線彈性斷裂力學(xué)基礎(chǔ)課件

1、線彈性斷裂力學(xué)基礎(chǔ)引言23第一節(jié)斷裂模式與裂紋類型4斷裂模式II型和III型斷裂 分別叫滑移型斷裂(slide mode)和撕裂型斷裂(twist mode) (也叫做滑開型和撕開型)，它們都和剪切應(yīng)力相關(guān)，因此也都屬于剪切斷裂(shear mode)。張開型滑移型撕裂型有關(guān)斷裂模式的討論可在任意一本斷裂力學(xué)的書籍中找到，這里僅僅做簡單的描述?；镜臄嗔涯Ｊ接蠭型、II型和III型，如下圖所示。I型斷裂 也叫張開型斷裂 (open mode)，是一種最為危險(xiǎn)同時(shí)也是研究最為深入的斷裂模式。5剪切斷裂最危險(xiǎn)面內(nèi)斷裂斷裂模式I型和II型屬于面內(nèi)斷裂 (in-plane fracture)，III型

2、則屬于面外斷裂(out-of-plane fracture)。面外斷裂拉伸斷裂在工程實(shí)際中，并不是所有的斷裂問題都可以簡化為上述三個(gè)基本模式之一。因此，存在著復(fù)合型斷裂(mixed mode)，即上述三種基本模式的組合。6通常用所處理問題的維數(shù)來說明裂紋在空間的構(gòu)型，如平面內(nèi)的裂紋叫做二維裂紋，空間內(nèi)的裂紋叫三維裂紋等等。裂紋類型本課件將只根據(jù)裂紋的形態(tài)將其分為線狀裂紋(line type crack)和面狀裂紋(surface type crack)兩大類，如下圖所示。7裂紋類型線狀裂紋通常由一個(gè)點(diǎn)，即裂紋尖端(crack tip)來描述。如果線狀裂紋處于一個(gè)平面內(nèi)，是二維問題；如果線狀裂紋

3、處于空間中，則是三維問題。沿管道方向擴(kuò)展的裂紋可簡化為線狀裂紋。許多工程結(jié)構(gòu)中的裂紋可簡化為線狀裂紋。8裂紋類型面狀裂紋又可以分為埋藏裂紋(embedded crack)、表面裂紋(surface crack)和角裂紋(corner crack)，見下圖。面狀裂紋通常由一條線，即裂紋前沿(crack front)來描述。如果一個(gè)面狀裂紋處于一個(gè)平面內(nèi)，則該面狀裂紋為平面裂紋或稱為片狀裂紋，否則則是一個(gè)曲面裂紋。而復(fù)合材料中的分層則是典型的面狀裂紋。9穿透裂紋未穿透裂紋裂紋類型10第二節(jié)裂紋尖端附近的應(yīng)力場和位移場11I 型斷裂裂紋長度:2a彈性力學(xué)不用考慮其主要作用12I 型斷裂13II 型斷

4、裂2axy邊界條件：Westerrgaard應(yīng)力函數(shù)形式：該應(yīng)力函數(shù)滿足雙調(diào)和方程：應(yīng)力表達(dá)形式：14II 型斷裂15II 型斷裂16III 型斷裂 17第三節(jié)應(yīng)力強(qiáng)度因子(Stress Intensity Factor-SIF)181. 問題b=a, 圓孔b0，裂紋應(yīng)力集中Stress Concentration應(yīng)力奇異Stress SingularityS.C.FS.I.F應(yīng)力集中與應(yīng)力奇異19應(yīng)力集中與應(yīng)力奇異20應(yīng)力強(qiáng)度因子I型斷裂的應(yīng)力強(qiáng)度因子(Stress Intensity Factor, SIF)對(duì)于無窮大板，有將此應(yīng)力分量代入定義，可得誰最先提出SIF的概念？21應(yīng)力強(qiáng)度因子

5、裂紋尖端應(yīng)力場用應(yīng)力強(qiáng)度因子來表達(dá)22解析方法(閉合解)手冊(cè)(公式、表格或曲線)數(shù)值方法有限元（商業(yè)軟件）邊界元、無網(wǎng)格、差分法等等實(shí)驗(yàn)方法光彈、散斑求解方法Rooke DP and Cartwright DJ (1976). Compendium of Stress Intensity Factors. Procurement Executive, Ministry of Defence. H.M.S.O. Tada H, Paris PC, and Irwin GR (1985). The Stress Intensity Factor Handbook. Hellertown, Phil

6、adelphia: Del Research CorporationMurakami Y (1987). Stress Intensity Factors Handbook. New York: Pergamon.23LEFM中應(yīng)力強(qiáng)度因子可疊加求解方法24應(yīng)力強(qiáng)度因子舉例-線狀裂紋取無限大板具有周期裂紋的解作為近似解Fedderson公式及其修正形式對(duì)Isida公式的最小二乘法擬合修正的Koiter25應(yīng)力強(qiáng)度因子舉例-線狀裂紋26應(yīng)力強(qiáng)度因子舉例-線狀裂紋27圓柱殼 應(yīng)力強(qiáng)度因子舉例-線狀裂紋28式中，f(a,W,.)稱為幾何修正系數(shù)，反映構(gòu)件和裂紋幾何尺寸對(duì)裂尖應(yīng)力場的影響。應(yīng)力強(qiáng)度因子

7、舉例-線狀裂紋29Newman-Raju解應(yīng)力強(qiáng)度因子舉例-面狀裂紋30應(yīng)力強(qiáng)度因子舉例-面狀裂紋31第四節(jié)材料的斷裂韌度(Fracture Toughness)32斷裂韌性當(dāng)KI增加至某一臨界值，從而使裂紋頂端區(qū)域內(nèi)足夠大的體積內(nèi)部都達(dá)到使材料分離的應(yīng)力而導(dǎo)致裂紋的迅速擴(kuò)展時(shí)，這時(shí)的KI就稱為應(yīng)力場強(qiáng)度因子的臨界值，記做KIc。并稱其為斷裂韌性，表示材料在此條件下抵抗裂紋失穩(wěn)擴(kuò)展的能力。33試驗(yàn)標(biāo)準(zhǔn)34試樣35先在試件的相應(yīng)位置用線切割機(jī)床切一切口，線切割使用的鉬絲直徑約0.1mm,左右，太粗的切口將不利于裂紋的預(yù)制。切口的尺寸應(yīng)小于預(yù)定裂紋尺寸，以留有用疲勞預(yù)制裂紋的余地。為避免切口的影響

8、，預(yù)制疲勞裂紋的長度應(yīng)不小于1.5mm。此外，施加疲勞載荷預(yù)制裂紋時(shí)，使用的載荷越小，裂紋尖端越尖銳，預(yù)制裂紋所需時(shí)間越長。為保證裂紋尖端 具有足夠的銳度，一般要求循環(huán)載荷中Kmax（2/3）K1c。試樣36試驗(yàn)裝置37PQ是裂紋開始擴(kuò)展時(shí)的載荷a/a=2%V/V=5%割線斜率低于曲線斜率 5%張開位移的增量P5P5P5數(shù)據(jù)處理38工具顯微鏡數(shù)據(jù)處理39PQaQQ若KQ滿足下述試驗(yàn)有效性條件： 則所測(cè)得的KQ即為材料的平面應(yīng)變斷裂韌性KIC。要求材料為脆性的要求試件滿足平面應(yīng)變條件有效性判斷40解答 裂紋長度為： a=(a2+a3+a4)/3=32mm (1) 裂紋長度檢查a-(a1+a5)/

9、2=0.15mm1.5mm滿足例題41解答 (2) 計(jì)算表征應(yīng)力強(qiáng)度因子KQQQa/W=32/60=0.533PQ=56kN例題42解答 (3) 有效性檢查滿足滿足例題43常用金屬材料KIC數(shù)據(jù)44常用金屬材料KIC數(shù)據(jù)45常用金屬材料KIC數(shù)據(jù)46平面應(yīng)力斷裂韌性47第五節(jié)斷裂判據(jù)與斷裂控制設(shè)計(jì)的基本思路48在線彈性條件下，低應(yīng)力脆性斷裂的判據(jù)為利用上述判據(jù)，可以類似于強(qiáng)度設(shè)計(jì)那樣，進(jìn)行抗斷設(shè)計(jì)。 49控制材料或結(jié)構(gòu)斷裂的，是下述三個(gè)主要因素： 裂紋尺寸和形狀作用應(yīng)力材料的斷裂韌性結(jié)構(gòu)分析實(shí)驗(yàn)測(cè)試KICKI斷裂判據(jù)50工作應(yīng)力 裂紋尺寸 斷裂韌度 已知已知選擇材料使其KIc值滿足斷裂判據(jù)，保

10、證不發(fā)生斷裂已知已知確定允許使用的工作應(yīng)力已知已知確定允許存在的最大裂紋尺寸515253545556575859第六節(jié)應(yīng)變能釋放率(Strain Energy Release Rate)60Griffith 能量平衡Griffith提出 斷裂能EDr Alan A. Griffith (1893-1963)總能量應(yīng)變能和外力功之和Griffith AA, The phenomena of rupture and flow in solids, Philosophical Transactions, Series A, 1920(221): 163-198. 61Griffith 能量平衡表面能

11、Inglis 對(duì)于無窮大板，有Inglis CE, Stress in a plate due to the presence of cracks and sharp corners, Transactions of the Institute of Naval Architects, 1913(55): 219-241. 62應(yīng)變能釋放率Irwin提出應(yīng)變能釋放率(Strain Energy Release Rate)外力功Dr George R. Irwin (1907-1998)由變形引起的應(yīng)變能UF對(duì)于無窮大板，有Irwin GR, Onset of fast crack propag

12、ation in high strength steel and aluminum alloys, Sagamore Research Conference Proceedings, 1956(2): 289-305. 63Father of Modern Fracture MechanicsDr George R. Irwin (1907-1998)After having received the A.B. in English and Physics from Knox College and the M.A. and Ph. D in Physics from the Universi

13、ty of Illinois, George Irwin began his career in 1937, at the U.S. Naval Research Lab (NRL) where he developed several new ballistics research techniques. As a result, the NRL Ballistics Branch, which was headed by Irwin, was able to develop non-metallic armors for fragment protection. These armors

14、received trial use in World War II and extensive use during the Korean and Vietnam Wars. The early years of this work led to an interest in brittle fracture and provided a basis for Irwins pioneering work in fracture mechanics. The basic concepts established by Irwin and his team from 1946 to 1960 a

15、re now used world wide for fracture control in aircraft, nuclear reactor vessels and other fracture- critical applicationsInducted in May 1993 in recognition of his pioneering efforts in creating the discipline of fracture mechanics and for his guidance to the technical community in helping to make

16、it a useful engineering design toolHis numerous awards include ASTM Honorary Member, Timoshenko Medal of ASME, Gold Medal of ASM, The Grand Medal of the French Metallurgical Society, Tetmajer Medal o the Technical University of Vienna, member of the National Academy of Engineering and foreign member

17、ship in the Royal Society of London. He was appointed to Boeing University Professor at Lehigh University in 1967. He later joined the University of Marylands Department of Engineering where he has been and active researcher and advisor of graduate students since 1972. 64應(yīng)變能釋放率Irwin 更進(jìn)一步將應(yīng)變能釋放率和裂紋簡單

18、應(yīng)力場和位移場聯(lián)系起來，提出對(duì)于無窮大板，有(平面應(yīng)力)xya65應(yīng)變能釋放率66應(yīng)變能釋放率與應(yīng)力強(qiáng)度因子的關(guān)系 平面應(yīng)力狀態(tài)平面應(yīng)變狀態(tài)67應(yīng)變能釋放率與柔度68應(yīng)變能釋放率與柔度柔度(Compliance)69應(yīng)變能釋放率與柔度載荷控制 (固定載荷): dP=0位移控制 (固定位移): d=070應(yīng)變能釋放率與柔度Mode I-Double Cantilever Beam (DCB)71應(yīng)變能釋放率與柔度Caddell RM (1980), Deformation and Fracture of Solids, Prentice-Hall, Inc., Englewood Cliffs,

19、 NJ72應(yīng)變能釋放率與柔度Caddell RM (1980), Deformation and Fracture of Solids, Prentice-Hall, Inc., Englewood Cliffs, NJ73應(yīng)變能釋放率與柔度Caddell RM (1980), Deformation and Fracture of Solids, Prentice-Hall, Inc., Englewood Cliffs, NJ74應(yīng)變能釋放率與柔度Caddell RM (1980), Deformation and Fracture of Solids, Prentice-Hall, In

20、c., Englewood Cliffs, NJ75第七節(jié)復(fù)合型斷裂問題(Mixed-Mode Fracture)76Practical structures are not only subjected to tension but they also experience shear and torsional loading. Cracks may therefore be exposed to tension and shear, which leads to mixed mode cracking. The combination of tension and shear gives

21、 a mixture of modes I and II. Several investigators have considered the mixed mode fracture problem, but a generally accepted analysis has not yet been developed. The discussion here will be limited to the III mixed mode. 77Mode II loading under an in-plane shear stress can be characterized by a str

22、ess intensity factor, analogous to mode I loading. Under these conditions fracture will occur when KII reaches a critical value KIIC. In mixed mode loading one has to deal with KI and KII, and fracture must be assumed to occur when a certain combination of the two reaches a critical value. 78When us

23、ing an energy balance criterion the total energy release rate GT, is given by: Fracture occurs when GT, is larger than the energy consumption rate, and hence the fracture condition is given byRT is assumed a constant here for simplicity.起裂的斷裂判據(jù)總量法79For I-II mixed mode loading, Mode I Mode II Hence,

24、the fracture condition would be 起裂的斷裂判據(jù)總量法80Consequently, it predicts that In practiceand that the locus for combined mode cracking is a circle with radius KIC. This is depicted in Figure 14.13.起裂的斷裂判據(jù)總量法81The fracture condition is more likely to be: 起裂的斷裂判據(jù)分量法82The locus of fracture is an ellipse (

25、Figure 14.13). Fracture occurs when KI and KII reach values sufficient to satisfy above condition起裂的斷裂判據(jù)分量法83起裂的斷裂判據(jù)實(shí)驗(yàn)數(shù)據(jù)與判據(jù)比較84The previous criterion is based on the assumption that the crack propagates in a self-similar manner. In other words, it is assumed that crack extension will be in the plane

26、 of the original crack. This is necessarily so, because the expressions for G and their relations to K were derived on this premise. In mixed mode experiments, it is usually observed that crack extension takes place under an angle with respect to the original crack (kinking) . This invalidates the s

27、tandard expression for G. self-similarkinking裂紋彎折的斷裂判據(jù)自相似裂紋與非自相似裂紋85Under general loading almost all theories for the direction of crack growth assume or predict that the continued crack growth will be with KII=0. Therefore, macroscopic cracks growing with continuously turning tangents will advance

28、straight ahead, presumably under Mode I conditions. The crack curvature will evolve in such a way as to maintain this in response to the loading. If the loading changes such that the local crack-tip stress field experiences a large change in local stress intensities, mixed-mode fracture will occur.裂

29、紋彎折的斷裂判據(jù)86Maximum Principal Stress CriterionMaximum Tangential Stress CriterionErdogan and Sih,1963Strain Energy Density CriterionSih, 1974Energy Release Rate CriterionNuismer, 1975KII=0 CriterionABAQUSABAQUSABAQUS裂紋彎折的斷裂判據(jù)87Erdogan and Sih (1963)Erdogan, F. and Sih. G. C. On the crack extension in

30、plates under plane loading and transverse shear, J. Basic Eng. 85(1963) pp.519-527. The maximum principal stress criterion postulates that crack growth will occur in a direction perpendicular to the maximum principal stress. If a crack is loaded in combined mode I and II, the stresses and r at the c

31、rack tip can be derived from the expressions in chapter 3, by adding the stresses due to separate mode I and mode II. ABAQUS：Maximum Tangential Stress Criterion-MTS裂紋彎折的斷裂判據(jù)Maximum Principal Stress Criterion88The result is as follows: The stress will be the principal stress if r=0. This is the case

32、for = m where m is found from equating the second equation to zero.裂紋彎折的斷裂判據(jù)Maximum Principal Stress Criterion89Can be solved by writing Which yields So that Method (1)裂紋彎折的斷裂判據(jù)Maximum Principal Stress Criterion90Can be solved by writing Which yields So that Method (2)：ABAQUS裂紋彎折的斷裂判據(jù)Maximum Principal Stress Criterion91ABAQUS裂紋彎折的斷裂判據(jù)Maximum Principal Stress Criterion92The strain