Balbi Valentina, Shearer Tom, Parnell William J
School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road, Galway, Republic of Ireland.
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK.
Proc Math Phys Eng Sci. 2018 Sep;474(2217):20180231. doi: 10.1098/rspa.2018.0231. Epub 2018 Sep 19.
The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.
准线性粘弹性（QLV）理论针对有限变形下的横观各向同性（TI）材料进行了修正和拓展。首次根据变形的物理模式，将不同的松弛响应纳入非线性粘弹性的积分公式中。该理论在小应变极限下与线性粘弹性一致，并且在时间/变形可分离性假设下，利用可从小应变实验确定的松弛函数。在考虑了适用于可压缩材料的一般本构形式之后，注意力集中在不可压缩介质上。这使得能够推导出本构关系的紧凑形式，用于说明模型在三种关键变形下的行为：单轴拉伸、横向剪切和纵向剪切。最后，结果表明，与经历相同变形的新胡克材料相比，在横向剪切下，TI、新胡克、修正QLV材料中存在坡印廷效应。其存在是由介质的各向异性松弛响应所解释的。
