裂尖具線性分布約束應(yīng)力的運(yùn)動(dòng)裂紋模型及其解析解

2015-12-30 03:19:47唐雪松,陳旻煒,高常輝

振動(dòng)與沖擊 2015年3期

第一作者唐雪松男，教授，博士，1964年生

唐雪松，陳旻煒，高常輝

(長(zhǎng)沙理工大學(xué)土木與建筑學(xué)院力學(xué)系，長(zhǎng)沙410114)

摘要：以恒定速度運(yùn)動(dòng)的Griffith裂紋解析解為著名的Yoffe解。靜止裂紋的條狀屈服模型即Dugdale模型，將其推廣到運(yùn)動(dòng)裂紋模型時(shí)發(fā)現(xiàn)，當(dāng)裂紋運(yùn)動(dòng)速度跨越Rayliegh波速時(shí)，裂紋張開(kāi)位移COD趨于∞，且表現(xiàn)為間斷。通過(guò)在裂尖引入一個(gè)約束應(yīng)力區(qū)及兩個(gè)速度效應(yīng)函數(shù)，假設(shè)約束應(yīng)力為線性分布，采用復(fù)變函數(shù)方法，求得動(dòng)態(tài)應(yīng)力強(qiáng)度因子SIF與裂紋張開(kāi)位移COD的解析解。新的結(jié)果，在Rayleigh波速下裂紋張開(kāi)位移連續(xù)且為有限值。給出裂紋張開(kāi)位移的一些數(shù)值結(jié)果，獲得了一些有意義的結(jié)論。

關(guān)鍵詞：運(yùn)動(dòng)裂紋；Ⅰ型裂紋；約束應(yīng)力；復(fù)變函數(shù)方法；應(yīng)力強(qiáng)度因子SIF；裂紋張開(kāi)位移COD

基金項(xiàng)目：973項(xiàng)目資助(2015CB057705);國(guó)家自然科學(xué)基金資助(51378081)

收稿日期：2013-09-30修改稿收到日期：2014-02-14

中圖分類(lèi)號(hào):O346.1文獻(xiàn)標(biāo)志碼：A

A model of moving crack with a linear distribution of restraining stresses in crack tip zone

TANGXue-song,CHENMin-wei,GAOChang-hui(Department of Mechanics, School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha 410114, China)

Abstract:The analytical solution of moving Griffith crack model with a constant speed is well known as Yoffe solution. For a static crack, its strip yielding model is well known as Dugdale model. It is found that when Dugdale model is generalized to the moving crack case, the crack opening displacement (COD) is discontinuous and approaches to positive and negative infinite at Rayleigh wave speed. Here, a restraining stress zone was attached to the crack tip while two speed effect functions were introduced assuming the restraining stress zone has a linear distribution. The complex function approach was employed to solve the problem. Analytical solutions of dynamic stress intensity factor (SIF) and crack opening displacement (COD) were then obtained. The new COD result was continuous and finite at Rayleigh wave speed. Some numerical results of COD were presented. Some valuable conclusions were obtained.

Key words:moving crack; model I crack; restraining stress; complex function approach; stress intensity factor (SIF); crack opening displacement (COD)

關(guān)于運(yùn)動(dòng)裂紋的研究與求解，不但具有重要學(xué)術(shù)意義，還具有一定實(shí)用參考價(jià)值。若僅考察運(yùn)動(dòng)裂紋尖端，可一定程度反映動(dòng)態(tài)裂紋擴(kuò)展的一些力學(xué)行為。Yoffe[1]最早獲得了以恒定速度運(yùn)動(dòng)的Griffith裂紋模型的解析解，即著名的Yoffe解。關(guān)于運(yùn)動(dòng)裂紋的研究仍然是當(dāng)前國(guó)際上的熱點(diǎn)研究問(wèn)題。近期的一些研究成果包括：電磁彈性材料反平面剪切下運(yùn)動(dòng)裂紋問(wèn)題的研究[2]，壓電雙材料中含接觸區(qū)的界面運(yùn)動(dòng)裂紋的研究[3]，壓電雙材料含有限穿透裂紋沿界面運(yùn)動(dòng)問(wèn)題的研究[4]，電磁彈性矩形板中運(yùn)動(dòng)裂紋問(wèn)題的研究[5]，正交各向異性雙材料中界面裂紋附近螺型運(yùn)動(dòng)位錯(cuò)的研究[6]，微觀、細(xì)觀與宏觀尺度下裂紋尺寸與速度相互影響的研究[7]，宏觀主裂紋運(yùn)動(dòng)速度接近剪切波速時(shí)微觀裂尖鈍化效應(yīng)的研究[8]等。國(guó)內(nèi)近期研究成果包括，關(guān)于宏微觀雙尺度運(yùn)動(dòng)裂紋解析求解的研究[9]，對(duì)Ⅲ型運(yùn)動(dòng)裂紋均布載荷與集中載荷作用下的斷裂動(dòng)力學(xué)問(wèn)題的研究[10]，對(duì)黏結(jié)于均勻材料基底上功能梯度材料涂層平面運(yùn)動(dòng)裂紋問(wèn)題的研究[11]，對(duì)功能梯度壓電板條中電絕緣型運(yùn)動(dòng)裂紋電彈性場(chǎng)的研究[12]，對(duì)不同壓電介質(zhì)界面上的反平面運(yùn)動(dòng)裂紋的研究[13]，等。

由于數(shù)學(xué)上的限制，對(duì)運(yùn)動(dòng)裂紋問(wèn)題一般限制速度在Rayleigh波速以下。實(shí)際上，裂紋運(yùn)動(dòng)速度不但可達(dá)到Rayleigh波速，還可超過(guò)剪切波速[14]。本文在運(yùn)動(dòng)裂紋尖端引入一個(gè)約束應(yīng)力區(qū)，約束應(yīng)力大小、分布與裂尖材料的損傷程度和運(yùn)動(dòng)速度有關(guān)。約束應(yīng)力區(qū)的概念由Sih等[15]提出，隨后基于約束應(yīng)力區(qū)建立了多種多尺度裂紋模型[15-19]，還建立了宏微觀跨尺度疲勞裂紋擴(kuò)展統(tǒng)一模型[20]。本文主要工作是在運(yùn)動(dòng)裂紋尖端引入約束應(yīng)力區(qū)后，裂紋張開(kāi)位移在Rayleigh波速下實(shí)現(xiàn)了連續(xù)，且直到剪切波速都是連續(xù)的。

1約束應(yīng)力區(qū)概念與分析模型

約束應(yīng)力區(qū)概念如圖1示，圖1(a)為無(wú)損傷的單向受拉板。切開(kāi)一個(gè)長(zhǎng)度為a的切口，則切口存在約束應(yīng)力，如圖1(b)所示。圖1(c)所示為一個(gè)長(zhǎng)度為a的裂紋。若σ0=σ∞，則情況(b)等價(jià)于情況(a)，表示材料無(wú)損傷。若σ0=0，則情況(b)等價(jià)于情況(c)，表示材料完全損傷。若0<σ0<σ∞，表示材料有一定程度損傷而又沒(méi)有完全分離，介于情況(a)與(c)之間。所以，可用約束應(yīng)力區(qū)描述材料的損傷狀態(tài)。

圖1　約束應(yīng)力區(qū)概念 Fig.1 Concept of restraining stress zone

圖2　裂尖具有約束應(yīng)力區(qū)的運(yùn)動(dòng)裂紋模型 Fig.2 Moving crack model with a restraining stress zone attached to the crack tip

ξ=x1-vt,η=x2

(1)

考慮到裂紋尖端材料有一定程度損傷，裂尖損傷區(qū)用一約束應(yīng)力區(qū)來(lái)描述，約束應(yīng)力區(qū)內(nèi)約束應(yīng)力的分布為R(χ)，χ為局部位置坐標(biāo)。分布函數(shù)R(χ)依賴(lài)于裂尖區(qū)材料的損傷及裂紋運(yùn)動(dòng)速度v，本文考慮線性分布的約束應(yīng)力，即

R(χ)=f(v)(a1χ+a2),a≤χ≤c

(2)

2基本控制方程

不計(jì)休力，二維彈性動(dòng)力學(xué)基本控制方程為Navier方程

(3)

式中:λ與μ為L(zhǎng)ame常數(shù)，ρ為質(zhì)量密度，ui為位移。引入體積變形勢(shì)函數(shù)φ(x1,x2,t)和剪切變形勢(shì)函數(shù)Ψ(x,y,t)，有

(4)

則式(3)成為

(5)

(6)

采用式(1)的運(yùn)動(dòng)坐標(biāo)系(ξ,η)，式(5)成為

(7)

(8)

引入復(fù)變量ζd和ζs

ζd=ξ+iαdη,ζs=ξ+iαsη

(9)

及解析函數(shù)F(ζd)與G(ζs)，兩個(gè)位移勢(shì)函數(shù)可表示

φ(ξ,η)=Re[F(ζd)],ψ(ξ,η)=Im[G(ζs)]

(10)

如此，位移與應(yīng)力解答可表示為[17]

u2(ξ,η)=-Im[αdF′(ζd)+G′(ζs)]

(10)

σ11(ξ,η)=

(11)

σ22(ξ,η)=

(12)

σ12(ξ,η)=

(13)

因此，關(guān)鍵是由邊界條件確定出兩個(gè)解析函數(shù)。

3問(wèn)題的求解

(14)

(15)

(16)

(17)

由式(15)知，D是速度v的函數(shù)，當(dāng)v=cR時(shí)，D=0，由此條件可得出Rayleigh波速計(jì)算公式為

(18)

圖3運(yùn)動(dòng)裂紋表面受兩對(duì)對(duì)稱(chēng)集中力作用 Fig.3 Two pairs of concentrated forces symmetrically applied to the moving crack surfaces

下面考慮約束應(yīng)力單獨(dú)作用的結(jié)果。如圖3示，對(duì)稱(chēng)位置上dχ上分布力的合力R(χ)dχ可看成是集中力。運(yùn)動(dòng)裂紋表面上兩對(duì)對(duì)稱(chēng)集中力作用下，解析函數(shù)的解答為[7]

(19)

式(2)代入式(19)中，并從a到c積分，即可得出約束應(yīng)力作用下的解答，有

(20)

(21)

式(20)代入式(21)中，可求出

(22)

遠(yuǎn)場(chǎng)應(yīng)力與約束應(yīng)力共同作用的結(jié)果為

F″(ζd)=F″r(ζd)+F″σ(ζd)

(23)

(24)

(25)

式(16)、(22)代入式(25)中，得

(26)

注意:c=a+b，由式(26)可確定出裂尖損傷區(qū)尺寸b。

裂紋表面張開(kāi)位移COD的定義為

(27)

式(10)、(14)、(20)、(23)、(24)代入式(27)中，求出

(28)

(29)

(30)

4數(shù)值結(jié)果與討論

假設(shè)材料為鋁合金LY12，材料參數(shù)見(jiàn)表1。

表1　鋁合金LY12的材料參數(shù)

當(dāng)取g(v)=f(v)=1，a1=0，a2=σ0時(shí)，此即運(yùn)動(dòng)裂紋Dugdale模型。取σ0=10 MPa,σ∞/σ0=0.4,ξ=0，圖4給出裂紋中點(diǎn)張開(kāi)位移隨運(yùn)動(dòng)速度的變化曲線。由圖4知，vcR時(shí)，COD趨于正無(wú)窮。v>cR時(shí)，COD為負(fù)值，且運(yùn)動(dòng)速度大于cR情況下逼近c(diǎn)R時(shí)，COD趨于負(fù)無(wú)窮。

為解決以上問(wèn)題，引入了兩個(gè)速度函數(shù)g(v)與f(v)，參照文獻(xiàn)[8]，可取

(31)

考慮約束應(yīng)力為三角形分布，取

(32)

即:χ=a時(shí)約束應(yīng)力為0，χ=c時(shí)約束應(yīng)力為σ0。由式(26)知，此時(shí)約束應(yīng)力區(qū)(即損傷區(qū))長(zhǎng)度b與速度v無(wú)關(guān)，僅取決于比值σ∞/σ0，數(shù)值結(jié)果見(jiàn)表2，裂尖損傷區(qū)隨作用荷載的增大而明顯增大。

圖4 裂紋中點(diǎn)張開(kāi)位移隨運(yùn)動(dòng)速度的變化曲線:運(yùn)動(dòng)裂紋Dugdale模型Fig.4Normalizedcrackopeningdisplacementatthemiddlepointversusnormalizedcrackmovingspeedcurves:movingcrackDugdalemodel圖5 不同位置裂紋張開(kāi)位移隨運(yùn)動(dòng)速度的變化曲線Fig.5Normalizedcrackopeningdisplacementversusnormalizedcrackmovingspeedcurvesatdifferentlocation圖6 不同應(yīng)力下裂紋中點(diǎn)張開(kāi)位移隨運(yùn)動(dòng)速度的變化曲線Fig.6Normalizedcrackopeningdisplacementatthemiddlepointversusnormalizedcrackmovingspeedfordifferentstressratio

表2　長(zhǎng)度比c/a隨應(yīng)力比σ ∞/σ 0的變化:σ 0=10 MPa

(33)

(34)

類(lèi)似地:v=cR時(shí)，式(28)中的f(v)/D成為0/0，由羅必塔法則，得

(35)

可見(jiàn)，引入速度函數(shù)f(v)后，COD在v=cR時(shí)連續(xù)且為有限值。圖5也證實(shí)，運(yùn)動(dòng)速度從0直到剪切波速cs，COD都是連續(xù)的，跨越Raylrigh波速時(shí)也是連續(xù)的。由圖5知，裂紋張開(kāi)位移隨運(yùn)動(dòng)速度增大而減小。

不同作用應(yīng)力下，裂紋中點(diǎn)張開(kāi)位移隨運(yùn)動(dòng)速度的變化曲線如圖6示，隨荷載的增大，張開(kāi)位移隨之明顯增大。圖7是不同速度下裂紋張開(kāi)位移隨橫坐標(biāo)位置的變化曲線。圖8是不同應(yīng)力下裂紋張開(kāi)位移隨橫坐標(biāo)位置的變化曲線，圖中顯示隨作用應(yīng)力的增加，裂紋張開(kāi)位移及裂尖損傷區(qū)長(zhǎng)度明顯增大。

圖7　不同速度下裂紋張開(kāi) 位移沿橫坐標(biāo)的變化曲線 Fig.7 Normalized crack opening displacement versus normalized coordinate ξ/a for different crack moving speed

圖8　不同應(yīng)力下裂紋張開(kāi)位移沿橫坐標(biāo)的變化曲線 Fig.8 Normalized crack opening displacement versus normalized coordinate ξ/a for different stress ratio

5結(jié)論

(1)研究運(yùn)動(dòng)裂紋問(wèn)題，裂尖損傷區(qū)用約束應(yīng)力區(qū)描述。設(shè)約束應(yīng)力為線性分布，采用復(fù)變函數(shù)解法，獲得了裂尖損傷區(qū)長(zhǎng)度與裂紋表面張開(kāi)位移的解析解。

(2)Dugdale運(yùn)動(dòng)裂紋模型，在Rayleigh波速下裂紋張開(kāi)位移為正、負(fù)無(wú)窮大。通過(guò)引入兩個(gè)速度效應(yīng)函數(shù)，成功實(shí)現(xiàn)了裂紋運(yùn)動(dòng)速度從0、到跨越Rayleigh波速、直到剪切波速，裂紋張開(kāi)位移的連續(xù)變化。

(3)數(shù)值計(jì)算結(jié)果表明，隨作用應(yīng)力的增大，裂尖損傷區(qū)長(zhǎng)度及裂紋張開(kāi)位移顯著增大。隨裂紋運(yùn)動(dòng)速度的增加，裂紋張開(kāi)位移明顯減小。

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