Abstract Indentation type dynamic mechanical analysis (IT-DMA) is an attractive approach to characterizing viscoelastic properties of soft matter under both quasi-static and dynamic loadings. Frequency-dependent viscoelastic properties are usually characterized by storage and loss moduli whereas extracting inherent viscoelastic model parameters from dynamic loadings remains unsolved. In this study, closed-form solutions to dynamic oscillation loadings were first derived and adapted to a variety of excitations for arbitrary shape indenters. A uniform theoretical framework for ramp-hold-oscillation-relaxation/creep indentation tests are established and Kelvin-Voigt fractional derivative (KVFD) solutions are presented to predict the viscoelastic behavior for both quasi-static loadings (creep, relaxation) and dynamic loadings (sinusoidal displacement/force cycles at 5,10,15,20,30 Hz). Four gelatin-cream hydrogel samples were investigated under ramp-hold-oscillation-relaxation and ramp-hold-oscillation-creep testing conditions. Viscoelastic quantities of elasticity E0, fluidity α, and viscosity τ were extracted by fitting the experimental curves to KVFD predictions. The results show that KVFD models can accurately depict the viscoelastic behavior under both dynamic and quasi-static loadings with high goodness of fit (R2>0.936). To assess the proposed method, comparison of the viscoelastic quantities from different experimental protocols were conducted. Statistical results revealed viscoelastic parameters from different loading are in close agreement. For dynamic loadings, significant differences were found for [E0, α] estimates at different frequencies. For different experimental protocols, estimations were consistent in that E0 showed maximum value on Gel5Cream15 sample, and α was increased with increasing cream percentage. Moreover, viscoelastic quantities extracted from dynamic relaxation were correlated with those of the dynamic creep (r = 0.977, p

中文翻译：

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通过 Kelvin-Voigt 分数阶导数建模评估水凝胶粘弹性的斜坡保持动态振荡压痕测试的解决方案

摘要 压痕型动态力学分析 (IT-DMA) 是表征软物质在准静态和动态载荷下的粘弹性特性的一种有吸引力的方法。频率相关的粘弹性特性通常以存储和损耗模量为特征，而从动态载荷中提取固有的粘弹性模型参数仍未解决。在这项研究中，动态振荡载荷的封闭形式解决方案首先被推导出并适用于任意形状压头的各种激励。建立了斜坡-保持-振荡-松弛/蠕变压痕测试的统一理论框架，并提出了 Kelvin-Voigt 分数阶导数 (KVFD) 解来预测两种准静态载荷（蠕变、蠕变、蠕变）的粘弹性行为。松弛）和动态载荷（5、10、15、20、30 Hz 下的正弦位移/力循环）。在斜坡保持振荡松弛和斜坡保持振荡蠕变测试条件下研究了四种明胶奶油水凝胶样品。通过将实验曲线拟合到 KVFD 预测来提取弹性 E0、流动性 α 和粘度 τ 的粘弹性量。结果表明，KVFD 模型可以准确地描述动态和准静态载荷下的粘弹性行为，拟合优度高（R2>0.936）。为了评估所提出的方法，进行了来自不同实验方案的粘弹性量的比较。统计结果显示来自不同载荷的粘弹性参数非常一致。对于动态载荷，[E0, α] 在不同频率下的估计。对于不同的实验方案，估计是一致的，因为 E0 在 Gel5Cream15 样品上显示最大值，并且 α 随着奶油百分比的增加而增加。此外，从动态松弛中提取的粘弹性量与动态蠕变的粘弹性量相关（r = 0.977，p