運(yùn)用改進(jìn)的無(wú)單元Galerkin方法分析機(jī)場(chǎng)道面斷裂力學(xué)問(wèn)題?

鄒詩(shī)瑩 席偉成 彭妙娟 程玉民?

1)(上海大學(xué)上海市應(yīng)用數(shù)學(xué)和力學(xué)研究所,上海 200072)

2)(上海大學(xué)土木工程系,上海 200444)

運(yùn)用改進(jìn)的無(wú)單元Galerkin方法分析機(jī)場(chǎng)道面斷裂力學(xué)問(wèn)題?

鄒詩(shī)瑩1)席偉成2)彭妙娟2)程玉民1)?

1)(上海大學(xué)上海市應(yīng)用數(shù)學(xué)和力學(xué)研究所,上海 200072)

2)(上海大學(xué)土木工程系,上海 200444)

(2017年2月26日收到;2017年3月26日收到修改稿)

在改進(jìn)的無(wú)單元Galerkin方法的基礎(chǔ)上,將能反映裂紋尖端附近應(yīng)力奇異性的特征項(xiàng)引入改進(jìn)的移動(dòng)最小二乘法的基函數(shù)中,將斷裂力學(xué)和改進(jìn)的無(wú)單元Galerkin方法結(jié)合,研究了線彈性斷裂力學(xué)的改進(jìn)的無(wú)單元Galerkin方法,并對(duì)含反射裂縫的機(jī)場(chǎng)復(fù)合道面層狀體系結(jié)構(gòu)進(jìn)行了數(shù)值分析.本文的理論為機(jī)場(chǎng)復(fù)合道面斷裂力學(xué)分析提供了一種新方法.

改進(jìn)的無(wú)單元Galerkin方法,機(jī)場(chǎng)復(fù)合道面,層狀體系,反射裂縫

1 引 言

無(wú)網(wǎng)格方法的研究已有二十多年的歷史,國(guó)際上目前將基于點(diǎn)的近似構(gòu)造試函數(shù)、不需要考慮節(jié)點(diǎn)與單元間關(guān)聯(lián)條件的各種數(shù)值方法稱為無(wú)網(wǎng)格方法[1,2].

程玉民等[3,4]在移動(dòng)最小二乘法的基礎(chǔ)上,建立了改進(jìn)的移動(dòng)最小二乘法. 引入復(fù)變量理論,程玉民等建立了復(fù)變量移動(dòng)最小二乘法[5,6]、改進(jìn)的復(fù)變量移動(dòng)最小二乘法[7]和復(fù)變量重構(gòu)核粒子法[8].在此基礎(chǔ)上,提出了改進(jìn)的無(wú)單元Galerkin方法[9,10]、插值型無(wú)單元Galerkin方法[11?14]、邊界無(wú)單元法[15?18]、復(fù)變量無(wú)單元Galerkin方法[19?23]、改進(jìn)的復(fù)變量無(wú)單元Galerkin方法[7,24?26]和復(fù)變量重構(gòu)核粒子法[8,27?29].

由于改進(jìn)的移動(dòng)最小二乘法形成的方程組易于求解并且不形成病態(tài)方程組,因而改進(jìn)的無(wú)單元Galerkin方法可以提高無(wú)單元Galerkin方法的計(jì)算精度和計(jì)算效率.

機(jī)場(chǎng)復(fù)合道面兼顧了剛性路面和柔性路面的優(yōu)點(diǎn).水泥混凝土板提供了一個(gè)穩(wěn)定、堅(jiān)實(shí)的基礎(chǔ),瀝青混凝土則作為表面功能特性優(yōu)良的磨耗層.然而,在水泥混凝土板間各種形式的接縫或裂縫使得加鋪的瀝青面層易產(chǎn)生反射裂縫.在瀝青混凝土加鋪層出現(xiàn)反射裂縫后,會(huì)使地表水進(jìn)入瀝青混合料內(nèi)部,從而降低集料與瀝青之間的黏結(jié)性能,形成松散、剝落等病害.特別是在飛機(jī)輪載的作用下容易導(dǎo)致應(yīng)力集中現(xiàn)象,反射裂縫周邊道面又會(huì)形成新的交叉次生裂縫,發(fā)展到一定程度后將使道面出現(xiàn)大面積剝落,嚴(yán)重影響飛機(jī)的運(yùn)行安全.反射裂縫是機(jī)場(chǎng)復(fù)合道面破壞的主要表現(xiàn)形式之一.

關(guān)于機(jī)場(chǎng)復(fù)合道面反射裂縫,Garzon等[30,31]利用三維廣義有限元法對(duì)荷載作用下機(jī)場(chǎng)道面的反射裂縫進(jìn)行了研究;李淑明等[32]采用有限元法分析了土工布、玻璃纖維網(wǎng)對(duì)舊水泥混凝土路面加鋪瀝青混凝土層復(fù)合路面結(jié)構(gòu)的內(nèi)力影響;凌建明等基于彈性層狀體系理論,利用有限元法對(duì)飛機(jī)荷載作用下機(jī)場(chǎng)道面地基附加應(yīng)力進(jìn)行了分析[33],提出了機(jī)場(chǎng)復(fù)合道面剩余壽命預(yù)估方法[34];馬翔等以B777-200飛機(jī)荷載為計(jì)算荷載,利用有限元法分析了復(fù)合機(jī)場(chǎng)道面荷載應(yīng)力[35],提出了以荷載與溫度綜合疲勞彎拉應(yīng)力和瀝青面層反射裂縫疲勞壽命為設(shè)計(jì)指標(biāo)的復(fù)合機(jī)場(chǎng)道面結(jié)構(gòu)設(shè)計(jì)方法[36].

綜上研究進(jìn)展,目前關(guān)于機(jī)場(chǎng)復(fù)合道面反射裂縫的計(jì)算均采用有限元法,尚未見(jiàn)到采用無(wú)網(wǎng)格方法分析機(jī)場(chǎng)復(fù)合道面反射裂縫,而無(wú)網(wǎng)格方法在求解斷裂問(wèn)題方面優(yōu)于有限元法.因而,對(duì)機(jī)場(chǎng)復(fù)合道面來(lái)說(shuō),研究應(yīng)用于反射裂縫分析的斷裂力學(xué)的無(wú)網(wǎng)格方法是非常必要的.

2 改進(jìn)的移動(dòng)最小二乘法

在移動(dòng)最小二乘法中,取函數(shù)u(x)的逼近函數(shù)為

其中m是基函數(shù)的個(gè)數(shù),pi(x)是基函數(shù),ai(x)是相應(yīng)的系數(shù).

對(duì)應(yīng)于(1)式的整體逼近,在點(diǎn)x鄰域內(nèi)的局部逼近定義為

其中xI為影響域覆蓋點(diǎn)x的節(jié)點(diǎn),n是影響域覆蓋點(diǎn)x的節(jié)點(diǎn)數(shù),uI=u(xI).

對(duì)J取極值,即得

可得

即

對(duì)于點(diǎn)集{xi}和權(quán)函數(shù){wi},若一組函數(shù)φ1(x),φ2(x),···,φm(x)滿足

則稱φ1(x),φ2(x),···,φm(x)是關(guān)于點(diǎn)集{xi}帶權(quán){wi}的正交函數(shù)族(k,j=1,2,···,m).

若{pi(x)},i=1,2,···,m,為Hilbert空間span(p)上的關(guān)于點(diǎn)集{xi}的帶權(quán)的正交基函數(shù)族,即

則方程(6)可寫成

這樣,我們可以直接得到系數(shù)ai(x),即

寫成矩陣形式

其中

將(10)式代入(2)式可得

其中Φ?(x)為形函數(shù),

這樣,系數(shù)ai(x)可以簡(jiǎn)單、直接地得到,不需要求矩陣A(x)的逆,避免了求解病態(tài)或奇異的方程組,既提高了效率,又提高了精度.

對(duì)如下基函數(shù)

利用Schmidt正交化方法得到的正交基函數(shù)為

3 裂紋尖端試函數(shù)

基函數(shù)擴(kuò)展法是通過(guò)在基函數(shù)中加入擴(kuò)展函數(shù)項(xiàng),來(lái)達(dá)到對(duì)試函數(shù)擴(kuò)展的目的.對(duì)于斷裂力學(xué)的改進(jìn)的無(wú)單元Galerkin方法,基函數(shù)中必須加入裂紋尖端位移場(chǎng)表達(dá)式中的一些重要項(xiàng)和一些重要的梯度項(xiàng).基函數(shù)擴(kuò)展的項(xiàng)數(shù)決定了數(shù)值解的精度,當(dāng)基函數(shù)中加入了能反映包含裂紋尖端位移場(chǎng)中的所有項(xiàng),即完全基函數(shù)擴(kuò)展時(shí),取如下形式:

類似于改進(jìn)的移動(dòng)最小二乘法的推導(dǎo),可得

類似(13)式可得裂紋尖端試函數(shù)為

由此可以看出基函數(shù)擴(kuò)展法沒(méi)有增加新的節(jié)點(diǎn)未知變量.但是由于基函數(shù)的項(xiàng)數(shù)增加了,形函數(shù)的計(jì)算量也會(huì)相應(yīng)地增加.為了提高計(jì)算效率,本文采用局部基函數(shù)擴(kuò)展法,即在基函數(shù)中加入即

這樣,可以得到裂紋尖端的試函數(shù)為

可得裂紋尖端節(jié)點(diǎn)的形函數(shù)

4 斷裂力學(xué)的改進(jìn)的無(wú)單元Galerkin方法

假設(shè)二維彈性體的求解域?yàn)?,邊界為Γ,彈性力學(xué)問(wèn)題的平衡方程為

其中σij為應(yīng)力分量,bi是單位體積上的體力分量,i,j=1,2.

幾何方程為

其中εij是應(yīng)變分量,ui是位移分量.

本構(gòu)方程為

式中D=(Dijkl)是彈性矩陣,對(duì)于平面應(yīng)力問(wèn)題,

對(duì)于平面應(yīng)變問(wèn)題,

其中E為材料的彈性模量.

對(duì)應(yīng)的邊界條件為

本節(jié)采用罰函數(shù)法施加本質(zhì)邊界條件,則彈性力學(xué)問(wèn)題對(duì)應(yīng)的泛函為

如果位移邊界條件是沿xi方向的,則si為0,否則為1.

將求解域離散為有限個(gè)節(jié)點(diǎn),節(jié)點(diǎn)總數(shù)為M.利用改進(jìn)的移動(dòng)最小二乘法建立逼近函數(shù),可得位移的逼近函數(shù)為

對(duì)非裂紋尖端節(jié)點(diǎn),形函數(shù)如(14)式所示;裂紋尖端節(jié)點(diǎn)的形函數(shù)見(jiàn)(24)式.

將(37)式代入(32)式,可得

矩陣Kα是根據(jù)節(jié)點(diǎn)信息得到的全局罰函數(shù)矩陣,

向量Fα由本質(zhì)邊界條件得到,

裂縫尖端的應(yīng)力強(qiáng)度因子采用以下方法進(jìn)行計(jì)算.對(duì)裂紋尖端選取適當(dāng)?shù)膢OC|=L得點(diǎn)C,|OB|=L/4得點(diǎn)B.由點(diǎn)O,B,C即構(gòu)成二維問(wèn)題裂紋尖端的虛擬奇異二次元,裂紋尖端局部坐標(biāo)如圖1所示,應(yīng)力強(qiáng)度因子表達(dá)式為

其中a為裂紋長(zhǎng)度的一半,vB和vC分別為點(diǎn)B和點(diǎn)C豎直方向的位移.

圖1 裂紋尖端虛擬奇異元Fig.1.The virtual singular element at the tip of a crack.

5 機(jī)場(chǎng)道面反射裂縫的數(shù)值分析

5.1 機(jī)場(chǎng)復(fù)合道面反射裂縫的計(jì)算模型

為了進(jìn)行反射裂縫影響因素分析,將復(fù)合機(jī)場(chǎng)道面結(jié)構(gòu)簡(jiǎn)化為平面應(yīng)變問(wèn)題的層狀體系,采用二維模型進(jìn)行數(shù)值計(jì)算.將荷載簡(jiǎn)化為二維平面應(yīng)變問(wèn)題下的條形均布荷載,采用B777-200B,輪壓為1.45 MPa,輪距為1.40 m.此外假定各結(jié)構(gòu)層都由線彈性的各向同性、均質(zhì)材料組成,地基由彈性半空間地基假設(shè),在模型中采用有限尺寸.模型的邊界條件為:地基底部完全約束,各結(jié)構(gòu)層兩側(cè)鉸接,具體幾何參數(shù)如圖2,各層的物理參數(shù)見(jiàn)表1,反射裂縫起裂位置如圖3,裂縫寬度取0.5 cm.

圖2 機(jī)場(chǎng)復(fù)合道面模型幾何參數(shù)(mm)Fig.2.The geometric parameters of the airport composite pavement(mm).

表1 結(jié)構(gòu)的物理參數(shù)Table 1.The physical parameters of the structure.

圖3 反射裂縫的位置Fig.3.The position of the re fl ective crack.

5.2 機(jī)場(chǎng)復(fù)合道面反射裂縫力學(xué)分析

首先考慮起裂位置處于接縫處的裂縫計(jì)算模型.無(wú)網(wǎng)格方法的節(jié)點(diǎn)布置采用5370個(gè)節(jié)點(diǎn),輪載下方區(qū)域節(jié)點(diǎn)進(jìn)行加密,并對(duì)起裂位置進(jìn)行局部加密,沿水泥板接縫位置布置2×27個(gè)加密節(jié)點(diǎn),具體節(jié)點(diǎn)分布和局部加密如圖4和圖5.

圖4 模型節(jié)點(diǎn)布置Fig.4.The node distribution of the model.

圖5 裂縫附近節(jié)點(diǎn)加密分布(2×27)Fig.5.The node re fi nement distribution at the tip of the crack(2×27).

計(jì)算結(jié)束后,選取裂紋尖端適當(dāng)計(jì)算點(diǎn),采用(47)式計(jì)算,得到此處應(yīng)力強(qiáng)度因子KI為9.42 MPa·mm1/2,裂縫尖端位于混凝土板層,回彈模量為30000 MPa.

在此基礎(chǔ)上,進(jìn)一步考慮裂縫反射到瀝青加鋪層的反射裂縫模型,如圖6取反射至瀝青面層的裂縫長(zhǎng)度為4 cm,并在裂紋區(qū)域進(jìn)行節(jié)點(diǎn)加密(3×31個(gè)加密節(jié)點(diǎn)),其節(jié)點(diǎn)布置局部如圖7所示.

通過(guò)計(jì)算得到的KI為0.36 MPa·mm1/2,裂縫尖端位于瀝青層,回彈模量為2000 MPa.

圖6 反射至瀝青層的裂縫Fig.6.The re fl ective crack in the asphalt layer.

圖7 反射裂縫附近節(jié)點(diǎn)加密分布(3×31)Fig.7.The node re fi nement distribution at the tip of the re fl ective crack(3×31).

圖8 瀝青層計(jì)算點(diǎn)豎向位移Fig.8.The vertical displacement at the points in the asphalt layer.

選取計(jì)算點(diǎn)后對(duì)模型的位移和應(yīng)力進(jìn)行計(jì)算,計(jì)算結(jié)果如圖8—圖10所示.從圖8可以看出,在荷載作用位置附近,豎向位移較大,與路面實(shí)際情況符合.從圖9可以看出,當(dāng)混泥土板開(kāi)裂后,在荷載作用下裂縫正上方面層的正應(yīng)力具有奇異性,從而使得路面反射裂縫繼續(xù)向上發(fā)展,以致最后貫通整個(gè)路面.可見(jiàn),當(dāng)基層出現(xiàn)裂縫后,由于裂縫尖端存在一定的應(yīng)力集中,這種應(yīng)力集中對(duì)于路面正常工作是不利的.所以,路面工程施工中要盡可能減少混凝土板的收縮裂縫,以防止反射裂縫發(fā)生.

從圖9和圖10可以看出,在對(duì)稱荷載作用下,裂縫尖端正應(yīng)力遠(yuǎn)大于剪應(yīng)力,裂縫為張開(kāi)型,即裂縫的擴(kuò)展受正應(yīng)力或KI的控制.

圖9 輪載下各層計(jì)算點(diǎn)正應(yīng)力σ11Fig.9.The normal stress σ11at the points in layers under the wheel load.

圖10 輪載下各層計(jì)算點(diǎn)剪應(yīng)力τ21Fig.10.The shear stress τ21at the points in layers under the wheel load.

6 結(jié) 論

本文針對(duì)機(jī)場(chǎng)復(fù)合道面的反射裂縫問(wèn)題,建立了斷裂力學(xué)改進(jìn)的無(wú)單元Galerkin方法.數(shù)值算例表明,該方法通過(guò)對(duì)試函數(shù)的擴(kuò)展能有效地反映裂紋尖端應(yīng)力場(chǎng)的奇異性,提高了無(wú)網(wǎng)格方法的求解精度.

在對(duì)試函數(shù)進(jìn)行改進(jìn)時(shí),采用基函數(shù)的局部擴(kuò)展法,不僅能有效地提高對(duì)裂紋尖端應(yīng)力場(chǎng)的計(jì)算精度,也比試函數(shù)外部擴(kuò)展法以及基函數(shù)完全擴(kuò)展法具有更好的求解效率.

使用改進(jìn)無(wú)單元Galerkin方法對(duì)機(jī)場(chǎng)復(fù)合道面模型進(jìn)行計(jì)算,在布置節(jié)點(diǎn)時(shí),在裂紋局部進(jìn)行加密布置.對(duì)于計(jì)算裂紋擴(kuò)展的模型時(shí),可以在擴(kuò)展路徑的區(qū)域內(nèi)增加布置相應(yīng)的節(jié)點(diǎn),而無(wú)需進(jìn)行有限元法的網(wǎng)格重構(gòu).

本文的理論為機(jī)場(chǎng)復(fù)合道面反射裂縫的分析提供了一種新的方法.

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PACS:02.60.Cb,02.60.Lj,46.50.+aDOI:10.7498/aps.66.120204

Analysis of fracture problems of airport pavement by improved element-free Galerkin method?

Zou Shi-Ying1)Xi Wei-Cheng2)Peng Miao-Juan2)Cheng Yu-Min1)?

1)(Shanghai Institute of Applied Mathematics and Mechanics,Shanghai University,Shanghai 200072,China)
2)(Department of Civil Engineering,Shanghai University,Shanghai 200444,China)

26 February 2017;revised manuscript

26 March 2017)

Using the improved element-free Galerkin(IEFG)method,in this paper we introduce the characteristic parameterwhich can re fl ect the singular stress near the crack tip into the basic function of the improved moving least-squares(IMLS)approximation.Combining fracture theory with the IEFG method,we present an IEFG method of treating the elastic fracture problems,and analyze a numerical example of two-dimensional layered system of airport composite pavement with re fl ective crack.

In the IEFG method,the IMLS approximation is used to form the shape function.The IMLS approximation is presented from the moving least-squares(MLS)approximation,which is the basis of the element-free Galerkin(EFG)method.Compared with the MLS approximation,the IMLS approximation uses the orthonormal basis functions to obtain the shape function,which leads to the fact that the matrices for obtaining the undetermined coefficients are diagonal.Then the IMLS approximation can obtain the solutions of the undetermined coefficients directly without the inverse matrices.The IMLS approximation can overcome the disadvantages of the MLS approximation,in which the ill-conditional or singular matrices are formed sometimes.And it can also improve the computational efficiency of the MLS approximation.

Because of the advantages of the IMLS approximation,the IEFG method has greater computational efficiency than the EFG method which is based on the MLS approximation,and can obtain the solution for arbitrary node distribution,even though the EFG method cannot obtain the solution due to the ill-conditional or singular matrices in the MLS approximation.

Paving the asphalt concrete layer on the cement concrete pavement is an e ff ective approach to improving the structure and service performance of an airport pavement,which is called airport composite pavement.The airport composite pavement has the advantages of rigid pavement and fl exible pavement,but there are various forms of joints or cracks of cement concrete slab,which makes the crack re fl ect into the asphalt overlay easily under the plane load and environmental factors.Re fl ective crack is one of the main failure forms of the airport composite pavement.Therefore,it is of great theoretical signi fi cance and engineering application to study the generation and development mechanism of re fl ective crack of the airport composite pavement.

For the numerical methods of solving the fracture problems,introducing the characteristic parameterwhich can re fl ect the singular stress near the crack tip into the basic function is a general approach.In this paper,we use this approach to obtain the IEFG method for fracture problems,and the layered system of airport composite pavement with re fl ective crack is considered.The numerical results of the displacements and stresses in the airport composite pavement are given.And at the tip of the crack,the stress is singular,which makes the crack of the airport composite pavement grow.

This paper provides a new method for solving the re fl ective crack problem of airport composite pavement.

improved element-free Galerkin method,airport composite pavement,layered system,re fl ective crack

10.7498/aps.66.120204

?國(guó)家自然科學(xué)基金委員會(huì)-中國(guó)民航局民航聯(lián)合研究基金(批準(zhǔn)號(hào):U1433104)資助的課題.

?通信作者.E-mail:ymcheng@shu.edu.cn

?2017中國(guó)物理學(xué)會(huì)Chinese Physical Society

http://wulixb.iphy.ac.cn

*Project supported by the National Natural Science Foundation of China(Grant No.U1433104).

?Corresponding author.E-mail:ymcheng@shu.edu.cn