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    任意荷載下連續排水邊界分數階黏彈性地基一維固結模型

    2020-04-17 14:45:26 《土木建筑與環境工程》 2020年1期

    王玨 童立紅 金立 徐長節

    摘 要:以基于Caputo分數階導數的彈壺元件修正Kelvin模型來描述飽和土體一維固結的力學行為,并引入連續排水邊界條件,通過Laplace變換,聯立求解得到任意荷載下連續排水邊界分數階黏彈性地基有效應力及固結沉降的解析解。采用Laplace逆變換,獲得了其時域內的理論解,并分析了梯形循環荷載及施工荷載作用下相關參數對固結沉降的影響。研究結果表明:循環荷載作用下,黏土地基的沉降變化呈振蕩增長,且振蕩幅值隨著邊界透水性的增大而增大;分數階次α增大,使固結前期沉降速率減慢,而在固結后期,α值對沉降的影響正好相反;循環荷載下沉降變化曲線的振蕩幅值隨著分數階次α的增大而減小。此外,一維固結沉降的發展還與土體力學參數及荷載參數相關,彈性模量E越大,最終沉降量越小;黏彈性體的延遲時間F越大,固結沉降變化越慢。

    關鍵詞:任意荷載;連續排水邊界;分數階導數;黏彈性;一維固結

    中圖分類號:TU431 文獻標志碼:A 文章編號: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固結理論中,土體被處理為線彈性模型,而流變特性是軟土的一種重要的工程特性[1]。因此,考慮軟黏土的流變特性,將土體視為黏彈性介質通常更符合實際工程[2]。Taylor等[3]首先引入Kelvin模型來描述土骨架的黏彈性變形;Tan[4]基于Maxwell模型對受側限土體的固結和滯流進行了研究。此后,金問魯等[5]提出了固結方程的一個近似解法,并給出了各種條件下簡單問題的解答;趙維炳[6]基于廣義Voigt模型,推導了飽和土體一維固結問題的普遍理論解答;Xie等[7-8]引入Merchant模型及四元件模型到固結理論中,分析了軟黏土的固結特性;蔡袁強等[9]求解了任意荷載下成層粘彈性地基一維變形問題。然而,上述經典流變模型不能很好地與實驗數據相吻合[10],主要是由于整數階微分算子的性質決定了經典流變模型的核函數通常是指數函數的組合,欲精確描述實驗數據,常常不得不取消高階的微分項或者以降低本構模型的應用范圍為代價[11]。

    Gement[12]首先提出了黏彈性材料的分數階導數本構模型,而后一些學者將其引入到固結理論中,并指出分數階導數流變模型可以有效克服經典模型的缺點。Koeller[13]用基于分數導數的彈壺元件替換牛頓黏壺,研究分析了多種模型的流變特性;孫海忠等[14]采用含分數導數的Kelvin模型對珠江三角洲南沙地區典型軟土的流變試驗數據進行擬合,得到很好的結果;Yin等[15]對分數階軟土蠕變過程中的力學性能進行了系統的研究;汪磊等[16]基于分數階導數理論引入Kelvin-Voigt模型,獲得了任意荷載情況下一維固結問題的半解析解;劉忠玉等[17]求得了恒載下基于分數階Kelvin模型飽和軟黏土一維固結理論解,并通過對比一維流變固結試驗曲線及整數階模型理論曲線,指出基于分數階Kelvin模型模擬的孔壓消散曲線更接近試驗曲線。

    另一方面,實際工程中土體的邊界往往是處于透水與不透水之間的一種中間狀態[18]。蔡袁強等[19]、汪磊等[20]研究了半透水邊界條件下一維固結問題。但是半透水邊界計算相對復雜,且不能嚴格滿足初始條件,限制了土體固結方程解的適用性[21]。基于此,梅國雄等[18]提出了一個從不透水到透水的雙面不對稱連續排水邊界。目前,關于變荷載、連續排水邊界及分數階導數黏彈性模型耦合的一維固結理論分析很少見諸于文獻。筆者針對Caputo分數階導數的彈壺元件修正Kelvin模型黏彈性地基,引入連續排水邊界條件,推導了任意荷載下連續排水邊界分數階黏彈性地基一維固結方程的半解析解,并分析了相關參數對軟黏土固結沉降特性的影響。

    4 結 論

    基于Caputo分數階導數的彈壺元件修正Kelvin模型,引入連續排水邊界條件,利用Laplace變換求得考慮連續排水邊界條件時分數階導數黏彈性地基在任意隨時間變化的荷載下有效應力及沉降的解析解,運用Laplace逆變換得到其時域內的數值解。通過系統的算例分析,可以得到如下結論:

    1)循環荷載作用下,黏土地基的沉降變化呈振蕩增長,但滯后于荷載的變化,且振蕩幅值隨著邊界透水性的增大而增大。

    2)分數階次α增大,使固結前期沉降發展速率減慢,但在固結后期,α值對沉降的影響正好相反,最終固結沉降達到穩定的時間隨著α的增大而縮短。另外,隨著分數階次α的增大,循環荷載下沉降變化曲線的振蕩幅值明顯減小。

    3)分數階黏彈性地基一維固結沉降的發展還與土體力學參數及荷載參數相關。彈性模量E越大,最終沉降量越小,固結沉降達到穩定的時間越短,且循環荷載下固結沉降的振蕩幅值越小;黏彈性體的延遲時間F越大,固結沉降變化速率越慢。

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    (編輯 胡玲)

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