On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

Abstract

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G ∞)/(G 0-G ∞) and the retardation function r(t) = (J ∞+t/η-J(t))/(J ∞-J 0) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (−(t/τ)β), can r(t) be represented as exp (−(t/λ)µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η−1 is finite for a fluid and zero for a solid), G ∞ is the equilibrium modulus G e for a solid or zero for a fluid, J ∞ is 1/G e for a solid or the steady-state recoverable compliance for a fluid, G 0= 1/J 0 is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.