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共振結構中的疲勞裂紋的擴展
摘要——已經研究了結構共振激發(fā)的疲勞裂紋的擴展。這一結構被離散，且由一個線性微分方程組表示。裂紋被模擬為局部柔度。應用一個環(huán)形裂紋的動態(tài)節(jié)點來說明滯止裂紋。裂紋的深度決定了由裂紋引入的局部剛度，這反過來又影響在共振頻率下擁有恒定振幅的外部激發(fā)的這一系統(tǒng)的動態(tài)響應。裂紋的擴展引入了額外的柔度，這使得擴展逐漸從共振處移開。在某些情況下，動態(tài)響應減弱，變得小于所考慮的材料的固有臨界值。被稱為動態(tài)滯止裂紋的這種現象，受材料的損耗系數的影響很大，是決定著開裂截面的荷載為最大時，裂紋生長的初始階段的裂紋擴展的主要因素。
1、 引言
在動態(tài)荷載作用下，一個傳播裂紋可能導致構件的破壞。因此，裂紋擴展速率的測定對機器的安全運行有著重大意義。Paris和Erdogan方程被用來確定傳播速率：
dadN=B?Km （1）
其中B，m是材料常數，而ΔK是應力強度因子范圍。方程（1）提到的是應力范圍而不是平均應力。它也限制裂縫的向前擴展。為了克服這些限制，Sih[4-6]提出了伸展的應變能密度因子的概念，用來預測疲勞裂紋的增長。
dadN=C?Sminn （2）
其中C，N是材料常數，ΔSmin是應變能密度因子范圍。裂紋形核和擴展在振動結構中的出現是很自然的，尤其是局部或整體的共振發(fā)生時。Chondros 和Dimarogonas[7-9]，Dimarogonas和massouros [10]表明，裂紋形貌對這種結構的振動特性影響相當大。由于這種變化，裂紋附近的應力場將會發(fā)生改變。Dentsoras和Dimarogonas[11-13]表明，對于簡單的結構而言，在這種情況下，裂紋擴展速率變的低于所考慮的材料的固有的臨界值是可能的。這個過程，稱為動態(tài)滯止裂紋，已被證明受系統(tǒng)中主要的阻尼機制中材料的阻尼的強烈影響。
在這里，這個原則是適用于被模擬成為一個多自由度系統(tǒng)的振動裂紋結構，而且它還適用于轉子在扭轉振動下的一個四階段的裂紋。
2、 分析
考慮一個質量集中，有n個自由度的振動系統(tǒng)[M]，[K]，其中[M]是對角質量矩陣而[K]是剛度矩陣。如果：
x=[x1tx2t…xn(t)]T （3）
為響應矢量，而且，
K=k0Ku,M=m0Mu （4）
其中，[Ku]，[Mu]分別是無量綱的質量矩陣和剛度矩陣，m0，k0是常數，
Ix+λ0Aux=[0] (5)
是自由運動微分方程，[Au] 是無量綱的動態(tài)矩陣，而且，λ0=k0/m0。從方程（5）中，我們可以獲得系統(tǒng)的特征值和特征向量。假設[Z]是模態(tài)矩陣，Dimarogonas[9]引入了標準化的特征向量，
φui=Zuiμiui (6)
其中，μiui是i型模式中無量綱的廣義質量，正規(guī)矩陣[Φu]將與矩陣[Mu]，[Ku]相關聯，通過以下方程
?uTMu?u=I （7）
和
[?u]TKu?u=diag(λui) （8）
其中，λui=λiλ0是無量綱的特征值。通過引入
x=?q （9）
方程（5）可以被修正為
Iqu+??qu=[0] （10）
其中，[∧]=diag(λ1)是特征值矩陣，方程（10）對應于非耦合振動。
如果系統(tǒng)的結構阻尼是占主導地位的阻尼機理，而且阻尼矩陣[C]可以標準化，那么[14]
[?u]TC?u=m0λ0diag(2γiλui) （11）
其中γi是i型模型的損耗系數。
更近一步來講，如果P=Feiwt是激發(fā)矢量，那么通過引入
F=F0Fu,Fc=F0λ0m0
方程（10）變?yōu)?
qu+λ0diag2γiλuiqu+??qu=Fc[?u]TFueiωt （12）
響應振幅為
Qur=Fc(i=1nφui(r)Fui)-ω2+λ0γrλurωi+λ0λur (13)
假設裂紋存在于所考慮系統(tǒng)的一部分中。由于裂紋的產生，剛度矩陣發(fā)生變化。如果矩陣[ΔK]表示的剛度矩陣[K]的變化，它已由Dimarogonas and Paipetis[15]證明，
?ωrωr=12[Z(r)]T[?K]Z(r)[Z(r)]T[K]Z(r) (14)
其中，Δwr是自然頻率r的變化。對于目前的情況下，如果ai是初始裂紋深度，（wr）ai是各自的修正的頻率，然后：
(ωr)aiωr=1-12ZurTKuZurZurTKuZur=1-12φurTKuφurφurTKuφur (15)
其中，[Ku]是無量綱的矩陣[Δk]。在共振中，
ω=(ωr)ai=ωr-?ωi
那么方程（13）變成
Qur=F0(i=1nφui(r)Fui)m0λurγr(φurTKuφurγrφurTKuφur)2+(2-φurTKuφurφurTKuφur )2-12 (16)
如果
Πr=λucγr(1γrφurTKuφurφurTKuφur)2+(2-φurTKuφurφurTKuφur )2-12 (17)
而
(Qur)c=F0(i=1nφui(r)Fui)m0 (18)
那么，最后，
Qur=(Qur)cΠr (19)
在坐標r中，應力為
Fr=[Kr]x (20)
其中，[Kr]是r方向上的剛度矩陣[K]。利用方程（9）
Fr=Kr?x=Kr?Qu (21)
應力與力的相關關系為
σr=S(Fr) （22）
或者根據方程（16）-（19）：
σr=S([Kr]?Qu) （23）
和
σur= S(K0[Kr]?Qu) （24）
是無量綱值的應力，其中K0是一個常數。應力強度因子的表達式：
Kr=σrπaF(a) （25）
根據方程（23）
Kr= S([Kr]?Qu) πaF(a) （26）
而
Kur=σurπaF(a) （27）
是無量綱應力強度因子。
如果ΔKr=Krmax-Krmin,ζ= Krmin/ Krmax,那么：
?Kr=Kr(1-ξ) （28）
最后，
?Kr=K0σurπaF(a)(1-ξ) （29）
根據方程（29），這個Paris方程可以寫成：
dadN=B[K0σurπaF(a)(1-ξ)]m （30）
所以最后，
N=B-1K01-ξ-maiafKur-mda （31）
方程（31）是通常使用的。它給出了裂紋深度從ai增加到af的必要周期數。Kur的測定，在積分上取決于特定的模型，這將在下面的應用中顯示。
3、 說明性的例子
考慮一個四級轉子的幾何特征d1，l1和一個周向裂紋，如圖1所示。在每一盤上，施加扭轉激勵
Pi=Tieiωt （32）
如果J是慣性量，而且
K=πd4G32l （33）
是每個單元的剛度常數，那么對于矩陣[J],[K]，在方程（4）中是合理的。對于一個測試轉子，
Ju= （34）
Ku= （35）
利用雅可比旋轉的方法發(fā)現了相應的特征值和特征向量：
λu1=0.0176 ,λu2=0.2482,λu3=0.5233,λu4=0.6905
Zu= （36）
如果Kr是構件r方向上沒有裂縫的剛度常數，而（Kr）a是由裂引入的剛度常數，那么等效剛度常數為
(Keq)r=(Kr)αKr(Kr)α+Kr (37)
根據Sih和MacDonald[l6] （Kr）a是
(Kr)α=πRr3GGIr(aRr) (38)
其中，Rr是截面半徑，并且：
Ir(aRr)=0.0351-aRr-4+0.011-aRr+0.00291-aRr2+0.00861-aRr3+0.00441-aRr4+0.00251-aRr5+0.00171-aRr6+0.0081-aRr7-0.092
由于對[Keq]r有一個分析表達式，所以矩陣[Ku]，可以用來計算每個裂紋深度且能作為一個結果，根據式（15），可以發(fā)現特征值的變化。在圖2中，這個圖表繪制了特征值與裂紋深度的比降。它可以很容易得出結論，裂紋深度的影響比系統(tǒng)的第一個特征值的影響強。
如果T=T0Tu，其中T是激發(fā)矢量，Tu是它的無量綱值，那么根據方程（19）：
(Qur)c=T0J0(i=1nφui(r)Tui) (39)
根據方程（23）:
τr=K0[Kr]?QuRIp (40)
正如Sih和MacDonald[15,16]所展示的那樣，對于第三型裂紋的擴展來講，
Kr=τraRr12Rr121-aRr-52I(aRr) (41)
其中，
IaRr=0.375+0.18751-aRr+0.140631-aRr2+0.117191-aRr3+0.102541-aRr4+0.0781-aRr5
根據方程（26）：
Kr=K0Kr?QuRIpaRr12Rr121-aRr-52IaRr (42)
和方程（40）：
Kur=τuraRr12Rr121-aRr-52IaRr (43)
因此，Kur，取決于裂紋的深度，這已在圖3中顯示。其中，損耗系數是參數。
最后，例如，通過公式（42），方程（30）可以進行修改，如下：
daRrdN=BCm1-RmRrKurm (44)
如果，
Cs=BCm1-RmRr
那么方程（44）整合后變?yōu)椋?
N=Cs-1aiRrafRrRur-mdaRr (45)
在圖4中，數值積分的結果顯示了，損失系數的各種值以及低碳鋼具有以下特點：
N=4.3×108N/m2
Su=4.3×108N/m2
m=2.51
B=2.832×10-26m2+3m2/cycle Nm
Kth=3.788×106N/m32
所示的制動線是所有點的軌跡，低速率的裂紋擴展在每一條曲線上發(fā)生轉變。
4、 結論
利用Paris和Erdogan的疲勞裂紋擴展模型來研究一個諧振系統(tǒng)的破裂行為。建立了裂紋深度與一般適用的疲勞周期值間的一般關系。在機械和結構共振處或接近共振處，經常出現裂紋的擴展，這是一個機器出現故障的常見原因。在這種情況下，裂紋的引入可以改變系統(tǒng)的動態(tài)特性，如動態(tài)分析是用來確定在一個恒定的外部激勵，開裂截面承載的任務剖面。這些信息被用來反饋到疲勞裂紋擴展模型中。
最初，在或非常接近共振處，裂紋擴展率很高。由于裂紋深度的變化，系統(tǒng)的固有頻率有一個相當大的減少，這使得裂紋從共振處“移走”，因此，降低了裂紋擴展速率。
這種情況，在某些特定環(huán)境下，強烈地依賴于阻尼因子，可導致動態(tài)滯止裂紋，如圖4所示。在這里，似乎總是發(fā)生裂紋滯止。在某種程度上，這是真的，因為如果裂紋擴展到一定程度，由于不穩(wěn)定的裂紋擴展或一些其他的機制，結構構件可能在某一方面破壞系統(tǒng)的結構完整性，使得裂紋擴展可以在滯止點之前發(fā)生停滯。
最后，可以很容易地得出結論，激勵的大小和系統(tǒng)的幾何形狀僅能定量的改變結果。止裂或許并不是主要取決于相應的節(jié)點的阻尼。
參考文獻
[1]P.C.Paris，The fracture mechanics approach to fatigue-an interdisciplinary approach.Proc.10th Materials Research Conf.，Syracuse University Press，New York(1964).
[2]P.C.Parts and F.A.Erdogan，Critical analysis of crack propagation laws.J.Bas.Engng 85，528(1963).
[3]F.A.Erdogan，Crack propagation theories，in Fracture (Edited by H.Liebowitx)，Vol.2.Academic Press，New York(1968).
[4]G.C.Sih，Some basic problems in fracture mechanics and new concepts.Engng Fracture Med.5，365-377(1973).
[5]G.C.Sih，Strain energy density factor applied to engineering problems.Inr.J.Fracture 10，305-321(1974).
[6]G.C.Sih and B.M.Barthelemy，Mixed mode fatigue crack growth predictions.Engng Fracture Mech.13.439-451(1980).
[7]T.G.Chondros and A.D.Dimarogonas，Identification of cracks in circular plates welded at the contour.979ASME Design Engineering Technical Conf..St.Louis，U.S.A.，Paper No.79-DET-106(September 1979).
[8]T.G.Chondros and A.D.Dimarogonas，Identification of cracks in welded joints of complex structures.J.Sound Vibration 69，531(1980).
[9]A.D.Dimarogonas，Vibration Engineering.West，St.Paul(1976).
[10]A.D.Dimarogonas and G.Masouros，Torsional vibration of a shaft with a circumferential crack Engng Fracture Mech.15，439444(1981).
[11]A.J.Dentsoras and A.D.Dimarogonas，Resonance controlled fatigue crack propagation.Engng Fracrure Mech.17， (1983).
[12]A.J. Dentsoras and A.D.Dimarogonas，Resonance controlled fatigue crack propagation in a beam under longitudinal vibration.Inr.J.Fracture 23，15-22(1983).
[13]A.J.Dentsoras and A.D.Dimarogonas，Resonance controlled fatigue crack propagation in cylindrical beams under combined loading.ASME Winter Annual Meeting，Boston(1983).
[14]S.Timoshenko，Vibration Problems in Engineering.Wiley，New York (1984).
[15]A.D.Dimarogonas and S.Paipetis，Rotor Dynamics.Applied Science，London(1983).
[16]G.C.Sih and B.McDonald，Fracture mechanics applied to engineering problems，strain energy density fracture criterion.Engng Fracrure Mech.6，493-507 (1974).
[17]A.P.Parker，The Mechanics of Fracture and Fatigue-An Introduction.E.F.N. Spon.Ltd.，London-New York(1981).
Engineering Fracture Mechanics Vol. 34, No. 3, pp. 721-728, 1989 Printed in Great Britain. 0013-7944/89 $3.00 + 0.00 0 1989 Perpmon Press pk. FATIGUE CRACK PROPAGATION IN RESONATING STRUCTURES A. J. DENTSORAS Lecturer, Machine Design Laboratory, University of Patras, Greece and A. D. DIMAROGONAS W. Palm Professor of Mechanical Design, Washington University, St. Louis, Missouri, U.S.A. Ahatraet-Fatigue crack propagation in structures excited at one of their resonances is investigated. The structure is discretixed and represented by a system of linear differential equations. Cracks are modelled as local flexibilities. Application on a 4-node rotor with a circumferential crack illustrates the dynamic crack arrest. The depth of the crack determines the local stiffness introduced by the crack, which in turn influences the dynamic response of the system under external excitation of constant amplitude at a resonant frequency. The propagation of the crack introduces additional flexibility which causes gradual shift from resonance. The dynamic response is reduced and, under certain circumstances, becomes less than a threshold value characteristic of the material considered. This phenomenon, known as dynamic crack arrest is drastically influenced by the loss coefficient of the material which is the main factor to determine the crack propagation rate at the initial stages of the crack growth when the loading of the cracked section becomes maximum. 1. INTRODUCTION UNDER the influence of the dynamic loads, a propagating crack might cause failure of a structural member. Consequently, the determination of the crack propagation rate is of major importance for the safe operation of machines. The equation of Paris and Erdoganl3 is used to determine the propagation rate: $ = B(AK)” (1) where B, m are material constants and AK is the stress intensity factor range. Equation (1) involves the stress range only and not the mean stress. It is also restricted to crack running ahead. To overcome these limitations, Sih4-6 proposed the extension of the strain energy density factor concept to predict the growth of fatigue cracks where S, n are material constants and AS, is the strain energy density factor range. Crack nucleation and propagation is natural to be expected in vibrating structures, in par- ticular when local or general resonance occurs. Chondros and Dimarogonas7-91 and Dimarogonas and Massouros lo have shown that the appearance of a crack influences considerably the vibration characteristics of such structures. Because of this variation, the stress field in the vicinity of the crack will change. Dentsoras and Dimarogonas 1 l-l 31 have shown for simple structures that, under such conditions, it is possible for the crack propagation rate to become lower than a threshold value characteristic of the material considered. This process, known as dynamic crack arrest, was shown that is strongly affected by material damping being the dominate damping mechanism of the system considered. Here this principle is applied for a vibrating cracked structure modelled as a multi- degree of freedom system and it was applied for a 4-stage cracked rotor under torsional vibrations. 721 722 A. J. DENTSORAS and A. D. DIMAROGONAS 2. ANALYSIS Consider a lumped mass vibrating system Ml, K with n-degrees of freedom, where M is the diagonal mass matrix and Kj the stiffness matrix. If: x = x,(r)x,(t) . . .x,(t) (3) is the response vector and: Kl = k3vGl Ml = wwul (4) where KU, MU are the dimensionless mass and stiffness matrices respectively and m, k. are constants, Zj2 + 0 i-eigenvalue. By introducing: x = MIS, (7) G-9 (9) eq. (5) can be modified as follows: mi” + ml” = PI (10) where A = diag(R,) is the eigenvalue matrix. Equation (10) corresponds to uncoupled vibration. If the structural damping of the system is the dominant damping mechanism and the damping matrix C can be normalized then 141 d,lCl d,l= moJllo diag( s (&,)-d(u/R,). (45) a,/& In Fig. 4, the results of the numerical integration are shown for various values of loss coefficient and for mild steel with the following characteristics: S, = 4.3 x lONlm* m = 2.51 B = 2.832 x 10-26m(2 +“)*/cycle Nm Kth = 3.788 x 106N/m3*. .l . 5 1 5 10 50 (iT,J, / IO3 Fig. 3. Variation of dimensionless stress intensity factor vs crack depth. Loss coefficient is the parameter. Crack propagation in resonating structures 121 a * 6 t : e * 3 . 2 II - 2 0 I 0 i : I 4 , I .4. * ,I, /I I / / 1 1 i&f: I / / R0 / / / 0 5 /- / : 4 0 0 / 0 r/ 0 0 : ,/ : 0 / / Arrest line . 1- 1 5 10 50 100 500 N/IO Fig. 4. Number of fatigue cycles vs crack depth for mild steel. Loss coefficient is the parameter. The arrest line shown is the locus of all points where transition to low rate crack propagation takes place on every curve. 4. CONCLUSIONS The Paris and Erdogan model for the fatigue crack propagation is used to study the behaviour of a cracked resonant system. A general relation is established relating crack depth with number of fatigue cycles applicable generally. Crack propagation at or near resonance appears very often in machines and structures and is one of the usual causes of machine failure. In situations where the introduction of the crack can alter the dynamic characteristics of the system, such dynamic analysis is used to determine the mission profile for the loading of the cracked section, for a certain constant external excitation. This information is used as feedback to the fatigue crack propagation model. Initially, crack propagates at or very closely to resonance with very high rates. Since the crack depth changes, there is a considerable reduction of the system natural frequency which causes a “shift away” from resonance and, as a consequence, a drop of the crack propagation rate. This situation, under certain circumstances, which strongly depend on the damping factor, can lead to dynamic crack arrest shown in Fig. 4. Here, it seems that crack arrest almost always occurs. This is true up to a certain point since if the crack propagates to a certain extent, which can happen before the arrest point, the structural member might fail by unstable crack growth or some other mechanism which could destroy the structural integrity of the system at hand. Finally, it can be easily concluded that the magnitude of the excitation and the geometry of the system can only alter the results quantitatively. Crack arrest or not depends primarily on the damping of the corresponding node. REFERENCES I P. C. Paris, The fracture mechanics approach to fatigue-an interdisciplinary approach. Proc. 10th Materials Research Conf., Syracuse University Press, New York (1964). 2 P. C. Parts and F. A. Erdogan, Critical analysis of crack propagation laws. J. Bas. Engng 85, 528 (1963). 3 F. A. Erdogan, Crack propagation theories, in Fracture (Edited by H. Liebowitx), Vol. 2. Academic Press, New York (1968). S G. C. Sih, Some basic problems in fracture mechanics and new concepts. Engng Fracture Med. 5, 365-377 (1973). S G. C. Sih, Strain energy density factor applied to engineering problems. Inr. J. Fracture 10, 305-321 (1974). 6 G. C. Sih and B. M. Barthelemy, Mixed mode fatigue crack growth predictions. Engng Fracture Mech. 13.439-451 (1980). A T. G. Chondros and A. D. Dimarogonas, Identification of cracks in circular plates welded at the contour. 1979 ASME Design Engineering Technical Conf. St. Louis, U.S.A., Paper No. 79-DET-106 (September 1979). 728 A. J. DENTSORAS and A. D. DIMAROGONAS 8 T. G. Chondros and A. D. Dimarogonas, Identification of cracks in welded joints of complex structures. J. Sound Vibration 69, 531 (1980). 9 A. D. Dimarogonas, Vibration Engineering. West, St. Paul (1976). IO A. D. Dimarogonas and G. Masouros, Torsional vibration of a shaft with a circumferential crack. Engng Fracture Mech. 15, 439444 (1981). I I A. J. Dentsoras and A. D. Dimarogonas, Resonance controlled fatigue crack propagation. Engng Fracrure Mech. 17, (1983). l2 A. J. Dentsoras and A. D. Dimarogonas, Resonance controlled fatigue crack propagation in a beam under longitudinal vibration. Inr. J. Fracture 23, 15-22 (1983). 13 A. J. Dentsoras and A. D. Dimarogonas, Resonance controlled fatigue crack propagation in cylindrical beams under combined loading. ASME Winter Annual Meeting, Boston (1983). 14 S. Timoshenko, Vibration Problems in Engineering. Wiley, New York (1984). 15 A. D. Dimarogonas and S. Paipetis, Rotor Dynamics. Applied Science, London (1983). 16 G. C. Sih and B. McDonald, Fracture mechanics applied to engineering problems, strain energy density fracture criterion. Engng Fracrure Mech. 6, 493-507 (1974). 17 A. P. Parker, The Mechanics of Fracture and Fatigue-An Introduction. E.F.N. Spon. 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