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    2. 任意荷载下连续排水边界分数阶黏弹性地基一维固结模型

      2020-04-17 14:45:26 《土木建筑与环境工程》 2020年1期

      王珏 童立红 金立 徐长节

      摘 要:以基于Caputo分数阶导数的弹壶元件修正Kelvin模型来描述饱和土体一维固结的力学行为,并引入连续排水边界条件,通过Laplace变换,联立求解得到任意荷载下连续排水边界分数阶黏弹性地基有效应力及固结沉降的解析解。采用Laplace逆变换,获得了其时域内的理论解,并分析了梯形循环荷载及施工荷载作用下相关参数对固结沉降的影响。研究结果表明:循环荷载作用下,黏土地基的沉降变化呈振荡增长,且振荡幅值随着边界透水性的增大而增大;分数阶次α增大,使固结前期沉降速率减慢,而在固结后期,α值对沉降的影响正好相反;循环荷载下沉降变化曲线的振荡幅值随着分数阶次α的增大而减小。此外,一维固结沉降的发展还与土体力学参数及荷载参数相关,弹性模量E越大,最终沉降量越小;黏弹性体的延迟时间F越大,固结沉降变化越慢。

      关键词:任意荷载;连续排水边界;分数阶导数;黏弹性;一维固结

      中图分类号:TU431 文献标志码:A 文章编号:2096-6717(2020)01-0056-08

      Abstract:The Kelvin constitutive model is modified by the spring-pot element based on the Caputo fractional derivative to describe the mechanical behavior of one-dimensional consolidation of saturated soil. After introducing the continuous drainage boundary condition, the analytical solutions of the effective stress and the settlement under time-dependent loading are derived by performing Laplace transformation. The Laplace inverse transformation is used to obtain the theoretical solutions in time domain, and the influences of relevant parameters on the settlement under trapezoidal cyclic loading and construction loading are studied. The results show that the settlement of viscoelastic soil under cyclic loading increases in an oscillating manner, and the amplitude of the oscillation increases with the boundary permeability. A higher value of the fractional order α slows the development of settlement in the early stage of consolidation. However, in the later stage of consolidation, the effect of α on settlement is reversed. The oscillation amplitude of the settlement under cyclic loading decreases with increase of α. Furthermore, detailed analysis indicates that the development of one-dimensional consolidation settlement is also related to mechanical properties of soil and loading parameters. The larger the elastic modulus E is, the smaller the final settlement, and the greater the delay time of viscoelastic is, the slower the settlement occurs.

      Keywords:time-dependent loading; continuous drainage boundary; fractional order derivative; viscoelastic; one-dimensional consolidation

      在Terzaghi固結理论中,土体被处理为线弹性模型,而流变特性是软土的一种重要的工程特性[1]。因此,考虑软黏土的流变特性,将土体视为黏弹性介质通常更符合实际工程[2]。Taylor等[3]首先引入Kelvin模型来描述土骨架的黏弹性变形;Tan[4]基于Maxwell模型对受侧限土体的固结和滞流进行了研究。此后,金问鲁等[5]提出了固结方程的一个近似解法,并给出了各种条件下简单问题的解答;赵维炳[6]基于广义Voigt模型,推导了饱和土体一维固结问题的普遍理论解答;Xie等[7-8]引入Merchant模型及四元件模型到固结理论中,分析了软黏土的固结特性;蔡袁强等[9]求解了任意荷载下成层粘弹性地基一维变形问题。然而,上述经典流变模型不能很好地与实验数据相吻合[10],主要是由于整数阶微分算子的性质决定了经典流变模型的核函数通常是指数函数的组合,欲精确描述实验数据,常常不得不取消高阶的微分项或者以降低本构模型的应用范围为代价[11]。

      Gement[12]首先提出了黏弹性材料的分数阶导数本构模型,而后一些学者将其引入到固结理论中,并指出分数阶导数流变模型可以有效克服经典模型的缺点。Koeller[13]用基于分数导数的弹壶元件替换牛顿黏壶,研究分析了多种模型的流变特性;孙海忠等[14]采用含分数导数的Kelvin模型对珠江三角洲南沙地区典型软土的流变试验数据进行拟合,得到很好的结果;Yin等[15]对分数阶软土蠕变过程中的力学性能进行了系统的研究;汪磊等[16]基于分数阶导数理论引入Kelvin-Voigt模型,获得了任意荷载情况下一维固结问题的半解析解;刘忠玉等[17]求得了恒载下基于分数阶Kelvin模型饱和软黏土一维固结理论解,并通过对比一维流变固结试验曲线及整数阶模型理论曲线,指出基于分数阶Kelvin模型模拟的孔压消散曲线更接近试验曲线。

      另一方面,实际工程中土体的边界往往是处于透水与不透水之间的一种中间状态[18]。蔡袁强等[19]、汪磊等[20]研究了半透水边界条件下一维固结问题。但是半透水边界计算相对复杂,且不能严格满足初始条件,限制了土体固结方程解的适用性[21]?;诖?,梅国雄等[18]提出了一个从不透水到透水的双面不对称连续排水边界。目前,关于变荷载、连续排水边界及分数阶导数黏弹性模型耦合的一维固结理论分析很少见诸于文献。笔者针对Caputo分数阶导数的弹壶元件修正Kelvin模型黏弹性地基,引入连续排水边界条件,推导了任意荷载下连续排水边界分数阶黏弹性地基一维固结方程的半解析解,并分析了相关参数对软黏土固结沉降特性的影响。

      4 结 论

      基于Caputo分數阶导数的弹壶元件修正Kelvin模型,引入连续排水边界条件,利用Laplace变换求得考虑连续排水边界条件时分数阶导数黏弹性地基在任意随时间变化的荷载下有效应力及沉降的解析解,运用Laplace逆变换得到其时域内的数值解。通过系统的算例分析,可以得到如下结论:

      1)循环荷载作用下,黏土地基的沉降变化呈振荡增长,但滞后于荷载的变化,且振荡幅值随着边界透水性的增大而增大。

      2)分数阶次α增大,使固结前期沉降发展速率减慢,但在固结后期,α值对沉降的影响正好相反,最终固结沉降达到稳定的时间随着α的增大而缩短。另外,随着分数阶次α的增大,循环荷载下沉降变化曲线的振荡幅值明显减小。

      3)分数阶黏弹性地基一维固结沉降的发展还与土体力学参数及荷载参数相关。弹性模量E越大,最终沉降量越小,固结沉降达到稳定的时间越短,且循环荷载下固结沉降的振荡幅值越小;黏弹性体的延迟时间F越大,固结沉降变化速率越慢。

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      (編辑 胡玲)

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