APPROXIMATION OF THE CONTINUOUS RELAXATION SPECTRUM OF PLANT VISCOELASTIC MATERIALS USING LAGUERRE FUNCTIONS
Journal Title: ELECTRONIC JOURNAL OF POLISH AGRICULTURAL UNIVERSITIES - Year 2008, Vol 11, Issue 1
Abstract
Relaxation spectrum is a very useful tool in the characterization of viscoelactic materials as the knowledge on them enables the calculation of any linear material functions such as the creep compliance, the Poisson’s ratio or shear and bulk modulus. The paper deals with the problem of recovery of continuous relaxation spectrum from discrete-time noise corrupted measurements of relaxation modulus obtained in stress relaxation test. An optimal orthogonal scheme of the least squares approximation of the spectrum of relaxation frequencies by the finite series of Laguerre functions is presented. The approach proposed allows us to reduce the primary infinite dimensional dynamic optimization problem to a static linear-quadratic programming task. The problem of relaxation spectrum identification is the practical ill-posed problem of reconstructing solution of Fredholm integral equation of the first kind from the measured data. Thus, Tikhonov regularization is used to guarantee the stability of the scheme. Generalized cross validation (GCV) is adopted for the optimal choice of the regularization parameter. The numerical realization of the scheme by using the singular value decomposition (SVD) is discussed and the resulting computer algorithm is outlined. An analysis of the model accuracy is conducted for noise measurements and the linear convergence of the approximations generated by the scheme is proved. It is also indicated that the accuracy of the spectrum approximation depends both on measurement noises and regularization parameter as well as on the proper selection of the basic orthogonal functions. The method combines effectiveness and accuracy and is general enough to cover both viscoelastic solids and liquids. Applying the scheme proposed, the relaxation spectrum of an unconfined cylindrical specimen of the beet sugar root is determined.
Authors and Affiliations
Anna Stankiewicz, Krzysztof Gołacki
Keywords
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