This paper studies how to improve the third-order shear deformation theories of isotropic plates, which is a question raised by late Reissner in 1985 (ASME Appl. Mech. Rev., 38, pp.1453–1464). It is demonstrated in this paper that a proper displacement field with the higher-order shear deformations given by the method of displacement assumption should satisfy the constraint on the consistence of the transverse shear strain energy a priori in addition to the traction conditions on plate surfaces. This additional constraint on the assumed displacement fields with the higher-order shear deformations is in line with Love's criterion of the consistent first approximation to the strain energy wherein the transverse shear strain energy is included. The constraint on the consistence of the transverse shear strain energy is physically similar to the requirement for the use of the shear coefficients in the first-order shear deformation plate theories proposed by Reissner and Mindlin, respectively. A procedure to improve the assumed displacement field with the third-order shear deformations is presented. The present study shows that the various displacement fields with the simple third-order shear deformations would be identical when the constraint on the consistence of the transverse shear strain energy is enforced.

References

1.
Reissner
,
E.
,
1985
, “
Reflection on the Theory of Elastic Plates
,”
ASME J. Appl. Mech. Rev.
,
38
, pp.
1453
1464
.10.1115/1.3143699
2.
Levinson
,
M.
,
1980
, “
An Accurate Simple Theory of Statics and Dynamics of Elastic Plates
,”
Mech. Res. Commun.
,
7
, pp.
340
350
.10.1016/0093-6413(80)90049-X
3.
Murthy
,
V. V.
,
1981
, “
An Improved Transverse Shear Deformation Theory for Laminate Anisotropic Plates
,” NASA Technical Paper No. 1903.
4.
Reddy
,
J. N.
,
1984
, “
A Simple Higher-Order Theory for Laminated Composite Plates
,”
ASME J. Appl. Mech.
,
51
, pp.
745
752
.10.1115/1.3167719
5.
Shi
,
G.
,
2007
, “
A New Simple Third-Order Shear Deformation Theory of Plates
,”
Int. J. Solids Struct.
,
44
, pp.
4399
4417
.10.1016/j.ijsolstr.2006.11.031
6.
Aydogdu
,
M.
,
2009
, “
A New Shear Deformation Theory for Laminated Composite Plates
,”
Compos. Struct.
,
89
. pp.
94
101
.10.1016/j.compstruct.2008.07.008
7.
Kapania
,
R. K.
, and
Raciti
,
S.
,
1989
, “
Recent Advances in Analysis of Laminated Beams and Plates, Part I: Shear Effects and Buckling
,”
AIAA J.
27
, pp.
923
934
.10.2514/3.10202
8.
Rohwer
,
K.
,
1992
, “
Application of Higher Order Theories to the Bending Analysis of Layered Composite Plates
,”
Int. J. Solids Struct.
,
29
, pp.
105
119
.10.1016/0020-7683(92)90099-F
9.
Timoshenko
,
S. P.
, and
Gere
,
J.
,
1972
,
Mechanics of Materials
,
Van Nostrand
,
New York
.
10.
Reissner
,
E.
,
1945
, “
The Effect of Transverse Shear Deformation on the Bending of Elastic Plates
,”
ASME J. Appl. Mech.
,
12
, pp.
66
77
.
11.
Mindlin
,
R. D.
,
1951
, “
Influence of Rotatory Inertia and Shear on Flexural Motion of Isotropic, Elastic Plates
,”
ASME J. Appl. Mech.
,
18
, pp.
31
38
.
12.
Pai
,
P. F.
, and
Schulz
,
M. J.
,
1999
, “
Shear Correction Factors and an Energy Consistent Beam Theory
,”
Int. J. Solids Struct.
,
36
, pp.
1523
1540
.10.1016/S0020-7683(98)00050-X
13.
Idlbi
,
A.
,
Karama
,
M.
, and
Touratie
,
M.
,
1997
, “
Comparison of Various Laminated Plate Theories
,”
Compos. Struct.
,
37
, pp.
173
184
.10.1016/S0263-8223(97)80010-4
14.
Carrera
,
E.
, and
Petrolo
,
M.
,
2011
, “
On the Effectiveness of Higher-Order Terms in Refined Beam Theories
,”
ASME J. Appl. Mech.
,
78
(
021013
), pp.
1
17
.10.1115/1.4002207
15.
Koiter
,
W. T.
,
1959
, “
A Consistent First Approximation in the General Theory of Thin Elastic Shells
,” in
Proc. IUTAM Symposium on the Theory of Thin Elastic Shells
, Delft, Holland, August 24–28, North Holland Publishing, Delft, Holland, pp.
12
33
.
16.
Reddy
,
J. N.
,
Wang
,
C. M.
,
Lim
,
G. T.
, and
Ng
,
K. Y.
,
2001
, “
Bending Solutions of Levinson Beam and Plates in Terms of the Classical Theories
,”
Int. J. Solids Struct.
,
38
, pp.
4701
4720
.10.1016/S0020-7683(00)00298-5
17.
Wang
,
C. M.
,
Kitipornchai
,
S.
,
Lim
,
C. W.
, and
Eisenberger
,
M.
,
2008
,
M., Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory
,
ASCE J. Eng. Mech.
,
134
, pp.
475
481
.10.1061/(ASCE)0733-9399(2008)134:6(475)
18.
Reissner
,
E.
,
1975
, “
On Transverse Bending of Plates, Including the Effect of Transverse Shear Deformation
,”
Int. J. Solids Struct.
,
11
, pp.
569
-
573
.10.1016/0020-7683(75)90030-X
19.
Panc
,
V.
,
1975
,
Theories of Elastic Plates
,
Noordhoff
,
Netherlands
.
20.
Karama
,
M.
,
Afaq
,
K. S.
, and
Mistou
,
S.
,
2003
, “
Mechanical Behavior of Laminated Composite Beam by the New Multi-Layered Laminated Composite Structures Model With Transverse Shear Stress Continuity
,”
Int. J. Solids Struct.
,
40
, pp.
1525
1546
.10.1016/S0020-7683(02)00647-9
21.
Voyiadjis
,
G. Z.
, and
Shi
,
G.
,
1991
, “
A Refined Two-Dimensional Theory for Thick Cylindrical Shells
,”
Int. J. Solids Struct.
,
27
, pp.
261
282
.10.1016/0020-7683(91)90082-Q
22.
Bickford
,
W. B.
,
1982
, “
A Consistent Higher Order Beam Theory
,” in
Developments in Theoretical and Applied Mechanics Vol. XI
,
University of Alabama
,
Huntsville
, AL, pp.
137
150
.
23.
Levinson
,
M.
,
1981
, “
A New Rectangular Beam Theory
,”
J. Sound Vib.
,
74
, pp.
81
87
.10.1016/0022-460X(81)90493-4
24.
Shi
,
G.
, and
Voyiadjis
,
G. Z.
,
2011
, “
A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions
,”
ASME J. Appl. Mech.
78
(
021019
), pp.
1
11
.10.1115/1.4002594
25.
Srinivas
,
C. V.
, and
Rao
,
A. K.
,
1970
, “
Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates
,”
Int. J. Solids Struct.
,
6
, pp.
1463
1481
.10.1016/0020-7683(70)90076-4
26.
Reddy
,
J. N.
,
1984
, “
A Refined Nonlinear Theory of Plates With Transverse Shear Deformations
,”
Int. J. Solids Struct.
,
20
, pp.
881
896
.10.1016/0020-7683(84)90056-8
27.
Shi
,
G.
, and
Voyiadjis
,
G. Z.
,
2012
, “
A Sixth-Order Beam Theory for Flexural Vibration Analysis of Beams With the Effects of Shear Flexibility and Rotary Inertia
,”
J. Sound Vib.
(submitted).
28.
Hutchinson
,
J.
,
1986
, “
On the Axisymmetric Vibrations of Thick Clamped Plates
,”
Proc. Int. Conf. on Vibration Problems in Engineering
,
Xi'an
,
China
, June 17–20, pp.
75
81
.
29.
Hutchinson
,
J.
,
1987
, “
A Comparison of Mindlin and Levinson Plate Theories
,”
Mech. Res. Commun.
,
14
, pp.
165
170
.10.1016/0093-6413(87)90070-X
30.
Levinson
,
M.
,
1987
, “
On Higher Order Beam and Plate Theories
,”
Mech. Res. Commun.
,
14
, pp.
421
424
.10.1016/0093-6413(87)90064-4
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