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Numerical determination of retardation and relaxation spectra optimalization of numerical process

  • J. Stanislav
  • F. A. Seyer
  • B. Hlaváček

Abstract

The behavior of all viscoelastic materials under a variety of linear stress strain conditions may be described by any of the functions such as relaxation modulus G(t), storage G′(ω) and loss modulus G″(ω), creep compliance J(t), storage J′(ω) and loss compliance J′(ω). All of them may be determined by either transient or dynamic types of experiment and carry the signature of the time function which produced them. The two functions — the relaxation H(ln τ) and retardation L(ln τ) spectra were derived in order to depict the behavior of a system regardless of the type of experiment. It should be noted that symbol H(τ) dlnτ, for the relaxation spectrum, will be used in this study to describe the contribution to rigidity associated with relaxation times whose logarithms lie in the range between lnτ and lnτ + dlnτ.

List of symbols

aij

elements of matrix t

bij

elements of matrix β

kij

elements of matrix x

mij

elements of matrix M

s

variable in eq. [1]

vi

elements of matrix V

X

variable in eq. [1]

γ

smoothing factor

λ

matrix eigenvalue

V

defined by eq. [6]

φ(x)

experimental measured function, defined by eq.[1]

F(s)

defined by eq. [1]

F*(s)

defined by eq. [13]

G′(x)

real part of dynamic modulus

H3(x)

third Schwarzl-Staverman approximation of relaxation spectrum

J(x)

creep compliance

K(x, s)

kernel of integral in eq. [1]

L(x)

second Schwarzl-Staverman approximation of retardation spectrum

R1

cumulative relative error of approximation of relaxation spectrum; defined by eq. [3]

R2

cumulative relative error of spectral relaxation function; defined by eq. [4]

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Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • J. Stanislav
    • 2
    • 1
    • 3
  • F. A. Seyer
    • 2
    • 1
    • 3
  • B. Hlaváček
    • 2
    • 1
    • 3
  1. 1.Department of Chemical EngineeringThe University of CalgaryCalgaryCanada
  2. 2.Department of Chemical and Petroleum EngineeringThe University of Alberta EdmontonCanada
  3. 3.Department of Chemical EngineeringEcole PolytechniqueMontrealCanada

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