”
Mech. Res. Commun.
,
72
, pp.
1
15
.10.1016/j.mechrescom.2015.12.002
17.
Thang
,
P. T.
, and
Nguyen-Thoi
,
T.
,
2016
, “
A New Approach for Nonlinear Dynamic Buckling of S-FGM Toroidal Shell Segments With Axial and Circumferential Stiffeners
,”
Aerosp. Sci. Tech
,
53
, pp.
1
9
.10.1016/j.ast.2016.03.008
18.
Bich
,
D. H.
,
Ninh
,
D. G.
, and
Thinh
,
T. I.
,
2016
, “
Non-Linear Buckling Analysis of FGM Toroidal Shell Segments Filled Inside by an Elastic Medium Under External Pressure Loads Including Temperature Effects
,”
Compos. Part B: Eng.
,
87
, pp.
75
91
.10.1016/j.compositesb.2015.10.021
19.
Bich
,
D. H.
,
Ninh
,
D. G.
,
Kien
,
B. H.
, and
Hui
,
D.
,
2016
, “
Nonlinear Dynamic Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments Surrounded by Elastic Foundation in Thermal Environment
,”
Compos. Part B: Eng.
,
95
, pp.
355
373
.10.1016/j.compositesb.2016.04.004
20.
Ninh
,
D. G.
, and
Bich
,
D. H.
,
2016
, “
Nonlinear Thermal Vibration of Eccentrically Stiffened ceramic-FGM-Metal Layer Toroidal Shell Segments Surrounded by Elastic Foundation
,”
Thin-Walled Struct.
,
104
, pp.
198
210
.10.1016/j.tws.2016.03.018
21.
Tung
,
H. V.
, and
Hieu
,
P. T.
,
2018
, “
Nonlinear Buckling of CNT-Reinforced Composite Toroidal Shell Segment Surrounded by an Elastic Medium and Subjected to Uniform External Pressure
,”
Vietnam J. Mech.
,
40
(
3
), pp.
285
301
.10.15625/0866-7136/12397
22.
Vuong
,
P. M.
, and
Duc
,
N. D.
,
2018
, “
Nonlinear Response and Buckling Analysis of Eccentrically Stiffened FGM Toroidal Shell Segments in Thermal Environment
,”
Aerosp. Sci. Technol.
,
79
, pp.
383
398
.10.1016/j.ast.2018.05.058
23.
Hieu
,
P. T.
, and
Tung
,
H. V.
, “
Thermomechanical Nonlinear Buckling of Pressure-Loaded Carbon Nanotube Reinforced Composite Toroidal Shell Segment Surrounded by an Elastic Medium With Tangentially Restrained Edges
,”
Proc. Inst. Mech. Eng., Part C
, 233(9), pp.
3193
3207
.10.1177/0954406218802942
24.
Ali
,
A. Y.
, and
Hasan
,
H. M.
,
2019
, “
Nonlinear Dynamic Stability of an Imperfect Shear Deformable Orthotropic Functionally Graded Material Toroidal Shell Segments Under the Longitudinal Constant Velocity
,”
Proc. Inst. Mech. Eng., Part C
,
233
(
19–20
), pp.
6827
6850
.10.1177/0954406219867991
25.
Phuong
,
N. T.
,
Nam
,
V. H.
,
Trung
,
N. T.
,
Duc
,
V. M.
,
Loi
,
N. V.
,
Thinh
,
N. D.
, and
Tu
,
P. T.
,
2019
, “
Thermomechanical Postbuckling of Functionally Graded Graphene-Reinforced Composite Laminated Toroidal Shell Segments Surrounded by Pasternak's Elastic Foundation
,”
J. Thermoplast. Compos. Mater
.10.1177/0892705719870593
26.
Vuong
,
P. M.
, and
Duc
,
N. D.
,
2019
, “
Nonlinear Vibration of FGM Moderately Thick Toroidal Shell Segment Within the Framework of Reddy's Third Order-Shear Deformation Shell Theory
,”
Int. J. Mech. Mater. Des.
, 16, pp.
245
264
.10.1007/s10999-019-09473-x
27.
Vuong
,
P. M.
, and
Duc
,
N. D.
,
2019
, “
Nonlinear Buckling and Postbuckling of a FGM Toroidal Shell Segment Under a Torsional Load in a Thermal Environment Within Reddy's Third-Order Shear Deformation Shell Theory
,”
Mech. Compos. Mater.
,
55
(
4
), pp.
467
482
.10.1007/s11029-019-09826-9
28.
Zhang
,
J.
,
Wang
,
X.
,
Tang
,
W.
,
Wang
,
F.
, and
Yin
,
B.
,
2020
, “
Experimental and Numerical Buckling Analysis of Toroidal Shell Segments Under Uniform External Pressure
,”
Thin-Walled Struct.
,
150
, p.
106689
.10.1016/j.tws.2020.106689
29.
Sofiyev
,
A. H.
,
2003
, “
Dynamic Buckling of Functionally Graded Cylindrical Thin Shells Under Non-Periodic Impulsive Loading
,”
Acta Mech.
,
165
(
3–4
), pp.
151
163
.10.1007/s00707-003-0028-3
30.
Sofiyev
,
A. H.
, and
Schnack
,
E.
,
2004
, “
The Stability of Functionally Graded Cylindrical Shells Under Linearly Increasing Dynamic Torsional Loading
,”
Eng. Struct.
,
26
(
10
), pp.
1321
1331
.10.1016/j.engstruct.2004.03.016
31.
Sofiyev
,
A. H.
,
2005
, “
The Stability of Compositionally Graded Ceramic–Metal Cylindrical Shells Under Aperiodic Axial Impulsive Loading
,”
Compos. Struct.
,
69
(
2
), pp.
247
257
.10.1016/j.compstruct.2004.07.004
32.
Sofiyev
,
A. H.
,
2010
, “
Buckling Analysis of FGM Circular Shells Under Combined Loads and Resting on the Pasternak Type Elastic Foundation
,”
Mech. Res. Commun.
,
37
(
6
), pp.
539
544
.10.1016/j.mechrescom.2010.07.019
33.
Shen
,
H. S.
,
2011
, “
Postbuckling of Nanotube-Reinforced Composite Cylindrical Shells in Thermal Environments: Part I—Axially-Loaded Shells
,”
Compos. Struct.
,
93
(
8
), pp.
2096
2108
.10.1016/j.compstruct.2011.02.011
34.
Shen
,
H. S.
,
2011
, “
Postbuckling of Nanotube-Reinforced Composite Cylindrical Shells in Thermal Environments: Part II—Pressure-Loaded Shells
,”
Compos. Struct.
,
93
(
10
), pp.
2496
2503
.10.1016/j.compstruct.2011.04.005
35.
Shen
,
H. S.
,
2012
, “
Thermal Buckling and Postbuckling Behavior of Functionally Graded Carbon Nanotube-Reinforced Composite Cylindrical Shells
,”
Compos. Part B: Eng.
,
43
(
3
), pp.
1030
1038
.10.1016/j.compositesb.2011.10.004
36.
Brush
,
D. O.
, and
Almorth
,
B. O.
,
1975
,
Buckling of Bars, Plates and Shells
,
Mc. Graw-Hill
,
New York
.
37.
Jabbari
,
M.
,
Mojahedin
,
A.
,
Khorshidvand
,
A. R.
, and
Eslami
,
M. R.
,
2014
, “
Buckling Analysis of Functionally Graded Thin Circular Plate Made of Saturated Porous Materials
,”
J. Eng. Mech.
,
140
(
2
), pp.
287
295
.10.1061/(ASCE)EM.1943-7889.0000663
38.
Reddy
,
J. N.
,
2003
,
Mechanics of Laminated Composite Plates and Shells, Theory and Application
,
CRC Press
,
Boca Raton, FL
.
39.
Eslami
,
M. R.
,
2018
,
Buckling and Postbuckling of Beams, Plates, and Shells
,
Springer
,
Switzerland
.
40.
Jabbari
,
M.
,
Farzaneh Joubaneh
,
E.
,
Khorshidvand
,
A. R.
, and
Eslami
,
M. R.
,
2014
, “
Buckling Analysis of Porous Circular Plate With Piezoelectric Actuator Layers Under Uniform Radial Compression
,”
Int. J. Mech. Sci.
, 70, pp.
50
56
.10.1016/j.ijmecsci.2013.01.031
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