Approximate J estimates for axial part-through surface-cracked pipes

This paper provides net‐section limit pressures and a reference stress based J estimation method for pipes with constant depth, internal axial surface cracks under internal pressure. Based on systematic small strain finite element (FE) limit analyses using elastic perfectly plastic materials, net‐section limit pressures are firstly determined, and based on FE results, a closed‐form limit pressure solution is proposed. Furthermore, based on the proposed limit pressure solution, a method to estimate elastic–plastic J is proposed based on the reference stress approach. When the reference stress is defined by the proposed (global) limit pressure, estimated J values based on the reference stress approach are overall slightly lower than FE results, implying that the method is non‐conservative. By re‐defining the reference using optimised reference loads, resulting J estimates agree well with FE results.     

Estimation of Field Reliability Based on Aggregate Lifetime Data

Because of the exponential distribution assumption, many reliability databases recorded data in an aggregate way. Instead of individual failure times, each aggregate data point is a summation of a series of collective failures representing the cumulative operating time of one component position from system commencement to the last component replacement. The data format is different from traditional lifetime data and the statistical inference is challenging. We first model the individual component lifetime by a gamma distribution. Confidence intervals for the gamma shape parameter can be constructed using a scaled χ² approximation to a modified ratio of the geometric mean to the arithmetic mean, while confidence intervals for the gamma rate and mean parameters, as well as quantiles, are obtained using the generalized pivotal quantity method. We then fit the data using the inverse Gaussian (IG) distribution, a useful lifetime model for failures caused by degradation. Procedures for point estimation and interval estimation of parameters are developed. We also propose an interval estimation method for the quantiles of an IG distribution based on the generalized pivotal quantity method. An illustrative example demonstrates the proposed inference methods. Supplementary materials for this article are available online.     

A framework for design optimization using surrogates

Design optimization is a computationally expensive process as it requires the assessment of numerous designs and each of such assessments may be based on expensive analyses (e.g. computational fluid dynamics method or finite element based method). One way to contain the computational time within affordable limits is to use computationally cheaper approximations (surrogates) in lieu of the actual analyses during the course of optimization. This article introduces a framework for design optimization using surrogates. The framework is built upon a stochastic, zero-order, population-based optimization algorithm, which is embedded with a modified elitism scheme, to ensure convergence in the actual function space. The accuracy of the surrogate model is maintained via periodic retraining and the number of data points required to create the surrogate model is identified by a k-means clustering algorithm. A comparison is provided between different surrogate models (Kriging, radial basis functions (Exact and Fixed) and Cokriging) using a number of mathematical test functions and engineering design optimization problems. The results clearly indicate that for a given fixed number of actual function evaluations, the surrogate assisted optimization model consistently performs better than a pure optimization model using actual function evaluations.     

Further developments in <Emphasis Type="Italic">J</Emphasis> evaluation procedure for growing cracks based on LLD and CMOD data

Laboratory testing of fracture specimens to measure resistance curves (J − Δa) have focused primarily on the unloading compliance method using a single specimen. Current estimation procedures (which form the basis of ASTM E1820 standard) employ load line displacement (LLD) records to measure fracture toughness resistance data incorporating a crack growth correction for J. An alternative method which potentially simplifies the test procedure involves the use of crack mouth opening displacement (CMOD) to determine both crack growth and J. However, while the J-correction for crack growth effects adopted by ASTM standard holds true for resistance curves measured using load line displacement (LLD) data, it becomes unsuitable for J-resistance measurements based upon the specimen response defined in terms of load-crack mouth opening displacement (CMOD). Consequently, direct application of the evaluation procedure for J derived from LLD records in laboratory measurements of resistance curves using CMOD data becomes questionable. This study provides further developments of the evaluation procedure for J in cracked bodies that experience ductile crack growth based upon the eta-method and CMOD data. The introduction of a constant relationship between the plastic components of LLD (Δ _p ) and CMOD (V _p ) drives the development of a convenient crack growth correction for J with increased loading when using laboratory measurements of P-CMOD data. The methodology broadens the applicability of current standards adopting the unloading compliance technique in laboratory measurements of fracture toughness resistance data (J resistance curves). The developed J evaluation formulation for growing cracks based on CMOD data provides a viable and simpler test technique to measure crack growth resistance data for ductile materials.     

Theory for Optimal Test Plans for the Random Fatigue-Limit Model

This article presents methodology for planning life tests based on the random fatigue-limit model (RFLM). The RFLM describes the relationship between fatigue life and the applied load in the presence of fatigue limits. A standardization of the RFLM is presented, and corresponding expressions for the elements of the Fisher information matrix are given. Different test plan criteria that express various objectives that the practitioner may have in conducting experiments under the RFLM are discussed. These criteria include estimation of life quantiles, estimation of stress/strain levels that yield a specific life quantile, estimation of fatigue-limit quantiles, and D and D_s optimality. Equivalence theorems used to verify optimality of test plans are also presented. The methods are sufficiently general so they can be applied to criteria based on any function of the model parameters. They are demonstrated using a nickel-base superalloy experiment.     

Evaluation of the ISO <Emphasis Type="Italic">J</Emphasis> initiation procedure using the EURO fracture toughness data set

The multiple specimen J _0.2/BL initiation fracture toughness test procedure from the ISO standard, ISO 12135:2002, is evaluated using the EURO fracture toughness data set. This standard is also compared with the ASTM standard, ASTM E 1820, multiple specimen J _Ic procedure. The EURO round robin data set was generated to evaluate the transition fracture toughness methods for steels. However, many of the tests resulted in ductile fracture behavior giving final J versus ductile crack extension points. This is the information that is measured in a multiple specimen J initiation fracture toughness test. The data set has more than 300 individual points of J versus crack extension with four different specimen sizes. It may be the largest data set of that type produced for one material. Therefore, its use to determine J initiation values can provide an important evaluation of the standard procedures. The results showed that a J _0.2/BL value could be determined from the ISO standard for three of the four specimen sizes, the smallest size did not meet the specimen size requirement on J. The construction line slopes in this method are very steep compared with the ASTM construction line slopes. This resulted in low J initiation values, about a factor of two lower than the one from the ASTM method. Of the various criteria imposed to determine a valid J _0.2/BL value, the one limiting the maximum J value was the most questionable. It had an effect of eliminating small specimen data that was identical to acceptable large specimen data.     

Effects of weld strength mismatch on <Emphasis Type="Italic">J</Emphasis> and CTOD estimation procedure for SE(B) specimens

This work examines the effect of weld strength mismatch on fracture toughness measurements defined by J and CTOD fracture parameters using single edge notch bend (SE(B)) specimens. A central objective of the present study is to enlarge on previous developments of J and CTOD estimation procedures for welded bend specimens based upon plastic eta factors (η) and plastic rotational factors (r _p ). Very detailed non-linear finite element analyses for plane-strain models of standard SE(B) fracture specimens with a notch located at the center of square groove welds and in the heat affected zone provide the evolution of load with increased crack mouth opening displacement required for the estimation procedure. One key result emerging from the analyses is that levels of weld strength mismatch within the range ±20% mismatch do not affect significantly J and CTOD estimation expressions applicable to homogeneous materials, particularly for deeply cracked fracture specimens with relatively large weld grooves. The present study provides additional understanding on the effect of weld strength mismatch on J and CTOD toughness measurements while, at the same time, adding a fairly extensive body of results to determine parameters J and CTOD for different materials using bend specimens with varying geometries and mismatch levels.     

Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function

This letter introduces a new approach for the demodulation of fringe patterns recorded in holographic interferometry using high-order ambiguity function (HAF). The proposed approach is capable of retrieving the phase from a single fringe pattern. The main advantage of this approach is that it directly provides an estimation of the continuous phase distribution and thereby avoids the necessity of using a cumbersome 2D phase unwrapping procedure. This method first computes the discrete-time analytic signal of the recorded fringe pattern. Then, by modelling this analytic signal as a polynomial phase signal embedded in additive complex white Gaussian noise, a parametric estimation procedure based on HAF is employed to directly estimate the unwrapped phase distribution. Numerical simulations and experimental results demonstrate the potential of the proposed approach.     

Accounting for randomness in measurement and sampling in studying cancer cell population dynamics

Knowing the expected temporal evolution of the proportion of different cell types in sample tissues gives an indication about the progression of the disease and its possible response to drugs. Such systems have been modelled using Markov processes. We here consider an experimentally realistic scenario in which transition probabilities are estimated from noisy cell population size measurements. Using aggregated data of FACS measurements, we develop MMSE and ML estimators and formulate two problems to find the minimum number of required samples and measurements to guarantee the accuracy of predicted population sizes. Our numerical results show that the convergence mechanism of transition probabilities and steady states differ widely from the real values if one uses the standard deterministic approach for noisy measurements. This provides support for our argument that for the analysis of FACS data one should consider the observed state as a random variable. The second problem we address is about the consequences of estimating the probability of a cell being in a particular state from measurements of small population of cells. We show how the uncertainty arising from small sample sizes can be captured by a distribution for the state probability.Inspec keywords: cancer, tumours, cellular biophysics, biomedical measurement, Gaussian distribution, maximum likelihood estimation, mean square error methods, hidden Markov models, fluorescence, random processes, convergence of numerical methodsOther keywords: cancer cell population dynamics, malignant tumours, tissue samples, normal tissue cells, disease, drugs, Markov process, cell population size measurement, hidden Markov model, noisy measurement, state transition probability, fluorescence‐activated cell sorting measurement, minimum mean square error estimator, maximum likelihood estimator, transition probability matrix, noise distributions, Gaussian distributions, MMSE, convergence mechanism, standard deterministic approach, stochastic phenomena, random variable     

Elastic-plastic fracture mechanics assessment for steam generator tubes with through-wall cracks

The present work provides an elastic‐plastic fracture mechanics (EPFM) assessment scheme for a steam generator tube with a through‐wall crack under internal pressure. Noting that the geometry and material are rather uniform for steam generator tubes, and furthermore the only loading to be considered is internal pressure, an engineering EPFM analysis method is proposed to assess through‐wall cracks in steam generator tubes. Important outcomes of the present work are closed‐form approximations for J and crack opening displacement (COD). Sufficient confidence in the proposed J and COD estimates is gained from good agreements with the finite element results over a wide range of the crack length and pressure magnitude. Another important element of the EPFM assessment is to determine relevant J‐resistance curve for steam generator tubes. To improve the accuracy of predicting tube failure, the present paper also proposes a new method to determine fracture toughness using an actual tubular specimen instead of using a standard specimen, from which J‐resistance curves of steam generator tubes are obtained. Using the proposed J and toughness estimates, maximum pressures of steam generator tubes with through‐wall crack are estimated based on EPFM analysis, which is compared with experimental results and predicted ones based on limit load approach.     

Semi-parametric second-order reduced-bias high quantile estimation

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.     

Reference Stress Approximations for J and COD of Circumferential Through-Wall Cracked Pipes

Reference stress approximations for the J integral and crack tip opening displacement (COD) for circumferential through-wall cracked pipes under tension and under bending are reported. The proposed J estimation equation is fully compatible with the existing reference stress based J estimation, currently embedded in the R6 assessment procedure, but involves a slightly different definition of the reference stress, using an optimised reference load instead of the limit load. This modification enhances the accuracy of the J estimation for circumferentially cracked pipes. Confidence in the proposed equation is gained from the significantly reduced hardening dependence of the plastic influence functions in the GE/EPRI method. The proposed COD estimation equation includes two further modifications. One is the use of a power-law fit to the plastic portion of the stress strain data, instead of the use of the actual stress strain data. In this context, a robust estimation equation for the strain hardening index is given. The other modification is to the plasticity correction term in contained yielding. A lower bound COD estimation equation is also given, similar to the R6 option 1 Jestimation curve, which is suitable when only limited tensile properties are available. The resulting estimation equations are simple to use. Comparisons with experimental pipe test data show that the proposed COD estimation equations provide overall good agreement, which gives confidence in applying them to Leak-before-Break (LBB) analyses.     

Non-linear fracture mechanics analyses of part circumferential surface cracked pipes

This paper provides engineering estimates of non-linear fracture mechanics parameters for pipes with part circumferential inner surface cracks, subject to internal pressure and global bending. Solutions are given in the form of two different approaches, the GE/EPRI approach and the reference stress approach. For the GE/EPRI approach, the plastic influence functions for fully plastic J solutions are tabulated based on extensive 3-D FE calculations using deformation plasticity, covering a wide range of pipe and crack geometries. The developed GE/EPRI-type fully plastic J estimation equations are then re-formulated using the concept of the reference stress approach for wider applications. The proposed reference stress based estimates are validated against detailed 3-D elastic-plastic and elastic-creep FE results. For a total of 26 cases considered in this paper, agreement between the proposed reference stress based J and C ^* estimates and the FE results is excellent. An important aspect of the proposed estimates is that they not only are simple and accurate but also can be used to estimate J and C ^* at an arbitrary point along the crack front.     

Models for Variable-Stress Accelerated Life Testing Experiments Based on Wener Processes and the Inverse Gaussian Distribution

Variable-stress accelerated life testing trials are experiments in which each of the units in a random sample of units of a product is run under increasingly severe conditions to get information quickly on its life distribution. We consider a fatigue failure model in which accumulated decay is governed by a continuous Gaussian process W(y) whose distribution changes at certain stress change points to < t _l < < … <t _k , Continuously increasing stress is also considered. Failure occurs the first time W(y) crosses a critical boundary ω. The distribution of time to failure for the models can be represented in terms of time-transformed inverse Gaussian distribution functions, and the parameters in models for experiments with censored data can be estimated using maximum likelihood methods. A common approach to the modeling of failure times for experimental units subject to increased stress at certain stress change points is to assume that the failure times follow a distribution that consists of segments of Weibull distributions with the same shape parameter. Our Wiener-process approach gives an alternative flexible class of time-transformed inverse Gaussian models in which time to failure is modeled in terms of accumulated decay reaching a critical level and in which parametric functions are used to express how higher stresses accelerate the rate of decay and the time to failure. Key parameters such as mean life under normal stress, quantiles of the normal stress distribution, and decay rate under normal and accelerated stress appear naturally in the model. A variety of possible parameterizations of the decay rate leads to flexible modeling. Model fit can be checked by percentage-percentage plots.     

Sequential approximate robust design optimization using radial basis function network

This paper proposes a sequential approximate robust design optimization (SARDO) with the radial basis function (RBF) network. In RDO, the mean and the standard deviation of objective should be minimized simultaneously. Therefore, the RDO is generally formulated as bi-objective design optimization. Our goal is to find a robust optimal solution with a small number of function evaluations, not identifying a set of Pareto-optimal solution using Multi-Objective Evolutionary Algorithms. The weighted sum is often used to find a robust optimal solution. In contrast, the weighted lp norm method is used in this paper. Through illustrative examples, some validations of the weighted lp norm method to the RDO are clarified. Next, SARDO with the RBF network is discussed. In general, the standard deviation of functions is obtained by using the finite difference method. Thus, in order to obtain the standard deviation of functions, the finite difference method is directly applied to the response surface. High accuracy of the finite difference method will leads to highly accurate robust optimal solution. In order to avoid the inaccurate numerical calculation, the standard deviation is expressed by only the Gaussian kernel. As the result, it is expected that a highly accurate robust optimal solution can be found with a small number of function evaluations. Through numerical examples, the validity of the proposed approach is examined. Finally, the variable blank holder force trajectory for reducing springback is examined.     

Fracture mechanics analysis of a crack in a residual stress field

The standard definition of the J integral leads to a path dependent value in the presence of a residual stress field, and this gives rise to numerical difficulties in numerical modelling of fracture problems when residual stresses are significant. In this work, a path independent J definition for a crack in a residual stress field is obtained. A number of crack geometries containing residual stresses have been analysed using the finite element method and the results demonstrate that the modified J shows good path-independence which is maintained under a combination of residual stress and mechanical loading. It is also shown that the modified J is equivalent to the stress intensity factor, K, under small scale yielding conditions and provides the intensity of the near crack tip stresses under elastic-plastic conditions. The paper also discusses two issues linked to the numerical modelling of residual stress crack problems-the introduction of a residual stress field into a finite element model and the introduction of a crack into a residual stress field.     

LQG optimal controller synthesis with robust optimization in multivariable systems

Abstract

This paper presents an algorithm to synthesize a controller for treating the problem of robustness optimization in an LQG (linear‐quadratic‐Gaussian) control system. The controller not only maximizes the excess stability margin in perturbed system but also minimizes the cost functional J in LQG problems by specifying two frequency dependent weighting matrices Q(s) and R(s) in the cost functional.

Our approach is based on Wiener‐Hopf's technique (frequency domain approach), and two weighting matrices in cost functional are shaped by inverse LQG method. The feature of this paper is that the plant of the system has no stable, proper, square, and minimum phase constraints.     

On the evaluation of the J-integral by a plastic crack tip element

Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J ₂ flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.     

Mode I–III Decomposition of the J‐integral from DIC Displacement Data

This paper presents the implementation of the decomposition method on digital image correlation (DIC) obtained displacement fields to obtain J‐integral results (J) and respective stress intensity factors (SIFs). DIC is increasingly used with the J‐integral approach in experimental mechanics to obtain J estimates from complex fracture processes. In this approach, the decomposition method is applied to DIC displacement fields for the first time. Here, displacement fields are separated before stresses and strains are computed, so that subsequent computation of separate J or SIF components may follow the classical full‐field J‐integral approach. The sensitivity of the decomposition method to experimental errors is investigated using synthetically generated errors imposed on crack tip displacement fields (Williams' series), from which improvements to the procedure are proposed. The method is experimentally tested on PMMA Arcan specimens under mode I, II, and III, and mixed‐mode I–III loading. Test results were compared to fracture toughness values obtained from ASTM tests and literature with close agreement.     

A response-modeling alternative to surrogate models for support in computational analyses

Often, the objectives in a computational analysis involve characterization of system performance based on some function of the computed response. In general, this characterization includes (at least) an estimate or prediction for some performance measure and an estimate of the associated uncertainty. Surrogate models can be used to approximate the response in regions where simulations were not performed. For most surrogate modeling approaches, however, (1) estimates are based on smoothing of available data and (2) uncertainty in the response is specified in a point-wise (in the input space) fashion. These aspects of the surrogate model construction might limit their capabilities.One alternative is to construct a probability measure, G(r), for the computer response, r, based on available data. This “response-modeling” approach will permit probability estimation for an arbitrary event, E(r), based on the computer response. In this general setting, event probabilities can be computed: prob(E)=∫_rI(E(r))dG(r) where I is the indicator function. Furthermore, one can use G(r) to calculate an induced distribution on a performance measure, pm. For prediction problems where the performance measure is a scalar, its distribution F_pm is determined by: F_pm(z)=∫_rI(pm(r)z)dG(r). We introduce response models for scalar computer output and then generalize the approach to more complicated responses that utilize multiple response models.