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将分数阶微积分理论引入Kelvin-Voigt本构模型,以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上述分数阶导数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,从而获得了物理空间一维固结问题的半解析解,并将其退化到弹性和黏弹性两种经典情形,分析表明,它与经典解析解完全相同,这证明了经典弹性和黏弹性解析解可视为本研究提出分数阶导数黏弹性解的特例。开展了参数研究,即分析了相关各种参数对固结沉降的影响。研究表明,瞬时荷载情形下分数阶导数黏弹性饱和土体一维固结最终沉降量与黏滞系数和分数阶次无关,而不同黏滞系数和分数阶次对固结时间有较大影响。其研究结果有助于深入认识黏弹性饱和土体的固结力学行为。
The fractional calculus theory is introduced into the Kelvin-Voigt constitutive model to describe the mechanical behavior of viscoelastic saturated soils. The Laplace transform is applied to the one-dimensional consolidation equation of saturated soil and the Kelvin-Voigt constitutive equation of fractional derivative described above, and the analytical solutions of effective stress and settlement in the transform domain are obtained by the simultaneous solution. The Laplace numerical inversion using the Crump method gives a semi-analytical solution to the one-dimensional consolidation problem in the physical space and degenerates into two classical cases of elasticity and viscoelasticity. The analysis shows that it is exactly the same as the classical analytical solution. It is proved that the classical elastic and viscoelastic analytical solution can be regarded as a special case of the viscoelastic solution of fractional derivative proposed in this study. The parameter study was carried out, that is, the influence of various parameters on the consolidation settlement was analyzed. The results show that the ultimate settlement of viscoelastic saturated soils with fractional derivative is not related to the viscosity coefficient and fractional order under transient loading conditions, while the different viscous coefficients and fractional orders have a significant effect on the consolidation time. The results of this study are helpful to understand deeply the consolidation mechanics behavior of viscoelastic saturated soil.