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

• Published in: Rheologica acta 1997-05, Vol.36 (3), p.320-329
• Main Authors: BERRY, G. C, PLAZEK, D. J
• Format: Article
• Language: English
• Subjects: Applied sciences; Creep (materials); Exact sciences and technology; Exponential functions; Organic polymers; Physicochemistry of polymers; Properties and characterization; Shear modulus; Solution and gel properties; Stress relaxation; Viscoelastic fluids; Viscoelasticity
• ISSN: 0035-4511; 1435-1528
• DOI: 10.1007/BF00366673

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G∞)/(G0-G∞) and the retardation function r(t) = (J∞+t/η-J(t))/(J∞-J0) 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 Ge for a solid or zero for a fluid, J∞ is 1/Ge for a solid or the steady-state recoverable compliance for a fluid, G0= 1/J0 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.