ALBERT

Notes: The unified beta theory of Connor (1984) requires that the market portfolio be well diversified in a given factor structure. Wei (1988) extended Connor's results without relying on this assumption. This note provides an alternative to Wei's result by assuming that residuals from the projection of asset return on a set of k factors follow a joint elliptical distribution.

Notes: We answer this question in the very general context of the n-factor Heath, Jarrow, and Morton model for the evolution of the term structure of interest rates, with nonrandom volatility. the answer is that a constraint is imposed on the behavior of the volatility structure. We explain the importance of this result for the design of efficient numerical algorithms for the valuation of options on the term structure.

Notes: This paper extends He and Pearson's (1991) martingale approach to the study of optimal intertemporal consumption and portfolio policies with incomplete markets and short-sale constraints to a framework in which no assumptions are made on the price process for the securities. We show how both their characterization of the budget-feasible set and duality result can be extended to account for an unbounded set II of Arrow-Debreu state prices compatible with the arbitrage-free assumption. We also supply a (fairly general) sufficient condition for II to be bounded, as required in their setting.

Notes: Given a sequence of discrete-time option valuation models in which the sequence of processes defining the state variables converges weakly to a diffusion, we prove that the sequence of American option values obtained from these discrete-time models also converges to the corresponding value obtained from the continuous-time model for the standard models in the finance/economics literature. the convergence proof carries over to the case when the limiting risky asset price process follows a diffusion, except it pays discrete dividends on some fixed dates.

Notes: The note deals with the pricing of American options related to foreign market equities. the form of the early exercise premium representation of the American option's price in a stochastic interest rate economy is established. Subsequently, the American fixed exchange rate foreign equity option and the American equity-linked foreign exchange option are studied in detail.

Notes: Working within the Heath-Jarrow-Morton framework and using the theory of stochastic equations in infinite dimensions, a useful multifactor Gauss-Markov model for the movement of the whole of the yield curve is derived. Swaptions are priced. They are hedged by eliminating random terms between the semimartingale representations of the swaption and hedging instruments. Hedging efficiency is analyzed. the model is fitted to the swap/cap strips in Australia. Computation times on a 20-MHz laptop computer are acceptable.

Notes: This paper develops a cross-market version of factor pricing models. It is shown that exact factor pricing holds across two submarkets with respect to their common factors if and only if the unique pricing operator for the first submarket is equal to that for the other submarket with probability 1. We define an APT measure as the squared distance between the two pricing operators. Then, testing whether this measure is zero is equivalent to testing exact factor pricing across the two submarkets. Since the estimation of this measure does not require parameterizing and extracting the underlying factors, one can test factor pricing models without knowing any factors. In addition, we present a randomization procedure so that one can use it to conduct a more comprehensive investigation on the empirical robustness of factor pricing models.

Notes: This article develops a general methodology that uses the observed prices of a derivative contract to compute maximum likelihood parameter estimates for an unobserved asset value process. the use of this estimation methodology is demonstrated in two applications: Vasicek's term structure model and deposit insurance pricing. This methodology can also be useful in the empirical analysis of complex financial contracts involving embedded options.

Notes: Threshold autoregressive (TAR) models condition the first moment of a time series on lagged information using a step-function-type nonlinear structure. TAR techniques are expected to be relevant in financial time-series modeling in situations where deviations of prices from equilibrium values depend on discrete transaction costs and where market regulators follow intervention rules based on threshold values of control variables. an important finance application is in modeling the difference in prices of equivalent assets in the presence of transaction costs. the focus of this paper is on motivating the use of TAR models in this context and on the statistical estimation and testing procedures. the procedures are illustrated by modeling the difference between the prices of an index futures contract and the equivalent underlying cash index. It is found that the hypothesis of linearity is conclusively rejected in favor of threshold nonlinearity and that the estimated thresholds are largely consistent with arbitrage-related transaction costs.

Notes: Let (St)tεI be an Rd-valued adapted stochastic process on (Ω, ?, (?t)tεI, P). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on ? equivalent to P such that (St)tεI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of “no arbitrage” and “no free lunch” in financial economics. We introduce the intermediate concept of “no free lunch with bounded risk.” This is a somewhat more precise version of the notion of “no free lunch.” It requires an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of “no free lunch.” We give an argument as to why the condition of “no free lunch with bounded risk” should be satisfied by a reasonable model of the price process (St)tεI of a securities market. We can establish the equivalence of the condition of “no free lunch with bounded risk” with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (1992) for continuous-time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous-time case when the process (St)tεR+ is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be “almost a martingale” in order to allow an equivalent martingale measure.

Notes: This article examines the sampling error of the lead of one political party over another as observed in a random sample of voters. The sample size needed to achieve a certain precision is also investigated.

Notes: This article presents a variation of the funnel experiment made famous by W. Edwards Deming. Ideally suited for classroom use, this exercise illustrates the disastrous consequences resulting from tampering with a stable process.

Notes: This article examines some of the difficulties associated with teaching probability. It is argued that a key difficulty is the lack of transferability of pupils' curriculum based knowledge.

Notes: The formal use of cooperative learning techniques developed originally in primary and secondary education proved effective in improving student performance and retention in a college freshman level statistics course. Lectures interspersed with group activities proved effective in increasing conceptual understanding and overall class performance.

Notes: Book reviewed in this Article: Introductory Statistics By Prem S. Mann Applied Statistics and Probability for Engineers By Douglas C. Montgomery and George C. Runger Handling data with Databases & Spreadsheets By Michael Hammond.

Notes: Examples are given which illustrate how independence enters into statistical problems and a demonstration of independence is presented. Coverage in statistical textbooks is examined

Notes: This artical presents examples of situations where a lack of understanding of probability leads to erroneous expectations.

Notes: This paper describes experiences of teaching statistics without mathematical theory but using computer-intensive re-sampling methods. The method is relevant to statistics teaching at all levels.

Notes: The instructor can help begining students master the numerous results of regression analysis by using helpful notation and by providing a framework for organising regression outputs. For the latter, model comparison is a useful operating notion.

Notes: Several estimates of an unknown population size are compared.

Notes: Book reviewed in this Article: Enterprising Mathematics for AS Level and A-Level Statistics core Heinemann Essentials of Statistical Methods in 41 pages By T P Hutchinson. Learning Data Analysis with DATA DESK Paul F Velleman

Notes: Some teaching material which invites a group of students to design and carry out a small experiment to investigate the working of short term memory is described. No equipment of any kind is required. These ideas have been found to be very helpful in making students think about some very basic and practical issues in designing and carrying out an experiment. The simple nature of the material makes it accessible to all levels of students.

Notes: The graphic presentation of random processes by Markov chains allows an easy access to both, recursive formulae for the distribution and the proof of suppositions about the probabilities appearing within these distributions.

Notes: This work demonstrates common elements correlation, its extension to negative correlation, and the production of bivariate normal samples for a specified correlation matrix.

Notes: For many physical processes it is extreme levels which are of greatest concern. This article gives a practical introduction to the problems and models involved.

Notes: Book reviewed in this Article: Practical Data Handling Book A Davies, G. Statistics for the Twenty-first Century. Sheldon and Florence Gordon, Editors. Statistics - A First Course Walker, McLean & Matthew.

Notes: This paper presents the results of fatigue crack growth and fatigue fracture toughness studies of a high-pressure vessel steel with particular emphasis on the influence of heat treatment, low temperatures, plastic prestraining, the stress ratio and specimen dimensions.It has been shown that steels in an embrittled state, caused primarily by thermal treatment and low-temperatures, exhibit unstable fatigue crack growth which is characterized by alternate crack jumps (cleavage zones) and zones of fatigue crack growth. The fatigue fracture toughness, which corresponds to the first crack jump, and final fracture can be appreciably lower (i.e. up to 50%) than the static fracture toughness under plane strain conditions at the corresponding temperature.An analysis has been performed of unstable and stable fatigue crack growth and a model of unstable crack propagation is proposed which accounts for the observed experimental behaviour.

Notes: Abstract— Elastic-plastic finite element analysis is used to study fatigue crack closure at three different crack length to width ratios for three plane stress specimen geometries: center-cracked plate, single-edge-cracked plate (tension), and single-edge-cracked plate (bend). The maximum stress to flow stress ratio, Smax/σO, which successfully describes closure results in many center-cracked plate configurations, does not correlate the effect of different geometries on the normalized opening stress, Sopen/Smax. Crack opening stresses for different geometries and crack lengths are successfully correlated by a normalized stress intensity parameter, Kmax/K0, where K0=σ0φa. The quality of the correlation is very high at small Kmax/K0, and gradually deteriorates as Kmax/K0 increases beyond the small-scale yielding regime.

Notes: Abstract— Fatigue crack propagation threshold values have been determined with two experimental methods, it., the constant R method and the constant Kmax method. Three materials, namely A17075-T7351 and Ti6A14V STA in the LT- and TL-orientations, and a Ti-turbine disk material (IMI 685) in the CR-orientation, were investigated.The paper is divided into 3 parts. In the first part the test conditions, the experimental results and the conclusions drawn from the experimental results are presented, namely that the three different functional dependencies of ΔKth on R cannot be reconciled with present continuum mechanics concepts. In the second part, some facts used in conjunction with the da/dN–ΔKeff methodology are applied to the non-propagation condition ΔKth. Parameters such as KOp, the threshold ΔKT, and a parameter “KLL” are investigated by numerical modelling of their individual influence on the ΔKth versus R curves. This modelling work shows that the individual ΔKth versus R curves are primarily dependent on the Kop behavior of the respective material. Further, it is shown that the threshold ΔKT is a constant value, independent of any particular cyclic loading condition. In the third part of the paper, the ΔKeff concept is applied to the experimental results obtained in the first part. Using either experimentally or semi-empirically determined Kop functions and the measured ΔKT values, the ΔKth versus R curves of the three materials investigated were accurately reconstructed. It follows that the ΔKth versus R curves of the individual materials are the natural consequence of the driving force for fatigue crack propagation, namely ΔKeff

Notes: Abstract— Anderson and Dodds have recently introduced a test specimen size requirement in the transition regime which they suggest must be satisfied for the cleavage fracture resistance to be size independent. This paper assesses the implications of imposing the size requirement in fracture toughness testing standards. It is shown that imposing the size requirement can lead to a size dependence in fracture toughness data. This observation raises the general question of whether even existing size requirements for the lower shelf and upper shelf regimes are capable of giving size independent data.

Notes: Abstract— A new method of analysis is proposed for an infinite solid containing an embedded plane crack of arbitrary shape. The analysis is fundamentally based on the body force method, but proper expressions of the body force densities are introduced and the stress conditions of the crack surface are replaced by the resultant force conditions in order to improve the accuracy and validity of the method. Numerical results for typical crack problems, based on coarse mesh patterns, are shown to be in remarkable agreement with known solutions. The method is also applied to the bending of circular and rectangular cracks generated from compressive surface contacts for which reliable solutions have not yet been proposed by any other methods. The results are found to be very different from conventional free-surface solutions.

Notes: Abstract— The cyclic stress-strain response of two heats of austenitic stainless steel 3 16L was experimentally studied under a wide range of cyclic plastic strain amplitudes. Three domains on the cyclic stress- strain curve were found. In the middle domain the plateau behaviour corresponds to fatigue softening. The domain at low plastic strain amplitudes is characterized by stabilized behaviour and the domain of the highest plastic strain amplitudes by continuous hardening. Individual domains were correlated with observed dislocation structures. The existence of the plateau can be related to the localization of the cyclic strain into persistent slip bands. A Manson-Coffin plot of the material over a wide cyclic amplitude range is curved and can be approximated by a double logarithmic dependence.

Notes: Abstract— Near crack tip stress and displacement fields are obtained for anisotropic bimaterial interface cracks. A contact zone model is used in order to get rid of the unphysical oscillatory interpenetration between the edges of the crack. Semi-infinite and the finite crack problems have been studied. Using the near crack tip results of this model crack branching angles can be predicted. These results are illustrated by numerical results for various materials.

Notes: Abstract— In order to investigate the fatigue strength and fracture mechanism of ceramic-sprayed steel, rotary bending fatigue tests were conducted at room temperature in air and 3% NaCl solution using specimens of a medium carbon steel (S45C) with sprayed coating layers of Ni-5% A1 (under-coating) and chromia (top-coating). The results obtained are discussed based on observations of fatigue cracks and experimental data on specimens subjected to individual treatments during the ceramic spraying process. It was found that at a very early stage of fatigue life, cracks were initiated at the interface between under- and top-coating layers, and grew rapidly into the ceramic-sprayed layer. However, these cracks did not propagate continuously into the substrate, and the final failure was led by the growth of a crack newly initiated at the surface of the substrate steel. Thus, the fatigue strength of the ceramic-sprayed steel in air could be evaluated due to the property of the substrate. The corrosion fatigue strength of ceramic- sprayed steel was improved when compared to that of the substrate steel. However, the coating layer contained many pores, through which NaCl solution was supplied from the specimen surface to the substrate. Corrosion pits were formed at the interface between the under-coating and the substrate. Subsequently, cracks initiated from the pits and grew into the substrate. Tests were also conducted on specimens whose pores were closed by a shielding treatment. In this case, NaCl solution was supplied to the substrate by cracks initiated in the top-coating layer. The shielding treatment was effective at low stress levels where fatigue life was more than 107 cycles, while it had little effect on improving corrosion fatigue strength at higher stress levels because of the many cracks initiated in the top-coating layer.

Notes: Abstract— The development of an elastic-plastic fracture mechanics model for predicting fatigue crack growth rates in notched specimens is presented in this paper. The model is based on experimental short and long crack growth results from un-notched specimens and the distribution of strain ahead of the notch root. This approach may be applied to the assessment of the lifetime of many high technology engineering components.

Notes: Abstract— By conducting creep tests and creep-fatigue tests at temperatures of 800°C, 900°C, and 1000°C, on centrifugal cast HK40 alloy (which is used for example for the steam reformer tubes of fuel cell plants) and varying the strain rate during the loading and unloading process, the influence of strain history on the life of the material has been clarified. Furthermore, the data obtained from these tests were subjected to analyses by means of life evaluation methods utilizing the life fraction rule, strain range partitioning method, and damage rate approach, and comparative studies were conducted on each life evaluation method. As a result, problematic points concerning the application of the life fraction rule, strain range partitioning method, and damage rate approach have been clarified and data which would be useful in establishing design guidelines for structures to be used under conditions such as involves the interaction of creep and fatigue have been obtained.

Notes: Abstract— By adopting a suggestion made by Thomason, a new failure criterion for the Gurson-Tvergaard model has been recently introduced by the authors. In this study, a method based on the Gurson-Tvergaard constitutive model and the new failure criterion is applied to the analysis of ductile fracture. The main features of the method are that the material failure is a natural process of the development of Thomason's dual dilational constitutive responses, and the void volume fraction corresponding to the failure by void coalescence is not necessarily a material constant and is not needed to be fitted beforehand. Furthermore, void nucleation parameter(s) can be numerically fitted from experimental tension results. This method has been implemented into the ABAQUS finite element program via a user material subroutine and is applied to the prediction of tension problems conducted by the authors. In the analyses, two strain-controlled void nucleation models have been studied and compared. The void nucleation parameters corresponding to the two models have been calibrated. The crack initiation of both smooth and notched axisymmetric tensile specimens are well predicted by the method. Finally, several critical issues in the analysis of ductile fracture are discussed.

Notes: Abstract— A computer program that creates time histories for use in fatigue tests in servo-hydraulic machines is presented. The time histories can be tailored from certain user specifications, namely the irregularity factor and the level-crossing spectrum. The program is based on the theory of stochastic time series, and the formulae needed for the program are derived in the paper. Problems of optimality are treated both with respect to fulfilment of the specifications and with respect to machine performance. Examples from two fatigue-testing projects, where the program has been applied, are presented.

Notes: Abstract— The interaction between fatigue damage (i.e., fatigue crack propagation) and internal grain boundary damage (i.e., cavity formation at grain boundaries) has been studied for the Alloy 800H at 750°C for constant plastic strain ranges but different experimental conditions. Most experiments were performed at constant ranges of alternating tensile/compression stresses. Symmetrical as well as asymmetrical tests (with larger compression stresses) were performed. In comparison to the former tests, asymmetrical tests led to shorter cyclic lifetimes mainly due to cavity formation which was not observed for symmetrical tests. It could be shown that a fast compressive and a slow tensile half cycle (at large compressive and low tensile stresses) are ideal conditions for the nucleation and growth of cavities. Based on quantitative measurements of the cavity density from interrupted fatigue tests, a physical model is presented which can predict the number of cycles to failure. This cycle number is determined only by fatigue crack growth which is controlled by (a) athermal plastic deformation, (b) creep deformation and (c) rate enhancement by cavitation.

Notes: Abstract— The fatigue behaviour of a Ni-Cr-base powder flame-spray coating on a 0.4% C steel is investigated. Fatigue tests were carried out using mild hour-glass profile specimens. Cracks were detected and measured using plastic replicas and an image analysis system. Coated specimens showed a slightly lower fatigue endurance than plain specimens under torsion loading, while the opposite was observed for push-pull loading. Microcracks in coated specimens invariably form at pores.Contrary to the usual case of stage I shear growth for a plain 0.4% C steel in tension or torsion loading, the coated specimens show initial crack growth from pores along directions perpendicular to the maximum tensile stress. The crucial behaviour of short cracks, and their growth rates, relative to the thickness of the coating, are discussed in some detail.

Notes: Abstract— The effect of stress state on the time and strain to failure has been considered in terms of currently accepted models of cavity growth. It is shown that the increasing contributions of compressive stress cause changes in cavity growth mechanisms which lead to increases in ductility. A tensile component of stress is necessary to provide the driving force for cavity growth by diffusion of vacancies and hence only strains in the presence of a tensile stress can lead to creep-dominated failure in creep-fatigue.Equivalent stress functions for isochronous stress rupture have been derived in terms of the cavity growth models and their corresponding relationships for calculating damage in terms of strain fractions developed. It is shown that it is difficult to discriminate between the various models on the basis of available experimental data. However, the analysis allows data to be assessed within the framework of physically based mechanisms and suggests methods which lead to conservative lower bound estimates of endurance.It is concluded that the shape of the isochronous creep rupture locus depends on the controlling process of cavity growth and that a detailed analysis of uniaxial creep ductility is necessary to obtain a complete description of the multiaxial behaviour. In many instances such an analysis will prove more valuable than simply performing creep tests over a limited range of stress states. Increasing contribution of principal stress to the failure process leads to a greater value for the equivalent stress in the presence of a compressive component compared with the von Mises equivalent value. However, the equivalent stress is reduced in the tensile quadrant of bi-axial stress. Hence the degree of conservatism arising from using the von Mises equivalent stress will vary with stress and may become slightly non-conservative. The relationship between equivalent stress functions for application in a time fraction assessment of creep and the calculation of creep damage by a strain fraction method has been demonstrated. Finally, guidance is given on how a limited data base of uniaxial rupture properties can be used to obtain a conservative estimate of behaviour under multiaxial loading.

Notes: Abstract— Different aspects of fatigue design problems are indicated and uncertainties are listed. Scatter as observed in many laboratory studies is analyzed. It is argued that scatter of crack initiation and crack growth are different issues. Various sources of scatter are discussed and illustrative examples are presented. Comments are given on statistical distribution functions, scatter under Variable-Amplitude loading, and scatter in service. The discussion touches upon the experience and meaning of scatter of laboratory test series related to practical problems.

Notes: Abstract— The notched strengths of four woven laminates (two orthotropic, one quasi-isotropic and one square symmetrical) under monotonic uniaxial loading, predicted with the point and average stress criteria, the two parameter criterion, the progressive degradation model, and the initial and improved point and minimal strength models, have been respectively compared with the experimental data. The damage mechanisms for the laminates were tentatively studied. The results show that, for minimal experimental information, the notched strengths predicted by the improved minimal strength models are the most precise and the progressive degradation model can numerically illustrate the damage mechanisms in detail. The damage mechanisms of the woven laminates were found to be very different from those of non-woven laminates.

Notes: Abstract— The dislocation substructures of ferrite in plain carbon, ferrite plus martensite, dual-phase steels in different stages of fatigue crack propagation were examined by transmission electron microscopy. The experimental results show that the dislocations are in random arrays in the ferrite before cyclic loading. At stages of low ΔK (near threshold) values, parallel clusters of dislocations and prolonged dislocation lines can be observed. At stages of intermediate ΔK values, the dislocations are rearranged into networks and loops while at stages of high ΔK (prior to failure) values, they are changed into dislocation cells or patches. It was also found that the volume fraction and carbon content of martensite have significant effects on the dislocation substructures of ferrite after cyclic deformation.

Notes: Abstract— Thermo-mechanical fatigue stress-strain data on ferritic/austenitic steels and superalloys from a variety of sources are analysed with regard to hysteresis loop stress asymmetry. This arises from a decoupling of the thermal and mechanical strain signals in the test technique so that many tension-compression load combinations are possible. Data from simplified isothermal and bithermal tests are also examined. Taking a typical example of an “out-of-phase” thermo-mechanical loop on a 1/2 CrMoV steel cycled between 200 and 550°C, isothermal stress-strain data were generated at 50°C intervals on material from the same cast and, used in conjunction with the elastic characteristics of the apparatus, an attempt was made to re-create this loop. The methods employed were (i) a graphical construction between appropriate isothermal yield contours (ii) a tangent modulus calculation (iii) a secant modulus calculation. Method (i) appeared to give the closest agreement in the present case.

Notes: Abstract— The plastic deformation of thin-walled cylinders has been experimentally examined for the loading conditions of ±1% axial strain with hoop stresses of approximately 0, 1/4, 1/2 and 3/4 of the initial uniaxial yield stress.Two materials similar to those used in the pipework of PWR nuclear plant in the U.K. have been tested, namely 304S11 stainless steel and En6 low-carbon steel. The results of the tests were to be compared with the allowable stresses and deformations specified in the ASME Boiler and Pressure Vessel Code, Section III. The code specifies that a prescribed combination of primary stresses must not exceed 1.5Sm, where Sm is a stress value defined for each material.The results indicate that the limit of 1.5Sm is excessively low for both materials and that in particular, the stainless steel could tolerate 5Sm. Although the En6 steel is more prone to ratchetting than the stainless steel, the results suggest that it too could tolerate a higher primary stress than the code allows. Both materials are shown to satisfy the proposed ASME ratchet strain limit of 5% hoop strain after 10 cycles of ±1% axial strain range, for any value of internal pressure.

Notes: Abstract— The susceptibility of an X65 line pipe steel to hydrogen-induced stress corrosion cracking (SCC) is investigated. SCC tests on the steel are carried out in three environments of different aggressivity based on a NACE TM-01-77 solution with dissolved gaseous H2S. The threshold stress intensity factor is calculated for each environment using the multiple specimen technique. The steel is tested in the as-received condition and after homogenisation. Permeation tests are carried out on specimens of the same steel in the same environment. The relationship between threshold stress intensity factor and hydrogen concentration is obtained. The results from these studies are used to generate the parameters in the micro-mechanic model of Akhurst and Baker. The validity and physical significance of the model is assessed.

Notes: Abstract— Measurements of the effect of stress ratio on the constant amplitude fatigue crack growth rates in four quenched and tempered steels in the Paris regime are reported. This data and published data for other alloys (including lower strength steels and non-ferrous alloys) are evaluated, and a correlation function suitable for practical fatigue life calculations is derived. In addition to stress intensity factor range and stress ratio, other significant parameters are the yield stress of the material and its thickness. For the four steels on which new measurements were made, the degree of dependence of the crack growth rate on stress ratio may be related to sensitivity to environmental conditions.

Notes: Abstract— A model based upon linear elastic bridging and fiber crack tip shielding is proposed for predicting fatigue crack growth in a SCS-6/Ti-6–4 composite. The model is characterized by the fiber/matrix debond length rather than the fiber/matrix interfacial frictional shear strength used in most current fatigue models. Finite elements combined with fracture mechanics are applied for computing the local stress intensity. The local stress intensity in the matrix is then utilized to predict crack growth in the composite via comparison to monolithic fatigue crack propagation data for a similar Ti-6–4 matrix material.

Notes: Abstract— The Modified Crack Closure Integral (MCCI) technique based on Irwin's crack closure integral concept is very effective for estimation of strain energy release rates G in individual as well as mixed-mode configurations in linear elastic fracture mechanics problems. In a finite element approach, MCCI can be evaluated in the post-processing stage in terms of nodal forces and displacements near the crack tip. The MCCI expressions are however, element dependent and require a systematic derivation using stress and displacement distributions in the crack tip elements.Earlier a general procedure was proposed by the present authors for the derivation of MCCI expressions for 3-dimensional (3-d) crack problems modelled with 8-noded brick elements. A concept of sub-area integration was proposed to estimate strain energy release rates at a large number of points along the crack front. In the present paper a similar procedure is adopted for the derivation of MCCI expressions for 3-d cracks modelled with 20-noded brick elements. Numerical results are presented for center crack tension and edge crack shear specimens in thick slabs, showing a comparison between present results and those available in the literature.

Notes: Abstract— Experiments have been performed on specimens subjected to strain cycles similar to those experienced by sub-surface elements of material in rolling/sliding contact. It has been observed that if the strain cycle is closed then failure takes place by low cycle fatigue and the Coffin-Manson relationship may be used to predict the number of cycles to failure. If however, the strain cycle is open, so that the material accumulates unidirectional plastic strain (the situation known as “ratchetting”) a different type of failure, which is termed ratchetting failure may occur. It occurs when the total accumulated plastic strain reaches a critical value which is comparable with the strain to failure in a monotonic tension test. The number of cycles to failure under these circumstances may be estimated by dividing this critical strain by the ratchetting strain per cycle. It is suggested that low cycle fatigue and ratchetting are independent and competitive mechanisms so that failure occurs by whichever of them corresponds to a shorter life. The results of both uniaxial and biaxial tests reported in the literature have been re-evaluated and these, together with new data on biaxial tests on copper, found to be consistent with this hypothesis.

Notes: Abstract— A hypothesis on the statistical relation between the local microscopic directions of fatigue crack growth and the macroscopic crack front shape is postulated. The method of fractographic reconstitution of the fatigue crack front shape from local microscopic directions of crack growth is proposed and tested in two different applications.

Notes: Abstract— Large scale, structurally representative, double tension crack arrest tests have been undertaken at temperatures between −99°C and −87°C. Applied stresses and the length of the embrittled crack starter sections were varied to give different applied stress intensity factors in the tests. The results indicate that crack arrest in structures is not governed solely by a so-called crack arrest temperature but that static linear elastic fracture mechanics can be used to describe it. The measured crack arrest toughness temperature curve of the 1.5%Ni TMCP steel investigated lies at the lower bound of published data.

Notes: Abstract— The criteria for determining whether ferritic material exhibits fully ductile behaviour are generally based on the fracture toughness vs temperature relationship determined from standard laboratory test pieces (e.g. Charpy V Impact tests or fracture toughness specimens). This relationship defines a ductile-brittle transition region. When fully ductile, microvoid coalescence behaviour is experienced, and fracture toughness is described as being on the “upper shelf”. At “off the shelf” temperatures brittle, cleavage fracture is experienced. On the lower shelf the material is entirely brittle, exhibiting 100% crystallinity on the fracture face. As the temperature increases, initiation of tearing by microvoid coalescence occurs and some stable tearing can occur prior to the cleavage event. Material toughness increases with temperature until the upper shelf condition is achieved.The characteristics of fracture toughness tests in terms of the toughness level exhibited and the extent of ductile tearing experienced have been used as a guide to whether the structural application (e.g. a pressure vessel) will behave in a brittle or a ductile manner. This paper reports on a feasibility study where various worked examples have been performed to examine the concept of using a “cut off” on the failure assessment diagram, determined from the conditions required to cause plastic collapse of a pressure vessel, as a criterion for defining effective “ductile” operation. Fracture assessment procedures (R6 revisions 2 and 3 and PD6493 levels 2 and 3) have been utilised to determine the influence on pressure vessel performance of the behaviour of fracture toughness test specimens.The procedure of plotting a structural collapse “cut off” on a failure assessment diagram enables the assessment of whether a particular flaw geometry would result in gross deformation of the structure at failure. The use of this procedure provides an unambiguous demarcation between “fracture dominated” and “collapse controlled” conditions. This procedure facilitates judgements on the level of toughness necessary to ensure ductile operation and whether a “tearing plus toughness” requirement is necessary.It is recommended that consideration be given to including structural collapse into fracture assessment procedures carried out using R6 revision 3 or PD 6493:1991 procedures in order to determine the conditions when enhanced toughness no longer influences structural performance (i.e. when effective “upper shelf” conditions are attained).

Notes: Abstract— The use of a previously presented general criterion of failure for high cycle multiaxial fatigue, τa/tA,B+σn.max/2σT= 1, is extended to cases where the shear and normal stress on the critical plane are non-proportional and also to give life predictions in the range of 104 to 106 cycles. The criterion takes account of whether case A cracks, growing along the surface, or case B cracks, growing in from the surface, occur.

Notes: Abstract— The successful implementation of a fracture mechanics approach to the in-service structural integrity assessment of welded and machined components requires an accurate and reliable NDT method for detecting and sizing surface breaking cracks. This paper discusses and compares the crack depth sizing abilities of two theoretically-based NDT techniques: alternating current potential difference (ACPD) and alternating current field measurement (ACFM). Surface fatigue cracks of high and low aspect ratio are used in the investigation. It is demonstrated that ACPD is capable of accurately sizing long shallow cracks and that ACFM provides accurate crack depth predictions for small deep cracks.

Notes: Abstract— Conventional mechanical tests and fracture mechanics experiments were carried out at – 196°C on a low alloy steel (A508 class 3) which was investigated under two different conditions: (i) a reference condition in which the failure mode was cleavage and (ii) an embrittled condition in which the fracture mode was either partly or predominantly intergranular fracture. These experiments, performed with a new specimen geometry, a ring specimen instrumented to measure also the crack velocity, were used to determine the fracture toughness at crack initiation (KIc) and at crack arrest (KIa). It is confirmed that the reduction in KIc measured in the embrittled material is associated with the appearance of intergranular fracture. It is also shown that KIa, determined by a static analysis decreases rapidly with crack velocity when the fracture mode is predominantly cleavage. On the other hand, KIa, corresponding to intergranular fracture seems to be much less dependent on crack speed. This difference in the sensitivity of both modes of brittle fracture to crack velocity is briefly discussed.

Notes: Abstract— Low cycle fatigue (LCF) testing of powder metallurgy (PM) superalloys has determined that LCF cracks initiate from defects such as pores and ceramic inclusions located either at the surface or internally. The ratio of surface-to-internal crack initiation sites depends upon experimental conditions. In the current work, a probabilistic model is presented that describes the competition between these two failure mechanisms from a single experimental parameter, μ. The model predicts the size distribution of defects which initiate LCF cracks, as well as the proportion of specimens which fail as a result of internally located defects. Model predictions were found to correlate well with experimental LCF data obtained from a N18 PM Ni-base superalloy. The model further enables evaluation of a “size effect”, as it predicts that internal crack initiation is favoured for larger components/specimens.

Notes: Abstract— A fatigue scatter band has been computed for laser welded austenitic stainless steel joints. These laser welded steels have a very small heat affected zone. The unified scatter band provided by standards for welded structural steels does not adequately describe the trend of the experimental data of laser welded steels and this makes their design parameters scarcely realistic. The scatter band proposed in this paper has been computed by re-sorting experimental data relative to joints with high stress concentration factors and has subsequently been assessed with data relevant to butt welded joints.

Notes: Abstract— Energy dissipation rate, D, and crack tip opening angle (CTOA) have been related for large amounts of ductile crack growth in a fully plastic two-dimensional model of real elastic plastic material. Several J-type rising R-curves have been constructed and related, some based on the cumulative energy dissipated and some as a characterising term that relates closely to the far-field J contour value. These two meanings co-exist but the relationship is geometry dependent. For deep notch bend (DNB) cases the non-dimensional group Lα/r* (where L is normalised load, α is CTOA and r* defines the position of the instantaneous centre of rotation) governs the value of dJ. Certain DNB cases where R-curves vary with size are resolved by this group rather than by CTOA itself whereas for centre-cracked tension there is a higher value of CTOA than for DNB.

Notes: Abstract— Failure of ceramic materials occurs when the stress intensity factor of the most serious crack in a component reaches a critical value KI,C, the fracture toughness of the material. In case of ideal brittle materials the fracture toughness is independent of the crack extension and, consequently, identical with the stress intensity factor KI,Onecessary for the onset of stable crack growth. It is a well-known fact that failure of several ceramics is influenced by an increasing crack-growth resistance curve. The effect of increasing crack resistance has consequences on many properties of ceramic materials. In this report the authors discuss some aspects of R-curve behaviour as represented by stress intensity factors or energies.

Notes: Abstract— The transferability of fracture results between homogeneous and cladded specimens was studied in single edge-notched bend specimens. The test material was of A533-B steel with a clad layer deposited by a commercial strip welding process. The fracture resistance properties were developed independently for cladding and base material using homogeneous specimens of each material. The experimental data from tests on cladded specimens were analysed with the finite element method. The 3D J-values were compared with values evaluated using the measured crack extension in the cladded specimen and the JR-data of the respective material provided from homogeneous specimens. A reasonably good agreement was obtained in this comparison for a limited amount of crack growth.

Notes: Abstract— Basic diagrams of the cyclic crack growth resistance of two of the most investigated titanium alloys, namely Ti-6A1–4V and Ti-6A1–6V-2Sn, are presented. Diagrams are plotted for, in-air, distilled water and 3.5% NaCl solution, which are necessary for lifetime calculations of structural elements made of these metals. The dependency of cyclic crack growth resistance on the yield strength is established. It is shown that cyclic crack growth resistance of titanium alloys in corrosive environments is determined not only by the stress-strain state but also by the electrochemical conditions at the corrosion fatigue crack tip, which for aqueous environments can be characterized integrally by the hydrogen index of the environment and the electrode potential of the metal. Therefore, cyclic corrosion crack growth resistance testing should be performed under constant electrochemical conditions at the corrosion fatigue crack tip or these conditions should be taken into account. A new method of plotting the basic cyclic corrosion crack growth resistance diagrams of titanium alloys is considered.

Notes: Abstract— A fretting fatigue crack growth model is proposed accounting for the effects of tribological parameters of the contacting materials and the biaxiality of contact together with bulk stresses, upon fretting fatigue crack growth rate and direction of growth in the fretting-zone subsurface layers. Based on this model a new technique is developed to determine fatigue life and predict the fatigue limit in an aluminium alloy, AMg6N, and a titanium alloy, VT9, under fretting conditions. For the above cases, fretting fatigue crack growth behaviour predicted by the proposed model is in good agreement with the experimental results.

Notes: Abstract— Mode I fatigue crack growth has been studied in notched specimens of 7017-T651 aluminium alloy subjected to fully compressive cyclic loads. The specimens were first subjected to a deliberate compressive preload which causes plastic deformation at the notch tip. On unloading, this region developed a residual tensile stress field and on subsequent compressive cyclic loading in laboratory air, a fatigue crack was nucleated at the notch and grew at a diminishing rate until it stopped. The final crack length increased with an increase in the value of the initial compressive preload and with an increase in the negative value of the applied cyclic mean load. To gain a better understanding of crack growth in residual stress fields, the magnitude and extent of residual stress induced from compressive preloads have been analysed. This was achieved when extending the notch by cutting while recording the change in the back face strain. From residual strain models it was found that the fatigue crack growth was confined to a region of tensile cyclic stress within the residual stress field. The effective stress intensity range was investigated at selected mean loads and amplitudes, for correlating purposes, using both the compliance technique and by invoking the crack growth rate behaviour of the alloy. Finally, a brief discussion of the fracture morphology of cracks subjected to cyclic compression is presented.

Notes: The cylindrical specimen is a common configuration adopted for biaxial fatigue testing. A rigorous yet practical analysis is presented here under the framework of incremental plasticity. The cyclic stress-strain curve is obtained from four different plasticity models, and correlated with twenty-six in-phase biaxial tests on En15R steel. Stabilized hysteresis loops can also be derived based on the cyclic stress-strain curve under a J2-kinematic hardening formulation with special consideration of the rounding transition at the instant of yielding. A correlation for the hysteretic strain hardening exponent N is required for materials that exhibit non-Masing behaviour; i.e. do not allow a full description of hysteresis loops from knowledge of the cyclic stress-strain curve. The solution is further generalized to out-of-phase loading, where the rotation of the active stress state around the yield surface is illustrated. This compares favourably with ten out-of-phase tests on En15R steel.

Notes: As is well known, the fatigue strength reduction factor due to a notch (Kf) in the modified Neuber's rule should not be considered a constant since it varies with different stress levels. In this paper, the definition and the variation of Kf are discussed on the basis of theoretical and experimental analyses, leading to the presentation of a variable Kf-Neuber's rule. Comparing the results of experimental stress analyses and the fatigue life tests of specimens, the variable Kf-Neuber's rule gives good predictions and is applied to the life prediction of an aircraft component.

Notes: Fatigue crack growth behaviour under intermittent overstressing was investigated in dry air with a low carbon steel under tension-compression and tension-tension loading with different mean stress levels. A very small number of cycles of overstress applied intermittently during a very large number of cycles of understress below threshold caused, in all cases, a significant acceleration in crack growth rate as compared to the case of steady cyclic stress. The acceleration for the case with tensile mean stresses was a little smaller than in the case without mean stress and it was related to a little higher crack closure level in the former case. Stress history in the precracking process had some effect on crack closure and, consequently, on the acceleration. A fairly good correlation was obtained between the acceleration and the effective stress intensity range. The acceleration under multi-level intermittent overstresses with different mean stresses could be predicted by linear summation from two-level intermittent overstress test data. No difference was recognized in the fracture surface morphology between the cases with different mean stress level including the cases of multi-level stresses, indicating that the micromechanism of crack growth acceleration would be the same in all cases.

Notes: In this paper the Rousselier ductile damage model is used to simulate the softening behaviour of notched bars and crack growth in a pre-cracked specimen of an A508 pressure vessel steel. The effect of the internal parameters needed to represent the damage on the calculated results is studied. The interactions between these parameters and their effects on the calibration process are discussed. The effect of the cell size on the values of the damage parameters obtained during calibration and on the response of the specimens is briefly reviewed.

Notes: Both high-cycle and low-cycle fatigue properties of hot-extruded powders of a Ni3Al-based alloy, IC218, have been evaluated. High cycle fatigue measurements were performed under stress controlled conditions at temperatures ranging from 25°C to 850°C. Tests were made in both laboratory air and vacuum environments. Low cycle fatigue tests were conducted under total strain control in a laboratory air environment at 650°C. In high cycle fatigue, high ratios of the fatigue limit (Δσ at 106 cycles) to monotonic yield strength (σys), of approximately Δσ/σys∼1, were obtained in the powder extruded IC218 alloy for temperatures ranging from 25°C to 650°C. In low cycle fatigue, a substantial decrease in fatigue life occurred at 650°C, compared to results obtained previously at 25°C. High cycle fatigue performance at low stress/strain amplitudes is better than expected when compared to precipitation strengthened superalloys. The improved performance is explained in terms of the cyclic hardening behavior of the alloy.

Notes: A theoretical study of the conditions for coalescence of two fatigue cracks in a multi-crack system is presented. As both cracks grow, the near-side plastic zone sizes of the interacting cracks increase more rapidly than the off-side zone sizes which results in higher overall crack propagation rates for an interacting crack growth system than the simple addition of individual crack growth rates. The use of a single crack growth model to predict failure in the presence of two or more cracks may therefore underestimate the interaction between cracks and lead to un-conservative predictions. Some of the theoretical predictions were confirmed by published results of experimental measurements of crack interaction distances in multi-crack systems.

Notes: Hydrogen induced cracking (HIC) initiation sites and their correlation with the critical stress intensity factors of hydrogen charged specimens were studied under combined I/II mode loading. Two series of tests, is. constant load (CL) tests and slow strain rate (SSR) tests, were carried out. Experimental results showed that in CL tests, irrespective of the ratio KIJKl, the HIC initiation sites always correspond to the point of maximum hydrostatic stress; which is located some distance ahead of the notch tip. However, for SSRT tests, when K11/K1〉 1, HIC started at the notch tip which corresponds to the point of maximum equivalent plastic strain. When K11/K1〈1 in SSR tests, HIC occurred initially some distance ahead of the notch tip.The relationship between the critical stress intensity factor for HIC and K11/K1 was shown to be different for the two types of test. Multiple effects of stress and strain on hydrogen redistribution and hence on HIC initiation sites, as well as critical stress intensity factors, are discussed.

Notes: Abstract— Two methods are employed, namely the Charpy impact test and a strain gauge technique, for the determination of the dynamic fracture toughnesses of CrNiMoV27 and CrNiMoV45 alloy steels under various impact loading rates and heat treatments. The results show that the dynamic fracture toughness KId is significantly affected by impact velocity and that tempering has a greater influence on KId than quenching. It is also shown that the alloy containing a greater amount of vanadium yields consistently a higher value of Kid.