Methods for estimation of the complex modulus generally produce data from which discrete results can be obtained for a set of frequencies. As these results are normally afflicted by noise, they are not necessarily consistent with the principle of causality and requirements of thermodynamics. A method is established for noise-corrected estimation of the complex modulus, subject to the constraints of causality, positivity of dissipation rate and reality of relaxation function, given a finite set of angular frequencies and corresponding complex moduli obtained experimentally. Noise reduction is achieved by requiring that two self-adjoint matrices formed from the experimental data should be positive semidefinite. The method provides a rheological model that corresponds to a specific configuration of springs and dashpots. The poles of the complex modulus on the positive imaginary frequency axis are determined by a subset of parameters obtained as the common positive zeros of certain rational functions, while the remaining parameters are obtained from a least squares fit. If the set of experimental data is sufficiently large, the level of refinement of the rheological model is in accordance with the material behavior and the quality of the experimental data. The method was applied to an impact test with a Nylon bar specimen. In this case, data at the 29 lowest resonance frequencies resulted in a rheological model with 14 parameters. The method has added improvements to the identification of rheological models as follows: (1) Noise reduction is fully integrated. (2) A rheological model is provided with a number of elements in accordance with the complexity of the material behavior and the quality of the experimental data. (3) Parameters determining poles of the complex modulus are obtained without use of a least squares fit.

References

References
1.
Duncan
,
J.
,
1999
, “
Dynamic Mechanical Analysis Techniques and Complex Modulus
,”
Mechanical Properties and Testing of Polymers
,
G. M.
Swallowe
, ed.,
Kluwer
,
Dordrecht
, pp.
43
48
.
2.
Garret
,
S. L.
,
1990
, “
Resonant Acoustic Determination of Elastic Moduli
,”
J. Acoust. Soc. Am.
,
88
(
1
), pp.
210
221
.10.1121/1.400334
3.
Guo
,
Q.
, and
Brown
,
D. A.
,
2000
, “
Determination of the Dynamic Elastic Moduli and Internal Friction Using Thin Rods
,”
J. Acoust. Soc. Am.
,
108
, pp.
167
174
.10.1121/1.429453
4.
Madigowski
,
W. M.
, and
Lee
,
G. F.
,
1983
, “
Improved Resonance Technique for Materials Characterization
,”
J. Acoust. Soc. Am.
,
73
(
4
), pp.
1374
1377
.10.1121/1.389242
5.
Pintelon
,
R.
,
Guillaume
,
P.
,
Vanlanduit
,
S.
,
De Belder
,
K.
, and
Rolain
,
Y.
,
2004
, “
Identification of Young's Modulus From Broadband Modal Analysis Experiments
,”
Mech. Syst. Signal Process.
,
18
, pp.
699
726
.10.1016/S0888-3270(03)00045-1
6.
Love
,
A. E. H.
,
1927
,
A Treatise on the Mathematical Theory of Elasticity
,
Cambridge University
,
London
, p.
428
.
7.
Blanc
,
R. H.
,
1971
, “
Détermination de l’Équation de Comportement des Corps Visco-élastiques Linéaires par une Méthode d'Impulsion
,“
Ph.D. thesis
,
l'Université d'Aix-Marseille
, Marsielle, France.
8.
Blanc
,
R. H.
,
1993
, “
Transient Wave Propagation Methods for Determining the Viscoelastic Properties of Solids
,”
J. Appl. Mech.
,
60
, pp.
763
768
.10.1115/1.2900870
9.
Lundberg
,
B.
, and
Blanc
,
R. H.
,
1988
, “
Determination of Mechanical Material Properties From the Two-Point Response of an Impacted Linearly Viscoelastic Rod Specimen
,”
J. Sound Vib.
,
137
, pp.
483
493
.10.1016/0022-460X(90)90813-F
10.
Lundberg
,
B.
, and
Ödeen
,
S.
,
1993
, “
In Situ Determination of the Complex Modulus From Strain Measurements on an Impacted Structure
,”
J. Sound Vib.
,
167
, pp.
413
419
.10.1006/jsvi.1993.1345
11.
Hillström
,
L.
,
Mossberg
,
M.
, and
Lundberg
,
B.
,
2000
, “
Identification of Complex Modulus From Measured Strains on an Axially Impacted Bar Using Least Squares
,”
J. Sound Vib.
,
230
, pp.
689
707
.10.1006/jsvi.1999.2649
12.
Othman
,
R.
,
2002
, “
Extension du Champ d'Application du Système des Barres de Hopkinson aux Essais à Moyennes Vitesses de Déformation
,”
Ph.D. thesis
,
Ecole Polytechnique
,
Palaiseau
, France.
13.
Mousavi
,
S.
,
Nicolas
,
D. F.
, and
Lundberg
,
B.
,
2004
, “
Identification of Complex Moduli and Poisson's Ratio From Measured Strains on an Impacted Bar
,”
J. Sound Vib.
,
277
, pp.
971
986
.10.1016/j.jsv.2003.09.053
14.
Zhao
,
H.
, and
Gary
,
G.
,
1995
, “
A Three Dimensional Analytical Solution of Longitudinal Wave Propagation in an Infinite Linear Viscoelastic Cylindrical Bar. Application to Experimental Techniques
,”
J. Mech. Phys. Solids
,
43
, pp.
1335
1348
.10.1016/0022-5096(95)00030-M
15.
Landau
,
L.
, and
Lifchitz
,
E.
,
1984
,
Electrodynamics of Continuous Media
,
2nd ed.
,
Pergamon
,
New York
, pp.
279
283
.
16.
Landau
,
L.
, and
Lifchitz
,
E.
,
1980
,
Statistical Physics
,
2nd ed.
,
Pergamon
,
New York
, pp.
377
384
.
17.
Hanyga
,
A.
2005
, “
Physically Acceptable Viscoelastic Models
,”
Trends in Applications of Mathematics to Mechanics
,
Y.
Wang
, and
K.
Hutter
, eds.,
Shaker Verlag
,
Aachen, Germany
, pp.
125
136
.
18.
Mossberg
,
M.
,
Hillström
,
L.
, and
Söderström
,
T.
,
2001
, “
Non-Parametric Identification of Viscoelastic Materials From Wave Propagation Experiments
,”
Automatica
,
37
(
4
), pp.
511
521
.10.1016/S0005-1098(00)00188-6
19.
Söderstrom
,
T.
,
2002
, “
System Identification Techniques for Estimating Material Functions From Wave Propagation Experiments
,”
Inverse Probl. Eng.
,
10
(
5
), pp.
413
439
.10.1080/1068276021000004715
20.
Golden
,
J.
,
2005
, “
A Proposal Concerning the Physical Rate of Dissipation in Materials With Memory
,”
Q. Appl. Math.
,
63
, pp.
117
155
.
21.
Bouleau
,
N.
,
1999
, “
Visco-élasticité et Processus de Levy
,”
Potential Anal.
,
11
, pp.
289
302
.10.1023/A:1008696219448
22.
Beris
,
A. N.
, and
Edwards
,
B. J.
,
1994
,
The Thermodynamics of Flowing Systems
,
Oxford University
,
New York
, pp.
289
299
.
23.
V Krein
,
M.
, and
Nudelman
,
A.
,
1998
, “
An Interpolation Approach in the Class of Stieltjes Functions and Its Connection With Other Problems
,”
Integral Equ. Oper. Theory
,
30
, pp.
251
278
.10.1007/BF01195584
24.
Flugge
,
W.
,
1975
,
Viscoelasticity
,
Springer-Verlag
,
Berlin
, pp.
17
31
.
25.
Gu
,
G.
,
Xiong
,
D.
, and
Zhou
,
K.
,
1993
, “
Identification in H∞ Using Pick's Interpolation
,”
Syst. Control Lett.
,
20
, pp.
263
272
.10.1016/0167-6911(93)90002-N
26.
Othman
,
R.
,
Blanc
,
R. H.
,
Bussac
,
M. N.
,
Collet
,
P.
, and
Gary
,
G.
,
2002
, “
Identification de la Relation de la Dispersion dans les Barres
,”
C. R. Mec.
,
330
, pp.
849
855
.10.1016/S1631-0721(02)01541-3
27.
Kolsky
,
H.
,
1963
,
Stress Waves in Solids
,
Clarendon
,
Oxford, UK
, p.
43
.
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