Skip to main navigation menu Skip to main content Skip to site footer
ASPTA Baner

Vol. 3 No. 1-2 (2004)

Artykuły

A scheme for identification of time-variable shear and bulk modulus of plant viscoelastic materials

DOI: https://doi.org/10.24326/aspta.2004.1-2.8
Submitted: March 22, 2023
Published: 2004-12-31

Abstract

The aim of the paper is to investigate a computer identification algorithm for determining the time-varying shear and bulk modulus of linear viscoelastic materials based on experimentally obtained stress and strain uniaxial relaxation functions of an cylindrical specimens of the tested material is proposed. On the first stage of the identification procedure the models of the uniaxial relaxation functions are determined and next, on the second stage, the shear and bulk modulus are computed. It is proved that if four parameter Maxwell models are used to approximate the uniaxial relaxation functions, then the shear and bulk modulus can be given as a linear combination of exponential functions and multiple convolution integrals of original and modified Bessel functions. For least squares identification of uniaxial relaxation functions by positive sums of exponentials the Prony's method is applied. Applying the scheme proposed the time-dependent shear and bulk modulus of the beet sugar root samples are computed using discrete time-measurements of the force relaxation of an unconfined and of a laterally constrained cylindrical specimens of the material obtained in double independent stress relaxation tests.

References

  1. Chen P., Chen S., 1986. Stress relaxation functions of apple under high loading rates. Transaction of the ASAE. 29, 1754-1759. DOI: https://doi.org/10.13031/2013.30384
  2. Chen P., Fridley R. B., 1972. Analytical method for Determining Viscoelastic Constants of Agricultural Materials. Transaction of the ASAE. 15, 1103-1106. DOI: https://doi.org/10.13031/2013.38080
  3. Christensen R. M., 1971. Theory of Viscoelasticity. An Introduction. Academic Press New York. De Baerdemeaker J. G., Segerlind L. J., 1976. Determination of the viscoelastic properties of the apple flesh. Transaction of the ASAE. 19, 346-353.
  4. Derski W., Ziemba S., 1968. Analiza modeli reologicznych. PWN Warszawa.
  5. Evans J. W., Gragg W.B., LeVeque R. J., 1980. On Least-Squares Exponential Sum Approximation with Positive Coefficients. Mathematics of Computations. 34(149), 203-211. DOI: https://doi.org/10.1090/S0025-5718-1980-0551298-6
  6. Fincan M., Dejmek P., 2003. Efect of osmotic pretreatment and pulsed electric field on the viscoelastic properties of potato tissue. Journal of Food Engineering, 59, 169–175 DOI: https://doi.org/10.1016/S0260-8774(02)00454-5
  7. Gołacki K., 1998. Charakterystyki lepkosprężyste korzeni marchwi w szerokim zakresie prędkości obciążeń mechanicznych. Rozprawy Naukowe Akademii Rolniczej w Lublinie. 216.
  8. Gołacki K., 2002. Lepkosprężyste charakterystyki korzeni buraków cukrowych. Acta Agrophysica. 78, 37-49.
  9. Gołacki K., Stankiewicz A., 2002. Algorytm obliczeniowy wyznaczania współczynnika Poissona lepkosprężystego materiału roślinnego. Acta Agrophysica. 78, 51-61.
  10. Gutenbaum J., 2003. Modelowanie matematyczne systemów. Akad. Ofic. Wyd. EXIT Warszawa.
  11. Hildebrand F. B., 1956. Introduction to Numerical Analysis. McGraw-Hill New York.
  12. Hughes H., Segerlind L. J., 1972. A rapid mechanical method for determining Poisson's ratio of biological materials. ASAE Paper No. 72-310, ASAE, St. Joseph, MI 49085.
  13. Kammler D. W., 1979. Least squares approximation of completely monotonic functions by sums of exponentials. SIAM J. Numer. Anal. 16(5), 801-818. DOI: https://doi.org/10.1137/0716060
  14. Lanczos C., 1956. Applied Analysis. Prentice Hall Englewood Clifs.
  15. Osborne, M. R., 1975. Some special nonlinear least squares problems. SIAM J. Num. Anal. 12, 571-592. DOI: https://doi.org/10.1137/0712044
  16. Osborne, M. R., Smyth, G. K., 1995. A modified Prony algorithm for fitting sums of exponential functions. SIAM J. Sci. Comput. 16, 119-138. DOI: https://doi.org/10.1137/0916008
  17. Petersson J., Holmström K., 1998. Initial values for two-classes of exponential sum least squares fitting problems. Research Report IMa-TOM-1998-07. Mälardalen University, Sweden.
  18. Prony G. R., 1795. Essai éxperimental et analytique: sur les lois de la dilatabilité de fluides élastique et sur celles de la force expansive de la vapeur de l'alkool, à différentes températures. J. École Polytechn. 1(22), 24-76.
  19. Rao M. A., Steffe J. F., 1992. Viscoelastic properties of foods. Elsevier Ltd London
  20. Ruhe A., 1980. Fitting empirical data by positive sums of exponentials. SIAM J. Sci. Stat. Comput. 1(4), 481-498. DOI: https://doi.org/10.1137/0901035
  21. Stankiewicz A., 2000. Algorytm wyznaczania oryginałów transformat Laplace'a wybranych funkcji. Praca niepubl. Inst. Podstaw Techn., AR Lublin.
  22. Stankiewicz A., 2003. Algorytm identyfikacji ciągłego spektrum czasów relaksacji biologicznych materiałów lepkosprężystych. Acta Sci. Pol., Technica Agraria. 2(2), 77-91.
  23. Stankiewicz A., 2004. Identyfikacja modeli matematycznych lepkosprężystych materiałów biologicznych metodą Prony'ego. Acta Sci. Pol., Technica Agraria (w druku).
  24. Szabatin J., 1990. Podstawy teorii sygnałów. WKŁ, Warszawa.
  25. Tichonow A. N., Samarski A. A., 1963. Równania fizyki matematycznej. PWN Warszawa.
  26. Varah J. M., 1985. On fitting exponentials by nonlinear least squares. SIAM J. Sci. Stat. Comput. 6 (1), 30-44. DOI: https://doi.org/10.1137/0906003
  27. Węgrzyn S., 1958. Przebiegi nieustalone w elektrycznych liniach i układach łańcuchowych. PWN Warszawa.

Downloads

Download data is not yet available.

Similar Articles

<< < 1 2 3 > >> 

You may also start an advanced similarity search for this article.