任意荷载下分数阶导数黏弹性饱和土体一维固结

将分数阶导数理论引入Kelvin–Voigt模型,来描述黏弹性饱和土体的力学行为。对饱和土体一维固结方程和分数阶导数Kelvin–Voigt本构方程实施Laplace变换,联立求解得到变换域内有效应力和沉降的解析解。采用Crump方法实现Laplace数值反演,获得了任意荷载情况下物理空间内一维固结问题的半解析解。并将指数荷载情况下分数阶导数模型退化到黏弹性情形,结果与已有文献解析解相同,验证了本研究提出任意荷载情况下分数阶导数黏弹性解的可靠性。最后,分析了相关参数对固结沉降的影响。研究表明,任意荷载情形下分数阶导数黏弹性饱和土体一维固结发展过程与黏滞系数和分数阶次有关,分数阶次越大,固结沉降发展越快;黏滞系数越大,固结沉降变化越慢;荷载变化趋势与由荷载参数变化引起的沉降变化规律是一致的,且最终沉降量一致。本研究有助于深入认识分数阶黏弹性饱和土体的固结行为。

One-dimensional consolidation of fractional order derivative viscoelastic saturated soils under arbitrary loading

The theory of fractional calculus is introduced into the Kelvin-Voigt constitutive model to describe the mechanical behavior of viscoelastic saturated soils. Applying the Laplace transform upon one-dimensional consolidation equation of saturated soils and the fractional order derivative Kelvin-Voigt constitutive equation, the analytical solutions of the effective stress and the settlement are derived in the Laplace domain. Then the semi-analytical solutions to one-dimensional consolidation problem under arbitrary loadings in physical space are obtained after implementing the Laplace numerical inverse transform using the Crump’s method. As the case of viscoelasticity, the simplified semi-analytical solutions under exponential loading in this study are the same as the available analytical solutions in literatures. It is indicated that the proposed solutions under arbitrary loading are reliable. Finally, parametric studies are conducted to analyze the effects of the related parameters on the consolidation settlement. The results show that the process of one-dimensional consolidation of viscoelastic saturated soils with fractional order derivative is related to viscosity coefficient and fractional order. The larger the fractional order is, the more quickly the consolidation settlement occurs; and the higher the viscosity coefficient is, the slower the consolidation settlement takes place. The trend of loadings is consistent with the variation pattern of soil settlement caused by the change of the load parameters, and the final settlement is identical. The present study can be of help to further understand the consolidation behavior of viscoelastic saturated soils.