張希萌， 齊 輝， 項(xiàng) 夢(mèng)， 丁曉浩

(哈爾濱工程大學(xué) 航天與建筑工程學(xué)院，哈爾濱 150001)

半空間雙相介質(zhì)垂直界面裂紋附近圓形襯砌和半圓凹陷對(duì)SH波的散射

張希萌， 齊 輝， 項(xiàng) 夢(mèng)， 丁曉浩

(哈爾濱工程大學(xué) 航天與建筑工程學(xué)院，哈爾濱 150001)

利用Green函數(shù)法、鏡像法與多級(jí)坐標(biāo)法，對(duì)半空間中半圓凹陷和圓形襯砌對(duì)SH波的散射進(jìn)行分析，得到其穩(wěn)態(tài)響應(yīng)。利用鏡像法得到了滿足水平邊界應(yīng)力自由、垂直邊界位移與應(yīng)力連續(xù)的波函數(shù)解析表達(dá)式。根據(jù)垂直邊界連續(xù)性條件,利用“裂紋切割法”和“契合法”建立起求解問題的第一類Fredholm型積分方程，得到了圓形襯砌周邊的動(dòng)應(yīng)力集中系數(shù)與裂紋尖端動(dòng)應(yīng)力強(qiáng)度因子的解析表達(dá)式。數(shù)值算例分析了入射波數(shù)、襯砌深度、半圓凹陷大小、裂紋長度等對(duì)動(dòng)應(yīng)力集中系數(shù)、裂紋尖端動(dòng)應(yīng)力強(qiáng)度因子與地表位移的影響,并與已有文獻(xiàn)進(jìn)行比較。

半空間；圓形襯砌；半圓凹陷；Green函數(shù)；動(dòng)應(yīng)力集中系數(shù)(DSCF)；動(dòng)應(yīng)力強(qiáng)度因子(DSIF)

隨著經(jīng)濟(jì)不斷提高，為了緩解交通壓力與維持國計(jì)民生，建立了大量生命線工程，例如供水、排水、燃?xì)?、石油管道與地下鐵道工程。很多復(fù)雜地形都有如孔洞、凹陷等各種缺陷，含缺陷的地下襯砌結(jié)構(gòu)受到由地震、爆破等彈性波作用后其應(yīng)力集中問題比靜態(tài)時(shí)更復(fù)雜，生命線工程受損將會(huì)導(dǎo)致整個(gè)城市或局部社區(qū)經(jīng)濟(jì)功能的癱瘓，因此生命線工程在復(fù)雜地形中要求有較高的抗震性與安全性，眾多學(xué)者對(duì)缺陷問題進(jìn)行研究并取得大量成果。近年來，Qi等[1-4]對(duì)直角域或半空間中襯砌與凹陷的動(dòng)力問題進(jìn)行了分析，韓峰等[5-6]對(duì)多個(gè)凸起與凹陷相連地形的動(dòng)力響應(yīng)問題給出了數(shù)值解，梁建文等[7]研究了地下圓形襯砌的動(dòng)應(yīng)力集中問題。孫苗苗[8]分析了空心管樁組成的陣列對(duì)SH波的散射問題。

本文利用Green函數(shù)法、“鏡像法”與多級(jí)坐標(biāo)法，構(gòu)造出波函數(shù)。根據(jù)連續(xù)性條件，利用“契合法”與“裂紋切割法”建立第一類Fredholm型積分方程組。分析討論入射角度、入射波數(shù)、襯砌埋深、半圓凹陷大小等對(duì)動(dòng)應(yīng)力集中系數(shù)、動(dòng)應(yīng)力強(qiáng)度因子與地表位移的影響。

1 問題的描述

如圖1，介質(zhì)Ⅰ為含半圓缺陷和圓形襯砌的直角域，其質(zhì)量密度與剪切模量分別為ρ1、μ1， 水平、垂直邊界分別為ΓH、ΓV，半圓缺陷中心位置與垂直邊界ΓV距離為d1， 半徑為c，其邊界為ΓC； 介質(zhì)Ⅱ?yàn)闊o缺陷的直角域，其質(zhì)量密度與剪切模量分別為ρ2、μ2； 介質(zhì)Ⅲ為圓形襯砌，其質(zhì)量密度與剪切模量分別為ρ3、μ3，中心位置與垂直邊界ΓV距離為d，與水平邊界ΓH距離為h，內(nèi)、外半徑分別為b、a，內(nèi)邊界、外邊界分別為ΓB、ΓA； 垂直界面裂紋長度為2A，尖端與水平邊界ΓH距離為h1。本文采用多級(jí)坐標(biāo)展開法，建立坐標(biāo)系xoy、x1o1y1、x′o′y′，所對(duì)應(yīng)的復(fù)坐標(biāo)系分別為：η=x+yi=reiθ、η1=x1+y1i=r1eiθ1、η′=x′+y′i=r′eiθ′。各坐標(biāo)系關(guān)系為

(1)

本文模型是對(duì)半空間中由兩種不同的介質(zhì)構(gòu)成的含垂直界面裂紋的復(fù)雜地形中凹陷與襯砌在SH波作用下動(dòng)應(yīng)力響應(yīng)這一生命線工程問題的簡化。

圖1 半空間垂直界面裂紋附近圓形襯砌和半圓形凹陷模型Fig.1 The model of a circular lining and a semi-circular canyon near vertical interface crack in half space

2 Green函數(shù)

(2)

(3)

本節(jié)研究的直角域介質(zhì)Ⅰ在線源荷載δ(η-η0)作用下的模型如圖2所示。其中η0=d+yi, (y≤h)， 表示η0在垂直邊界ΓV上。

圖2 受線源荷載作用的直角域模型Fig.2 The right-angle plane model impacted by a line source force

介質(zhì)Ⅰ邊界條件可以表示為

(4)

由線源荷載δ(η-η0)產(chǎn)生并的擾動(dòng)，可視為已知的入射波Gi，應(yīng)滿足直角域水平邊界ΓH上應(yīng)力自由，利用“鏡像法”， 構(gòu)造出入射波表達(dá)式

(5)

對(duì)于半圓凹陷形成的散射波Gs1和圓形襯砌所形成的散射波Gs2，均滿足直角域中直線邊界應(yīng)力自由條件,利用“鏡像法”，構(gòu)造出其表示式

(6)

(7)

式中：

其中：η=x+yi,η1=x1+y1i,η1=η+η0,

η0=(d1-d)-hi,η2=η-2hi,

η3=η-2d,η4=η2-2d

對(duì)于介質(zhì)Ⅲ圓形襯砌內(nèi)所形成駐波Gst，按文獻(xiàn)[3]中思路，構(gòu)造其表達(dá)式如下：

(8)

由以上推導(dǎo)可知：

(9)

由邊界條件(3)建立方程組，

(10)

式中：

ξ(3)=ξ(4)=0

其中：

對(duì)于介質(zhì)Ⅱ，其Green函數(shù)表達(dá)式為

(11)

3 SH波的散射

入射波w(i,e)、散射波w(r,e)和折射波w(f,e)均滿足直角域中水平邊界ΓH上應(yīng)力自由，垂直邊界ΓV上的連續(xù)性條件，按與求解Green函數(shù)相同思路，利用“鏡像法”構(gòu)造其表達(dá)：

(12)

(13)

(14)

其中：β0=π-α0,各參數(shù)關(guān)系式：

k1sinα0=k2sinα2,c1sinα2=c2sinα0

經(jīng)驗(yàn)證，式(12)、式(13)、式(14)滿足垂直邊界ΓV連續(xù)性條件：

(15)

式中：α0是入射角度；α2是折射角度。在SH波作用下產(chǎn)生的波場與上節(jié)中Green函數(shù)作用下產(chǎn)生的波場具有相同的形式：

(16)

(17)

(18)

其中未知系數(shù)Rm、Tm、Pm、Qm根據(jù)邊界條件(4)確定，所列方程組中已知系數(shù)與求解Green函數(shù)所列方程組中已知系數(shù)相同。

4 契 合

在介質(zhì)Ⅰ中：

(19)

在介質(zhì)Ⅱ中：

(20)

WⅠ+W1+Wc1=WⅡ+W2+Wc2,

(21)

其中：

(22)

圖3 半空間雙相介質(zhì)垂直界面的契合Fig.3 Conjunction of vertical interface in bi-material half space

利用式(15)對(duì)式(21)進(jìn)行簡化，得到關(guān)于外力系的積分方程：

(23)

5 動(dòng)應(yīng)力集中系數(shù)

在SH波作用下環(huán)向剪切應(yīng)力可以表示為

(24)

動(dòng)應(yīng)力系數(shù)可表示為

6 動(dòng)應(yīng)力強(qiáng)度因子

在裂紋尖端外力系f1具有平方根奇異性。引入動(dòng)應(yīng)力強(qiáng)度因子

(25)

為在定解積分方程組(18)中直接包含動(dòng)應(yīng)力強(qiáng)度因子kⅢ，對(duì)被積函數(shù)進(jìn)行變換：

(26)

求解變換后的積分方程組(23)，裂紋尖端對(duì)應(yīng)的值即為動(dòng)應(yīng)力強(qiáng)度因子kⅢ。在計(jì)算中,通常定義一個(gè)無量綱的動(dòng)應(yīng)力強(qiáng)度因子k3。

(27)

7 具體算例

圖4 本文方法的驗(yàn)證Fig.4 The vertifying of the present method

圖5 SH波低頻水平入射時(shí)圓形襯砌周邊DSCF隨k*與μ*的分布Fig.5 Distribution of DSCF around circular lining edge vs. k* and μ* by low frequency SH-wave horizontally

(a)

(b)圖6 SH波高頻水平入射時(shí)圓形襯砌周邊DSCF隨k*與μ*的分布Fig.6 Distribution of DSCF around circular lining edge vs. k* and μ* by high frequency SH-wave horizontally

圖7 圓形襯砌周邊DSCF隨ka的分布Fig.7 Distribution of DSCF around circular lining edge vs. ka

圖8 圓形襯砌周邊DSCF隨α0的分布Fig.8 Distribution of DSCF around circular lining edge vs. α0

由圖5～圖8可知，當(dāng)SH波高頻水平入射襯砌相對(duì)于基體越軟，危害越大。

圖9 圓形襯砌周邊DSCF隨h*的分布Fig.9 Distribution of DSCF around circular lining edge vs. h*

圖10 圓形襯砌周邊DSCF隨c*的分布Fig.10 Distribution of DSCF around circular lining edge vs. c*

圖11 圓形襯砌周邊DSCF隨A*的分布Fig.11 Distribution of DSCF around circular lining edge vs. A*

圖12(b)給出裂紋尖端動(dòng)應(yīng)力因子k3隨A*分布情況。由圖12(b)可知，k3呈振蕩變化，A*=2，ka=1.4時(shí)k3最大值為9.89， 比A*=0.5，ka=0.6時(shí)k3最大值8.08提高了22%。因此裂紋長度A*對(duì)k3存在影響。

由以上可知，裂紋長度對(duì)裂紋尖端動(dòng)應(yīng)力因子k3影響顯著。

圖12 DSIF隨ka的變化Fig.12 Variation of DSIF vs. ka

圖13 SH波低頻入射時(shí)|W|隨k*與μ*的分布Fig.13 Distribution of |W| vs. k* and μ* by low frequency SH-wave horizontally

(a)

(b)圖14 SH波高頻入射時(shí)|W|隨k*與μ*的分布Fig.14 Distribution of |W| vs. k* and μ* by high frequency SH-wave horizontally

圖15給出了地表位移|W|隨ka分布情況。由圖15可知，地表位移|W|受ka影響較大，當(dāng)x>40時(shí)|W|逐漸穩(wěn)定。當(dāng)ka=1時(shí)，在x=6處|W|達(dá)到最大值6.5。

圖15 |W|隨ka的分布Fig.15 Distribution of |W| vs. ka

8 結(jié) 論

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ScatteringofSH-wavebyacircularliningandasemi-circularcanyonnearverticalinterfacecrackinthebi-materialhalfspace

ZHANG Ximeng, QI Hui, XIANG Meng, DING Xiaohao

(College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China)

The scattering problem of SH-wave by a circular lining and a semi-circular canyon in the bi-material half space was analyzed by the Green function method, the mirror method and the multi-level coordinate method to obtain the steady state response. The analytical expression of the wave function which satisfies the stress free on the horizontal boundaries, displacement and stress continuity on the vertical boundaries was obtained by the image method. According to the continuity condition on the vertical boundary, the first kind of Fredholm integral equation was set up to obtain analytical expression of dynamic stress concentration factor around the edge of circular lining and dynamic stress intensity factor at crack tip by “the conjunction method” and “the crack-division method”. The influence of the incident wave number, the ground depth of circular lining and the size of semi-circular canyon and the length of crack on the dynamic stress concentration factor, the dynamic stress intensity factor and the displacement along horizontal surface was analyzed and compared with the existed literature through a numerical example.

half space; circular lining; semi-circular canyon; Green function; dynamic stress concentration factor (DSCF); dynamic stress intensity factor (DSIF)

黑龍江省自然科學(xué)基金資助項(xiàng)目(A201404)

2016-03-10 修改稿收到日期： 2016-07-21

張希萌 男，博士生，1989年生

齊輝 男，教授，博士生導(dǎo)師，1963 年生

E-mail:qihui205@sina.com

O343.1; O347.3

A

10.13465/j.cnki.jvs.2017.20.022