Research Papers

Noise-Corrected Estimation of Complex Modulus in Accord With Causality and Thermodynamics: Application to an Impact Test

[+] Author and Article Information
P. Collet

Centre de Physique Théorique
Ecole Polytechnique
91128 Palaiseau, France
e-mail: Pierre.Collet@cpht.polytechnique.fr

G. Gary

Laboratoire de Mécanique des Solides
Ecole Polytechnique
91128 Palaiseau, France
e-mail: gary@lms.polytechnique.fr

B. Lundberg

Laboratoire de Mécanique des Solides
Ecole Polytechnique
91128 Palaiseau, France
The Ångström Laboratory
Uppsala University
Box 534, SE-751 21 Uppsala, Sweden
e-mail: bengt.lundberg@angstrom.uu.se

1Corresponding author.

Manuscript received June 29, 2011; final manuscript received June 27, 2012; accepted manuscript posted July 6, 2012; published online November 19, 2012. Assoc. Editor: Krishna Garikipati.

J. Appl. Mech 80(1), 011018 (Nov 19, 2012) (7 pages) Paper No: JAM-11-1208; doi: 10.1115/1.4007081 History: Received June 29, 2011; Revised June 27, 2012; Accepted July 06, 2012

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.

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Grahic Jump Location
Fig. 1

Rheological model with 2(p + 1) parameters

Grahic Jump Location
Fig. 2

Impact test with a uniform viscoelastic bar specimen

Grahic Jump Location
Fig. 8

Real part E' and imaginary part E"of the complex modulus E at low frequencies determined from servo-hydraulic tests

Grahic Jump Location
Fig. 3

Recorded strain in the Nylon bar specimen. (a) Long-time record of strain pulses ɛb∞(t). (b) Primary strain pulse ɛb1(t) followed by strain pulses which have undergone one and two free-end reflections.

Grahic Jump Location
Fig. 4

Spectrum |ɛ∧b∞| of the recorded strain in the Nylon bar specimen

Grahic Jump Location
Fig. 5

Details of the spectrum |ɛ∧b∞|2. The thin and thick curves are based on the left and the right member of Eq. (30), respectively. (a) 1st resonance peak, n = 1. (b) 25th resonance peak, n = 25.

Grahic Jump Location
Fig. 6

(a) Phase velocity c and (b) damping coefficient α versus frequency. Open circles: discrete experimental results. Continuous curves: results based on the rheological model given by Eqs. (12) and (14).

Grahic Jump Location
Fig. 7

(a) Real part E' and (b) imaginary part E" of the complex modulus E versus frequency. Open circles: discrete experimental results. Filled circles: discrete noise-corrected results. Continuous curves: results obtained from the rheological model given by Eqs. (12) and (14).



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