The analytic response for the Cauchy extra stress in large amplitude oscillatory shear (LAOS) is computed from a constitutive model for isotropic incompressible materials, including viscoelastic contributions, and relaxation time. Three cases of frame invariant derivatives are considered: lower, upper, and Jaumann. In the first two cases, the shear stress at steady-state includes the first and third harmonics, and the difference of normal stresses includes the zeroth, second, and fourth harmonics. In the Jaumann case, the stress components are obtained in integral form and are approximated with a Fourier series. The behavior of the coefficients is studied parametrically, as a function of relaxation time and constitutive parameters. Further, the shear stress and the difference of normal stresses are studied as functions of shear strain and shear rate, and are visualized by means of the elastic and viscous Lissajous–Bowditch (LB) plots. Sample results in the Pipkin plane are reported, and the influence of the constitutive parameters in each case is discussed.

References

References
1.
Truesdell
,
C.
, and
Noll
,
W.
,
1965
, “
The Non-Linear Field Theories of Mechanics
,”
Handbuch der Physik
, Vol. 3,
Springer
,
Berlin
.
2.
Schiessel
,
H.
,
Metzler
,
R.
,
Blumen
,
A.
, and
Nonnenmacher
,
T.
,
1995
, “
Generalized Viscoelastic Models: Their Fractional Equations With Solutions
,”
J. Phys. A: Math. Gen.
,
28
(
23
), pp.
6567
6584
.
3.
Meyers
,
M. A.
, and
Chawla
,
K. K.
,
2009
,
Mechanical Behavior of Materials
, Vol.
2
,
Cambridge University Press
,
Cambridge, UK
.
4.
Philippoff
,
W.
,
1966
, “
Vibrational Measurements With Large Amplitudes
,”
Trans. Soc. Rheol.
,
10
(
1
), pp.
317
334
.
5.
Harris
,
J.
, and
Bogie
,
K.
,
1967
, “
The Experimental Analysis of Non-Linear Waves in Mechanical Systems
,”
Rheol. Acta
,
6
(
1
), pp.
3
5
.
6.
Dodge
,
J. S.
, and
Krieger
,
I. M.
,
1971
, “
Oscillatory Shear of Nonlinear Fluids I. Preliminary Investigation
,”
Trans. Soc. Rheol.
,
15
(
4
), pp.
589
601
.
7.
Tee
,
T.-T.
, and
Dealy
,
J.
,
1975
, “
Nonlinear Viscoelasticity of Polymer Melts
,”
Trans. Soc. Rheol.
,
19
(
4
), pp.
595
615
.
8.
Rajagopal
,
K. R.
, and
Saccomandi
,
G.
,
2003
, “
Shear Waves in a Class of Nonlinear Viscoelastic Solids
,”
Q. J. Mech. Appl. Math.
,
56
(
2
), pp.
311
326
.
9.
Rajagopal
,
K. R.
,
2010
, “
On a New Class of Models in Elasticity
,”
Math. Comput. Appl.
,
15
(
4
), pp.
506
528
.
10.
Ewoldt
,
R. H.
,
Hosoi
,
A.
, and
McKinley
,
G. H.
,
2008
, “
New Measures for Characterizing Nonlinear Viscoelasticity in Large Amplitude Oscillatory Shear
,”
J. Rheol.
,
52
(
6
), pp.
1427
1458
.
11.
Cho
,
K. S.
,
Hyun
,
K.
,
Ahn
,
K. H.
, and
Lee
,
S. J.
,
2005
, “
A Geometrical Interpretation of Large Amplitude Oscillatory Shear Response
,”
J. Rheol.
,
49
(
3
), pp.
747
758
.
12.
Ewoldt
,
R. H.
, and
McKinley
,
G. H.
,
2010
, “
On Secondary Loops in LAOS Via Self-Intersection of Lissajous–Bowditch Curves
,”
Rheol. Acta
,
49
(
2
), pp.
213
219
.
13.
Sousa
,
P.
,
Carneiro
,
J.
,
Vaz
,
R.
,
Cerejo
,
A.
,
Pinho
,
F.
,
Alves
,
M.
, and
Oliveira
,
M.
,
2013
, “
Shear Viscosity and Nonlinear Behavior of Whole Blood Under Large Amplitude Oscillatory Shear
,”
Biorheology
,
50
(
5–6
), pp.
269
282
.
14.
Hyun
,
K.
, and
Wilhelm
,
M.
,
2008
, “
Establishing a New Mechanical Nonlinear Coefficient Q From FT-Rheology: First Investigation of Entangled Linear and Comb Polymer Model Systems
,”
Macromolecules
,
42
(
1
), pp.
411
422
.
15.
Hyun
,
K.
, and
Kim
,
W.
,
2011
, “
A New Non-Linear Parameter Q From FT-Rheology Under Nonlinear Dynamic Oscillatory Shear for Polymer Melts System
,”
Korea-Aust. Rheol. J.
,
23
(
4
), pp.
227
235
.
16.
Hyun
,
K.
,
Kim
,
S. H.
,
Ahn
,
K. H.
, and
Lee
,
S. J.
,
2002
, “
Large Amplitude Oscillatory Shear as a Way to Classify the Complex Fluids
,”
J. Non-Newtonian Fluid Mech.
,
107
(
1
), pp.
51
65
.
17.
Hyun
,
K.
,
Wilhelm
,
M.
,
Klein
,
C. O.
,
Cho
,
K. S.
,
Nam
,
J. G.
,
Ahn
,
K. H.
,
Lee
,
S. J.
,
Ewoldt
,
R. H.
, and
McKinley
,
G. H.
,
2011
, “
A Review of Nonlinear Oscillatory Shear Tests: Analysis and Application of Large Amplitude Oscillatory Shear (LAOS)
,”
Prog. Polym. Sci.
,
36
(
12
), pp.
1697
1753
.
18.
Giacomin
,
A. J.
, and
Dealy
,
J. M.
,
1993
, “
Large-Amplitude Oscillatory Shear
,”
Techniques in Rheological Measurement
, A. A. Collyer, ed.,
Springer
,
Dordrecht, The Netherlands
, pp.
99
121
.
19.
Thompson
,
R. L.
,
Alicke
,
A. A.
, and
de Souza Mendes
,
P. R.
,
2015
, “
Model-Based Material Functions for SAOS and LAOS Analyses
,”
J. Non-Newtonian Fluid Mech.
,
215
, pp.
19
30
.
20.
Poulos
,
A. S.
,
Renou
,
F.
,
Jacob
,
A. R.
,
Koumakis
,
N.
, and
Petekidis
,
G.
,
2015
, “
Large Amplitude Oscillatory Shear (LAOS) in Model Colloidal Suspensions and Glasses: Frequency Dependence
,”
Rheol. Acta
,
54
(
8
), pp.
715
724
.
21.
Garinei
,
A.
, and
Pucci
,
E.
,
2016
, “
Constitutive Issues Associated With LAOS Experimental Techniques
,”
J. Rheol.
,
60
(
4
), pp.
705
714
.
22.
Pipkin
,
A. C.
,
1986
,
Lectures on Viscoelasticity Theory
(Applied Mathematical Sciences), Vol.
7
,
Springer Science & Business Media
,
New York
.
23.
Filograna
,
L.
,
Racioppi
,
M.
,
Saccomandi
,
G.
, and
Sgura
,
I.
,
2009
, “
A Simple Model of Nonlinear Viscoelasticity Taking Into Account Stress Relaxation
,”
Acta Mech.
,
204
(
1
), pp.
21
36
.
24.
Gordon
,
R.
, and
Schowalter
,
W.
,
1972
, “
Anisotropic Fluid Theory: A Different Approach to the Dumbbell Theory of Dilute Polymer Solutions
,”
Trans. Soc. Rheol.
,
16
(
1
), pp.
79
97
.
25.
Goddard
,
J.
, and
Miller
,
C.
,
1966
, “
An Inverse for the Jaumann Derivative and Some Applications to the Rheology of Viscoelastic Fluids
,”
Rheol. Acta
,
5
(
3
), pp.
177
184
.
26.
Giacomin
,
A.
,
Bird
,
R.
,
Johnson
,
L.
, and
Mix
,
A.
,
2011
, “
Large-Amplitude Oscillatory Shear Flow From the Corotational Maxwell Model
,”
J. Non-Newtonian Fluid Mech.
,
166
(
19
), pp.
1081
1099
.
27.
Antman
,
S.
,
1989
,
Nonlinear Problems of Elasticity
,
Springer
,
New York
.
28.
Beatty
,
M. F.
,
1987
, “
Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues-With Examples
,”
ASME Appl. Mech. Rev.
,
40
(
12
), pp.
1699
1734
.
29.
Hyun
,
K.
,
Nam
,
J. G.
,
Wilhellm
,
M.
,
Ahn
,
K. H.
, and
Lee
,
S. J.
,
2006
, “
Large Amplitude Oscillatory Shear Behavior of PEO-PPO-PEO Triblock Copolymer Solutions
,”
Rheol. Acta
,
45
(
3
), pp.
239
249
.
30.
Miller
,
R.
,
Lee
,
E.
, and
Powell
,
R.
,
1991
, “
Rheology of Solid Propellant Dispersions
,”
J. Rheol.
,
35
(
5
), pp.
901
920
.
31.
Citerne
,
G. P.
,
Carreau
,
P. J.
, and
Moan
,
M.
,
2001
, “
Rheological Properties of Peanut Butter
,”
Rheol. Acta
,
40
(
1
), pp.
86
96
.
32.
Giacomin
,
A. J.
, and
Bird
,
R. B.
,
2011
, “
Normal Stress Differences in Large-Amplitude Oscillatory Shear Flow for the Corotational ‘ANSR' Model
,”
Rheol. Acta
,
50
(
9–10
), pp.
741
752
.
33.
Giacomin
,
A. J.
,
Saengow
,
C.
,
Guay
,
M.
, and
Kolitawong
,
C.
,
2015
, “
Padé Approximants for Large-Amplitude Oscillatory Shear Flow
,”
Rheol. Acta
,
54
(
8
), pp.
679
693
.
34.
Nam
,
J. G.
,
Hyun
,
K.
,
Ahn
,
K. H.
, and
Lee
,
S. J.
,
2008
, “
Prediction of Normal Stresses Under Large Amplitude Oscillatory Shear Flow
,”
J. Non-Newtonian Fluid Mech.
,
150
(
1
), pp.
1
10
.
35.
Nam
,
J. G.
,
Ahn
,
K. H.
,
Lee
,
S. J.
, and
Hyun
,
K.
,
2010
, “
First Normal Stress Difference of Entangled Polymer Solutions in Large Amplitude Oscillatory Shear Flow
,”
J. Rheol.
,
54
(
6
), pp.
1243
1266
.
36.
Ince
,
E. L.
,
1956
,
Integration of Ordinary Differential Equations
,
Oliver and Boyd
,
London
.
37.
Giacomin
,
A. J.
,
Bird
,
R. B.
,
Johnson
,
L. M.
, and
Mix
,
A. W.
,
2012
, “
Corrigenda: ‘Large-Amplitude Oscillatory Shear Flow From the Corotational Maxwell Model’ [Journal of Non-Newtonian Fluid Mechanics 166 (2011) 1081–1099]
,”
J. Non-Newtonian Fluid Mech.
,
187–188
, p.
48
.
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