半空間中裂紋邊緣界面圓孔對(duì)SH波的散射

趙春香,　齊　輝,　南景富,　陳冬妮,　蔡立明

(1.黑龍江科技大學(xué) 理學(xué)院, 哈爾濱 150022; 2.哈爾濱工程大學(xué) 航天與建筑工程學(xué)院, 哈爾濱 150001)

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半空間中裂紋邊緣界面圓孔對(duì)SH波的散射

趙春香1,齊輝2,南景富1,陳冬妮2,蔡立明2

(1.黑龍江科技大學(xué) 理學(xué)院, 哈爾濱 150022; 2.哈爾濱工程大學(xué) 航天與建筑工程學(xué)院, 哈爾濱 150001)

研究界面波動(dòng)力學(xué)對(duì)保障結(jié)構(gòu)安全具有重要工程意義。應(yīng)用“剖分-契合”法和“裂紋切割”技術(shù),研究半空間雙相介質(zhì)界面上同時(shí)嵌有圓孔和裂紋時(shí)SH波對(duì)圓孔的動(dòng)力作用。利用“鏡像法”,將半空間的水平界面問(wèn)題變換為全空間的水平界面問(wèn)題,構(gòu)造出能自動(dòng)滿足半空間自由表面邊界條件的散射波;由保障水平界面應(yīng)力位移連續(xù)的“剖分-契合”條件,導(dǎo)出求解該問(wèn)題的定解積分方程組;對(duì)界面圓孔邊的動(dòng)應(yīng)力集中情況進(jìn)行討論分析。結(jié)果表明:工程應(yīng)用中必須考慮自由邊界、裂紋長(zhǎng)度、介質(zhì)材料參數(shù)和入射波數(shù)等對(duì)孔邊動(dòng)應(yīng)力集中程度的綜合影響。應(yīng)該避免較大的波數(shù)比、較高的入射波波數(shù)、較長(zhǎng)的裂紋長(zhǎng)度與垂直入射的同時(shí)出現(xiàn),這對(duì)降低孔邊的動(dòng)應(yīng)力集中系數(shù)效果明顯,對(duì)減輕結(jié)構(gòu)動(dòng)力效應(yīng)具有一定的工程價(jià)值。

界面圓孔; SH波散射; 剖分-契合法; 裂紋切割技術(shù); 動(dòng)應(yīng)力集中

分層和斷層的天然介質(zhì)分布廣泛,人為制造的各種地下工程結(jié)構(gòu)日趨頻繁,在天然介質(zhì)材料中不可避免地存在著連續(xù)介質(zhì)界面和分段連續(xù)介質(zhì)界面。彈性波在傳播過(guò)程中遇到界面后,不僅會(huì)改變波的傳播方向,也會(huì)對(duì)介質(zhì)材料內(nèi)的結(jié)構(gòu)產(chǎn)生動(dòng)力作用。在該動(dòng)力作用下,工程結(jié)構(gòu)的強(qiáng)度和剛度等固體力學(xué)性質(zhì)會(huì)發(fā)生變化,工程結(jié)構(gòu)的安全性會(huì)受到威脅。因此,研究界面波動(dòng)力學(xué)對(duì)保障地下結(jié)構(gòu)的安全具有重要的工程意義[1-8]。

對(duì)于斷層、裂隙等非連續(xù)介質(zhì)界面動(dòng)力性能的研究,推動(dòng)了界面斷裂動(dòng)力學(xué)的發(fā)展[9],但對(duì)于半空間內(nèi)介質(zhì)分層和斷層界面上的圓形孔洞對(duì)SH波散射時(shí)的動(dòng)力響應(yīng)問(wèn)題,還有待深入研究。筆者應(yīng)用“剖分” “契合” “裂紋切割”等理論分析手段,研究半空間雙相介質(zhì)界面上同時(shí)嵌有圓孔和裂紋時(shí)圓孔對(duì)SH波的動(dòng)力響應(yīng);在獲得其解析解基礎(chǔ)上,再運(yùn)用編程計(jì)算手段,尋求SH波動(dòng)力作用下半空間雙相介質(zhì)界面圓孔邊緣的動(dòng)應(yīng)力依半空間自由界面、介質(zhì)材料參數(shù)、入射波數(shù)及裂紋長(zhǎng)度的變化規(guī)律,以期為廣泛存在于天然介質(zhì)中的工程結(jié)構(gòu)的強(qiáng)度設(shè)計(jì)提供參考。

1　力學(xué)模型

當(dāng)天然或人為的應(yīng)力自由的鉛垂界面右側(cè)的天然介質(zhì)或工程結(jié)構(gòu)材料中存在近似水平的分界面,除了分界面上某處的接近圓形的孔洞外,靠近孔洞兩側(cè)的分界面呈有限長(zhǎng)非連續(xù),且在該界面兩側(cè)各自的材料介質(zhì)仍呈現(xiàn)均勻、連續(xù)、各向性質(zhì)相同。當(dāng)該雙相介質(zhì)材料的尺寸相對(duì)于分界面的孔洞和裂紋尺寸很大時(shí),可視該孔洞和裂紋位于半無(wú)限大雙相介質(zhì)界面上,即可將該半無(wú)限大區(qū)域抽象成鉛垂半空間內(nèi)雙相介質(zhì)水平界面上并存圓形孔洞與裂紋的理想化力學(xué)模型,如圖1所示。

圖1　力學(xué)模型Fig. 1　Mechanical model

由圖1可見(jiàn),坐標(biāo)系下,沿y=0相接的兩種不同介質(zhì)組成的鉛垂半空間內(nèi),在y=0,-A-a≤x≤a+A處有兩個(gè)長(zhǎng)度分別為A的直線界面裂紋;在y<0直角平面區(qū)域Ⅰ中介質(zhì)的密度和剪切模量分別為ρ1和G1;在y>0直角平面區(qū)域Ⅱ中相應(yīng)的值為ρ2和G2;水平界面上含有圓心位于水平界面上半徑為a的圓形孔洞,圓形孔洞的圓心到鉛垂界面的距離為d。穩(wěn)態(tài)SH波沿入射角α0由第四象限入射。

2　理論分析

2.1SH波入射下的總波場(chǎng)和總應(yīng)力場(chǎng)

(1)

和

(2)

(3)

(4)

(5)

在介質(zhì)I中產(chǎn)生的散射波構(gòu)造為

(6)

在介質(zhì)Ⅱ中產(chǎn)生的散射波構(gòu)造為

(7)

區(qū)域I和區(qū)域Ⅱ中的總波場(chǎng)和總應(yīng)力場(chǎng)為:

(8)

(9)

(10)

確定未知數(shù)An1和An2的邊界條件為:

(11)

由式(2)～(11),即可確定區(qū)域I和區(qū)域Ⅱ中的總波場(chǎng)和總應(yīng)力場(chǎng)。

2.2定解積分方程組

由文獻(xiàn)[11]的“剖分”“契合”思想和“裂紋切割”技術(shù)及文獻(xiàn)[12]已獲得的Green函數(shù),則可確定保障界面應(yīng)力位移連續(xù)條件的未知外力系f1(r0,θ0)和f2(r0,θ0)的定解積分方程組,即

(12)

(13)

其中,GⅠ和GⅡ分別為區(qū)域Ⅰ和區(qū)域Ⅱ的Green函數(shù)。

2.3孔邊動(dòng)應(yīng)力集中

通常用動(dòng)應(yīng)力集中系數(shù)(DSCF)來(lái)描述圓形孔洞邊緣動(dòng)應(yīng)力集中程度,其表達(dá)式為

(14)

在平面區(qū)域Ⅰ和區(qū)域Ⅱ中,r=a上的兩半圓孔邊的環(huán)向剪應(yīng)力分別為:

(15)

(16)

3　數(shù)值分析

下面以具體算例分析SH波作用下,半空間內(nèi)雙相介質(zhì)分界面上圓形孔洞和裂紋共存時(shí)孔邊的動(dòng)應(yīng)力集中問(wèn)題。

3.1SH波水平入射

圖2　SH波水平入射時(shí)孔邊隨k1a的變化

3.2SH波垂直入射

圖3　SH波垂直入射時(shí)孔邊受裂紋的影響

圖4　SH波垂直入射時(shí)孔邊隨A/a的變化

圖5　SH波垂直入射時(shí)孔邊隨d/a的變化

4　結(jié)　論

(1) SH波入射方向與裂紋平行時(shí),圓形孔洞孔邊的DSCF不受裂紋的影響。

(3) 介質(zhì)材料的波數(shù)比、入射波數(shù)及裂紋長(zhǎng)度對(duì)圓孔的DSCF的綜合影響必須重視,適當(dāng)避開(kāi)較大的波數(shù)比、較高的入射波波數(shù)及較長(zhǎng)的裂紋長(zhǎng)度與垂直入射的同時(shí)出現(xiàn),對(duì)降低孔邊的DSCF的效果明顯,對(duì)減輕結(jié)構(gòu)動(dòng)力效應(yīng)具有工程價(jià)值。

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(編輯徐巖)

Scattering of SH-wave by interface circular cavity of crack edge in a half space

ZHAOChunxiang1,QIHui2,NANJingfu1,CHENDongni2,CAILiming2

(1.School of Sciences, Heilongjiang University of Science & Technology, Harbin 150022, China;2. College of Aerospace & Civil Engineering, Harbin Engineering University, Harbin 150001, China)

This paper builds on the insight that the study of interface wave mechanics is of greater engineering significance to ensuring the structure safety and introduces the application of ‘partitioning-conjunction’ method and‘crack-division’ technique to study the dynamic effect of SH-wave on interface circular cavity, as occurs when bi-material interfaces in a half-space are embedded with a circular hole and crack. The research is best achieved by transforming the horizontal interface problem from half space to whole space using the method of ‘image’ and thereby constructing the scattering waves capable of automatically satisfying the free boundary condition on the half space surface; deriving the integral equations for solving the problem from continuous‘partitioning-conjunction’ conditions of stress and displacement on horizontal interface; and ultimately analyzing the dynamic stress concentrations of the interface circular cavity. The research results demonstrate that engineering application necessitates not only the consideration of the comprehensive effects, such as free boundary, crack length, material parameter, and incident wave number on circular cavity edge stress concentration degree, but also the prevention of the simultaneous occurrence of such influence factors as a larger wave number, a higher incident wave number, a longer the crack length， and SH-waves incident in vertical angle and this potentially contributes significantly to decreasing dynamic stress concentration factor of hole edge and thereby offers some engineering value for reducing the dynamic effect of SH-waves on engineering structure.

interface circular cavity; scattering of SH-wave; partitioning-conjunction method; crack-division technique; dynamic stress concentration

2014-11-19

黑龍江省自然科學(xué)基金項(xiàng)目(A201307)

趙春香(1967-),女,黑龍江省雞西人,副教授,博士,研究方向:彈性波散射理論及應(yīng)用研究,E-mail:zhaocx2005@sohu.com。

10.3969/j.issn.2095-7262.2015.01.019

O343.4; O347.3

2095-7262(2015)01-0087-05

A