王玨 童立紅 金立 徐長(zhǎng)節(jié)
摘 要:以基于Caputo分?jǐn)?shù)階導(dǎo)數(shù)的彈壺元件修正Kelvin模型來(lái)描述飽和土體一維固結(jié)的力學(xué)行為,并引入連續(xù)排水邊界條件,通過(guò)Laplace變換,聯(lián)立求解得到任意荷載下連續(xù)排水邊界分?jǐn)?shù)階黏彈性地基有效應(yīng)力及固結(jié)沉降的解析解。采用Laplace逆變換,獲得了其時(shí)域內(nèi)的理論解,并分析了梯形循環(huán)荷載及施工荷載作用下相關(guān)參數(shù)對(duì)固結(jié)沉降的影響。研究結(jié)果表明:循環(huán)荷載作用下,黏土地基的沉降變化呈振蕩增長(zhǎng),且振蕩幅值隨著邊界透水性的增大而增大;分?jǐn)?shù)階次α增大,使固結(jié)前期沉降速率減慢,而在固結(jié)后期,α值對(duì)沉降的影響正好相反;循環(huán)荷載下沉降變化曲線的振蕩幅值隨著分?jǐn)?shù)階次α的增大而減小。此外,一維固結(jié)沉降的發(fā)展還與土體力學(xué)參數(shù)及荷載參數(shù)相關(guān),彈性模量E越大,最終沉降量越小;黏彈性體的延遲時(shí)間F越大,固結(jié)沉降變化越慢。
關(guān)鍵詞:任意荷載;連續(xù)排水邊界;分?jǐn)?shù)階導(dǎo)數(shù);黏彈性;一維固結(jié)
中圖分類號(hào):TU431 文獻(xiàn)標(biāo)志碼:A 文章編號(hào):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(jié)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(zhì)通常更符合實(shí)際工程[2]。Taylor等[3]首先引入Kelvin模型來(lái)描述土骨架的黏彈性變形;Tan[4]基于Maxwell模型對(duì)受側(cè)限土體的固結(jié)和滯流進(jìn)行了研究。此后,金問(wèn)魯?shù)萚5]提出了固結(jié)方程的一個(gè)近似解法,并給出了各種條件下簡(jiǎn)單問(wèn)題的解答;趙維炳[6]基于廣義Voigt模型,推導(dǎo)了飽和土體一維固結(jié)問(wèn)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(jié)理論中,分析了軟黏土的固結(jié)特性;蔡袁強(qiáng)等[9]求解了任意荷載下成層粘彈性地基一維變形問(wèn)題。然而,上述經(jīng)典流變模型不能很好地與實(shí)驗(yàn)數(shù)據(jù)相吻合[10],主要是由于整數(shù)階微分算子的性質(zhì)決定了經(jīng)典流變模型的核函數(shù)通常是指數(shù)函數(shù)的組合,欲精確描述實(shí)驗(yàn)數(shù)據(jù),常常不得不取消高階的微分項(xiàng)或者以降低本構(gòu)模型的應(yīng)用范圍為代價(jià)[11]。
Gement[12]首先提出了黏彈性材料的分?jǐn)?shù)階導(dǎo)數(shù)本構(gòu)模型,而后一些學(xué)者將其引入到固結(jié)理論中,并指出分?jǐn)?shù)階導(dǎo)數(shù)流變模型可以有效克服經(jīng)典模型的缺點(diǎn)。Koeller[13]用基于分?jǐn)?shù)導(dǎo)數(shù)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?jǐn)?shù)導(dǎo)數(shù)的Kelvin模型對(duì)珠江三角洲南沙地區(qū)典型軟土的流變?cè)囼?yàn)數(shù)據(jù)進(jìn)行擬合,得到很好的結(jié)果;Yin等[15]對(duì)分?jǐn)?shù)階軟土蠕變過(guò)程中的力學(xué)性能進(jìn)行了系統(tǒng)的研究;汪磊等[16]基于分?jǐn)?shù)階導(dǎo)數(shù)理論引入Kelvin-Voigt模型,獲得了任意荷載情況下一維固結(jié)問(wèn)題的半解析解;劉忠玉等[17]求得了恒載下基于分?jǐn)?shù)階Kelvin模型飽和軟黏土一維固結(jié)理論解,并通過(guò)對(duì)比一維流變固結(jié)試驗(yàn)曲線及整數(shù)階模型理論曲線,指出基于分?jǐn)?shù)階Kelvin模型模擬的孔壓消散曲線更接近試驗(yàn)曲線。
另一方面,實(shí)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(tài)[18]。蔡袁強(qiáng)等[19]、汪磊等[20]研究了半透水邊界條件下一維固結(jié)問(wèn)題。但是半透水邊界計(jì)算相對(duì)復(fù)雜,且不能?chē)?yán)格滿足初始條件,限制了土體固結(jié)方程解的適用性[21]?;诖?,梅國(guó)雄等[18]提出了一個(gè)從不透水到透水的雙面不對(duì)稱連續(xù)排水邊界。目前,關(guān)于變荷載、連續(xù)排水邊界及分?jǐn)?shù)階導(dǎo)數(shù)黏彈性模型耦合的一維固結(jié)理論分析很少見(jiàn)諸于文獻(xiàn)。筆者針對(duì)Caputo分?jǐn)?shù)階導(dǎo)數(shù)的彈壺元件修正Kelvin模型黏彈性地基,引入連續(xù)排水邊界條件,推導(dǎo)了任意荷載下連續(xù)排水邊界分?jǐn)?shù)階黏彈性地基一維固結(jié)方程的半解析解,并分析了相關(guān)參數(shù)對(duì)軟黏土固結(jié)沉降特性的影響。
4 結(jié) 論
基于Caputo分?jǐn)?shù)階導(dǎo)數(shù)的彈壺元件修正Kelvin模型,引入連續(xù)排水邊界條件,利用Laplace變換求得考慮連續(xù)排水邊界條件時(shí)分?jǐn)?shù)階導(dǎo)數(shù)黏彈性地基在任意隨時(shí)間變化的荷載下有效應(yīng)力及沉降的解析解,運(yùn)用Laplace逆變換得到其時(shí)域內(nèi)的數(shù)值解。通過(guò)系統(tǒng)的算例分析,可以得到如下結(jié)論:
1)循環(huán)荷載作用下,黏土地基的沉降變化呈振蕩增長(zhǎng),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?jǐn)?shù)階次α增大,使固結(jié)前期沉降發(fā)展速率減慢,但在固結(jié)后期,α值對(duì)沉降的影響正好相反,最終固結(jié)沉降達(dá)到穩(wěn)定的時(shí)間隨著α的增大而縮短。另外,隨著分?jǐn)?shù)階次α的增大,循環(huán)荷載下沉降變化曲線的振蕩幅值明顯減小。
3)分?jǐn)?shù)階黏彈性地基一維固結(jié)沉降的發(fā)展還與土體力學(xué)參數(shù)及荷載參數(shù)相關(guān)。彈性模量E越大,最終沉降量越小,固結(jié)沉降達(dá)到穩(wěn)定的時(shí)間越短,且循環(huán)荷載下固結(jié)沉降的振蕩幅值越小;黏彈性體的延遲時(shí)間F越大,固結(jié)沉降變化速率越慢。
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(編輯 胡玲)