Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime and scaling

Engineering structures must be designed for an extremely low failure probability such as 10⁻⁶, which is beyond the means of direct verification by histogram testing. This is not a problem for brittle or ductile materials because the type of probability distribution of structural strength is fixed and known, making it possible to predict the tail probabilities from the mean and variance. It is a problem, though, for quasibrittle materials for which the type of strength distribution transitions from Gaussian to Weibullian as the structure size increases. These are heterogeneous materials with brittle constituents, characterized by material inhomogeneities that are not negligible compared to the structure size. Examples include concrete, fiber composites, coarse-grained or toughened ceramics, rocks, sea ice, rigid foams and bone, as well as many materials used in nano- and microscale devices.This study presents a unified theory of strength and lifetime for such materials, based on activation energy controlled random jumps of the nano-crack front, and on the nano-macro multiscale transition of tail probabilities. Part I of this study deals with the case of monotonic and sustained (or creep) loading, and Part II with fatigue (or cyclic) loading. On the scale of the representative volume element of material, the probability distribution of strength has a Gaussian core onto which a remote Weibull tail is grafted at failure probability of the order of 10⁻³. With increasing structure size, the Weibull tail penetrates into the Gaussian core. The probability distribution of static (creep) lifetime is related to the strength distribution by the power law for the static crack growth rate, for which a physical justification is given. The present theory yields a simple relation between the exponent of this law and the Weibull moduli for strength and lifetime. The benefit is that the lifetime distribution can be predicted from short-time tests of the mean size effect on strength and tests of the power law for the crack growth rate. The theory is shown to match closely numerous test data on strength and static lifetime of ceramics and concrete, and explains why their histograms deviate systematically from the straight line in Weibull scale.Although the present unified theory is built on several previous advances, new contributions are here made to address: (i) a crack in a disordered nano-structure (such as that of hydrated Portland cement), (ii) tail probability of a fiber bundle (or parallel coupling) model with softening elements, (iii) convergence of this model to the Gaussian distribution, (iv) the stress-life curve under constant load, and (v) a detailed random walk analysis of crack front jumps in an atomic lattice. The nonlocal behavior is captured in the present theory through the finiteness of the number of links in the weakest-link model, which explains why the mean size effect coincides with that of the previously formulated nonlocal Weibull theory. Brittle structures correspond to the large-size limit of the present theory. An important practical conclusion is that the safety factors for strength and tolerable minimum lifetime for large quasibrittle structures (e.g., concrete structures and composite airframes or ship hulls, as well as various micro-devices) should be calculated as a function of structure size and geometry.     

海洋平台K型管节点的疲劳裂纹扩展分析Ⅰ:试验测试

对承受疲劳载荷的海洋平台K型管节点首先进行了静力测试,确定了沿着焊缝的热应力区的应力分布及热应力区最大应力点的位置,从而判断裂纹产生的位置;然后通过专用测试设备提供循环疲劳载荷,用ACPD(Alternating Current Potential Drop,即交流电流势能落差法)技术检测裂纹的产生和增长过程,得到裂纹最深点,用S-N曲线估算其疲劳寿命。对已有裂纹的K型管节点,用应力强度因子估计其剩余寿命。同时用测试的结果验证了S-N的准确性和可靠性。     

自紧身管临界裂纹尺寸的概率断裂力学研究

基于概率断裂力学理论和Mome Caflo模拟方法，本文进行了自紧身管临界裂纹尺寸的可靠性研究。自紧身管内表面的疲劳裂纹考虑为半椭圆形式。裂纹尖端处的应力强度因子由内压和自紧残余应力共同产生。自紧残余应力采用了符合身管材料具有强化和包辛格效应性能推导的公式计算，它产生的应力强度因子通过权函数方法得到。根据断裂准则，可计算出自紧身管的临界裂纹尺寸。实例分析表明，对数正态分布为临界裂纹尺寸的最佳分布，同时给出了在各种置信度和可靠度下自紧身管的临界裂纹尺寸。     

Fatigue failure of polyethylene methacrylate in adhesive assembly under unsymmetrical bending

In this paper, a special bending fatigue experiment was firstly performed to investigate the fatigue behavior of polyethylene methacrylate in adhesive assembly. Fatigue lifetime property (S–N curve) was obtained. Finite element calculations on the whole structures also gave the same results with the testing. Based on the experimental data and finite element analysis, a local stress law of predicting bending fatigue lifetime was put forward. The predication lifetime for the polyethylene methacrylate agreed well with the experimental results. Following the strain energy density (SED) criterion was applied to predict the crack initiation and growth path of the adhesive assembly. The predicted results were in good agreement with the optical microscopy (OM) failure image of the failure specimen. SEM image of fracture further showed that there were lots of parallel fatigue lines with perpendicularity to the direction of crack, and an obvious boundary from the crack propagation failure to final brittle fracture.     

机轮轮毂模拟疲劳寿命分析

本文利用模拟试件对机轮轮毂危险细节的制造质量和几何条件进行模拟,并在使危险细节局部应力梯度尽可能与构件一致的条件下确定模拟件加载方式;由模拟件疲劳试验结果和极值分布理论建立了描述构件初始疲劳质量的TTCI 分布,并由此直接导出了构件在任一时刻的裂纹尺寸分布和裂纹超过数概率,为进一步进行耐久性经济寿命分析提供了基础.最后,用实物验证试验的结果说明了上述方法的合理性.     

SAPH440汽车薄板高低周疲劳特性试验研究

本文通过设计防屈曲装置,对2.3mm厚度汽车薄板SAPH440在室温下进行了单轴拉伸、低周疲劳和高周疲劳等一系列试验．给出该材料静态力学参数,低周疲劳曲线（E-N曲线）、高周疲劳曲线（S-N曲线）以及相关疲劳性能参数．分别得到了在应力比R=0.1和R= -1状态下的疲劳寿命公式．通过考察低周循环应力-应变曲线,发现材料在试验控制的循环应变幅范围内发生了循环软化,具有明显Masing特性．此外,通过对疲劳断口形貌分析,初步给出了疲劳裂纹的扩展机理．     

TC4钛合金超高周次轴向振动疲劳试验

应用基于压电超声疲劳试验技术开发的20kHz轴向振动疲劳试验系统,完成了室温下TC4钛合金超高周疲劳试验,获得了TC4合金在107～109周次范围内的轴向振动疲劳寿命曲线(S-N曲线);运用C.Paris推导公式预测了TC4合金材料的寿命,得到各应力水平下破坏率为50％、95％、99％的安全寿命.结果表明:在疲劳循环大于107周次时,试件仍会发生疲劳断裂,疲劳强度随循环次数的增加而下降,并不存在明显的疲劳极限.TC4合金的S-N曲线在107～109周次的范围内呈连续下降型.在轴向振动超高周疲劳试验中,试件的裂纹扩展寿命只占其在50％破坏率下疲劳安全寿命的一小部分,其疲劳寿命主要由试件的裂纹萌生寿命决定.     

Statistical distribution and size effect of residual strength of quasibrittle materials after a period of constant load

In preceding studies, the type of cumulative probability distribution functions (cdf) of strength and of static lifetime of quasibrittle structures, including their tails, was mathematically derived from atomistic scale arguments based on nano-scale cracks propagating by many small, activation energy-controlled, random breaks of atomic bonds in the nanostructure. It was shown that a quasibrittle structure (of positive geometry) must be modeled by a finite (rather than infinite) weakest-link model, and that the cdf of structural strength as well as lifetime varies from nearly Gaussian to Weibullian as a function of structure size and shape. Excellent agreement with the observed distributions of structural strength and static lifetime was demonstrated. Based on the same theoretical framework, the present paper formulates the statistics of the residual structural strength, which is the strength after the structure has been subjected to sustained loading. A strength degradation equation is derived based on Evans' law for static crack growth during sustained loading. It is shown that the rate of strength degradation is not constant but continuously increasing. The cdf of residual strength of one RVE is shown to be closely approximated by a graft of Weibull and Gaussian (normal) distributions. In the left tail, the cdf is a three-parameter Weibull distribution consisting of the (n+1)th power of the residual strength, where n is the exponent of the Evans law and the threshold is a function of the applied load and load duration. The finiteness of the threshold, which is typically very small, is a new feature of quasibrittle residual strength statistics, contrasting with the previously established absence of a threshold for strength and lifetime. Its cause is that there is a non-zero probability that some specimens fail during the static preloading, and thus are excluded from the statistics of the overload. The predictions of the theory are validated by available test data on glass–epoxy composites and on borosilicate and soda-lime silicate glasses. The size effect on the cdf of residual strength is also determined. The size effect on the mean residual strength is found to be as strong as the size effect on the mean initial strength.     

常幅载荷下结构元件断裂可靠度估算的应力强度因子模型

给出了一个估算结构元件疲劳可靠度的应力强度因子模型,系统阐述了元件在常幅载荷下疲劳可靠性的分析方法。该模型研究了常幅载荷作用下材料瞬时裂纹长度和应力强度因子的分布形式,建立了应力强度因子与断裂韧性之间的干涉关系。对7075-T7351铝合金中心裂纹试件试验数据分析的结果表明：裂纹的瞬时扩展长度和可靠度的预测结果均与试验结果符合很好,本文给出的基于应力强度因子的可靠性分析模型是合理的。     

谐振载荷作用下工程结构振动疲劳寿命预估的损伤力学-有限元法

建立了预估谐振载荷作用下结构振动疲劳寿命的损伤力学-有限元方法。首先根据损伤热力学原理,构建了损伤演化方程,建立光滑试件在恒幅应变交变载荷作用下寿命预估方法;进一步由损伤力学守恒积分原理,得到恒幅重复载荷作用下应力与寿命的关系式;然后根据标准件疲劳试验结果,拟合得到损伤演化方程中的材质参数;最后利用APDL语言编程对ANSYS软件进行了二次开发,借助ANSYS软件对谐振载荷作用下结构振动疲劳裂纹萌生寿命进行预估。作为算例,本文利用该方法预估了LC9CgS1铝合金梁谐振载荷作用下疲劳裂纹萌生寿命。     

随机超载对疲劳裂纹扩展迟滞效应的模拟

考虑超载的迟滞效应,对随机超载作用下的疲劳裂纹扩展进行了模拟计算．载荷谱为在基本恒幅循环载荷基础上加入一以泊松流发生的随机超载序列,超载的大小为均匀分布．采用裂纹闭合模型考虑超载的迟滞效应,认为裂纹张开应力在超载引起的塑性区内按线性规律衰减．循环续循环模拟计算出裂纹从初始长度一直到疲劳破坏的扩展曲线．据此,计算了各种超载发生强度和大小下的疲劳裂纹扩展寿命的平均值与标准差。     

TC21钛合金显微组织对超长寿命疲劳行为的影响

本文主要研究钛合金的网篮尺寸对于超声疲劳行为的影响。研究发现,对于TC21钛合金,高应力下疲劳裂纹萌生于试样表面;低应力下疲劳裂纹萌生于试样亚表面。最大应力-循环次数(S-N)曲线显示,在两段连续下降的曲线之间有一个表面萌生向内部萌生的转变平台。对于网篮尺寸为60μm的钛合金,转变平台对应的应力幅为540MPa,疲劳极限为430MPa。网篮尺寸为40μm的转变平台应力幅为600MPa,疲劳极限为530MPa。分析得知网篮尺寸为60μm的TC21钛合金疲劳性能低于网篮尺寸为40μm的TC21钛合金。     

Lifetime distribution of notched components containing void defects

The lifetime distribution of a component containing notches of various shapes is determined both in a fatigue test and using numerical simulation. The simulation model is based on the ideas of probabilistic fracture mechanics with an underlying simple fracture mechanics model for void defects. A postprocessing tool is used to determine the failure probability given the stress field in the component.     

Two-parameter failure criterion for elastoplastic bodies with mode I cracks

The generalized Dugdale crack model is used to formulate two-parameter failure criteria for the cases of quasibrittle state and developed plastic zones at a mode I crack tip. The failure criteria relate the fracture strength characteristics and the stress mode at the crack tip through the plastic constraint factor. The critical state of bodies with cracks under uni-and biaxial loading is analyzed in the cases of plane stress and plane strain using the Tresca and von Mises yield criteria. A small-scale yield criterion, which is an analytic relation between the critical stress intensity factor and T-stresses, is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 47–57, July 2007.     

基于随机点过程理论的疲劳裂纹随机扩展模型

本文根据疲劳裂纹扩展特性，运用随机点过程理论导出了求取裂纹扩展随机过程低阶矩的公式。同时根据二阶矩近似法用Ｗｅｉｂｕｌｌ分布近似给定寿命下的疲劳纹长度分布。给出了求取给定寿命下裂纹超值概率分布和给定裂纹长度下疲劳寿命分布公式。最后，用小样本孔边短裂纹扩展数据对模型适用性进行了初步验证。     

Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture

Because the uncertainty in current empirical safety factors for structural strength is far larger than the relative errors of structural analysis, improvements in statistics offer great promise. One improvement, proposed here, is that, for quasibrittle structures of positive geometry, the understrength factors for structural safety cannot be constant but must be increased with structures size. The statistics of safety factors has so far been generally regarded as independent of mechanics, but further progress requires the cumulative distribution function (cdf) to be derived from the mechanics and physics of failure. To predict failure loads of extremely low probability (such as 10^-6 to 10^-7) on which structural design must be based, the cdf of strength of quasibrittle structures of positive geometry is modelled as a chain (or series coupling) of representative volume elements (RVE), each of which is statistically represented by a hierarchical model consisting of bundles (or parallel couplings) of only two long sub-chains, each of them consisting of sub-bundles of two or three long sub-sub-chains of sub-sub-bundles, etc., until the nano-scale of atomic lattice is reached. Based on Maxwell-Boltzmann distribution of thermal energies of atoms, the cdf of strength of a nano-scale connection is deduced from the stress dependence of the interatomic activation energy barriers, and is expressed as a function of absolute temperature T and stress-duration τ (or loading rate 1/τ). A salient property of this cdf is a power-law tail of exponent 1. It is shown how the exponent and the length of the power-law tail of cdf of strength is changed by series couplings in chains and by parallel couplings in bundles consisting of elements with either elastic-brittle or elastic-plastic behaviors, bracketing the softening behavior which is more realistic, albeit more difficult to analyze. The power-law tail exponent, which is 1 on the atomistic scale, is raised by the hierarchical statistical model to an exponent of m=10 to 50, representing the Weibull modulus on the structural scale. Its physical meaning is the minimum number of cuts needed to separate the hierarchical model into two separate parts, which should be equal to the number of dominant cracks needed to break the RVE. Thus, the model indicates the Weibull modulus to be governed by the packing of inhomogeneities within an RVE. On the RVE scale, the model yields a broad core of Gaussian cdf (i.e., error function), onto which a short power-law tail of exponent m is grafted at the failure probability of about 0.0001-0.01. The model predicts how the grafting point moves to higher failure probabilities as structure size increases, and also how the grafted cdf depends on T and τ. The model provides a physical proof that, on a large enough scale (equivalent to at least 500 RVEs), quasibrittle structures must follow Weibull distribution with a zero threshold. The experimental histograms with kinks, which have so far been believed to require the use of a finite threshold, are shown to be fitted much better by the present chain-of-RVEs model. For not too small structures, the model is shown to be essentially a discrete equivalent of the previously developed nonlocal Weibull theory, and to match the Type 1 size effect law previously obtained from this theory by asymptotic matching. The mean stochastic response must agree with the cohesive crack model, crack band model and nonlocal damage models. The chain-of-RVEs model can be verified and calibrated from the mean size effect curve, as well as from the kink locations on experimental strength histograms for sufficiently different specimen sizes.     

高强度钢的超高周疲劳裂纹扩展模型研究

当疲劳寿命在106或107周时,Wöhler S-N 曲线被看作渐近于水平轴,107的疲劳强度被看成是疲劳极限。现代应用要求延长零件的工作寿命,实际齿轮部件应用超过107循环的疲劳失效。本文应用压电超声疲劳试验机对经过热处理和渗碳处理后的低铬合金钢材料进行研究,采用红外摄像仪观测试件表面的温度场随疲劳裂纹萌生和扩展的过程。试验条件是室温,应力比为0.1（R=0.1）,频率为20KHz。通过对表面渗碳处理后试件的断口分析,探讨表面渗碳处理、微观结构和与杂质有关的断裂机理,根据Paris公式建立超高周疲劳裂纹扩展模型。对裂纹扩展过程中裂纹尖端的塑性区的分析结果,结合传热学原理,建立热耗散模型,有限元方法的数值解结果较好地符合红外摄像仪的观测的试验结果。     

An Investigation of Fatigue Characteristics of Copper Film

Tensile and fatigue behaviors of the copper film coated by tin (Sn) were investigated considering S-N relationships and scanning electron microscope (SEM) observation of fracture surfaces. The fatigue behavior was investigated considering the effect of load ratio, R. The specimen of 2000 μm width, 8000 μm length and 15.26 μm thickness was fabricated by etching process. Tensile properties were measured using the micro-tensile testing system and in-plane electronic speckle pattern interferometric (ESPI) system for measuring the tensile strain during the test. The fatigue tests of the film were carried out in load-control mode with 40 Hz at three different stress ratios of 0.05, 0.3 and 0.5. The S-N curves, including the slope of the curve and fatigue limit, at the respective stress ratios were obtained. These curves were dependent on the load ratio. Empirical relationships indicating the dependency of the fatigue limit and S-N curve on the load ratio were suggested in this study. SEM observation of the tensile fracture surface showed that the cross-sectional area of the testing section was necked in the direction of the film thickness (i.e. parallel to the substrate surface normal) and some ductile dimples in the fracture surface were present. The fracture of the copper film under cyclic loading was progressed in the transgranular fracture mode.     

核电690合金传热管微动疲劳试验及损伤机理分析

核电蒸汽发生器传热管在微幅磨损与交变载荷的作用下形成微动疲劳,导致其表面裂纹萌生和扩展乃至破裂,从而影响反应堆的安全. 为研究径向载荷以及轴向交变应力对690合金管微动疲劳寿命的影响规律,开展690合金管管材的微动疲劳试验,获得690合金管管材的微动疲劳寿命曲线,并与相关研究数据进行对比分析,以便探讨材料在微动疲劳下的寿命模型. 对不同载荷下的690合金管试样的磨痕表面进行三维形貌和扫描电镜观测,分析磨损表面的损伤机理;对不同载荷下的690合金管试样断口的宏观与微观形貌进行表征,分析裂纹萌生、起裂过程及其失效机理. 结果表明690合金管与403不锈钢(SS)抗振条间的磨损机理为剥层及磨粒磨损;690合金管在径向载荷作用下于微动磨损处产生裂纹源,裂纹在轴向交变应力的作用下不断向内部扩展,最终导致断裂;其断裂形式为解理疲劳断裂.