Chapter Five Fracture criterion for mixed mode crackIn 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 criterionMaximum stress criterion can be applied to the mixed mode crack of mode I and mode II.The asymptotic stress solution is)23cos 2cos 2(2sin 2)23sin 2sin 1(2cos 2θθ+θπ-θθ-θπ=σrK r K II I xx 23cos 2cos 2sin 2)23sin 2sin 1(2cos 2θθθπ+θθ+θπ=σr K r K II Iyy )23sin 2sin 1(2cos 223cos 2sin 2cos 2θθ-θπ+θθθπ=σrK r K II I xy By application of the coordinate transformation formulas, we can obtain the expressions of three stress components in the polar coordinates (r , θ). The circumferential normal stress is]sin 3)cos 1([2cos 2121θ-θ+θπ=σθθII I K K r The circumferential normal stress intensity factor θθK is defined as]sin 3)cos 1([2cos 212lim 0θ-θ+θ=σπ=θϑ→θθII I r K K r KHence, θθσ can be written asr K π=σθθθθ2Assumptions:(1) Crack initiation direction is the direction of the maximum θθK ;(2) When θθK reaches its critical value c K θθ, break occurs. c K θθ is a material constant.The crack initiation angle 0θ can be determined from0=θ∂∂θθK , 022<θ∂∂θθK The result is0)1cos 3(sin 00=-θ+θII I K K0)5cos 9(2sin )cos 31(2cos 0000<+θθ+θ-θII I K K The critical condition is c II I K K K K θθθ=θ-θθ=)sin 232cos (2cos0020maxDetermination of c K θθ:For mode I crack, 0≠I K , 0=II K , 00=θ, the critical condition reduces to c Ic K K K θθθ==maxNote that c K θθ is a material constant. When, 0≠I K and 0≠II K , there still prevails Ic c K K =θθ. The maximum stress criterion is expressed asIc II I K K K ≤θ-θθ)sin 232cos (2cos 0020Application to mode II crack:For a mode II crack, 0=I K and 0≠II K . The crack initiation angle can be solved,o 5.700-=θ. FromIc c II I K K K K K ==θ-θθ=θθθ)sin 232cos (2cos 0020max one can obtain thatIc IIc K K =149.1, Ic IIc K K 87.0=,The fracture criterion for Mode II crack can be derived from the maximum stress criterion thatIIc II K K ≤It is convenient for the engineering application. However, there is no difference between the plane stress and plane strain.§5-2 Maximum normal strain criterionNear the crack tip, the circumferential normal strain is]}2sin )1cos 3(2cos sin 3[2cos )]cos 3()cos 1[({2121)(111111θ-θν+θθ-θθ-ν-θ+π=σν-σ=εθθθθII I rr K K E r E E E =1, νν=1, for plane stress; 211ν-=E E , ννν-=11, for plane strain. The circumferential normal strain intensity factor *θθK is defined as]}2sin )1cos 3(2cos sin 3[2cos )]cos 3()cos 1[({212lim 1110*θ-θν+θθ-θθ-ν-θ+=επ=θθ→θθII I r K K E r K Then,r K π=εθθθθ2*Assumptions:(1) Crack initiation direction is the direction of the maximum *θθK ;(2) When *θθK reaches its critical value *c K θθ, break occurs. *c K θθ is a materialconstant.The cracking angle 0θ satisfies0*=θ∂∂θθK , 02*2<θ∂∂θθK The critical value *c K θθ can be determined by Ic K . For Mode I, 0=II K , 00=θ. It can be obtained thatIc c K E K 11*1ν-=θθ The maximum normal strain criterion isIc II I K K K ≤θ-θν+θθ-θθ-ν-θ+ν-}2sin )]1cos 3(2cos sin 3[2cos )]cos 3()cos 1[({)1(210010000101Now the plane stress and plane strain can be distinguished.§5-3 Strain energy density factor theoryStrain energy density factor theory was proposed by Prof. G . C. Sih that can be applied to the three dimensional problem.When, 0≠I K , 0≠II K , 0≠III K , the asymptotic stress solution is)23cos 2cos 2(2sin 2)23sin 2sin 1(2cos 2θθ+θπ-θθ-θπ=σrK r K II I xx 23cos 2cos 2sin 2)23sin 2sin 1(2cos 2θθθπ+θθ+θπ=σr K r K II Iyy )23sin 2sin 1(2cos 223cos 2sin 2cos 2θθ-θπ+θθθπ=σrK r K II I xy 2sin 222cos 22θπν-θπν=σr K r K II I zz 2cos 2θπ=σrK III yz , 2sin 2θπ-=σr K III zx The strain energy density w is)(21)()(21222222zx yz xy xx zz zz yy yy xx zz yy xx E E w σ+σ+σμ+σσ+σσ+σσν-σ+σ+σ= The strain energy density w can be expressed in the form ofrS w = where233222122112IIIII II I I K a K a K K a K a S +++=, strain energy density factor )cos )(cos 1(16111θ-κθ+πμ=a )1cos 2(sin 16112+κ-θθπμ=a )]1cos 3)(cos 1()cos 1)(1[(16122-θθ++θ-+κπμ=a πμ=4133a Assumptions: it is physics, not mathematics.(1) Crack initiation direction is the direction of the minimum S ;(2) When S reaches its critical value c S , break occurs. c S is a material constant.The cracking angle 0θ can be solved from0=θ∂∂S , 022>θ∂∂S The critical condition isc S S S =θ=)(0minDetermination of S c :For mode I, it can be derived that2421Ic c K S πμν-= The minimum strain energy density factor criterion can be expressed asS ≤S c , i.e.,223322212211]2[214Ic III II II I I K K a K a K K a K a ≤+++ν-πμ.Mode II crack: 0==III I K K , )321arccos(0ν-=θ, Ic IIc K K 2)1(2)21(3ν-ν-ν-= Take 31=ν. There is 7383o 0'-=θ, Ic IIc K K 9.0=Recall that for the maximum normal stress criterion, there iso 05.70-=θ, Ic IIc K K 87.0=Two results have little difference.§5-4 Modified maximum normal stress criterionSometime 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 ofrS w = a strain energy density factor S is defined. For the mixed mode of mode I and II, S canbe written as222122112IIII I I K a K K a K a S ++= Let constant ==C w .)(]2[122212211θ=++===F K a K K a K a CC S w S r II II I I For different values of C , we can obtain a group of curves called as isolines of strain energy density.The circumferential normal stress is]sin 3)cos 1([2cos 2121θ-θ+θπ=σθθII I K K r Let )cos 1(2cos 21)(θ+θ=θI f , θθ-=θsin 2cos 23)(II f . )]()([21θ+θπ=σθθII II I I f K f K rLet )]()([21)(θ+θπ=θII II I I f K f K S f . This gives )(θ=σθθf rS On the isolines of the strain energy density, C S r =, the circumferential normal stress is)(θ=σθθf CThe circumferential normal stress intensity factor θθK is identical with §5-1. )()(2lim 0θ+θ=σπ=θθ→θθII II I I r f K f K r K Assumptions:(1) Crack initiation direction is the direction of the maximum θθK on the isoline of the strain energy density. The crack initiation angle 0θ can be determined from0=θ∂∂θθK , 022<θ∂∂θθK(2) When θθK reaches its critical value c K θθ, break occurs.c II II I I K f K f K K θθθθ=θ+θ=)()(00maxIt can be derived from Mode I problem thatIc c K K =θθThe fracture criterion isIc II II I I K f K f K ≤θ+θ)()(00§5-5 Energy release rate theoryNear the crack tip, the stresses in the polar coordinates are]sin 3)cos 1([2cos 2121θ-θ+θπ=σθθII I K K r )]1cos 3(sin [2cos 2121-θ+θθπ=σθII I r K K r Let]sin 3)cos 1([2cos 21θ-θ+θ=θII I I K K K )]1cos 3(sin [2cos 21-θ+θθ=θII I II K K K There resultsr K I π=σθθθ2, r K II r π=σθθ2Energy release rate θG along the angle θ:G denotes the energy release rate along the direction θ=0. Now we need to know the energy release rate θG along the direction θ.It is known that002lim =θ→σπ=yy r I r K , 002lim =θ→σπ=yx r II r K , )(8122II I K K G ++=μκRecall the definitions of θI K and θII K . It is known thatθθ→θσπ=r K r I 2lim 0, θ→θσπ=r r II r K 2lim 0Comparing two cases, we know that θI K and θII K 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 θG along angle θ for the real crack is equal to the energy release rate G along its own direction for the virtual crack. Hence, we have )(8122θθθ+μ+κ=II I K K GAssumptions:(1) Crack initiation direction is the direction of the maximum θG . The crack initiation angle 0θ can be determined from0=θ∂∂θG , 022<θ∂∂θG (2) When θG reaches its critical value c G θ, the break occurs.In a same way, it is obtained that281Ic c K G μ+κ=θ The cracking angle 0θ satisfies the equation0)cos 31(sin )cos cos (sin 2)cos 1(sin 00200202002=θ-θ+θ-θ-θ-θ+θII II I I K K K K The fracture criterion is2020020)]cos 35(sin 4)cos 1()[cos 1(41Ic II II I I K K K K K ≤θ-+θ-θ+θ+§5-6 Fatigue crack propagation problemFatigue process:(1) Fatigue crack initiation period: empirical formula (Miner ’s liner damage accumulation theory) or damage mechanics;(2) Fatigue crack propagation period: fracture mechanics.max σ, maximum stress; min σ, minimum stress; )(21min max σ+σ=σm , mean stress; min max σ-σ=σ∆, stress amplitude; maxmin σσ=R , cyclic stress ratio. In a fatigue process, the stress intensity factor )(t K I also varies with time t. a K K K I I I πσ∆=-=∆min maxThe fatigue crack propagation rate dN da / depends on the amplitude of SIF. )(I K f dNda ∆=Experimental result:Fracture and Damage Mechanics Chapter Five Fracture criterion for mixed mode crack 77Region I: small crack, microscopic effect is important.Region II: crack stable propagation.Region III: crack instable propagation to failure.Paris equation: 1960s, Lehigh University, USA For the region II, the relation can be given byn I K C dNda )(∆= )log(log )log(I K n C dNda ∆+=, straight line Parameters C and n can be determined by the experimental data, which depend on the stress ratio R , material property, temperature and so on.The fatigue crack growth life can be calculated by using the Paris equation. There are many improvements for Paris equation.第五章完。
第六章 二维脆性断裂§6.1 引 言破裂判据是断裂力学的核心问题, 这需要从微观、亚微观、宏观三个层次进行研究。
所谓微观就是涉及物体的终极结构单元发生相对运动时其间内聚力的破坏。
亚微观涉及颗粒及粒间界面这一水平上的破坏。
宏观涉及肉眼可以看得见的破坏。
破裂判据是针对某一特定尺度、特定层次提出的, 做为一个完整的破裂判据，至少应该能够回答两个问题: ① 破裂在什么可测条件下起始或继续？ ② 破裂向什么方向扩展? 岩石微观、亚微观破裂机制与宏观不同, 因而破裂判据也不同。
实际上, 迄今为止并不存在一种万能判据, 能够同时包括这三个层次。
为有所区分, 本文仍沿袭惯例, 对于微观、亚微观的 Griffth 裂纹, 按照Ⅰ型、Ⅱ型、Ⅲ型命名, 对于宏观断层模型化的裂纹,按照张破裂、平面内剪切裂纹、反平面剪切裂纹命名。
一些共用名词, 例如内聚力, 内聚区等, 在宏观中的含义也与微观不同。
对于岩石中的断裂机理的研究, 最早可以追溯到Griffith(1921)提出的脆性破坏理论, 该理论认为, 当裂纹端部扩展一小段长度时, 弹性势能的释放率如果大于或等于表面能的增加率时, 裂纹才能持续扩展。
在这之后, 发展了两种受压闭合裂纹模式, 即扁椭圆裂纹模式和Griffith 裂纹模式(Jaeger 和Cook, 1979)。
扁椭圆裂纹模式在第四章中已经介绍。
这里讨论的是Griffith 裂纹模式(也叫做数学裂缝), 是在Irwin(1957)引入应力强度因子的概念之后发展起来的, 它以断裂韧性作为材料抗脆断能力的指标, 也叫做K 判据。
断裂力学的其它模式和判据都是在这个模式的基础上加以修正或发展起来的, 也是断裂动力学的基础模式。
K 判据不能回答破裂方向问题, 特别是复合型裂纹问题, 因此产生了一系列脆性断裂理论。
线弹性断裂力学中关于脆性断裂的理论可分为两类：一类是应力场参数法，以应力场的某一特征量为参数。
脆性材料复合型裂纹断裂准则杨军;李强【摘要】基于假设:裂纹沿最小应变能密度S的方向扩展,并且当其达到临界值开裂(SED准则),文章提出一个新的断裂准则(MSED准则),适用于混凝土和岩石等脆性材料的Ⅰ-Ⅱ混合型断裂.该准则与试件加载情况及几何形状有关,它的关键值不仅包含Ⅰ型断裂韧度KIc,且其中亦考虑了Ⅱ型断裂韧度KIIc.为了验证MSED准则的有效性及预测精度,利用文献中的混凝土断裂实验结果进行比较.相较于其他传统的准则,文献的实验结果与本文提出的准则吻合得更好,能更加精确地预测裂纹的起裂及扩展.【期刊名称】《四川建筑》【年(卷),期】2018(038)005【总页数】3页(P224-226)【关键词】混凝土;混合型裂纹;断裂准则【作者】杨军;李强【作者单位】四川省地质工程勘察院,四川成都 610072;四川省地质工程勘察院,四川成都 610072【正文语种】中文【中图分类】O346.1为了研究裂纹的起裂和扩展,许多学者从应力、应变能密度、能量等多种角度分析建立了相应的断裂准则。
Erdogan 和 Sih[1]首次提出了适用于复合型裂纹的最大周向应力准则(即MTS准则),该准则假定裂纹沿最大周向应力σmax扩展。
Sih[2]提出最小应变能密度准则(SED 准则)，该准则的基本假定为：当材料的应变能密度S达到临界应变能密度Scr裂纹开始扩展。
Theocaris 和 Andrianopoulos[3]基于von Mises屈服准则，认为裂纹沿着最大弹性应变能密度方向开裂，提出T准则，该准则对于金属材料非常适用。
Ukadgaonkera和Awasare[4]提出修正的T准则，即通过应力张量第一不变量(I1)和应力偏量第二不变量(J2)预测裂纹开裂。
Yehia等[5]提出最大体积应变能密度准则，即 NT 准则，此外，Yehia[6]讨论了在塑性核心区的基础上定义断裂准则的可行性，并且克服了修正的T 准则的一致性问题，提出了Y 准则。
