Complex viscoelastic materials exhibit power law (PL) relaxations, as opposed to simple materials described by exponential decays. Other interesting materials, like living cells, hold a universal double PL behavior whose exponents depend on the health and type of the cells. Usually, only dynamic assays are considered capable to study such viscoelastic relaxation mechanisms. In this work, we propose analytical responses with single or multiple power-law relaxation behavior by generalizing classical viscoelastic models in terms of fractional derivatives of arbitrary order α (0 1). In addition, we demonstrate that simple atomic force microscopy force curves are powerful methods to directly observe the viscoelastic relaxation of such complex materials. In order to validate our findings, we compare the viscoelastic relaxation exponents measured directly from simple force curves (SFCs) with those measured with dynamic techniques in both living cells and polyacrylamide gels. We believe the fractional models unveiled here describe a variety of complex materials and may be used (with SFCs) to explore sophisticated viscoelastic phenomena.

复杂的粘弹性材料表现出幂律 (PL) 弛豫，这与由指数衰减描述的简单材料相反。其他有趣的材料，如活细胞，具有普遍的双 PL 行为，其指数取决于细胞的健康状况和类型。通常，只有动态分析被认为能够研究这种粘弹性松弛机制。在这项工作中，我们通过将经典粘弹性模型推广到任意阶α (01）。此外，我们证明了简单的原子力显微镜力曲线是直接观察这种复杂材料的粘弹性松弛的有力方法。为了验证我们的发现，我们将直接从简单力曲线 (SFC) 测量的粘弹性松弛指数与在活细胞和聚丙烯酰胺凝胶中使用动态技术测量的粘弹性松弛指数进行了比较。我们相信这里公布的分数模型描述了各种复杂的材料，并且可以（与 SFC 一起）用于探索复杂的粘弹性现象。