This paper examines the effects of relaxing the assumption of classical linear elasticity that the loads act in their entirety on the undeformed shape. Instead, loads here are applied incrementally as deformation proceeds, and resulting fields are integrated. A formal statement of the attendant integrated elasticity theory is provided. A class of problems is identified for which this formulation is amenable to solution in closed form. Some results from these configurations are compared with linear elasticity and experimentally measured data. The comparisons indicate that, as deformation increases, integrated elasticity is capable of tracking the physical response better than linear elasticity.

1.
Frisch-Fay
,
R.
, 1962,
Flexible Bars
,
Butterworth, Inc.
Washington.
2.
Griffith
,
A. A.
, 1920, “
The Phenomena of Rupture and Flow in Solids
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
221
, pp.
163
198
.
3.
Truesdell
,
C.
, 1952, “
The Mechanical Foundations of Elasticity and Fluid Mechanics
,”
J. Rational Mech. Anal
,
1
pp.
125
300
.
4.
Murnaghan
,
F. D.
, 1949, “
The Foundations of the Theory of Elasticity
,” in:
Non-Linear Problems in Mechanics of Continua
,
American Mathematical Society
, New York, pp.
158
174
.
5.
Jaumann
,
G.
, 1911, “
Geschlossenes System Physikalischer und Chemischer Differentialgesetze
,”
Akad.Wiss. Wien Sitzber. (IIa)
,
120
, pp.
385
530
.
6.
Hencky
,
H.
, 1929, “
Das Superpositionsgesetz Eines Endlich Deformierten Relaxasionsfähigen Elastischen Kontinuums und Seine Bedeutung für Eine Exakte Ableitung der Gleichungen für die Zähe Flüssigkeit in der Eulerschen Form
Ann. Phys.
0003-3804,
5
, pp.
617
630
.
7.
Fung
,
Y. C.
, 1965,
Foundations of Solid Mechanics
,
Prentice–Hall, Inc.
Englewood Cliffs, NJ.
8.
Sokolnikoff
,
I. S.
, 1956,
Mathematical Theory of Elasticity
,
2nd ed.
,
McGraw–Hill
, New York.
9.
Ludwik
,
P.
, 1909
Elemente der Technologischen Mechanik
,
Springer
, Berlin.
10.
Green
,
A. E.
, 1956, “
Simple Extension of a Hypo-Elastic Body of Grade Zero Mechanics
,”
J. Rational Mech. Anal.
5
, pp.
637
642
.
11.
Rivlin
,
R. S.
, and
Saunders
,
D. W.
, 1951, “
Large Elastic Deformations of Isotropic Materials VII. Experiments on the Deformation of Rubber
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
243
, pp.
251
288
.
12.
Assaad
,
A. T.
, and
Sinclair
,
G. B.
, 2005, “
An Experimental Study of the Applicability of Integrated Elasticity to the Tension and Torsion of Rubber Materials
,” Report No. ME-MA7-05, Department of Mechanical Engineering,
Louisiana State University
, Baton Rouge.
13.
Treloar
,
L. R. G.
, 1975,
The Physics of Rubber Elasticity
,
3rd ed.
,
Clarendon Press
, Oxford.
14.
Ward
,
I. M.
, and
Sweeney
,
J.
, 2004,
The Mechanical Properties of Solid Polymers
,
2nd ed.
,
Wiley
, Sussex, UK.
15.
Hart-Smith
,
L. J.
, 1966, “
Elasticity Parameters for Finite Deformations of Rubber-Like Materials
,”
Z. Angew. Math. Phys.
0044-2275
17
, pp.
608
626
.
16.
Assaad
,
A. T.
, and
Sinclair
,
G. B.
, 2000, “
Some Experiments on the Size-Versus-Pressure Response of Balloons
,” Report No. SM 00-2, Department of Mechanical Engineering,
Carnegie Mellon University
, Pittsburgh.
17.
Mansfield
,
E. H.
, 1967, “
On the Stresses Near a Crack in an Elastic Sheet
,” Technical Report No. 67030,
Royal Aircraft Establishment
, Cranfield, UK.
18.
Kolossoff
,
G.
, 1910,
On an Application of the Theory of Complex Variables to the Two-Dimensional Problem of Elasticity Theory
,” Ph.D. dissertation, St. Petersburg;
See also
Kolossoff
,
G.
, 1914,
Z. Angew. Math. Phys.
62
, pp.
384
409
.
19.
Inglis
,
C. E.
, 1913, “
Stresses in a Plate due to the Presence of Cracks and Sharp Corners
,”
Trans. INA
55
, pp.
219
241
.
20.
Kondo
,
M.
, and
Sinclair
,
G. B.
, 1982, “
Stress and Displacement Fields for an Elliptical Hole in a Thick Elastic Plate Under tension
,” Report No. SM 82-15, Department of Mechanical Engineering,
Carnegie Mellon University
, Pittsburgh.
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