D. Gary Harlow

The Chain-of-Bundles Probability Model For the Strength of Fibrous Materials I: Analysis and Conjectures

Analytical results are discussed for the chain-of-bundles probability model for the strength of fibrous materials. Two load sharing rules are considered for failed and nonfailed fibers in a bundle. The first is the equal load sharing rule of classical analysis, and the second is a local load sharing rule which is more realistic for composite materials. A rather detailed discussion of past statistical analysis is given. From a careful study of previous results, several conjectures and key questions about the behavior of the strength are generated. Also, an exact analysis of failure is per formed so that the properties of the strength distribution can be studied. Difficulties of a general analysis are discussed in detail. The sequel will contain a thorough numerical investigation of the model with emphasis on studying the convergence of certain transformed distributions and on answering key questions raised in this study.

Approximations for the strength distribution and size effect in an idealized lattice model of material breakdown

Abstract Various random network models have been developed recently to explain certain features of fracture development in materials, including the character of ‘cracks’, the form of the strength distribution, the size effect, and the connection to percolation theory. Applications include fibrous composites, random fuse networks, superconducting networks, dielectrics and elastic lattices. Because of extreme analytical difficulties researchers have relied on Monte Carlo simulation to validate various scaling hypotheses and approximations. Since only small network sizes are presently accessible, features which eventually emerge at the largest scale may not be uncovered. To shed light on this issue we consider a simple, idealized model where elements have strength zero or one with probabilities α or 1 − α, respectively. The load of a failed element is redistributed equally onto the nearest surviving neighbors, and open boundary conditions are considered for simplicity of calculation. Various exact, asymptotic and numerical results are obtained including a careful evaluation of any errors. Features of the results are in conflict with some of those in the literature for more complex stress redistribution situations. For α close to one, the mean strength of the network is dominated by small-scale bm boundary effects which may persist up to relatively large network sizes (1000 × 1000) before large-scale effects ultimately dominate.

PROBABILITY DISTRIBUTIONS FOR THE STRENGTH OF FIBROUS MATERIALS UNDER LOCAL LOAD SHARING I: TWO-LEVEL FAILURE AND EDGE EFFECTS

The focus of this paper is on obtaining a conservative but tight bound on the probability distribution for the strength of a fibrous material. The model is the chain-of-bundles probability model, and local load sharing is assumed for the fiber elements in each bundle. The bound is based upon the occurrence of two or more adjacent broken fiber elements in a bundle. This event is necessary but not sufficient for failure of the material. The bound is far superior to a simple weakest link bound based upon the failure of the weakest fiber element. For large materials, the upper bound is a Weibull distribution, which is consistent with experimental observations. The upper bound is always conservative, but its tightness depends upon the variability in fiber element strength and the volume of the material. In cases where the volume of material and the variability in fiber strength are both small, the bound is believed to be virtually the same as the true distribution function for material strength. Regarding edge effects on composite strength, only when the number of fibers is very small is a correction necessary to reflect the load-sharing irregularities at the edges of the bundle.

Mechanistically based probability modelling, life prediction and reliability assessment

Life prediction and reliability assessment are essential components for the life-cycle engineering and management (LCEM) of modern engineered systems. These systems can range from microelectronic and bio-medical devices to large machinery and structures. To be effective, the underlying approach to LCEM must be transformed to embody mechanistically based probability modelling, vis-a-vis the more traditional experientially based statistical modelling, for predicting damage evolution and distribution. In this paper, the probability and statistical approaches are compared and differentiated. The process of model development on the basis of mechanistic understanding derived from critical experiments is illustrated through selected examples. The efficacy of this approach is illustrated through an example of the evolution and distribution of corrosion and corrosion fatigue damage in aluminium alloys in relation to aircraft that had been in long-term service.

Probability versus statistical modeling: examples from fatigue life prediction

Probability analyses are increasingly being used for reliability and durability assessments for life prediction of engineered components and systems. Nevertheless, many of the current analyses are predominately statistical rather than probabilistic. Fatigue life prediction has historically been based on the safe-life or the crack growth approaches, both of which are empirically based. Consequently, they do not adequately reflect long-term operating conditions, or identify the sources and extent of their contributions to variability. A comparison between probability and statistical approaches for fatigue life prediction is developed herein. Using simple crack growth models, the variability inherent in S-N response can be related to key random variables that are readily identified in the models. The identification and quantification of these variables are paramount for predicting fatigue lives. The effectiveness of probability modeling compared to statistical methodologies is shown through the analysis of an extensive set of S-N data for 2024-T4 aluminum alloy. Subsequently, the probability approach is demonstrated with S-N data for SUJ2 steel, in which two distinct failure modes are operative. Variability associated with manufacturing and material variables are considered. The adoption of this technique to put life prediction on a sound scientific and probabilistic basis is recommended.

Corrosion-Enhanced Fatigue and Multiple-Site Damage

Multiple-site (fatigue) damage, (MSD) and its impact on the structural integrity (or safety of flight) of aging aircraft have been well recognized. Research to date has focused on fracture-mechanics-based analysis and experimentation for the consequences of MSD. The impact of corrosion on the early onset of MSD and the need for quantitative methodologies to predict the evolution and distribution of damage and MSD, on the other hand, are not fully appreciated. The mechanism for pitting corrosion in airframe aluminum alloys and the influence of pitting on the onset of fatigue cracking are briefly reviewed, and the influence of localized corrosion on the evolution of MSD is discussed. A mechanistically based probability model for corrosion and corrosion-enhanced fatigue crack growth and its application in predicting the probability of occurrence (PoO) of damage are summarized. The use of the PoO in a methodology to assess the onset and severity of MSD is demonstrated using teardown data from a Boeing 707 and two AT-38B aircraft.

Probabilistic considerations of creep crack growth

Abstract A tensile ligament instability model for creep controlled crack growth, based on Krafft, and Hart and Li, is used to illustrate the approach in formulating a mechanistically based probability model for life prediction. Such a model is needed for making statistically accurate estimates of service life for conditions not included within typical supporting design data. The approach and methodology for model development are described. The contributions and significance of selected fundamental variables for the deformation and failure processes are discussed.

A critical comparison between mechanistically based probability and statistically based modeling for materials aging

Probability analyses are increasingly used for reliability and durability assessments in the life-cycle design and management of engineered systems. This paper provides a critical comparison between the mechanistically based probability and statistically based approaches. The crucial differences between the two approaches are highlighted and demonstrated through modeling of the creep crack growth response of a high-strength steel. The impact of these differences on structural reliability and durability analyses for life-cycle design and management is discussed.

The effect of statistical variability in material properties on springback

A major concern with manufacturing processes using metals is the accurate prediction of the response of the material to complex forming operations that shape a blank into a component. One of the more difficult problems is springback, which is the tendency of a metal to not maintain the desired shape after the forming process is terminated. Springback is more pronounced in high-strength steel and aluminium alloys than other metals. Unless springback is accurately estimated, manufactured parts may deviate excessively from the design specifications. The purpose of this paper is to use mechanistically based probability modelling to accurately estimate variability in the dimensions of a component and the adherence to design tolerances. It is assumed that the variability in the component dimensions is due to randomness in the material properties that directly affect the magnitude of the springback. The yield stress is the primary source of randomness and is most critical in the springback prediction. In order to assess the effects of variability, a simple model for elastic springback in a straight rectangular bar bent around a circular mandrel is considered. Explicit computations are made for a commercially available aluminium alloy for which material properties are readily available. The stress–strain behaviour of the alloy is assumed to be bilinear. The effects of the stress–strain behaviour, the model parameters, and the statistical variability in material properties on springback are assessed.

Statistical and Probabilistic Analysis of Thermal-Fatigue Test Data Generated Using V-Shape Specimen Testing Method

V-shape specimen testing is a relatively new, simple and useful technique to characterize the thermal-fatigue resistance of materials subjected to combined thermal/mechanical loadings, and to rank and select materials. However, the V-shape specimen test data, similar to many other life test data, always contain an inherent scatter not only because of material non-uniformity but also of the difficulties in operating control, such as loading, boundary conditions, and environment. Therefore, statistical and probabilistic approaches have to be used to interpret the test data in order to implement the observations into new product designs. In this paper, the V-shape specimen test data are selected, analyzed and the scatter properties of the test data are fitted using several continuous probability distribution functions. The results are compared, and the root failure mechanisms of the V-specimens are also discussed. Finally, the main observations are summarized, and a recommendation is provided.Copyright