基于分数阶流变模型的饱和软土一维流变固结

One-dimensional Rheological Consolidation of Soft Clay Based on Fractional Order Derivative Rheological Model

在太沙基一维固结理论的基础上,引入分数阶导数Merchant流变模型来描述土体的变形特性,通过对微分型流变方程和一维固结方程进行Laplace变换,推导得到了Laplace变换域内饱和软黏土的一维流变固结方程,采用FlexPDE软件对方程进行求解,并通过Durbin反Laplace变换获得了平均固结度的时域解答。在此基础上,研究了分数阶导数流变模型的微积分阶数和流变模型参数对饱和软土一维流变固结特性的影响规律。计算结果表明:基于整数阶和基于分数阶导数流变模型的流变固结理论在固结前期基本相同,但在固结后期两者差异显著,基于分数阶导数流变模型的软黏土Up要高于整数阶的情况,而Us的规律则截然相反。此外,微分阶数β对两种定义的平均固结度均有较大影响,β越大,Up越小而Us就越大,软黏土的流变效应越显著。随着流变模型参数F的逐渐增大,同一时刻软黏土的Up逐渐减小,而Us则逐渐增大;随着参数β1的逐渐增大,同一时刻软黏土的Up和Us均随之减小。

A fractional order Merchant’s rheological model is introduced into Terzaghi’consolidation theory to describe the mechanical behavior of viscoelastic saturated soil.By applying Laplace transforms upon the one-dimensional consolidation equation and the fractional order derivative Merchant constitutive equation,a onedimensional rheological consolidation equation in Laplace transform domain is established.The equation is solved by using Flex PDE software and the average degree of consolidation in time domain is obtained after implementing Laplace numerical inversion by using Durbin method.Then the behaviors of soft soil during rheological consolidation are investigated and the effects of the fractional order and the parameters of rheological model on rheological consolidation are analyzed and discussed in detail.The results show that the average degree of consolidation in terms of settlement Upand that associated with settlement Usof the consolidation theory base on fractional order derivation rheological model are as same as those of integral order derivation derivation model at the beginning of consolidation.While there are big differences between two rheological consolidation theories in the late stage of consolidation,and Upcalculated by the consolidation theory based on fractional order derivation rheological model is bigger than that of integral order derivation rheological model,while the conditions of Usare on the contrary.Further,the fractional orderβcan effect the average degree of consolidation dramatically,and the biggerβis,the smaller Upis,while the higher Us,indicating that the rheological effect strengthens.Moreover,Updecreases with the increase of the rheological model parameter F,while Usis on the contrary.Upand Usboth decreases with the decreasing of the rheological model parameterβ1.