Pseudo global energy released locally by crack extension involving multiscale reliability

The energy release rate criterion, being mono scale by definition, is incompatible with the failure behavior of solids that are inherently dual, if not, multiscale. Time span of reliability is scale sensitive and can be addressed with consistency only by use of transitional functions that are designed to transform a function from one scale to another. A pseudo transitional energy release rate G^∗ is defined to address the cross-scaling properties of energy release rate. The reliability of such a function is found to fall quickly when the scale range deviates from that of micro-macro. In general, the time span of reliability based on G^* shortens considerably within the nano-micro and pico-nano scale ranges, resulting in fast turnover of system usability. Prediction accuracy tends to be scale range specific. Stress or strain based criteria are also mono scale. They may be adequate for some situations at the macroscopic scale, but can be ambiguous for multiscale problems. These situations are analyzed by application of the principle of least variance in conjunction with the R-integrals.Accelerated test data for the equivalent of 20 years’ fatigue crack growth in 2024-T3 aluminum panels were analyzed using the mutliscale reliability model. A time span plateau within the micro-macro range is from 8 to 17 years. This corresponds to the reliable portion of prediction, while the terminal 3 years are regarded as unreliable. A similar time span plateau were also found from 4 to 6 years within the nano-micro scale range. And an even smaller plateau hovering around 1.2 years were found for the pico-nano scale range. Time span of reliable prediction narrows with down sized scale range. The overlapping ends of the scale ranges are rendered unreliable as anticipated. These regions can be suppressed by the addition of meso scale ranges. Reference can be made to past discussions related to multiscaling and mesomechanics.     

Principle of least variance for dual scale reliability of structural systems

A R-integral is defined to account for the evolution of the root functions from Ideomechanics. They can be identified with, though not limited to, the fatigue crack length or velocity. The choice was dictated by the available validated data for relating accelerated testing to real time life expectancy. The key issue is to show that there exists a time range of high reliability for the crack length and velocity that correspond to the least variance of the time dependent R-integrals. Excluded from the high reliability time range are the initial time span where the lower scale defects are predominant and the time when the macrocrack approaches instability at relatively high velocity. What remains is the time span for micro-macro cracking. The linear sum (ls) and root mean square (rms) average are used to delineate two different types of variance. The former yields a higher reliability in comparison with that for the latter. The results support the scale range established empirically by in-service health monitoring for the crack length and velocity. The principle of least variance can be extended to multiscale reliability analysis and assessment for multi-component and multi-function systems.     

Scale shifting laws from pico to macro in consecutive segments by use of transitional functions

Systems with parts that vary in size from pico to macro inclusive are vulnerable of being incapacitated when a single part fails owing to deterioration of material properties. The majority of system failure can be attributed to incompatibility of integrated parts that were designed individually for general purpose. Total reliability calls for all parts, small and large, to be compatible in life spans. Mass, when regarded as energized matter, can vary as a function of time. This, in retrospect, explains why non-equilibrium and non-homogeneity cannot be avoided for multiscale shifting laws. A consistent and scale invariant definition of energy dissipation gives rise to mass pulsation, a common mechanism that seems to be applicable to living and non-living organisms.Scale shifting laws are developed from the use of transitional functions that stand for the mass ratios related to absorption energies and dissipation energies . The notations j and j + 1 stand for two successive scales: pi-na, na-mi, and mi-ma. Hence, the mass ratios , and can be referred to as the transitional inhomogeneity coefficients. They make up the scale shifting laws . Connection of the accelerated test data at the different scales, say from pico to nano to micro to macro, can be made by application of the definition of a scale invariant energy density dissipation function.On physical grounds, the segmented non-equilibrium and non-homogeneous test data can be connected through a velocity dependent mass and energy relation. Energy and power efficiency are defined to explore the macroscopic experiences to those at the lower scales. The time evolution properties of the material can also be derived as a package to include the accelerated test data, a procedure normally referred to as validation. The separation of derive-first and test-later, can never be abridged without ambiguities. Hence, total reliability of a system with many parts is advocated by judiciously matching the nine primary variables consisting of the initial disorder sizes, the time rates, and increments of the absorbed and dissipated energy density. The nine controllable variables consisting of life span distribution, energy, and power efficiencies for the three scale ranges are of secondary consideration.     

Assurance of reliable time limits in fatigue depending on choice of failure simulation: Energy density versus stress intensity

The principle of least variance is applied to evaluate the reliability of the design conditions of the Runyang cable-stayed bridge. Monitored fatigue load in service data are analyzed in conjunction with the specimen fatigue crack growth data for bridge steel. Aside from size differences, the interactive effects of material behavior with load amplitude and frequency would vary with the depicted physical model for the reliability of life prediction. Based on the same crack growth history in time or cycle, the two choice selected for comparison are stress intensity factor (SIF) range, and the strain energy density (SED) range. Reliability is found to depend on the trade off between load amplitude and frequency. Considered are high-amplitude; low-frequency and low-amplitude; high-frequency. In each case, the chances are the reliable time span of fatigue crack growth will not coincide with the useful portion of bridge life, simply because the load frequency must be anticipated as an educated estimate. It is subject to change. Conversion of the crack length fatigue cycle history to the corresponding time history requires the specification of load frequency that can set the time span of the useful life. This is demonstrated for the Runyang bridge, where approximately 30 MPa and 8 MPa would correspond to the high and low fatigue load, respectively.Significant variances were found for the SIF and SED models. The difference can be attributed to the inclusion of the mean stress in the SED that is more forgiving since it accounts for both the stress and strain effects, in contrast to the SIF model that leaves out the strain and the mean stress. Since the principle of least variance refers to the average of the R-integrals, the results based on the linear sum (LS) and root mean square (RMS) will differ quantitatively, but not qualitatively. The obvious mismatch of the fatigue load used to determine the material property and that for the bridge design can be adjusted and absorbed into the appropriate choice for the load frequency, a compensating factor not realized up to now. To this end, the weighted functions in the R-integrals further emphasize long run effects of the least variance reliability analysis. Attention is called to Changeability in addition to determinability and probability for predicting the time to failure. That is to better anticipate the change in the fatigue load frequency, to which the assistance of health monitoring should provide.     

Fatigue crack growth model for assessing reliability of box-girders for cable-stayed bridge combining SHMS with strain data

Enhancement of the computer algorithm developed for the Runyang cable-stayed bridge (RYCB) in China is made by incorporating the fatigue crack growth analysis in contrast to the SN curve approach. Strain data obtained from the structural health monitoring system (SHMS) and finite element calculations are used. This provides the application of a deterministic method in addition to the probabilistic approach with the added feature of crack growth. The choice of selecting the two-parameter fatigue crack growth criterion was based on the definition of reliability index β such that the new results can be compared with those using the SN curves. A gradual drop of the reliability index β with time with an upper limit was obtained for the crack growth model in contrast to the linear time relation for the SN curve model that had no upper limit. This difference is significant and reveals the importance for selecting the fatigue failure criterion. Deterministic and probabilistic crack growth models are used to assess the differences. The results are based on the box-girder component strain history data of the Runyang Cable-stayed Bridge (RYCB) in China, stress history recorded by structural health monitoring system (SHMS) is analyzed using the monitored stress amplitude, mean stress and stress ratio. Finite element calculations are used to supplement data at locations not accessible for measurements. Additional improvement with reference to damage accumulation and the physical meaning of the reliability index will be studies in relation to the fatigue damage of box-girder of long cable-stayed bridges.     

基于可靠性映射函数的结构优化Structural optimization based on reliability mapping functions

基于可靠性分析理论,构建可靠性映射函数,将基于可靠性的结构优化由双层优化转化为单层优化,提高优化效率,解决基于可靠性结构优化的计算量大、不利用于工程应用的问题。可靠性映射函数具有明确理论依据,能够确保将其应用到基于可靠性的结构优化是可行的。为提高失效概率求解精度,以设计点为基础,提出一种改进响应面方法,并将其用于可靠性映射函数的求解。算例表明,该方法具有较好计算精度,功能函数评估次数明显少于其他方法,计算效率高,能够获得满意的优化结果。     

Why the stress trajectories in the Dean-Hutchinson plastic sector of the growing mode III crack are an unfocused fan

The plastic zone of the growing mode III crack in an elastic perfectly plastic solid consists of two sectors in contact with each other. The sector closer to the crack plane, first studied analytically by Chitaley and McClintock (CM), consists of a fan of straight maximum shear stress trajectories that are focused on the crack tip. The other sector, first analyzed numerically by Dean and Hutchinson (DH), is a ‘radial’ fan of straight lines that are not focused at the crack tip or at another common point. In this paper it is shown with use of the dislocation density field that the need that the stress magnitude in the plastic wake be below the yield stress requires the existence of an unfocused fan in the DH sector. It appears unlikely that this result can be obtained without explicit use of dislocations.     

An engineering predictive approach of fatigue crack growth behavior: The case of the lug-type joint

The present attempt proposes a predictive approach of the fatigue crack growth (FCG) behavior of a lug-type joint used in an aeronautic context. The crack tip residual stress distribution and material dispersions are considered. The developed approach was implemented by coupling the Extended Finite Element Method (XFEM), the Residual Corrected Stress Intensity Factor (RC-SIF), developed by the authors, and the Monte Carlo simulation (MCS) method. The Lemaitre–Chaboche model, developed upon the ABAQUS commercial code, was considered for characterizing material behavior. The developed approach treats FCG life by considering the stochastic behavior of material parameters and the crack tip residual stress field during propagation. Comparing with experimental data, the proposed approach exhibits a good ability in evaluating the FCG reliability of a cracked lug-type joint subjected to different loading conditions. The iso-probabilistic P–a–N curves can be used as an efficient tool for ensuring the safety behavior of cracked components.     

A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature

Viscoplastic crack-tip deformation behaviour in a nickel-based superalloy at elevated temperature has been studied for both stationary and growing cracks in a compact tension (CT) specimen using the finite element method. The material behaviour was described by a unified viscoplastic constitutive model with non-linear kinematic and isotropic hardening rules, and implemented in the finite element software ABAQUS via a user-defined material subroutine (UMAT). Finite element analyses for stationary cracks showed distinctive strain ratchetting behaviour near the crack tip at selected load ratios, leading to progressive accumulation of tensile strain normal to the crack-growth plane. Results also showed that low frequencies and superimposed hold periods at peak loads significantly enhanced strain accumulation at crack tip. Finite element simulation of crack growth was carried out under a constant ΔK-controlled loading condition, again ratchetting was observed ahead of the crack tip, similar to that for stationary cracks.A crack-growth criterion based on strain accumulation is proposed where a crack is assumed to grow when the accumulated strain ahead of the crack tip reaches a critical value over a characteristic distance. The criterion has been utilized in the prediction of crack-growth rates in a CT specimen at selected loading ranges, frequencies and dwell periods, and the predictions were compared with the experimental results.     

稳态循环应力下结构断裂可靠性设计方法

对含初始缺陷（宏观裂纹）结构进行无限寿命断裂可靠性设计，给出了疲劳裂纹扩展应力强度因子的二维概率密度函数及其门槛值分布函数公式，通过应力强度因子，门槛值干涉模型可求得裂纹不扩展的可靠度和指定可靠度下不扩展裂纹的最大尺寸，并确定含裂纹构件的检修周期。     

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom using stochastic methods

In the paper several stochastic methods for detection and identification of cracks in the shafts of rotating machines are proposed. All these methods are based on the Monte Carlo simulations of the rotor-shaft lateral-torsional-longitudinal vibrations mutually coupled by transverse cracks of randomly selected depths and locations on the shaft. For this purpose there is applied a structural hybrid model of a real cracked rotor-shaft. This model is characterized by a high practical reliability and great computational efficiency, so important for hundreds of thousands numerical simulations necessary to build databases used in solving the inverse problem, i.e. crack parameter identifications. In order to ensure a good identification accuracy, for creating the Monte Carlo samples of data points there are proposed special probability density functions for locations and depths of the crack. Such an approach helps in enhancing databases corresponding to the most probable faults of the rotor-shaft system of the considered rotor machine. In the presented study six different database sizes are considered to compare identification efficiency and accuracy of considered methods. A sufficiently large database enables us to estimate almost immediately (usually in less than one second) the crack parameters with precision that is in most of the cases acceptable in practice. Then, as a next stage, one of the proposed fast improvement algorithms can be applied to refine identification results in a reasonable time. The proposed methods seem to provide very convenient diagnostic tools for industrial applications.     

A generalized Paris’ law for fatigue crack growth

An extension of the celebrated Paris law for crack propagation is given to take into account some of the deviations from the power-law regime in a simple manner using the Wöhler SN curve of the material, suggesting a more general “unified law”. In particular, using recent proposals by the first author, the stress intensity factor K(a) is replaced with a suitable mean over a material/structural parameter length scale Δa, the “fracture quantum”. In practice, for a Griffith crack, this is seen to correspond to increasing the effective crack length of Δa, similarly to the Dugdale strip-yield models. However, instead of including explicitly information on cyclic plastic yield, short-crack behavior, crack closure, and all other detailed information needed to eventually explain the SN curve of the material, we include directly the SN curve constants as material property. The idea comes as a natural extension of the recent successful proposals by the first author to the static failure and to the infinite life envelopes. Here, we suggest a dependence of this fracture “quantum” on the applied stress range level such that the correct convergence towards the Wöhler-like regime is obtained. Hence, the final law includes both Wöhler's and Paris’ material constants, and can be seen as either a generalized Wöhler's SN curve law in the presence of a crack or a generalized Paris’ law for cracks of any size.     

Energy dissipation mechanisms in ductile fracture

The objective is to investigate energy dissipation mechanisms that operate at different length scales during fracture in ductile materials. A dimensional analysis is performed to identify the sets of dimensionless parameters which contribute to energy dissipation via dislocation-mediated plastic deformation at a crack tip. However, rather than using phenomenological variables such as yield stress and hardening modulus in the analysis, physical variables such as dislocation density, Burgers vector and Peierls stress are used. It is then shown via elementary arguments that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms at different length scales from the crack tip; the energy dissipations mechanisms are cleavage, crack tip dislocation nucleation and also dislocation nucleation from a Frank-Read source. Therefore, the material behavior is classified into three groups. The first two groups are the well-known intrinsic brittle and intrinsic ductile behavior. The third group is designated to be extrinsic ductile behavior for which Frank-Read dislocation nucleation is the initial energy dissipation mechanism. It is shown that a material is predicted to exhibit extrinsic ductility if the dimensionless parameter bρ_disl^1/2 (b is Burgers vector, ρ_disl is dislocation density) is within a certain range defined by other dimensionless parameters, irrespective of the competition between cleavage and crack tip dislocation nucleation. The predictions compare favorably to the documented behavior of a number of different classes of materials.     

Mode III effects on interface delamination

For crack growth along an interface between dissimilar materials the effect of combined modes I, II and III at the crack-tip is investigated. First, in order to highlight situations where crack growth is affected by a mode III contribution, examples of material configurations are discussed where mode III has an effect. Subsequently, the focus is on crack growth along an interface between an elastic-plastic solid and an elastic substrate. The analyses are carried out for conditions of small-scale yielding, with the fracture process at the interface represented by a cohesive zone model. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and this solution is applied as boundary conditions on the outer edge of the region analyzed. For several combinations of modes I, II and III crack growth resistance curves are calculated numerically in order to determine the steady-state fracture toughness. For given values of K_I and K_II the minimum fracture toughness corresponds to K_III=0 in most of the range analyzed, but there is a range where the minimum occurs for a nonzero value of K_III.     

Crack-compliance method for assessing residual stress due to loading/unloading history: Numerical and experimental analysis

The understanding of how materials fail is still today a fundamental research problem for scientist and engineers. The main concern is the assessment of the necessary conditions to propagate a crack that will eventually lead to failure. Nevertheless, this kind of analysis tends to be more complicated, when a prior loading history in the material is taken into consideration and it will be extremely important to recognize all the factors involved in this process. In this work, a numerical simulation and experimental evaluation of the induction of residual stresses, which change the crack initiation conditions, in a modified compact tensile specimen is presented. Several analyses were carried out; an initial evaluation (numerical and experimental) was performed in a specimen without a crack and this was used for the estimation of a residual stress field produced by an overload; three more cases were simulated and a crack was introduced in each specimen (1 mm, 5 mm and 10 mm long, respectively). The overload was then applied to set up a residual stress field into the component; furthermore, in each case the Crack Compliance Method (CCM) was applied to measure the induced residual stress field. By performing this numerical simulation, the accuracy of the CCM can be evaluated and later corroborated by experimental procedure. On the other hand, elastic-plastic finite element analysis was utilized for the residual stress estimation. The analyses were based on the mechanical properties of a biocompatible material (AISI 316L). The obtained results provided significant data about diverse factors, like; the manner in which a residual stress field could modify the crack initiation conditions, the convenient set up for the induction of a beneficial residual stresses field, as well as useful information that can be applied for the experimental implementation in this research. Finally, some beneficial aspects of residual stresses are discussed.     

Combined use of SHMS and finite element strain data for assessing the fatigue reliability index of girder components in long-span cable-stayed bridge

The design of long-span cable-stayed bridge involves a large number of loads, geometric and material parameters, all of which can interact in a random fashion. It is desirable to have a total measure of the operational reliability and safety of the structural components. Based on the box-girder component strain history data of the Runyang Cable-stayed Bridge (RYCB) in China, a computer algorithm is developed to evaluate the fatigue damage that is assumed to occur in increments, according to a lognormal distribution. The corresponding probability density function is then found to obtain a fatigue reliability index β for ranking the integrity of the girders. Emphases are placed on the overall scheme of structural reliability evaluation such that the different fatigue damage criteria, probability density functions, and strain measurement techniques can be made. Finite element calculations are also used to provide strain data at locations that are not conducive for installing strain gauges, while the compatibility of measured and calculated data is made empirically. Each of the subroutine in the fatigue reliability algorithm can be altered for improvement. The flexibility allows up-dating the prediction as the monitored strains are changed by the environmental conditions. Preliminary results are first obtained to test the selected damage increments in relation to the probability function and fatigue damage criterion. Particular attention has been given to test the sensitivity of the combined governing parameters. The highly non-linear behavior of numerical calculations related to fatigue failure necessitates an in-depth understanding of the physical model. The condition under which fatigue damage accumulation is needed in contrast to the linear sum of fatigue cycles will be left for the future. Justification should be given to include the more complex issues. The aim here is to seek a simple, and yet reliable index that can account for the fatigue damage of box-girder of long cable-stayed bridges.     

Probability of random events of inspection and repair and maintaining reliability of structures

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常幅载荷下结构元件断裂可靠度估算的应力强度因子模型

给出了一个估算结构元件疲劳可靠度的应力强度因子模型,系统阐述了元件在常幅载荷下疲劳可靠性的分析方法。该模型研究了常幅载荷作用下材料瞬时裂纹长度和应力强度因子的分布形式,建立了应力强度因子与断裂韧性之间的干涉关系。对7075-T7351铝合金中心裂纹试件试验数据分析的结果表明：裂纹的瞬时扩展长度和可靠度的预测结果均与试验结果符合很好,本文给出的基于应力强度因子的可靠性分析模型是合理的。