We show, by considering a special class of nonlinear viscoelastic materials, that consistency of a mechanical model with classical linear viscoelasticity, may be a fundamental condition to ensure a mathematical and physical well-posedness behavior. To illustrate our arguments we use a rectilinear class of shear motions that we investigate in the static and quasistatic case in the framework of a simple boundary value problem and the classical recovery phenomenon.

1.
Truesdell
,
C.
, and
Noll
,
W.
, 2004,
The Non-Linear Field Theories of Mechanics
,
3rd ed.
,
Springer-Verlag
, Berlin.
2.
Beatty
,
M. F.
, 1996, “
Introduction to Nonlinear Elasticity
,” in
Nonlinear Effects in Fluids and Solids
,
M. M.
Carroll
and
M. A.
Hayes
, eds.,
Plenum Press
, New York, pp.
16
112
.
3.
Weinberger
,
H. F.
, 1965,
A First Course in Partial Differential Equations
,
Waltham
, Blaisdell.
4.
Antman
,
S. S.
, 1995,
Nonlinear Problems of Elasticity
,
Springer-Verlag
, New York.
5.
Takamizawa
,
K.
, and
Hayashi
,
K.
, 1987, “
Strain Energy Density Function and Uniform Strain Hypothesis for Arterial Mechanics
,”
J. Biomech.
0021-9290,
20
, pp.
7
17
.
6.
Horgan
,
C. O.
, and
Saccomandi
,
G.
, 2003, “
A Description of Arterial Wall Mechanics Using Limiting Chain Extensibility Constitutive Models
,”
Biomech. Modeling in Mechanobiology
,
1
, pp.
251
266
.
7.
Gasser
,
T. C.
,
Ogden
,
R. W.
, and
Holzapfel
,
G. A.
, 2006, “
Hyperelastic Modelling of Arterial Layers with Distributed Collagen Fibre Orientations
,”
J. R. Soc., Interface
1742-5689,
3
, pp.
15
35
.
8.
Pioletti
,
D. P.
,
Rakotomanana
,
L. R.
,
Benvenuti
,
J.-F.
, and
Leyvraz
,
P.-F.
, 1998, “
Viscoelastic Constitutive Law in Large Deformations: Application to Human Knee Ligaments and Tendons
,”
J. Biomech.
0021-9290,
31
, pp.
753
757
.
9.
Zhang
,
J. P.
, and
Rajagopal
,
K. R.
, 1992, “
Some Inhomogeneous Motions and Deformations Within the Context of a Non Linear Elastic Solid
,”
Int. J. Eng. Sci.
0020-7225,
30
, pp.
919
938
.
10.
Pucci
,
E.
, and
Saccomandi
,
G.
, 1999, “
Some Remarks on the Gent Model of Rubber Elasticity
,”
CanCNSM Proceedings
,
E. M.
Croitoro
, ed.,
University of Victoria Press
, Victoria, Canada, pp.
163
172
.
11.
Bailey
,
P. B.
,
Shampine
,
L. F.
, and
Waltman
,
P. A.
, 1968,
Nonlinear Two Point Boundary Value Problems
,
Academic Press
, New York.
12.
Horgan
,
C. O.
,
Saccomandi
,
G.
, and
Sgura
,
I.
, 2003, “
A Two-Point Boundary Value Problem for the Axial Shear of Hardening Isotropic Incompressible Nonlinearly Elastic Materials
,”
SIAM J. Appl. Math.
0036-1399,
65
, pp.
1712
1727
.
13.
Serrin
,
J.
, 1969, “
The Problem of Dirichlet for Quasilinear Elliptic Differential Equations with Many Independent Variables
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
264
, pp.
413
496
.
14.
MacCamy
,
R. C.
, 1970, “
Existence, Uniqueness and Stability of Solutions of the Equation utt=∂∕∂x(σ(ux)+λ(ux)uxt)
,”
Indiana Univ. Math. J.
0022-2518,
20
, pp.
231
238
.
15.
Georgiev
,
V.
,
Rubino
,
B.
, and
Sampalmieri
,
R.
, 2005, “
Global Existence for Elastic Waves with Memory
,”
Arch. Ration. Mech. Anal.
0003-9527,
176
, pp.
303
330
.
16.
Ciarlet
,
P. G.
, 1988,
Mathematical Elasticity: Vol. 1
,
North Holland, Amsterdam.
17.
Nicholson
,
D. W.
, 2003,
Finite Element Analysis: Thermomechanics of Solids
,
CRC Press
, Boca Raton, FL.
You do not currently have access to this content.