Interconversions between linear viscoelastic functions by using relaxation-creep duality representation

Abstract

The interconversions between relaxation moduli and creep compliances, including stretch, shear, bulk parts, and the time-dependent Poisson's ratio, are derived by using the relaxation-creep duality representation. The relaxation-creep duality representation for the viscoelastic functions introduced in this paper is composed of an exponential function that characterizes the relaxation behavior and a complementary one that characterizes the creep behavior. All viscoelastic functions can be represented as the same form. The new sets of coefficients, called the modulating constants, between viscoelastic functions obey the elastic-like interconversions, and do not involve the characteristic times. The relationships of characteristic times between those functions are also derived. These interconversion formulas can then be calculated easily. Three literatures are referenced to calculate the consistency of the viscoelastic functions via the new interconversions introduced in this work. The Young's relaxation modulus in one literature is not consistent to the shear one in another literature. By assuming a constant bulk modulus, the modified Young's relaxation modulus and time-dependent Poisson's ratio that was derived by the new interconversions can meet the measured curves and can be consistent to the shear creep compliance in the literatures. The fitted data from experiments can then be checked via the new mathematical interconversions for the consistency.

A unified procedure for rapidly determining asphalt concrete discrete relaxation and retardation spectra

“…Other methods were reported to resolve the ill-posed (i.e., non-unique) issues during the spectral identification, e.g., the Tikhonov regularization method [14], the quadratic programming regularization method [15] and the maximum entropy method [16]. In addition, given that the discrete relaxation and retardation spectra are mathematically equivalent [4], some researchers indirectly obtained one discrete spectrum from the other by using different inter-conversion methods [17][18][19][20][21][22].…”

Interconversions between linear viscoelastic functions with a time-dependent bulk modulus

“…It had been proven in a previous work [2] that for the constant bulk modulus, the characteristic times of the time-dependent Poisson’s ratio are the same as those of Young’s relaxation modulus. From equation (24), it can be seen that even for the time-dependent bulk modulus, the characteristic times of the time-dependent Poisson’s ratio are still the same as those of Young’s relaxation modulus.…”

“…If K ( t ) is time-independent, then all viscoelastic functions can be expressed as the relaxation-creep duality representation (a variant of Prony series representation) [2]…”

“…It should be noted that A ( t ) in equation (15) is monotonic if all 0 ≤ a n A ≤ 1 , a constant if A 0 = A ∞ , a relaxation modulus if A 0 > A ∞ , a creep compliance if A 0 < A ∞ [2], non-monotonic if some of a n A < 0 or a n A > 1 [10], and oscillatory if some of τ n A are complex-valued [11].…”

“…then equation (1) can be translated to the time-differential constitutive equation (TDCE) [2] € s(t)…”