The benefit of fractional derivatives in modelling the dynamics of filler-reinforced rubber under large strains: a comparison with the Maxwell-element approach
The benefit of fractional derivatives in modelling the dynamics of filler-reinforced rubber under large strains: a comparison with the Maxwell-element approach
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DOI：
10.1007/s00466-013-0946-4
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发表时间：
2014-05
影响因子：
4.1
通讯作者：
D. Wollscheid;A. Lion
中科院分区：
文献类型：
--
作者：
D. Wollscheid;A. Lion
橡胶类材料的动态特性与预变形和频率有很大的依赖关系。本文的重点是用分数导数的概念来描述一种填充量为40份的炭黑填充丁苯橡胶的频率和预变形相关的动态行为。因此，我们引入了一种有限分数粘弹性本构方法，该方法适用于关于储能和损耗模数的动态材料性质的近似。本构方法是在文献[18]的基础上提出的，在前人的工作[46]中引入了变形相关松弛函数来表示动弹模对预变形和频率的依赖关系。文[46]中的本构方法是基于经典的有限粘弹性理论，在频域中建立的。在这项工作中，将利用分数导数的概念来扩展[46]的方法，并将其与经典的方法进行比较。为此，首先介绍了经典和扩展的分数阶本构模型，并推导了两种模型的复模张量。值得一提的是，这两种本构方法都是首先在时间域中制定的。这个公式是满足热力学一致性所必需的。为了计算效率高的弹性体结构的振动分析，将方程转换到频域。为此，本构模型在大的和时间上恒定的预变形附近被几何线性化。增量应变张量是简谐变化的，其幅值必须很小。此外，通过对炭黑填充丁苯橡胶的准静态和动态研究，对两种方法进行了参数辨识。从所需材料参数的个数和逼近的质量两方面比较了经典模型和分数模型参数识别的数值结果。最后，在文献[28]的基础上，给出了频域公式在有限元程序MSC Marcon中的数值实现。
The dynamic properties of rubber-like materials are characterised by a significant dependence on the predeformation and the frequency. The focus of this paper is to represent the frequency and predeformation dependent dynamic behaviour of a carbon-black filled SBR rubber with 40 phr amount of filler using the concept of fractional derivatives. Thus, we introduce a constitutive approach of finite fractional viscoelasticity which is suitable to approximate the dynamic material properties with respect to the storage and the loss modulus. The constitutive approach is based on a proposal of [18] which was modified by a deformation dependent relaxation function in a previous work [46] to represent the dependence of the dynamic modulus on the predeformation and the frequency. The constitutive approach in [46] is based on the classical theory of finite viscoelasticity and formulated in the frequency domain. In this work, the approach of [46] will be extended by the concept of fractional derivatives and compared to the classical one. Thus, the classical and the extended fractional constitutive models are firstly introduced and the complex modulus tensors of both models are derived. It should be mentioned that both constitutive approaches are firstly formulated in the time domain. This formulation is necessary to satisfy the thermodynamical consistency. In order to conduct vibration analyses of elastomer structures with high computational efficiency, the equations are then transferred to the frequency domain. To this end, the constitutive model is geometrically linearised in the neighbourhood of a large and temporally constant predeformation. The incremental strain tensor varies harmonically and its amplitude has to be small. Furthermore, parameter identification of both approaches is done on the basis of quasi-static and dynamic investigations of the carbon-black filled SBR rubber. The numerical results of the parameter identification of the classical and the fractional model are compared to each other with respect to the number of necessary material parameters and the quality of the approximation. Finally, the numerical implementation of the frequency domain formulation into the finite element codeMSC Marcon the basis of the proposal of [28] will be presented.