Several constitutive theories have been proposed in the literature to model the viscoelastic response of materials, including widely used rheological constitutive models. These models are characterized by certain parameters (“time constants”) that define the time scales over which the material relaxes. These parameters are primarily obtained from stress relaxation experiments using curve-fitting techniques. However, the question of how best to estimate these time constants remains open.

As a step towards answering this question, we propose an optimal experimental design approach based on ideas from information geometry, namely Fisher information and Kullback–Leibler divergence. The material is modeled as a spring element in parallel with multiple Maxwell elements and described using a one- or two-term Prony series. Treating the time constants as unknowns, we develop expressions for the Fisher information and Kullback–Leibler divergence that allow us to maximize information gain from experimental data. Based on the results of this study, we propose that the largest time constant estimated from a stress relaxation experiment for a linear viscoelastic material should be at most one-fifth of the total time of the experiment in order to maximize information gain. Our results also provide confirmation that the equilibrium modulus of the material cannot be reliably determined from curve-fitting to data from a stress relaxation experiment.

文献中已经提出了几种本构理论来模拟材料的粘弹性响应，包括广泛使用的流变本构模型。这些模型的特征在于某些参数（“时间常数”），这些参数定义了材料松弛的时间尺度。这些参数主要从使用曲线拟合技术的应力松弛实验中获得。但是，如何最好地估计这些时间常数的问题仍然存在。

为了回答这个问题，我们基于信息几何学的思想，即Fisher信息和Kullback-Leibler散度，提出了一种最佳的实验设计方法。该材料被建模为与多个Maxwell元素平行的弹簧元素，并使用一或两个项的Prony系列进行描述。将时间常数视为未知数，我们开发了Fisher信息和Kullback-Leibler散度的表达式，使我们能够从实验数据中获得最大的信息收益。根据这项研究的结果，我们建议从应力松弛实验中估计的线性粘弹性材料的最大时间常数最多应为实验总时间的五分之一，以最大程度地增加信息增益。