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Interpretation of Zerodur® Strength Data

Recent, detailed fractographic analysis of Zerodur® strength test specimens prepared by linear grinding and etching with a proprietary process indicated low frequency damage to be the strength limiting defects Recent, detailed fractographic analysis of Zerodur® strength test specimens prepared by linear grinding and etching with a proprietary process indicated low frequency damage to be the strength limiting defects [1]. The 2-parameter Weibull distribution is usually assumed when working with ceramic and glasses such as Zerodur®, although 3-parameter behavior is occasionally considered [2]. Detailed statistical modeling of the Zerodur® strength data with the was initial perspective of a 3-parameter Weibull distribution gave unsatisfying results [3]. The subsequent fractographic investigation indicated [1] that the usual assumption of many small, random, noninteracting flaws [4] was not represented, but instead, less frequent, aligned flaws along etching ridges parallel to the grinding direction were present, Figure 1. It was thus concluded that the strength of Zerodur® as prepared was not Weibull distributed. The flaws represent a sparse flaw population relative to typical grinding damage, but an extensive occurrence (high frequency) of handling damage. We examine the type of distribution present and explain the appearance of a 3-parameter distribution.

Glass, ceramic, strength, Weibull, distribution, m↗

A Multiscale Progressive Failure Modeling Methodology for Composites that Includes Fiber Strength Stochastics

A multiscale modeling methodology was developed for continuous fiber composites that incorporates a statistical distribution of fiber strengths into coupled multiscale micromechanics/finite element (FE) analyses. A modified two-parameter Weibull cumulative distribution function, which accounts for the effect of fiber length on the probability of failure, was used to characterize the statistical distribution of fiber strengths. A parametric study using the NASA Micromechanics Analysis Code with the Generalized Method of Cells (MAC/GMC) was performed to assess the effect of variable fiber strengths on local composite failure within a repeating unit cell (RUC) and subsequent global failure. The NASA code FEAMAC and the ABAQUS finite element solver were used to analyze the progressive failure of a unidirectional SCS-6/TIMETAL 21S metal matrix composite tensile dogbone specimen at 650 degC. Multiscale progressive failure analyses were performed to quantify the effect of spatially varying fiber strengths on the RUC-averaged and global stress-strain responses and failure. The ultimate composite strengths and distribution of failure locations (predominately within the gage section) reasonably matched the experimentally observed failure behavior. The predicted composite failure behavior suggests that use of macroscale models that exploit global geometric symmetries are inappropriate for cases where the actual distribution of local fiber strengths displays no such symmetries. This issue has not received much attention in the literature. Moreover, the model discretization at a specific length scale can have a profound effect on the computational costs associated with multiscale simulations.models that yield accurate yet tractable results.

Generalized Method of Cells↗

Ceramics Analysis and Reliability Evaluation of Structures (CARES). Users and programmers manual

This manual describes how to use the Ceramics Analysis and Reliability Evaluation of Structures (CARES) computer program. The primary function of the code is to calculate the fast fracture reliability or failure probability of macroscopically isotropic ceramic components. These components may be subjected to complex thermomechanical loadings, such as those found in heat engine applications. The program uses results from MSC/NASTRAN or ANSYS finite element analysis programs to evaluate component reliability due to inherent surface and/or volume type flaws. CARES utilizes the Batdorf model and the two-parameter Weibull cumulative distribution function to describe the effect of multiaxial stress states on material strength. The principle of independent action (PIA) and the Weibull normal stress averaging models are also included. Weibull material strength parameters, the Batdorf crack density coefficient, and other related statistical quantities are estimated from four-point bend bar or unifrom uniaxial tensile specimen fracture strength data. Parameter estimation can be performed for single or multiple failure modes by using the least-square analysis or the maximum likelihood method. Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests, ninety percent confidence intervals on the Weibull parameters, and Kanofsky-Srinivasan ninety percent confidence band values are also provided. The probabilistic fast-fracture theories used in CARES, along with the input and output for CARES, are described. Example problems to demonstrate various feature of the program are also included. This manual describes the MSC/NASTRAN version of the CARES program.

Nemeth, Noel N.↗

Design of ceramic components with the NASA/CARES computer program

The ceramics analysis and reliability evaluation of structures (CARES) computer program is described. The primary function of the code is to calculate the fast-fracture reliability or failure probability of macro-scopically isotropic ceramic components. These components may be subjected to complex thermomechanical loadings, such as those found in heat engine applications. CARES uses results from MSC/NASTRAN or ANSYS finite-element analysis programs to evaluate how inherent surface and/or volume type flaws component reliability. CARES utilizes the Batdorf model and the two-parameter Weibull cumulative distribution function to describe the effects of multiaxial stress states on material strength. The principle of independent action (PIA) and the Weibull normal stress averaging models are also included. Weibull material strength parameters, the Batdorf crack density coefficient, and other related statistical quantities are estimated from four-point bend bar or uniform uniaxial tensile specimen fracture strength data. Parameter estimation can be performed for a single or multiple failure modes by using a least-squares analysis or a maximum likelihood method. Kolmogorov-Smirnov and Anderson-Darling goodness-to-fit-tests, 90 percent confidence intervals on the Weibull parameters, and Kanofsky-Srinivasan 90 percent confidence band values are also provided. Examples are provided to illustrate the various features of CARES.

Nemeth, Noel N.↗

Improved silicon carbide for advanced heat engines

The development of high strength, high reliability silicon carbide parts with complex shapes suitable for use in advanced heat engines is studied. Injection molding was the forming method selected for the program because it is capable of forming complex parts adaptable for mass production on an economically sound basis. The goals were to reach a Weibull characteristic strength of 550 MPa (80 ksi) and a Weibull modulus of 16 for bars tested in four-point loading. Statistically designed experiments were performed throughout the program and a fluid mixing process employing an attritor mixer was developed. Compositional improvements in the amounts and sources of boron and carbon used and a pressureless sintering cycle were developed which provided samples of about 99 percent of theoretical density. Strengths were found to improve significantly by annealing in air. Strengths in excess of 550 MPa (80 ksi) with Weibull modulus of about 9 were obtained. Further improvements in Weibull modulus to about 16 were realized by proof testing. This is an increase of 86 percent in strength and 100 percent in Weibull modulus over the baseline data generated at the beginning of the program. Molding yields were improved and flaw distributions were observed to follow a Poisson process. Magic angle spinning nuclear magnetic resonance spectra were found to be useful in characterizing the SiC powder and the sintered samples. Turbocharger rotors were molded and examined as an indication of the moldability of the mixes which were developed in this program.

Whalen, Thomas J.↗

Fracture mechanics concepts in reliability analysis of monolithic ceramics

Basic design concepts for high-performance, monolithic ceramic structural components are addressed. The design of brittle ceramics differs from that of ductile metals because of the inability of ceramic materials to redistribute high local stresses caused by inherent flaws. Random flaw size and orientation requires that a probabilistic analysis be performed in order to determine component reliability. The current trend in probabilistic analysis is to combine linear elastic fracture mechanics concepts with the two parameter Weibull distribution function to predict component reliability under multiaxial stress states. Nondestructive evaluation supports this analytical effort by supplying data during verification testing. It can also help to determine statistical parameters which describe the material strength variation, in particular the material threshold strength (the third Weibull parameter), which in the past was often taken as zero for simplicity.

Manderscheid, Jane M.↗

Probabilistic Analysis for Comparing Fatigue Data Based on Johnson-Weibull Parameters

Probabilistic failure analysis is essential when analysis of stress-life (S-N) curves is inconclusive in determining the relative ranking of two or more materials. In 1964, L. Johnson published a methodology for establishing the confidence that two populations of data are different. Simplified algebraic equations for confidence numbers were derived based on the original work of L. Johnson. Using the ratios of mean life, the resultant values of confidence numbers deviated less than one percent from those of Johnson. It is possible to rank the fatigue lives of different materials with a reasonable degree of statistical certainty based on combined confidence numbers. These equations were applied to rotating beam fatigue tests that were conducted on three aluminum alloys at three stress levels each. These alloys were AL 2024, AL 6061, and AL 7075. The results were analyzed and compared using ASTM Standard E739-91 and the Johnson-Weibull analysis. The ASTM method did not statistically distinguish between AL 6010 and AL 7075. Based on the Johnson-Weibull analysis confidence numbers greater than 99 percent, AL 2024 was found to have the longest fatigue life, followed by AL 7075, and then AL 6061. The ASTM Standard and the Johnson-Weibull analysis result in the same stress-life exponent p for each of the three aluminum alloys at the median or L(sub 50) lives.

Hendricks, Robert C.↗

Joint aircraft loading/structure response statistics of time to service crack initiation

A reliability analysis for predicting the statistical distribution of time to fatigue crack initiation for aircraft structures in service is presented. The present analysis utilizes the statistical data of the specimen fatigue tests, the full-scale structure tests, and the statistical dispersion of aircraft service loads. The statistical distribution of the time to fatigue crack initiation of the full-scale structure under laboratory loading spectrum is assumed to be Weibull. The service loads for gust turbulences are modeled as Poisson processes for transport-type aircraft, while the maneuver loads are modeled as compound Poisson processes for fighter and training aircraft. It is found that the statistical distribution of time to fatigue crack initiation for aircraft structures in service is not Weibull and that the prediction on the basis of the Weibull distribution is unconservative, in particular in the early service time.

Yang, J.-N.↗

Test Population Selection from Weibull-Based, Monte Carlo Simulations of Fatigue Life

Fatigue life is probabilistic and not deterministic. Experimentally establishing the fatigue life of materials, components, and systems is both time consuming and costly. As a result, conclusions regarding fatigue life are often inferred from a statistically insufficient number of physical tests. A proposed methodology for comparing life results as a function of variability due to Weibull parameters, variability between successive trials, and variability due to size of the experimental population is presented. Using Monte Carlo simulation of randomly selected lives from a large Weibull distribution, the variation in the L10 fatigue life of aluminum alloy AL6061 rotating rod fatigue tests was determined as a function of population size. These results were compared to the L10 fatigue lives of small (10 each) populations from AL2024, AL7075 and AL6061. For aluminum alloy AL6061, a simple algebraic relationship was established for the upper and lower L10 fatigue life limits as a function of the number of specimens failed. For most engineering applications where less than 30 percent variability can be tolerated in the maximum and minimum values, at least 30 to 35 test samples are necessary. The variability of test results based on small sample sizes can be greater than actual differences, if any, that exists between materials and can result in erroneous conclusions. The fatigue life of AL2024 is statistically longer than AL6061 and AL7075. However, there is no statistical difference between the fatigue lives of AL6061 and AL7075 even though AL7075 had a fatigue life 30 percent greater than AL6061.

Vlcek, Brian L.↗

Test Population Selection from Weibull-Based, Monte Carlo Simulations of Fatigue Life

Fatigue life is probabilistic and not deterministic. Experimentally establishing the fatigue life of materials, components, and systems is both time consuming and costly. As a result, conclusions regarding fatigue life are often inferred from a statistically insufficient number of physical tests. A proposed methodology for comparing life results as a function of variability due to Weibull parameters, variability between successive trials, and variability due to size of the experimental population is presented. Using Monte Carlo simulation of randomly selected lives from a large Weibull distribution, the variation in the L10 fatigue life of aluminum alloy AL6061 rotating rod fatigue tests was determined as a function of population size. These results were compared to the L10 fatigue lives of small (10 each) populations from AL2024, AL7075 and AL6061. For aluminum alloy AL6061, a simple algebraic relationship was established for the upper and lower L10 fatigue life limits as a function of the number of specimens failed. For most engineering applications where less than 30 percent variability can be tolerated in the maximum and minimum values, at least 30 to 35 test samples are necessary. The variability of test results based on small sample sizes can be greater than actual differences, if any, that exists between materials and can result in erroneous conclusions. The fatigue life of AL2024 is statistically longer than AL6061 and AL7075. However, there is no statistical difference between the fatigue lives of AL6061 and AL7075 even though AL7075 had a fatigue life 30 percent greater than AL6061.

Vlcek, Brian L.↗

Statistical structural analysis of rotor impact ice shedding

The statistical characteristics of impact ice shear strength are analyzed, with emphasis placed on the most probable shear strength and statistical distribution of an ice deposit. Several distribution types are considered: the Weibull, two-parameter Weibull, and exponential distributions, as well as the Gumbell distribution of the smallest extreme and the Gumbell distribution of the largest extreme. It is concluded that the Weibull distribution yields the best results; however, the expected life, shape parameter, and scale parameter should be determined separately for each case of varying wind speed and droplet size. The theoretical predictions of shear stresses in a specific rotating ice shape are compared, and it is noted that when the effects of lift are added to the theoretical model and the interference is calculated with a new mean and standard deviation, the probability of ice shed is computed as 36.64 pct.

Kellacky, C. J.↗

Structural Design Parameters for Germanium

The fracture toughness and slow crack growth parameters of germanium supplied as single crystal beams and coarse grain disks were measured. Although germanium is anisotropic (A* 1.7), it is not as anisotropic as SiC, NiAl, or Cu. Thus the fracture toughness was similar on the 100, 110, and 111 planes, however, measurements associated with randomly oriented grinding cracks were 6 to 30 higher. Crack extension in ring loaded disks occurred on the 111 planes due to both the lower fracture energy and the higher stresses on stiff 111 planes. Germanium exhibits a Weibull scale effect, but does not exhibit significant slow crack growth in distilled water. (n 100), implying that design for quasi static loading can be performed with scaled strength statistics. Practical values for engineering design are a fracture toughness of 0.69 0.02 MPam (megapascals per square root meter) and a Weibull modulus of m 6 2. For well ground and reasonable handled coupons, average fracture strength should be greater than 40 megapascals. Aggregate, polycrystalline elastic constants are Epoly 131 gigapascals, vpoly 0.22.

fracture toughness↗

SCARE: A post-processor program to MSC/NASTRAN for the reliability analysis of structural ceramic components

A computer program was developed for calculating the statistical fast fracture reliability and failure probability of ceramic components. The program includes the two-parameter Weibull material fracture strength distribution model, using the principle of independent action for polyaxial stress states and Batdorf's shear-sensitive as well as shear-insensitive crack theories, all for volume distributed flaws in macroscopically isotropic solids. Both penny-shaped cracks and Griffith cracks are included in the Batdorf shear-sensitive crack response calculations, using Griffith's maximum tensile stress or critical coplanar strain energy release rate criteria to predict mixed mode fracture. Weibull material parameters can also be calculated from modulus of rupture bar tests, using the least squares method with known specimen geometry and fracture data. The reliability prediction analysis uses MSC/NASTRAN stress, temperature and volume output, obtained from the use of three-dimensional, quadratic, isoparametric, or axisymmetric finite elements. The statistical fast fracture theories employed, along with selected input and output formats and options, are summarized. An example problem to demonstrate various features of the program is included.

Gyekenyesi, J. P.↗

SCARE - A postprocessor program to MSC/NASTRAN for reliability analysis of structural ceramic components

A computer program was developed for calculating the statistical fast fracture reliability and failure probability of ceramic components. The program includes the two-parameter Weibull material fracture strength distribution model, using the principle of independent action for polyaxial stress states and Batdorf's shear-sensitive as well as shear-insensitive crack theories, all for volume distributed flaws in macroscopically isotropic solids. Both penny-shaped cracks and Griffith cracks are included in the Batdorf shear-sensitive crack response calculations, using Griffith's maximum tensile stress or critical coplanar strain energy release rate criteria to predict mixed mode fracture. Weibull material parameters can also be calculated from modulus of rupture bar tests, using the least squares method with known specimen geometry and fracture data. The reliability prediction analysis uses MSC/NASTRAN stress, temperature and volume output, obtained from the use of three-dimensional, quadratic, isoparametric, or axisymmetric finite elements. The statistical fast fracture theories employed, along with selected input and output formats and options, are summarized. An example problem to demonstrate various features of the program is included.

Gyekenyesi, J. P.↗

The Influence of Roughness on Gear Surface Fatigue

Gear working surfaces are subjected to repeated rolling and sliding contacts, and often designs require loads sufficient to cause eventual fatigue of the surface. This research provides experimental data and analytical tools to further the understanding of the causal relationship of gear surface roughness to surface fatigue. The research included evaluations and developments of statistical tools for gear fatigue data, experimental evaluation of the surface fatigue lives of superfinished gears with a near-mirror quality, and evaluations of the experiments by analytical methods and surface inspections. Alternative statistical methods were evaluated using Monte Carlo studies leading to a final recommendation to describe gear fatigue data using a Weibull distribution, maximum likelihood estimates of shape and scale parameters, and a presumed zero-valued location parameter. A new method was developed for comparing two datasets by extending the current methods of likelihood-ratio based statistics. The surface fatigue lives of superfinished gears were evaluated by carefully controlled experiments, and it is shown conclusively that superfinishing of gears can provide for significantly greater lives relative to ground gears. The measured life improvement was approximately a factor of five. To assist with application of this finding to products, the experimental condition was evaluated. The fatigue life results were expressed in terms of specific film thickness and shown to be consistent with bearing data. Elastohydrodynamic and stress analyses were completed to relate the stress condition to fatigue. Smooth-surface models do not adequately explain the improved fatigue lives. Based on analyses using a rough surface model, it is concluded that the improved fatigue lives of superfinished gears is due to a reduced rate of near-surface micropitting fatigue processes, not due to any reduced rate of spalling (sub-surface) fatigue processes. To complete the evaluations, surface inspection were completed. The surface topographies of the ground gears changed substantially due to running, but the topographies of the superfinished gears were essentially unchanged with running.

Krantz, Timothy↗

Rolling Bearing Life Prediction-Past, Present, and Future

Comparisons were made between the life prediction formulas of Lundberg and Palmgren, Ioannides and Harris, and Zaretsky and full-scale ball and roller bearing life data. The effect of Weibull slope on bearing life prediction was determined. Life factors are proposed to adjust the respective life formulas to the normalized statistical life distribution of each bearing type. The Lundberg-Palmgren method resulted in the most conservative life predictions compared to Ioannides and Harris, and Zaretsky methods which produced statistically similar results. Roller profile can have significant effects on bearing life prediction results. Roller edge loading can reduce life by as much as 98 percent. The resultant predicted life not only depends on the life equation used but on the Weibull slope assumed, the least variation occurring with the Zaretsky equation. The load-life exponent p of 10/3 used in the American National Standards Institute (ANSI)/American Bearing Manufacturers Association (ABMA)/International Organization for Standardization (ISO) standards is inconsistent with the majority roller bearings designed and used today.

Zaretsky, E V↗

Simulation of flight maneuver-load distributions by utilizing stationary, non-Gaussian random load histories

Random numbers were generated with the aid of a digital computer and transformed such that the probability density function of a discrete random load history composed of these random numbers had one of the following non-Gaussian distributions: Poisson, binomial, log-normal, Weibull, and exponential. The resulting random load histories were analyzed to determine their peak statistics and were compared with cumulative peak maneuver-load distributions for fighter and transport aircraft in flight.

Leybold, H. A.↗

Computing Reliabilities Of Ceramic Components

CARES/PC computer program performs statistical analysis of data obtained from fracture of simple, uniaxial tensile or flexural specimens of ceramics and estimates Weibull and Batdorf material parameters from these data. CARES/PC is subset of Ceramics Analysis and Reliability Evaluation of Structures (CARES) program (LEW-15168), which calculates fast-fracture reliabilities or failure probabilities of ceramic components by use of Batdorf and Weibull models to describe effects of multiaxial stress states on strengths of materials. CARES/PC written and compiled with the Microsoft FORTRAN v5.0 compiler.

Szatmary, S. A.↗