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

The use of the stretched-exponential function to represent both the relaxation function g(t) = (G(t)-G(infinity)/(G(0)-G(infinity)) and the retardation function r(t) = (J(i)nfinity+t/eta-J(t))/(J(x)-J(0)) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (-(t/tau)(beta), can r(t) be represented as exp (-(t/lambda)(mu)) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, g is the viscosity (eta(-1) is finite for a fluid and zero for a solid), G(infinity) is the equilibrium modulus G(e) for a solid or zero for a fluid, J(infinity) 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 beta = mu in the range 0.5 to 0.6, with the correspondence being very close with beta = mu = 0.5 and lambda = 2 tau. 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.