复合型裂纹的断裂准则.doc

Fracture and Damage Mechanics Chapter Five Fracture criterion for mixed mode crackChapter Five Fracture criterion for mixed mode crack In the material mechanics, for the multiaxial stress state, four strength theories have been developed. In the fracture mechanics for the mixed mode crack problem, we need to develop the fracture theory accordingly. Many fracture theories have been developed. Two key questions must be answered.(1) What direction does a crack propagate along?(2) What is the critical case? In what follows, five theories will be introduced.5-1 Maximum normal stress criterion Maximum stress criterion can be applied to the mixed mode crack of mode I and mode II. The asymptotic stress solution is By application of the coordinate transformation formulas, we can obtain the expressions of three stress components in the polar coordinates (r, q). The circumferential normal stress is The circumferential normal stress intensity factor is defined asHence, can be written asAssumptions:(1) Crack initiation direction is the direction of the maximum ;(2) When reaches its critical value , break occurs. is a material constant. The crack initiation angle can be determined from, The result isThe critical condition isDetermination of : For mode I crack, , , , the critical condition reduces to Note that is a material constant. When, and , there still prevails . The maximum stress criterion is expressed asApplication to mode II crack: For a mode II crack, and . The crack initiation angle can be solved, . Fromone can obtain that, , The fracture criterion for Mode II crack can be derived from the maximum stress criterion that It is convenient for the engineering application. However, there is no difference between the plane stress and plane strain.5-2 Maximum normal strain criterion Near the crack tip, the circumferential normal strain is, , for plane stress; , , for plane strain. The circumferential normal strain intensity factor is defined asThen,Assumptions:(1) Crack initiation direction is the direction of the maximum ;(2) When reaches its critical value , break occurs. is a material constant. The cracking angle satisfies, The critical value can be determined by . For Mode I, , . It can be obtained that The maximum normal strain criterion is Now the plane stress and plane strain can be distinguished. 5-3 Strain energy density factor theory Strain energy density factor theory was proposed by Prof. G. C. Sih that can be applied to the three dimensional problem. When, , , , the asymptotic stress solution is, The strain energy density w is The strain energy density w can be expressed in the form ofwhere, strain energy density factorAssumptions: it is physics, not mathematics.(1) Crack initiation direction is the direction of the minimum S;(2) When reaches its critical value , break occurs. is a material constant. The cracking angle can be solved from , The critical condition isDetermination of Sc: For mode I, it can be derived that The minimum strain energy density factor criterion can be expressed asSSc, i.e., .Mode II crack: , , Take . There is, Recall that for the maximum normal stress criterion, there is, Two results have little difference. 5-4 Modified maximum normal stress criterion Sometime the maximum normal stress criterion is not so good. A modified maximum normal stress criterion has been proposed. It has been known that in view ofa strain energy density factor S is defined. For the mixed mode of mode I and II, S can be written as Let .For different values of C, we can obtain a group of curves called as isolines of strain energy density. The circumferential normal stress isLet , .Let . This gives On the isolines of the strain energy density, , the circumferential normal stress is The circumferential normal stress intensity factor is identical with 5-1.Assumptions: (1) Crack initiation direction is the direction of the maximum on the isoline of the strain energy density. The crack initiation angle can be determined from, (2) When reaches its critical value , break occurs. It can be derived from Mode I problem that The fracture criterion is 5-5 Energy release rate theory Near the crack tip, the stresses in the polar coordinates are LetThere results, Energy release rate along the angle q: G denotes the energy release rate along the direction q=0. Now we need to know the energy release rate along the direction q. It is known that, , Recall the definitions of and . It is known that, Comparing two cases, we know that and are the stress intensity factors of the virtual crack. The stress fields for two cracks are completely same. The conclusion is that the energy release rate along angle for the real crack is equal to the energy release rate G along its own direction for the virtual crack. Hence, we haveAssumptions: (1) Crack initiation direction is the direction of the maximum . The crack initiation angle can be determined from, (2) When reaches its critical value , the break occurs. In a same way, it is obtained that The cracking angle satisfies the equationThe fracture criterion is5-6 Fatigue crack propagation problemFatigue process:(1) Fatigue crack initiation period: empirical formula (Miners liner damage accumulation theory) or damage mechanics;(2) Fatigue crack propagation period: fracture mechanics., maximum stress; , minimum stress; , mean stress;, stress amplitude; , cyclic stress ratio. In a fatigue process, the stress intensity factor also varies with time t.