An application of the J-Q model for estimating cleavage stress in the brittle-to-ductile transition

A recent model was proposed by the authors to predict cleavage failure for steels based on a weak link mechanism and a crack tip stress field modified for planar constraint by the J – Q theory. The model uses the distribution of toughness results at a single temperature to predict a toughness distribution at a different temperature and/or geometry. In this model a material cleavage stress is needed to predict when the weak link fracture is triggered. This cleavage stress is a key input for the application of the model but it is not a property that is routinely measured and it is hence not available for most steel alloys. In this paper, a method to estimate the average value of the cleavage stress is presented, based on a characteristic of the model to predict cleavage failure. Examples of cleavage stress are given for several steels and these results are used to predict the toughness distributions for structural component models.     

Fatigue Property of Nano-grained Delaminated Low-carbon Steel Sheet

Tension-tension fatigue life tests on nano-grained delaminated low-carbon steel sheet under different fatigue loads are carried out to study the fatigue properties of the steel.The three-dimensional microstructures of the steel are observed by TEM.In addition,the morphology of the fatigue fracture of the specimen under different loads is observed by SEM.The results show that micro-cracks form on the weak interface of the nano-grained steel under low-stress conditions,which hinders the propagation of the main cracks and reduces the fatigue crack propagation rate,resulting in the extending fatigue life of the steel.     

Incorporation of statistical length scale into Weibull strength theory for composites

In this paper an extension of Weibull theory by the introduction of a statistical length scale is presented. The classical Weibull strength theory is self-similar; a feature that can be illustrated by the fact that the strength dependence on structural size is a power law (a straight line on a double logarithmic graph). Therefore, the theory predicts unlimited strength for extremely small structures. In the paper, it is shown that such a behavior is a direct implication of the assumption that structural elements have independent random strengths. By the introduction of statistical dependence in the form of spatial autocorrelation, the size dependent strength becomes bounded at the small size extreme. The local random strength is phenomenologically modeled as a random field with a certain autocorrelation function. In such a model, the autocorrelation length plays the role of a statistical length scale. The focus is on small failure probabilities and the related probabilistic distributions of the strength of composites. The theoretical part is followed by applications in fiber bundle models, chains of fiber bundle models and the stochastic finite element method in the context of quasibrittle failure.     

Experimental and finite element analysis of fracture criterion in general yielding fracture mechanics

Efforts made over the last three decades to understand the fracture behaviour of structural materials in elastic and elasto-plastic fracture mechanics are numerous, whereas investigations related to fracture behaviour of materials in thin sheets or general yielding fracture regimes are limited in number. Engineering simulative tests are being used to characterize formability and drawability of sheet metals. However, these tests do not assure consistency in quality of sheet metal products. The prevention of failure in stressed structural components currently requires fracture mechanics based design parameters like critical load, critical crack-tip opening displacement or fracture toughness. The present attempt would aim to fulfill this gap and generate more information thereby increased understanding on fracture behaviour of sheet metals. In the present investigation, using a recently developed technique for determining fracture criteria in sheet metals, results are generated on critical CTOD and fracture toughness. Finite element analysis was performed to support the results on various fracture parameters. The differences are within 1 to 4%. At the end it is concluded that magnitude of critical CTOD and/or critical load can be used as a fracture criterion for thin sheets.     

Limiting state of two-phase composites under uniaxial tension

We propose a model based on the statistical analysis of fracture processes and a concept of effective medium when a system of structural elements of different kinds is regarded as a quasihomogeneous medium with certain effective properties. This approach enables one to compute parameters of the stable stage of fracture of composite materials. The suggested model is used for the prediction of critical values of the parameters corresponding to the transition to unstable fracture under uniaxial tension. We also compare two-phase composites with brittle and plastic phases with strong and weak interfaces between the structural elements, respectively.Frantsevich Institute for Problems in Materials Science, Ukrainian Academy of Sciences, Kiev. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 6, pp. 51–58, November–December, 1995.     

Deterministic method for predicting the strength distribution of a fibre bundle

Owing to the non-strain hardening plastic behaviour of the aluminium matrix and the weak fibre/matrix interface, it has been shown that the strength of a carbon fibre-reinforced aluminium matrix composite made by diffusion bonding of prepreg layers can be derived from the corresponding fibre bundle strength. Application of Coleman's model to predict bundle strength leads to the conclusion that the composite must break when 15% of the fibres are broken. This greatly overestimates the experimental composite strength. Overestimations made by using the Coleman model are due to some implicit assumptions which are not valid in the case under consideration and which may consequently not describe our material. A new approach is proposed for the calculation of the strength distribution of a fibre bundle, based on the same fracture mechanism (fibres fracture progressively until the catastrophic fracture) but without restrictive assumptions. The real interpolated experimental fibre strength distribution (and not the Weibull distribution) is taken into account to predict bundle strength. The proposed method clearly shows the limit of strength prediction, in term of bundle size (number of fibres and gauge length). The risk of making predictions following the Weibull distribution out of the range of the observations (through single-fibre tensile tests) is demonstrated.     

Computational modeling of size effects in concrete specimens under uniaxial tension

The paper presents a follow-up study of numerical modeling of complicated interplay of size effects in concrete structures. The major motivation is to identify and study interplay of several scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material and also fracture-plastic material laws to show that a portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the combined size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments could be explained either by the presence of a weak surface layer of constant thickness (caused e.g. by drying, surface damage, aggregate size limitation at the boundary, or other irregularities) or three dimensional effects incorporated by out-of-plane flexure of specimens. The latter effect is examined by comparison of 2D and 3D models with the same material laws. All three named sources (deterministic-energetic, statistical size effects and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complicated quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very complicated for quasibrittle materials such as concrete.     

Failure mode of laser welds in lap‐shear specimens of high strength low alloy (HSLA) steel sheets

In this paper, the failure mode of laser welds in lap‐shear specimens of non‐galvanized SAE J2340 300Y high strength low alloy steel sheets under quasi‐static loading conditions is examined based on experimental observations and finite element analyses. Laser welded lap‐shear specimens with reduced cross sections were made. Optical micrographs of the cross sections of the welds in the specimens before and after tests are examined to understand the microstructure and failure mode of the welds. Micro‐hardness tests were also conducted to provide an assessment of the mechanical properties in the base metal, heat‐affected and fusion zones. The micrographs indicate that the weld failure appears to be initiated from the base metal near the boundary of the base metal and the heat‐affected zone at a distance away from the pre‐existing crack tip, and the specimens fail due to the necking/shear of the lower left load carrying sheets. Finite element analyses based on non‐homogenous multi‐zone material models were conducted to model the ductile necking/shear failure and to obtain the J integral solutions for the pre‐existing cracks. The results of the finite element analyses are used to explain the ductile failure initiation sites and the necking/shear of the lower left load carrying sheets. The J integral solutions obtained from the finite element analyses based on the 3‐zone finite element model indicate that the J integral for the pre‐existing cracks at the failure loads are low compared to the fracture toughness and the specimens should fail in a plastic collapse or necking/shear mode. The effects of the sheet thickness on the failure mode were then investigated for laser welds with a fixed ratio of the weld width to the thickness. For the given non‐homogenous material model, the J integral solutions appear to be scaled by the sheet thickness. With consideration of the plastic collapse failure mode and fracture initiation failure mode, a critical thickness can be obtained for the transition of the plastic collapse or necking/shear failure mode to the fracture initiation failure mode. Finally, the failure load is expressed as a function of the sheet thickness according to the governing equations based on the two failure modes. The results demonstrate that the failure mode of welds of thin sheets depends on the sheet thickness, ductility of the base metal and fracture toughness of the heat‐affected zone. Therefore, failure criteria based on either the plastic collapse failure mode or the fracture initiation failure mode should be used cautiously for welds of thin sheets.     

Random load sequences and stochastic crack growth based on measured load data

This paper addresses the modeling of random fatigue load sequences based on real measured loads, which represents a key step in stochastic damage tolerance. Discrete-time Markov chains and hidden Markov chains are investigated to model sequences of max–min stresses. The parameters of these models are inferred from in-flight measured loads of a given fleet of fighter aircrafts. The accuracy of the two models is assessed based on the statistical properties of a parameter representative of the crack growth under variable amplitude loads, which accounts for load interactions and retardation/acceleration effects. This parameter obtained with the PREFFAS crack closure model represents the maximum stress of a constant amplitude load sequence supposedly equivalent in terms of crack extension to the variable amplitude sequence. The statistical properties of this parameter corresponding to variable amplitude load sequences generated with these models are compared to those of sequences randomly sampled from the measured loads. The Hidden Markov Model appears to be the most appropriate and accurate model. It is then used in a stochastic analysis of a crack initiating at a hole of a skin-stringer panel in a fighter aircraft. This illustrative example explores the effects of the scatter in both loads and material properties (here through the crack growth rate) on the stochastic crack growth.     

QP980板材边缘开裂的实验研究和数值模拟

目的 冲裁加工后的第3代先进高强钢QP980板材在成形中会因边缘开裂而显著影响汽车结构件安全性,针对这一问题,对QP980板材边缘开裂行为进行研究。方法 使用QP980板材通过钻孔和冲孔2种方法制备不同边缘状态的试样,并进行扩孔和中心孔拉伸试验。分析不同边缘状态试样的扩孔率和断裂应变演化规律。采用DF2015断裂模型对QP980板材的韧性断裂行为进行预测。结果 钻孔试样的扩孔率约为33%,冲孔试样的扩孔率约为24%。与钻孔试样的试验结果相比,DF2015断裂模型的模拟结果显示出了良好的预测性,但DF2015断裂模型无法准确预测冲孔试样的载荷–位移响应、扩孔率和断裂应变。结论 不同的预加工工艺导致QP980板材表现出不同的边缘开裂行为。中心孔拉伸试验结果与扩孔试验结果趋势一致,因此中心孔拉伸试验是研究边缘开裂的良好方法。钻孔预加工工艺可以保持板材的原始性能,而冲裁预加工工艺会导致板材边缘发生严重的预损伤。由于DF2015断裂模型未考虑预损伤,因此无法准确预测冲孔试样的边缘开裂行为。     

A probabilistic analysis on thickness effect in fracture toughness

Fracture toughness values are often influenced by specimen thickness and they indicate generally decreasing toughness with increasing thickness. In the present paper, a probabilistic analysis has been carried out by using various kinds of toughness data in order to clarify the applicability of the weakest link model to thickness effect in fracture toughness. Moreover, a new statistical method is proposed for determining fracture toughness distribution, which is necessary for the above analysis, with taking the temperature dependency of fracture toughness into account. Thickness effect in fracture toughness is brought about by its statistical nature and the weakest link model can be applied to evaluate the thickness effect for both steel plate and its welds with heterogeneity in toughness. This thickness effect is considerably affected by Weibull shape parameter and the probability of cleavage fracture for the material. The statistical method proposed newly in this paper is sufficiently applicable and superior to the conventional method. By using this new method, Weibull parameters at a temperature of interest can be determined with much the same reliability as in the conventional method, and also Weibull parameters at lower and higher temperatures can be obtained with a certain confidence depending on the number of specimens tested.     

Statistical analysis of fracture in graphite

A statistical model is proposed to study the fracture of graphite. This model, based on a more general one proposed by She et al., uses a local fracture criterion for a microcrack, a distribution function for microcracks and the weakest link principle to predict failure probability as a function of applied loading and distribution of microcracks. The model considers the effect of the three-dimensional stress state and therefore gives a general representation of both the effects of complex loading and microcrack distribution. The inputs to the model can be determined by either studying the microstructural features of the graphite or by choosing inputs to make the model prediction fit a set of experimental results. The latter method is used to apply the model to failure results of Rose and Tucker. One set of results is used to calibrate the model which is then applied to predict the failure behavior of a second set of experimental results. Finally, the effect of the various input variables on failure probability is studied first by considering a graphical representative of failure probability as a function of input variables and second by writing the equations in terms of nondimensional variables.     

The asymptotic stochastic strength of bundles of elements exhibiting general stress–strain laws

The fiber bundle model is widely used in probabilistic modeling of various phenomena across different engineering fields, from network analysis to earthquake statistics. In structural strength analysis, this model is an essential part of extreme value statistics that governs the left tail of the cumulative probability density function of strength. Based on previous nano-mechanical arguments, the cumulative probability distribution function of strength of each fiber constituting the bundle is assumed to exhibit a power-law left tail. Each fiber (or element) of the bundle is supposed to be subjected to the same relative displacement (parallel coupling). The constitutive equations describing various fibers are assumed to be related by a radial affinity while no restrictions are placed on their particular form. It is demonstrated that, even under these most general assumptions, the power-law left tail is preserved in the bundle and the tail exponent of the bundle is the sum of the exponents of the power-law tails of all the fibers. The results have significant implications for the statistical modeling of strength of quasibrittle structures.     

A statistical model for cleavage fracture in notched specimens of C–Mn steel

A statistical model for cleavage fracture in notched specimens of C-Mn steel has been proposed. This model is based on a recently suggested physical model. This statistical model satisfactorily describes the distributions of the cumulative failure probability and failure probability density of 36 notched specimens fractured at various loads at test temperature of −196 °C. The minimum notch toughness has also been discussed.     

THE SIZE EFFECT OF MICROSTRUCTURAL IMPLICATIONS OF THE WEAKEST LINK MODEL

Abstract— CT specimens made of a reactor pressure vessel steel were loaded at—40°C until final failure occurred by cleavage fracture. The samples of J _lcl values obtained in these tests are analysed using the weakest link model. The size effect observed with specimens of different thicknesses is compared with the predictions of the weakest link model. A formula is derived for the distribution of the locations of fracture origins which have been determined for almost all specimens with a scanning electron microscope. The distribution of the size of the "weak spots" is calculated from the distribution of the fracture origins using two different models for the stress field ahead of the crack tip. These fractographic results and the J _lcl data confirm the basic ideas of the weakest link model. The deviations observed between the quantitative predictions of the weakest link model and the data can partly be explained by the change in the stress state ahead of the crack tip caused by a change in the specimen thickness.     

微机模拟法研究纤维增强金属基复合材料拉伸断裂过程

本研究在统计分析及材料断裂力学分析的基础上,建立了长纤维增强金属基复合材料板材拉伸断裂过程的微计算机模拟方法。用该方法得到的模拟结果与已有实测结果吻合良好,从而初步证实了分析模型及其模拟程序系统的有效性。     

The residual strength characteristics of stiffened panels containing fatigue cracks

In heavy structural members, where plane strain conditions prevail, linear fracture mechanics can be used for predicting residual strength. Aircraft structures consist largely of sheet structures with plane stress conditions where linear fracture mechanics do not seem to apply. Yet it is in the aircraft main structure that large fatigue cracks can develop and that has to be designed fail-safe. The present paper describes a method to predict the residual strength of a cracked sheet structure.Contrary to an unstiffened sheet, the sheet structure contains stiffening elements that can act as crack stoppers. This crack arresting action and its consequences for the residual strength are considered in the analysis.The paper proposes a method that relates the crack resistance of a stiffened panel to that of an unstiffened sheet. It takes full account of sheet-stringer interaction in the cracked region. A criterion for crack arrest is put forward. Ultimate panel failure after crack arrest is initiated either by subsequent unstable crack growth or by stiffener failure. Critical load conditions for both failure modes are presented. In case crack arrest does not occur, the residual strength of the unstiffened panel constitutes a safe lower bound.Computational results of the interacting rivet forces by both analytical and numerical (finite element) methods are presented. From these the load concentration in the stiffener and the reduction of the stress intensity at the crack tip can be determined. This enables the complete residual strength characteristics to be predicted.The results of residual strength tests on bonded and riveted panels with symmetric strip stiffeners or eccentric Z-stringers fully substantiate the method proposed for residual strength calculations.     

Closed-form structural stress and stress intensity factor solutions for spot welds in commonly used specimens

Closed-form new structural stress and stress intensity factor solutions for spot welds in lap-shear, square-cup, U-shape, cross-tension and coach-peel specimens are obtained based on elasticity theories and fracture mechanics. The loading conditions for spot welds in the central parts of the five types of specimens are first examined. The resultant loads on the weld nugget and the self-balanced resultant loads on the lateral surface of the central parts of the specimens are then decomposed into various types of symmetric and anti-symmetric parts. Closed-form structural stress and stress intensity factor solutions for spot welds under various types of loading conditions are then adopted from the recent work of Lin and Pan to derive new closed-form structural stress and stress intensity factor solutions for spot welds in the five types of specimens. The selection of a geometric factor for square-cup specimens and the decompositions of the loads on the central parts of the U-shape, cross-tension and coach-peel specimens are based on the corresponding three-dimensional finite element analyses of these specimens. The new closed-form solutions are expressed as functions of the spot weld diameter, the sheet thickness, the width and the length of the five types of specimens. The closed-form solutions are also expressed as functions of the angular location along the nugget circumference of spot welds in the five types of specimens in contrast to the limited available solutions at the critical locations in the literature. The new closed-form solutions at the critical locations of spot welds in the five types of specimens are listed or can be easily obtained from the general closed-form solutions for fatigue life predictions.