Abstract
Toupin’s Theorem plays the most influential role in the history of development concerning Saint-Venant’s Principle. We now review the history and the previous works, distinguishing Saint-Venant type decay from Toupin-type decay and arguing that Toupin’s Theorem is not a formulation of Saint-Venant’s Principle; Saint-Venant’s Principle stated by Love can be disproved mathematically from Toupin’s Theorem, so Toupin’s Theorem is mathematically inconsistent with Saint-Venant’s Principle; Modified Saint-Venant’s Principle can be proved or formulated, though Saint-Venant’s Principle in its general form stated by Boussinesq and Love is not true.
Issue Section:
Review Articles
Keywords:
elasticity
References
1.
de Saint-Venant
, A-J-C B
, 1855,
“Mémoire sur la Torsion des Prismes
,” Mémoires présentes pars divers Savants à l’ Académie des Sciences de l’ Institut Impérial de France
, 14
, pp.233
–560
.2.
de Saint-Venant
, A-J-C B
, 1855, “Mémoire sur la Flexion des Prismes
,” J Math Pures Appl
, 1
, pp.89
–189
.3.
Boussinesq
, M. J.
, 1885, "Application des Potentiels à l’ étude de l’ équilibre et des Mouvements des Solides élastiques
" Gauthier-Villars, Paris.4.
Love
, A. E. H.
, 1927, A Treatise on the Mathematical Theory of Elasticity
,
4th ed., The University Press
, Cambridge, England
.5.
Mises
, R.v.
, 1945,
“On Saint-Venant's Principle
,” Bull
Am. Math Soc
, 51
, pp.
555
–562
.6.
Sternberg
,
E.
,
1954, “On Saint-Venant’s Principle
,” Q. Appl. Math.
,
11
, pp. 393
–402
.7.
Zanaboni
,
O.
,
1937, “Dimostrazione Generale del Principio del De Saint-Venant
,” Atti Acad Naz dei Lincei, Rendiconti
, 25
, pp.
117
–121
.8.
Zanaboni
,
O.
,
1937, “Valutazione
Dell’errore Massimo cui dà Luogo L’applicazione del Principio del De
Saint-Venant in un Solido Isotropo
,” Atti Acad Naz
dei Lincei, Rendiconti
, 25
, pp.
595
–601
.9.
Zanabolni
,
O.
,
1937,
“Sull’approssimazione Dovuta al Principio del De Saint-Venant
nei Solidi Prismatici Isotropi
,” Atti Acad Naz dei
Lincei, Rendiconti
, 26
, pp.
340
–345
.10.
Truesdell
,
C.
,
1959, “The Rational
Mechanics of Materials - Past, Present, Future
,”
Appl. Mech. Rev.
, 12
, pp.
75
–80
.11.
Toupin
, R.
A.
, 1965,
“Saint-Venant’s Principle
,” Archive
for Rational Mech and Anal
, 18
, pp.
83
–96
.12.
Toupin
, R.
A.
, 1965,
“Saint-Venant and a Matter of Principle
,”
Trans N. Y. Acad. Sci.
, 28
, pp.
221
–232
.13.
Berdichevskii
,
V. L.
, 1974,
“On the Proof of the Saint-Venant’s Principle for Bodies of
Arbitrary Shape
,” Prikl. Mat. Mekh.
,
38
, pp.
851
–864
Berdichevskii
,
V. L.
, [J. Appl. Math. Mech.
,
38
, 799
–813
(1975)].14.
Horgan
, C.
O.
, and Knowles
, J.
K.
, 1983,
“Recent Developments Concerning Saint-Venant’s
Principle
,” Advances in Applied Mechanics
,
T.
Y.
Wu
and J.
W.
Hutchinson
, eds.,
Academic
, New
York
, Vol. 23
, pp.
179
–269
.15.
Horgan
, C.
O.
, 1989,
“Recent Developments Concerning Saint-Venant’s Principle: An
Update
,” Appl. Mech. Rev.
,
42
, pp. 295
–303
.16.
Horgan
, C.
O.
, 1996,
“Recent Developments Concerning Saint-Venant’s Principle: A
Second Update
,” Appl. Mech. Rev.
,
49
, pp. S101
–S111
.17.
Timoshenko
, S.
P.
, and Goodier
, J.
N.
, 1970,
Theory of Elasticity
, 3rd ed.,
McGraw-Hill
, New
York
, pp.
62
–283
.18.
Flavin
, J.
N.
, 1978,
“Another Aspect of Saint-Venant’s Principle in
Elasticity
,” ZAMP
, 29
, pp.
328
–332
.19.
Flavin
, J.
N.
, 1982,
“Saint-Venant Pointwise Decay Estimates
,”
J. Elasticity
, 12
, pp.
313
–316
.20.
Roseman
,
J.
,
1966, “A Pointwise
Estimate for the Stress in a Cylinder and its Application to Saint-Venant’s
Principle
,” Archive for Rational Mech. and
Anal.
, 21
, pp.
23
–48
.21.
Fung
, Y.
C.
, 1965,
Foundations of Solid Mechanics
,
Prentice-Hall
, Englewood
Cliffs, NJ
, pp.
300
–303
.22.
Zhao
,
J.-Z.
,
1986,
“Toupin-Berdichevskii Theorem Can’t be Considered as a
Mathematical Expression of Saint-Venant’s Principle
,”
Appl. Math. Mech., English ed.
, 7
, pp.
971
–974
.23.
Knowles
, J.
K.
, 1966,
“On Saint-Venant’s Principle in the Two-Dimensional Linear
Theory of Elasticity
,” Archive for Rational Mech and
Anal
, 21
, pp.
1
–22
.24.
Knowles
, J.
K.
, and Sternberg
E.
,
1966, “On Saint-Venant’s
Principle and the Torsion of Solids of Revolution
,”
Archive for Rational Mech and Anal
, 22
,
pp. 100
–120
.25.
Knowles
, J.
K.
, 1967,
“A Saint-Venant’s Principle for a Class of Second-Order
Elliptic Boundary Value Problems
,” ZAMP
,
18
, pp. 473
–490
.26.
Flavin
, J.
N.
, 1974,
“On Knowles’ Version of Saint-Venant’s Principle in
Two-Dimensional Elastostatics
,” Archive for Rational
Mech and Anal
, 53
, pp.
366
–375
.27.
Flavin
, J.
N.
, and Knops
, R.
J.
, 1988,
“Some Convexity Considerations for a Two-Dimensional Traction
Problem
,” ZAMP
, 39
, pp.
166
–176
.28.
Flavin
, J.
N.
, and Knops
, R.
J.
, 1988,
“Some Decay and Other Estimates in Two-Dimensional Linear
Elastostatics
,” Q. J. Mech. Appl. Math.
,
41
, pp. 223
–238
.29.
Horgan
, C.
O.
, 1989,
“Decay Estimates for the Biharmonic Equation With
Applications to Saint-Venant’s Principles in Plane Elasticity and Stokes
Flows
,” Q. Appl. Math.
,
47
, pp. 147
–157
.30.
Miller
, K.
L.
, and Horgan
, C.
O.
, 1995,
“End Effects for Plane Deformations of an Elastic Anisotropic
Semi-Infinite Strip
,” J. Elasticity
,
38
, pp. 261
–316
.31.
Knops
, R.
J.
, and Villaggio
,
P.
,
2000, “Spatial Behavior
in Plane Incompressible Elasticity on a Half-Strip
,”
Q. Appl. Math.
, 58
, pp.
355
–367
.32.
Flavin
, J.
N.
, 2003,
“Spatial-Decay Estimates for a Generalized Biharmonic
Equation in Inhomogeneous Elasticity
,” J.
Engineering Math
, 46
, pp.
241
–252
.33.
Gregory
, R.
D.
, and Wan
, F. Y.
M.
, 1985,
“On Plate Theories and Saint-Venant’s
Principle
,” Inter. J. Solids Structures
,
10
, pp. 1005
–1024
.34.
Flavin
, J.
N.
, Knops
R. J.
, and Payne
, L.
E.
, 1989,
“Decay Estimates for the Constrained Elastic Cylinder of
Variable Cross Section
,” Q. Appl. Math.
,
47
, pp.
325
–350
.35.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1993,
“On the Asymptotic Behavior of Solutions of Linear
Second-Order Boundary-Value Problems on a Semi-Infinite
Strip
,” Archive for Rational Mech. and
Anal.
, 124
, pp.
277
–303
.36.
Horgan
, C.
O.
, 1995,
“Anti-Plane Shear Deformations in Linear and Nonlinear Solid
Mechanics
,” SIAM Rev.
, 37
,
pp. 53
–81
.37.
Lin
,
C.
,
1994, “Exponential Decay
Estimates for Solutions of the Polyharmonic Equation in a Semi-Infinite
Cylinder
,” J. Math Anal. Appl.
,
181
, pp. 626
–647
.38.
Knops
, R.
J.
, and Payne
, L.
E.
, 1996,
“The Effect of a Variation in the Elastic Moduli on
Saint-Venant’s Principle for a Half-Cylinder
,” J.
Elasticity
, 44
, pp.
161
–182
.39.
Quintanilla
,
R.
,
1997, “Spatial Decay
Estimates and Upper Bounds in Elasticity for Domains With Unbounded
Cross-Sections
,” J. Elasticity
,
46
, pp. 239
–254
.40.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1997,
“Saint-Venant’s Principle in Linear Isotropic Elasticity for
Incompressible or Nearly Incompressible Materials
,”
J. Elasticity
, 46
, pp.
43
–52
.41.
Knops
, R.
J.
, 2000,
“Alternative Spatial Behaviour in the Incompressible Linear
Elastic Prismatic Constrained Cylinder
,” J.
Engineering Math
, 37
, pp.
111
–128
.42.
Quintanilla
,
R.
,
2002, “On the Spatial
Decay for the Dynamical Problem of Thermo-Microstretch Elastic
Solids
,” Inter. J Engineering Science
,
40
, pp. 109
–121
.43.
Flavin
, J.
N.
, and Gleeson
,
B.
,
2005, “Decay and Other
Estimates for an Annular Elastic Cylinder in an Axisymmetric State of
Stress
,” Math Mech. of Solids
,
10
, pp. 213
–225
.44.
Roseman
,
J.
,
1967, “The Principle of
Saint-Venant in Linear and Non-Linear Plane Elasticity
,”
Archive for Rational Mech and Anal
, 26
,
pp. 142
–162
.45.
Breuer
,
S.
, and
Roseman
,
J.
,
1977, “On Saint-Venant’s
Principle in Three-Dimensional Nonlinear Elasticity
,”
Archive for Rational Mech and Anal
, 63
,
pp. 191
–203
.46.
Knops
, R.
J.
, and Payne
, L.
E.
, 1983,
“A Saint-Venant Principle for Nonlinear
Elasticity
,” Archive for Rational Mech and
Anal
, 81
, pp.
1
–12
.47.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1984,
“Decay Estimates for Second-Order Quasilinear Partial
Differential Equations
,” Adv. Appl. Math.
,
5
, pp. 309
–332
.48.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1984,
“Decay Estimates for a Class of Second-Order Quasilinear
Equations in Three Dimensions
,” Archive for Rational
Mech and Anal
, 86
, pp.
279
–289
.49.
Galdi
, G.
P.
, Knops
, R.
J.
, and Rionero
S.
,
1985, “Asymptotic
Behaviour in the Nonlinear Elastic Beam
,” Archive
for Rational Mech and Anal
, 87
, pp.
305
–318
.50.
Vafeades
,
P.
, and
Horgan
, C.
O.
, 1988,
“Exponential Decay Estimates for Solutions of the von Karman
Equations on a Semi-Infinite Strip
,” Archive for
Rational Mech and Anal
, 104
, pp.
1
–25
.51.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1989,
“On the Asymptotic Behavior of Solutions of Inhomogeneous
Second-Order Quasilinear Partial Differential Equations
,”
Q. Appl. Math.
, 47
, pp.
753
–771
.52.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1992,
“A Saint-Venant Principle for a Theory of Nonlinear Plane
Elasticity
,” Q. Appl. Math.
,
50
, pp.
641
–675
.53.
Flavin
, J.
N.
, Knops
, R.
J.
, and Payne
, L.
E.
, 1992,
“Asymptotic Behaviour of Solutions to semi-Linear Elliptic
Equations on the Half-Cylinder
,” ZAMP
,
43
, pp. 405
–421
.54.
Breuer
,
S.
, and
Roseman
,
J.
,
1982, “Saint-Venant’s
Principle in Nonlinear Plane Elasticity With Sufficiently Small
Strains
,” Archive for Rational Mech and
Anal
, 80
, pp.
19
–37
.55.
Breuer
,
S.
, and
Roseman
,
J.
,
1986, “Phragmen-Lindelof
Decay Theorems for Classes of Nonlinear Dirichlet Problems in a Circular
Cylinder
,” J Math Anal Appl
,
113
, pp. 59
–77
.56.
Breuer
,
S.
, and
Roseman
,
J.
,
1986, “Decay Theorems
for Nonlinear Dirichlet Problems in Semi-Infinite
Cylinders
,” Archive for Rational Mech and Anal
,
94
, pp. 363
–371
.57.
Horgan
, C.
O.
, 1985,
“A Note on the Spatial Decay of a Three-Dimensional Minimal
Surface over a Semi-Infinite Cylinder
,” J Math Anal
Appl
, 107
, pp.
285
–290
.58.
Horgan
, C.
O.
, and Olmstead
,
W. E.
, 2003,
“A Saint-Venant Principle for Shear Band
Localization
,” ZAMP
, 54
,
pp. 807
–814
.59.
Horgan
, C.
O.
, and Knowles
, J.
K.
, 1981,
“The Effect of Nonlinearity on a Principle of Saint-Venant
Type
,” J. Elasticity
, 11
,
pp. 271
–291
.60.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1990,
“On Saint-Venant’s Principle in Finite Anti-Plane Shear: An
Energy Approach
,” Archive for Rational Mech. and
Anal.
, 109
, pp.
107
–137
.61.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1993,
“The Effect of Constitutive Law Perturbations on Finite
Antiplane Shear Deformations of a Semi-Infinite Strip
,”
Q. Appl. Math.
, 51
, pp.
441
–465
.62.
Horgan
, C.
O.
, and Payne
, L.
E.
, 1997,
“Spatial Decay Estimates for a Coupled System of Second-Order
Quasilinear Partial Differential Equations Arising in Thermoelastic Finite
Anti-Plane Shear
,” J. Elasticity
,
47
, pp. 3
–21
.63.
Borrelli
,
A.
,
Horgan
, C.
O.
, and Patria
, M.
C.
, 2002,
“Saint-Venant’s Principle for Antiplane Shear Deformations of
Linear Piezoelectric Materials
,” SIAM J. Appl.
Math.
, 62
, pp.
2027
–2044
.64.
Horgan
, C.
O.
, and Abeyaratne
,
R.
,
1983, “Finite Anti-Plane
Shear of a Semi-Infinite Strip Subject to a Self-Equilibrated End
Traction
,” Q. Appl. Math.
,
40
, pp.
407
–417
.65.
Horgan
, C.
O.
, Payne
, L.
E.
, and Philippin
,
G. A.
, 1995,
“Poiniwise Gradient Decay Estimates for Solutions of the
Laplace and Minimal Surface Equations
,” Diff.
Integral Eq
. 8
, pp.
1761
–1773
.66.
Stephen
, N.
G.
, and Wang
, P.
J.
, 1996,
“Saint-Venant’s Principle and the Anti-Plane Elastic
Wedge
,” J Strain Anal
. 31
,
pp. 231
–234
.67.
Scalpato
, M.
R.
, and Horgan
, C.
O.
, 1997,
“Saint-Venant Decay Rates for an Isotropic Inhomogeneous
Linearly Elastic Solid in Anti-Plane Shear
,” J.
Elasticity
, 48
, pp.
145
–166
.68.
Chan
, A.
M.
, and Horgan
, C.
O.
, 1998,
“End Effects in Anti-Plane Shear for an Inhomogeneous
Isotropic Linearly Elastic Semi-Infinite Strip
,” J.
Elasticity
, 51
, pp.
227
–242
.69.
Horgan
, C.
O.
, and Quintanilla
,
R.
,
2001, “Saint-Venant End
Effects in Antiplane Shear for Functionally Graded Linearly Elastic
Materials
,” Math. Mech. Solids
6
, pp. 115
–132
.70.
Knops
, R.
J.
, and Lupoli
,
C.
,
1997, “End Effects for
Plane Stokes Flow Along a Semi-Infinite Strip
,”
ZAMP
, 48
, pp.
905
–920
.71.
Horgan
, C.
O.
, 1978,
“Plane Entry Flows and Energy Estimates for the Navier-Stokes
Equations
,” Archive for Rational Mech.
and
Anal. 68
, pp.
359
–381
.72.
Horgan
, C.
O.
, and Wheeler
, L.
T.
, 1978,
“Spatial Decay Estimates for the Navier-Stokes Equations With
Application to the Problem of Entry Flow
,” SIAM J.
Appl. Math.
35
, pp. 97
–116
.73.
Ames
, K.
A.
, and Payne
, L.
E.
, 1989,
“Decay Estimates in Steady Pipe Flow
,”
SIAM J. Math. Anal.
20
, pp. 789
–815
.74.
Ames
, K.
A.
, Payne
, L.
E.
, and Schaefer
,
P. W.
, 1993,
“Spatial Decay Estimates in Time-Dependent Stokes
Flow
,” SIAM J. Math. Anal.
24
, pp. 1395
–1413
.75.
Chadam
,
J.
, and
Qin
,
Y.
,
1997, “Spatial Decay
Estimates for Flow in a Porous Medium
,” SIAM J.
Math. Anal.
28
, pp. 808
–830
.76.
Payne
, L.
E.
, and Song
, J.
C.
, 2000,
“Spatial Decay for a Model of Double Diffusive Convection in
Darcy and Brinkman Flow
,” ZAMP
,
51
, pp. 867
–889
.77.
Song
, J.
C.
, 2002,
“Spatial Decay Estimates in Time-Dependent Double-Diffusive
Darcy Plane Flow
,” J. Math. Anal. Appl.
267
, pp. 76
–88
.78.
Song
, J.
C.
, 2003,
“Improved Decay Estimates in Time-Dependent Stokes
Flow
,” J. Math. Anal. Appl
,
288
, pp. 505
-517
.79.
Lin
,
C.
, and
Payne
, L.
E.
, 2004,
“Spatial Decay Bounds in Time Dependent Pipe Flow of an
Incompressible Viscous Fluid
,” SIAM J. Appl.
Math
, 65
, pp.
458
–474
.80.
Boley
, B.
A.
, 1958,
“Some Observations on Saint-Venant’s
Principle,”
Proceedings of the Third U. S. National Congress in Applied
Mechanics
, ASME
,
New York
, pp.
259
–264
.81.
Boley
, B.
A.
, 1960,
“Upper Bounds and Saint-Venant’s Principle in Transient Heat
Conduction
,” Q. Appl. Math.
18
, pp. 205
–208
.82.
Knowles
, J.
K.
, 1971,
“On the Spatial Decay of Solutions of the Heat
Equation
,” ZAMP
, 22
, pp.
1050
–1056
.83.
Edelstein
, W.
S.
, 1969,
“A Spatial Decay Estimate for the Heat
Equation
,” ZAMP
, 20
, pp.
900
–905
.84.
Horgan
, C.
O.
, Payne
, L.
E.
, and Wheeler
, L.
T.
, 1984,
“Spatial Decay Estimates in Transient Heat
Conduction
,” Q. Appl. Math.
42
, pp. 119
–127
.85.
Payne
, L.
E.
, and Song
, J.
C.
, 1996,
“Phragmen-Lindelof and Continuous Dependence Type Results in
Generalized Heat Conduction
,” ZAMP
,
47
, pp. 527
–538
.86.
Quintanilla
,
R.
,
1996, “A Spatial Decay
Estimate for the Hyperbolic Heat Equation
,” SIAM J.
Math. Anal
, 27
, pp.
78
–91
.87.
Horgan
, C.
O.
, and Quintanilla
,
R.
,
2001, “Spatial Decay of
Transient End Effects in Functionally Graded Heat Conducting
Materials
,” Q. Appl. Math.
59
, pp. 529
–542
.88.
Payne
, L.
E.
, and Song
, J.
C.
, 2004,
“Spatial Decay Estimates for the Maxwell-Cattaneo Equations
With Mixed Boundary Conditions
,” ZAMP
,
55
, pp. 962
–973
.89.
Payne
, L.
E.
, and Song
, J.
C.
, 2005,
“Improved Spatial Decay Bounds in Generalized Heat
Conduction
,” ZAMP
, 56
, pp.
805
–820
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
