Establishing continuous relaxation spectrum based on complex modulus tests to construct relaxation modulus master curves in compliance with linear viscoelastic theory

Abstract

A number of methods have been developed to determine the relaxation modulus, which is a fundamental material property that characterizes the rheological behavior of viscoelastic materials. However, the relaxation modulus master curves constructed using the current methods are likely to be accompanied by evident deficiencies. To correct these deficiencies, this study developed a method of using complex modulus tests to construct the master curve model of the relaxation modulus with a continuous relaxation spectrum that complied with the linear viscoelastic theory and allowed possible asymmetry in the relaxation spectrum curve. Using the data of the complex modulus tests, master curve models of the storage modulus and loss modulus were developed according to an approximate Kramers-Kronig relation. Based on the relationship between the relaxation modulus and the complex modulus, a specific model form of the continuous relaxation spectrum was established in terms of the same model parameters as those of the master curve models of the storage modulus and loss modulus. The established spectrum was found to be asymmetric and was ensured to be compliant with the linear viscoelastic theory. With the established relaxation spectrum, the master curve model of the relaxation modulus was formulated, and its numerical solution was determined as a function of the loading time; the master curves of the relaxation moduli were therefore constructed. The accuracy of the constructed master curves of the relaxation moduli were evaluated in a fairly wide time range and at the boundaries. The evaluation results demonstrated that the numerical method used to obtain the numerical solution of the relaxation modulus model produced negligible errors and that the approximate Kramers-Kronig relation provided a satisfactory representation of the mathematical interrelation of the viscoelastic parameters.The method developed in this study can be utilized to construct an accurate master curve of the relaxation modulus of any linear viscoelastic material. The constructed master curve of the relaxation modulus complies with the linear viscoelastic theory; specifically, it satisfies the approximate Kramers-Kronig relation and addresses the complete linear viscoelastic information delivered by both storage modulus and loss modulus.