Paris Model Research Articles

Maintenance and Rehabilitation Programming of the Portuguese Road Network: Development of a Cracking Prediction Model

This paper describes the development of a cracking prediction model for Portuguese conditions which is expected to integrate the Pavement Management System (PMS) of Estradas de Portugal. The World Bank's highway development and management (versions III and 4) and PARIS models are used as reference for the development of a deterministic (mechanistic-empirical) model, using pavement condition data from sections of the main road network. A two-phase distress evolution model is proposed where the initiation of cracking (1st phase) is ruled by a different equation than the progression of cracking (2nd phase). Cracking initiation is predicted on a traffic basis, from the annual traffic load and the structural capacity of the pavement. An absolute model is presented and recommended for the maintenance and rehabilitation (M&R) programming in the long-term and for the analysis of non-cracked segments. Absolute and relative type models were obtained for cracking progression. The relative model shows better agreement to data and is proposed for short- to medium-term analysis on segments with cracking history, while the absolute model is proposed for the M&R programming in the long-term and the analysis of non-cracked segments. Finally, the recommended model is evaluated based on the application to a set of pavement structures defined in the Portuguese pavement design guide.

Study of TBM cutterhead fatigue damage mechanisms based on a segmented comprehensive failure criterion

The cutterhead endures strong, multi-point distributed impact loads when the full face rock tunnel boring machine (TBM) tunnels, which leads to intense vibration and fatigue fracture. Accordingly the structure should meet the requirement of service life considering the fracture failure at weak locations. Hence, cutterhead damage mechanism is crucial to predict and analyze residual life of a operational cutterhead under impact loads. This paper combines with the characteristics of cutterhead life, Paris model and damage cumulative theory, proposes a new failure criterion for the long thick plate in complex stress states, and establishes a segmented crack propagation life prediction model, based on the existing multi-degree-of-freedom coupling dynamic TBM cutterhead model and fatigue crack propagation prediction method for large and complex structures. Meanwhile, a physical fatigue damage test of cutterhead specimens is carried out to validate the theoretical model. Then a TBM cutterhead in a water tunnel project is taken as an example, to calculate the fatigue crack propagation and analyze how different parameters affect fatigue life. The results of the analyses reveal that with fatigue crack propagation, life of actual cutterhead is 36.5km, which gives a meaningful reference for cutterhead design. In addition, the cutterhead life decreases nearly 45.2% with the initial crack depth increasing from 0.5mm to 2mm, and it is inferred that the value of initial crack depth should be controlled below 1.6mm, to ensure the service life. On the other side, the larger the crack shape ratio (a/c), the longer the fatigue life, and the cutterhead life with a round crack (a/c=1) is about 3 times with a narrow crack (a/c=0.1). Furthermore, the crack growth rate parameter m has greater influence on cutterhead life than parameter C, with C increasing 10%, the life decreases about 9%; however, the life decreases nearly 13% when m increases 1%. Then, the critical threshold values of cutterhead material parameters to meet life requirement are obtained. From the results of fatigue damage test, it can be showed that the specimens fracture quickly when the crack depth exceeds 50% of the panel thickness, and the crack shape ratio gradually increases with the loading cycles, which is consistent with the theoretical assumption, as well as the influence rules of parameters.

Microstructure and fatigue crack growth behavior in tungsten inert gas welded DP780 dual-phase steel

Abstract The purpose of this study was to evaluate microstructural and mechanical change of DP780 steel after tungsten inert gas (TIG) welding and the influence of notch locations on the fatigue crack growth (FCG) behavior. The tempering of martensite in the sub-critical heat affected zone (HAZ) resulted in a lower hardness (~ 220 HV) compared to the base material (~ 270 HV), failure was found to originate in the soft HAZ during tensile test. The fusion zone (FZ) consisted of martensite and some acicular ferrite. The joint showed a superior tensile strength with a joint efficiency of 94.6%. The crack growth path of HAZ gradually deviated towards BM due to the asymmetrical plastic zone at the crack tip. The FCG rate of the crack transverse to the weld was fluctuant. The Paris model can describe the FCG rate of homogeneous material rather well, but it cannot precisely represent the FCG rate of heterogeneous material. The fatigue fracture surface showed that the stable expanding region was mainly characterized by typical fatigue striations in conjunction with secondary cracks; the rapid expanding region contained quasi-cleavage morphology and dimples. However, ductile fracture mechanism predominated with an increasing stress intensity factor range (Δ K ). The final unstable failure fractograph was subtotal dimples.

Fatigue crack growth rate of Ti-6Al-4V considering the effects of fracture toughness and crack closure

Fatigue fracture is one of the main failure modes of Ti-6Al-4V alloy, fracture toughness and crack closure have strong effects on the fatigue crack growth(FCG) rate of Ti-6Al-4V alloy. The FCG rate of Ti-6Al-4V is investigated by using experimental and analytical methods. The effects of stress ratio, crack closure and fracture toughness on the FCG rate are studied and discussed. A modified prediction model of the FCG rate is proposed, and the relationship between the fracture toughness and the stress intensity factor(SIF) range is redefined by introducing a correcting coefficient. Notched plate fatigue tests (including the fracture toughness test and the FCG rate test) are conducted to investigate the influence of affecting factors on the FCG rate. Comparisons between the predicted results of the proposed model, the Paris model, the Walker model, the Sadananda model, and the experimental data show that the proposed model gives the best agreement with the test data particularly in the near-threshold region and the Paris region, and the corresponding calculated fatigue life is also accurate in the same regions. By considering the effects of fracture toughness and crack closure, the novel FCG rate prediction model not only improves the estimating accuracy, but also extends the adaptability of the FCG rate prediction model in engineering.

Study on the fatigue crack growth rates of Ti–5Al–5Mo–5V–1Cr-1Fe titanium alloy with basket-weave microstructure

In the present paper, the fatigue crack growth behavior of Ti–5Al–5Mo–5V–1Cr–1Fe titanium alloy (Ti-55511) with different basket-weave microstructure features prepared by quasi-β forging and subsequent double annealing treatments are studied. The results show that the fatigue crack growth rates (FCGR) for five microstructures of Ti-55511 alloy vary obviously at low stress intensity factor range levels and then tend to be similar at high stress intensity factor range levels. According to analyses, the difference of the crack deflection effect caused by different basket-weave features is responsible for above phenomenon. It׳s found that the α platelet length and thickness are two critical basket-weave features influencing the crack deflection effect of Ti-55511 alloy. The long and thick α platelets are most favorable to deflecting the main crack and increasing the crack path tortuosity, which can obviously improve the fatigue crack growth resistance. In addition, at the end of the article, a generalized Paris model based on tensile properties is established for Ti-55511 alloy. Results show that this model can fairly describe the FCGR for Ti-55511 alloy when tested at a stress ratio of 0.1.

Effects of microstructure on fatigue crack growth behavior in cold-rolled dual phase steels

Fatigue crack growth behaviors of cold-rolled dual phase steels with different microstructures were investigated at room temperature. The ferrite–martensite dual-phase microstructure was obtained by intercritical annealing. Fatigue crack growth (FCG) behaviors were described by both the Paris model and a new exponential model; fatigue fractography and surface morphology near the fracture were arrested by scanning electron microscopy (SEM); the relationship between macroscopic and microcosmic FCG rate was analyzed quantificationally. The results showed that both the models can be used to describe the fatigue crack growth rate of the samples rather well; fatigue striations and secondary cracks were observed in the fracture surface at stable expanding region (II), while the fracture at rapid expanding region (III) combined dimple and quasi-cleavage morphology; the roughness of fracture surface and the degree of secondary cracking increased with an increase in martensite content, leading to a higher threshold value. Moreover, the changes of microcosmic FCG rate were smoother than that of the macroscopic FCG rate.

Fatigue crack growth prediction in a gas turbine casing

Some gas turbines casings are very susceptible to involve cracking in critical points. According to studies, thermal fatigue specified as the main reason of creation of cracks. This approach is to predict the fatigue life of the detected crack on the edge of eccentric pin hole of a heavy duty gas turbine casing, which is one of the most probable positions for crack initiation and propagation. The crack position and direction was determined, by using non-destructive tests. Distribution of outer surface temperature was measured by using thermography method and for the measurement of inner surface ten thermocouples were installed inside the casing wall. The temperature data was applied as boundary condition in finite element simulation by using Abaqus software. Stress distribution of the casing was calculated in accord to engine work cycle and then, fatigue crack growth was predicted by using ZENCRACK fracture mechanics code. The analysis results showed that the crack growth rate will decrease gradually and after periods of time while stress intensity factor values will reach the Threshold value, the crack propagation will stop. CTOD and J-Integral methods were implemented for calculating the fracture mechanics quantities and Paris model was used to estimate the fatigue life.

Study on Critical Moisture Content of Unsaturated Evaporation

Thin layer evaporation tests of three types of soil were conducted by a newly designed humidity-controllable evaporation and penetration measuring system and lasted for 8 days. The whole process of sample mass variation from wet to dry was recorded in the laboratory. The critical moisture content and air-dried moisture content were obtained from evaporation curve, which divided the evaporation process of thin layer unsaturated soil into three stages, including stable rate stage, reducing rate stage and residual stage. The soil water characteristic curves of soils were predicted by Arya and Paris model, the results showed that the critical moisture contents of evaporation process were the same with the water contents corresponding to residue values of SWCC, it is significant to studying on unsaturated evaporation process.

Eddy Current Sensor Modeling for the Nondestructive Evaluation of Stress Intensity Factor

In this paper, a nondestructive evaluation by sensor Eddy current is used as a tool to control cracks and microcracks in materials. A simulation by a numerical approach such as the finite element method is employed to detect cracks in materials and eventually to study their propagation using a crucial parameter such as a Stress Intensity Factor (SIF). This method has emerged as one of the most efficient techniques for prospectin cracks in materials, evaluating SIFs and analyzing crack's growth in the context of linear elastic fracture mechanics (LEFM). This technique uses extrapolation of displacements from results compared with those obtained by the integral interaction. On the other hand, crack'sgrowth is analyzed as a model by combining the maximum circumferential stress criteria with the critical plane for predicting the direction of crack growth. In this research, a constant crack growth increment is determined using the classical Paris's model, or the so-called modified Paris's model. It is also shown herein that stress intensity factors needed for these models are calculated using the domain form of the J-integral interactions.

Chaboche-Paris Full-life Prediction Model Considering Multi-level Loading and Non-linear Continuum Damage

Structural fatigue life contains crack initiation stage and crack propagation stage. Generally, crack initiation life and crack propagation life are calculated separately. Extended Paris model to a whole life prediction model based on Chaboche non-linear damage model, and build a relational model between nonlinear damage accumulation and crack length to analyze how a multi-level loading affect the damage accumulation. The results show that:When there is no initial crack, the Chaboche-Paris model analysis results are in good agreement with the S-N experimental results; when there is a macroscopic long crack, the results are in good agreement with the Paris model, which proves the accuracy and applicability of the model to predict both the two stages of the structural life. When add a two level loading, the Chaboche-Paris model can reflect the fatigue damage cumulative effect of the loading order, in the case of low-high loading, the sum of recycle is more than 1; in the case of high-low loading, the sum of recycle is less than 1, the trend is consistent with the test results.

Sedimentary and hydraulic characterization of a heterogeneous glaciofluvial deposit: Application to the modeling of unsaturated flow

Flows in the unsaturated zones of alluvial deposits are often poorly understood at the scale of structures such as stormwater infiltration basins. The heterogeneous nature of the sediment inherent to this type of formation can lead to preferential flows. The difficulty of quantifying them lies in obtaining hydrodynamic parameters representative of the soil structure in-situ, when the latter is inaccessible. This study proposes a method of estimating the hydrodynamic parameters of the lithofacies composing a glaciofluvial deposit (east Lyon, France) and their integration in a 2D flow model to model flow in the unsaturated zone. The approach developed comprises a sedimentological study based on the Miall code, carried out to determine the analogy between the lithofacies of the study site and those of two glaciofluvial reference sites currently in the process of formation (Chamonix, France; Breidamerkurjökull, Iceland), thereby permitting easy access to all the lithofacies identified on the study site. The water retention and hydraulic conductivity curves were obtained on the reference sites by using Beerkan type infiltration tests for finely textured materials, completed by the Arya and Paris model and data from the literature for coarser materials. Thus hydrofacies corresponding to all the lithofacies were characterized. These hydrofacies were used to model a drainage phase on the study site at the scale of a trench (13.50m long×2.50m deep). The model highlighted the importance of the role played by the spatial distribution of hydrofacies on flows, with a pronounced capillary barrier effect of the very coarse lithofacies and a sand lens leading to funneled flows. By assuming that there is a direct link between the sedimentary lithological heterogeneity and the unsaturated hydrodynamic properties of the materials composing its different structures, it was possible to perform a detailed hydrodynamic characterization of the unsaturated zone that could then be used to model flows, and in particular highlight preferential flows.

A simplified approach to estimate water retention for Sicilian soils by the Arya–Paris model

Application of the Arya and Paris (AP) model to estimate the soil water retention curve requires a detailed description of the particle-size distribution (PSD) because the scale factor α, relating the pore length of an ideal soil to that of the natural one, depends on the particle-size distribution parameters. For a dataset of 140 Sicilian soils that were grouped in five texture groups, the logistic and linear models were applied to evaluate α, and the water retention values predicted by the AP model were compared with the measured ones. Using the parameters proposed by Arya et al. (1999), the two models yielded similar unsystematic root mean error of estimate (RMSEu). Therefore, their potential accuracy was considered comparable. However, the water retention data predicted by the logistic model were more biased (higher systematic root mean error of estimate, RMSEs) than those predicted by the linear model. A calibration was conducted for the logistic model to obtain five sets of parameters specifically developed for Sicilian soils. The calibrated logistic model only minimally improved the prediction accuracy of the AP model. This result also supported Arya et al.'s (1999) procedure for soils that were not included in their original calibration dataset. With the aim to simplify AP model application, an alternative procedure was developed by optimizing a soil-specific α value in the range of measured water content values. For Sicilian soils, the optimized values of the scale parameter (αOPT) were significantly correlated with clay content and bulk density. The empirical relationship that was obtained for the calibration dataset allowed estimation of the water retention data of the validation dataset (N=70) with an estimation error (RMSE=0.042cm3cm−3) lower than that of the traditional approach based on the logistic model. Therefore, it can be considered as a reasonable alternative to the more complex logistic model for estimating the water retention curve of Sicilian soils.

Investigation on corrosion fatigue crack growth rate in 7075 aluminum alloy

The corrosion fatigue crack growth behaviors of 7075 alloy in seawater were investigated. The results showed that corrosion solution concentration, pH value, temperature as well as load frequency have great influences on the corrosion fatigue crack growth rate. The corrosion fatigue crack growth rate in 7075 alloy accelerates with increasing the corrosion solution concentration, raising the temperature, or lowering the pH value. The above three factors can be simply expressed by the surface current intensity, Icor. On the other hand, the corrosion fatigue crack growth rate increases with decreasing the load frequency, due to the longer exposure of the new-generated specimen surface to the corrosive environment. The fracture surface indicated that the corrosion pits extending along the direction of the crack growth are responsible for the acceleration of the crack growth rates, since the corrosion pits may act as physical defects and stress concentrations around the crack-tip. Furthermore, the electrochemical reaction occurring between the new-generated surface and the corrosion solution makes the crack closure difficult, which also facilitates the crack growth. In addition, a modified Paris model has been proposed on the basis of the experimental results, consisting of the surface current intensity Icor and the load frequency f.

Applicability of Equivalent Constant Amplitude Loading for Assessing the Fatigue Life of Pipelines and Risers and the Influence of Compressive Stress Cycles

Fatigue life assessment of pipelines and risers is a complex process, involving various uncertainties. The selection of an appropriate fatigue model is important for establishing the inspection intervals and maintenance criteria. In offshore structures, the vortex-induced vibration (VIV) could cause severe fatigue damage in risers and pipelines, resulting in leakage or even catastrophic failure. The industry has customarily used simple fatigue models for fatigue life assessment of pipelines and risers (such as the Paris or Walker models); however, these models were developed based on constant amplitude loading scenarios. In contrast, VIV-induced stress-time history has a variable amplitude nature. The use of the simplified approach (which is inherently non conservative), has necessitated the implementation of large safety factors for fatigue design of pipelines and risers. Moreover, most of the experimental investigations conducted to date with the aim of characterizing the fatigue response of pipelines and risers have been done based on incorporation of constant amplitude loading (CAL) scenarios (which is unrealistic), or converting the variable amplitude loading (VAL) scenarios to an equivalent CAL. This study demonstrates that the use of such approaches would not be lead to accurate assessment of the fatigue response of risers subject to VIV-induced VAL. The experimental investigation performed in this study will also clarify the underlying reasons for the use of large safety factors by the industry when assessing the fatigue life of pipelines and risers. In addition, an experimental investigation was also conducted to highlight the influence of the compressive portion of VIV stress-time history on the fatigue life of such components. It is shown that the compressive stress cycles significantly influence the fatigue crack growth response of risers, and their presence should not be ignored.

Spatial and temporal patterns of water loss in heterogeneous landscapes: using plaster models as amphibian analogues

Water balance is critical to survival, growth, and performance of many terrestrial organisms because it can influence foraging time, limit dispersal, and curtail courtship activities. Water loss can vary in time and space across the landscape, and can also be modulated by behavior. Amphibians are particularly sensitive to water loss because their skin provides little to no resistance to evaporative water loss. Our study sought to quantify rates of water loss across a heterogeneous landscape using plaster of Paris models as analogues for the Western Slimy Salamander (Plethodon albagula Grobman, 1944). Models were validated within a controlled laboratory setting prior to field deployment, and were shown to approximate rates and magnitudes of water loss observed in living salamanders. In the field, we tested both adult- and juvenile-sized models, and found that juvenile-sized models lost water at a greater rate under all contexts. The rates of water loss measured at night (1.5%/h–4.5%/h) was nearly half of those measured during the day (2%/h–10%/h). Rates of water loss were greatest on ridges with southwestern aspects during the day (5%/h–10%/h) and lowest in ravines with northeast aspects at night (1.50%/h–3.75%/h). The results of spatial and temporal patterns of water loss corroborate field observations of salamander activity patterns and distribution across the landscape, providing a physiological mechanism driving fine-scale habitat use and distribution. Although we tested plaster models as analogues for salamanders, this approach should be generalizable to other amphibian taxa, providing an efficient means of measuring rates of water loss in the field under biologically meaningful contexts.

PREDICTION OF SOIL WATER CHARACTERISTIC CURVE USING PHYSICALLY BASED SCALING TECHNIQUE

The soil water characteristic curve (SWCC) is an important hydraulic parameter for modeling water flow and contaminant transport in the vadose zone. However, direct measurement of the SWCC is still difficult. The Arya and Paris (AP) model estimates the SWCC from particle-size distribution curve (PSD) based on the shape similarity of the two curves. They introduced an empirical parameter, α, used to scale pore attributes from hypothetical formations to natural structures. Several approaches have been used to derive α. However, the calculation procedures of these approaches are either quite complicated or are developed without paying much attention to the physical significance of the soil properties. In the present paper the physically based scaling technique (PBS) was applied to derive α for the AP model. Fifty soil samples, representing a range of textures that include sand, sandy loam, loam, silt loam, and clay, were selected from UNSODA hydraulic property database for calculating α using PBS approach. In addition, nineteen soil samples with different textures were used to verify effectiveness of proposed α values. The results compared with other approaches show that the PBS technique combine with the AP model is a more useful and easier approach to predict SWCC from PSD.

Simplified estimation of unsaturated soil hydraulic conductivity using bulk electrical conductivity and particle size distribution

Unsaturated soil hydraulic conductivity K is a fundamental transfer property of soil but its measurement is costly, difficult, and time-consuming due to its large variations with water content (θ) or matric potential (h). Recently, C. Doussan and S. Ruy proposed a method/model using measurements of the electrical conductivity of soil core samples to predict K(h). This method requires the measurement or the setting of a range of matric potentials h in the core samples—a possible lengthy process requiring specialised devices. To avoid h estimation, we propose to simplify that method by introducing the particle-size distribution (PSD) of the soil as a proxy for soil pore diameters and matric potentials, with the Arya and Paris (AP) model. Tests of this simplified model (SM) with laboratory data on a broad range of soils and using the AP model with available, previously defined parameters showed that the accuracy was lower for the SM than for the original model (DR) in predicting K (RMSE of logK = 1.10 for SM v. 0.30 for DR; K in m s–1). However, accuracy was increased for SM when considering coarse- and medium-textured soils only (RMSE of logK = 0.61 for SM v. 0.26 for DR). Further tests with 51 soils from the UNSODA database and our own measurements, with estimated electrical properties, confirmed good agreement of the SM for coarse–medium-textured soils (<35–40% clay). For these textures, the SM also performed well compared with the van Genuchten–Mualem model. Error analysis of SM results and fitting of the AP parameter showed that most of the error for fine-textured soils came from poorer adequacy of the AP model’s previously defined parameters for defining the water retention curve, whereas this was much less so for coarse-textured soils. The SM, using readily accessible soil data, could be a relatively straightforward way to estimate, in situ or in the laboratory, K(h) for coarse–medium-textured soils. This requires, however, a prior check of the predictive efficacy of the AP model for the specific soil investigated, in particular for fine-textured/structured soils and when using previously defined AP parameters.

Testing the shape-similarity hypothesis between particle-size distribution and water retention for Sicilian soils

Application of the Arya and Paris (AP) model to estimate the soil water retention curve requires a detailed description of the particlesize distribution (PSD) but limited experimental PSD data are generally determined by the conventional sieve-hydrometer (SH) method. Detailed PSDs can be obtained by fitting a continuous model to SH data or performing measurements by the laser diffraction (LD) method. The AP model was applied to 40 Sicilian soils for which the PSD was measured by both the SH and LD methods. The scale factor was set equal to 1.38 (procedure AP1) or estimated by a logistical model with parameters gathered from literature (procedure AP2). For both SH and LD data, procedure AP2 allowed a more accurate prediction of the water retention than procedure AP1, confirming that it is not convenient to use a unique value of for soils that are very different in texture. Despite the differences in PSDs obtained by the SH and LD methods, the water retention predicted by a given procedure (AP1 or AP2) using SH or LD data was characterized by the same level of accuracy. Discrepancies in the estimated water retention from the two PSD measurement methods were attributed to underestimation of the finest diameter frequency obtained by the LD method. Analysis also showed that the soil water retention estimated using the SH method was affected by an estimation bias that could be corrected by an optimization procedure (OPT). Comparison of a-distributions and water retention shape indices obtained by the two methods (SH or LD) indicated that the shape-similarity hypothesis is better verified if the traditional sieve-hydrometer data are used to apply the AP model. The optimization procedure allowed more accurate predictions of the water retention curves than the traditional AP1 and AP2 procedures. Therefore, OPT can be considered a valid alternative to the more complex logistical model for estimating the water retention curve of Sicilian soils.

Testing the shape-similarity hypothesis between particle-size distribution and water retention for Sicilian soils

Application of the Arya and Paris (AP) model to estimate the soil water retention curve requires a detailed description of the particlesize distribution (PSD) but limited experimental PSD data are generally determined by the conventional sieve-hydrometer (SH) method. Detailed PSDs can be obtained by fitting a continuous model to SH data or performing measurements by the laser diffraction (LD) method. The AP model was applied to 40 Sicilian soils for which the PSD was measured by both the SH and LD methods. The scale factor was set equal to 1.38 (procedure AP1) or estimated by a logistical model with parameters gathered from literature (procedure AP2). For both SH and LD data, procedure AP2 allowed a more accurate prediction of the water retention than procedure AP1, confirming that it is not convenient to use a unique value of for soils that are very different in texture. Despite the differences in PSDs obtained by the SH and LD methods, the water retention predicted by a given procedure (AP1 or AP2) using SH or LD data was characterized by the same level of accuracy. Discrepancies in the estimated water retention from the two PSD measurement methods were attributed to underestimation of the finest diameter frequency obtained by the LD method. Analysis also showed that the soil water retention estimated using the SH method was affected by an estimation bias that could be corrected by an optimization procedure (OPT). Comparison of a-distributions and water retention shape indices obtained by the two methods (SH or LD) indicated that the shape-similarity hypothesis is better verified if the traditional sieve-hydrometer data are used to apply the AP model. The optimization procedure allowed more accurate predictions of the water retention curves than the traditional AP1 and AP2 procedures. Therefore, OPT can be considered a valid alternative to the more complex logistical model for estimating the water retention curve of Sicilian soils.

A Comparison Study of Methods for Parameter Estimation in the Physics-based Prognostics

Prediction of remaining useful life of a system is important for safety and maintenance scheduling. In the physics-based prognostics, the accuracy of predicted remaining useful life is directly related to that of estimated model parameters. It, however, is not a simple task to estimate the model parameters because most real systems have multivariate model parameters, which are often correlated each other. This paper mainly discusses the difference in estimating model parameters among different prognostics methods: the particle filter method, the overall Bayesian method, and the incremental Bayesian method. These methods are based on the same theoretical foundation, Bayesian inference, but they are different from each other in the sampling scheme and/or uncertainty analysis process. A simple analytical example and the Paris model for crack growth are used to demonstrate the difference among the three methods in terms of prognostics metrics. The numerical results show that particle filter and overall Bayesian methods outperform the incremental Bayesian method. Even though the particle filter shows slightly better results in terms of prognostics metrics, the overall Bayesian method is efficient when batch data exist.