Fourth International Conference on Material and Component Performance under Variable Amplitude Loading
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Probabilistic and Energy Based Methods

Session chair: Lagoda, Tadeusz, Professor (Opole University of Technology, Department of Mechanics and Machine Design, Opole, Poland)
Shortcut: L
Date: Tuesday, 31. March 2020, 14:00
Room: Hall B
Session type: Oral


14:00 L-01

Energy based model for fatigue damage accumulation in materials under random loading (#40)

T. Łagoda1, M. Kurek1

1 Opole University of Technology, Opole, Poland

The main aim of this paper is to present a new model of fatigue damage accumulation, which is another development of Palmgren-Miner hypothesis, as well as a modification of the Serensen-Kogayev hypothesis. The authors' proposal is based on a new way of taking into account the characteristics of the material load history. The proposed model of damage accumulation is verified on the basis of experimental data obtained from aluminum alloy 2017A-T4 tests under random proportional bending with torsion.

Keywords: fatigue damage accumulation, random loading, aluminium alloy
14:20 L-02

A strain energy density approach to fatigue crack growth under spectrum block loading (#123)

E. Amsterdam1

1 NLR, Gas Turbine & Structural Integrity, Marknesse, Netherlands

The effective block approach is a concept where the crack growth rates are quantified and modeled by treating a block of variable amplitude (VA) loading as being similar to a single cycle of constant amplitude (CA) loading, but with a greater amount of energy compared to a single CA cycle [1]. The fatigue crack growth rate (FCGR) per block can be determined for a certain VA spectrum and shows an almost linear relationship with the reference stress (i.e. maximum spectrum stress) and a square root relationship with the crack length both to the power n [2]. There are pivot points in the VA FCGR curve where the exponent, n, changes. This is similar to the Paris equation for CA loading, where the stress intensity factor depends linearly on the stress and square root on the crack length, and where pivot points are present in the FCGR curve where the exponent, n, changes [3]. This indicates that it is justified to treat a block of VA loading as being similar to a single CA cycle, but with a greater amount of energy compared to a single CA cycle. However, in general there has been little success in relating the crack growth rates from one VA spectrum to another. In the current paper a strain energy density based approach is used relate the crack growth rate per VA block for different VA spectrums. Different types of spectrums and small variations to spectrums are used to determine the load interaction or sequence effects on the strain energy density calculation of a spectrum. The influence of the reference stress on the VA crack growth rate per block of cycles is used to determine the contribution of the stress range of individual cycles in the VA block on the total energy of the VA block. This ensures a scale invariance when, for example, two spectrum blocks are considered as one.


[1] L. Molent, M. McDonald, S. Barter, R. Jones. Evaluation of spectrum fatigue crack growth using variable amplitude data. International Journal of Fatigue 30 (2008) 119–137.

[2] E. Amsterdam. Effect of crack length and reference stress on variable amplitude fatigue crack growth rate. 30th ICAF Symposium, Kraków, 5-7 June 2019.

[3] E. Amsterdam, F. Grooteman. The influence of stress state on the exponent in the power law equation of fatigue crack growth. International Journal of Fatigue 82 (2016) 572–578.

Keywords: Strain energy density, Effective block approach, Fracture mechanics, Power law exponent, Pivot points
14:40 L-03

PhyBaLSL – short time procedure for the determination of the fatigue lifetime of metallic materials (#11)

B. Blinn1, T. Beck1, B. Jost1, M. Klein1, D. Eifler1

1 TU Kaiserslautern, Institute of Materials Science and Engineering, Kaiserslautern, Rhineland-Palatinate, Germany

For an efficient and reliable design of cyclically loaded components, a sound knowledge about the materials fatigue behavior is indispensable. Because components are usually not loaded with periodical constant amplitudes, but with temporally variable, cyclic loadings, fatigue testing under service near loading conditions is essential for optimization of the components design. However, this kind of fatigue tests requires high efforts in material, time and costs. Consequently, the application of short-time procedures to reduce this effort can be very helpful.

In the present work, fatigue tests under near service condition based on the Car Loading Standard (CARLOS) were performed at cast iron EN-GJV-400. The first 5 000 cycles of CARLOS were used for periodically repeated loading intervals (LI) with defined maximum stress σmax, CARLOS, which was variated for realization of different loading levels in single step tests (SST). In addition to the SSTs, load increase tests (LIT) were conducted to determine the materials cyclic properties. In LITs, after each LI σmax, CARLOS was increased by 20 MPa. To characterize the cyclic deformation behavior, measuring intervals (MI) of 150 cycles with constant amplitude loading were applied between LIs (see Fig. 1) to determine the plastic strain amplitude εa,p at different fatigue states. Additionally, the change in electrical resistance ΔR was measured in the gauge length during the MIs as well as the LIs. To investigate the influence of σa in the MI on the resulting fatigue lifetime, stress amplitudes lower (σa = 160MPa) and higher (σa = 220MPa) than the endurance limit were applied in separate LITs and SSTs.

In the resulting fatigue lifetime no significant influence of the stress amplitude in the MI could be observed (see Fig. 2), whereas a higher resolution of εa,p­ could be achieved at σa = 220MPa and hence, this conditions for MI were used for determination of the cyclic deformation behavior. Based on the results of the LIT and two additional SSTs at different σmax, CARLOS, a modification of physically based lifetime calculation (PhyBaL™) [1] for service loading conditions, i.e. PhyBaLSL was used for S-N*f curve calculation. Therefore, the results of εa,p­ and ΔR in the MIs of LIT as well as SSTs were used, leading to a good accordance of calculated S-N*f curve to experimental data. However, the MIs only provide an integral view on the changes in materials behavior during the previous LIs, whereas the measurement of ΔR enables the in-situ detection of fatigue processes even within the CARLOS intervals. Therefore, the results of ΔR-N* curve were additionally used for PhyBaLSL method, leading to an excellent correlation between calculated S-N*f curves and obtained lifetimes in SSTs. Moreover, the fatigue tests with MIs with σa = 160MPa and σa = 220MPa result in similar calculated S-N*f curves (see Fig. 2), which underlines the assumption, that the different stress levels in MIs have no significant influence on fatigue behavior during the LIs.

The present results show, that the PhyBaLSL method can be used to efficiently calculate the materials fatigue lifetime for near service loading conditions. Furthermore, the detection of ΔR enables an application of this method for structural health monitoring.

Recent research extends the approach outlined above to anisothermal loadings [2]. For this, isothermal MIs were applied at defined cycle numbers of Out-of-Phase thermomechanical fatigue (TMF) tests at GJS-600 ductile cast iron. The evolution of εa,p in the MIs correlated well with that observed during the constant amplitude TMF cycles.


1.         Starke P., Walther F., Eifler D. Adv Eng Mater 2010. 12: p. 276-282.

2.         Jost B., Klein M., Beck T., Eifler D. Int J Fatigue 2017. 96: p. 102-113.

Fig. 1
Schematic description of the implementation of measuring intervals (MIs) for determination of the cyclic deformation behavior under random loading conditions (CARLOS)
Fig. 2
Results of the fatigue tests under random loading conditions (CARLOS) as well as with S-N*f curves determined by PhyBaLSL based on measurements of ΔR and εa,p in measuring intervals (MIs) and ΔR-N* curves
Keywords: fatigue behavior, service near loading, short time procedure, cyclic deformation behavior
15:00 L-04

Probabilistic Fatigue and Reliability Simulation (#73)

A. Halfpenny1, M. Bonato1, S. Vervoort1, A. Chabod1

1 HBM United Kingdom Ltd., nCode Products, Catcliffe, United Kingdom

The fatigue design of mechanical systems has historically followed a ‘deterministic’ process. That means, for a given set of inputs they will return a consistent set of fatigue life results with no scatter. In practice, the designer will apply a safety factor to each input parameter to account for the uncertainty. A final safety factor is also applied to the resulting damage to account for modelling errors. In most cases, the engineer is fairly certain that the simulation results are conservative, but cannot state with any confidence what the final safety margin, reliability or failure rate will be.

In comparison, a ‘Probabilistic Fatigue Simulation’ method is ‘stochastic’ in nature. That means inputs can be expressed using an expected value along with a probability distribution. This design process helps to avoid poor in-service reliability whilst reducing over-design. Furthermore, it is easy to see which uncertainties contribute most to the overall design conservatism allowing justification to study these uncertainties more thoroughly.

This paper address the three stages of Probabilistic Fatigue and Reliability Simulation:

  1. Uncertainty Quantification (UQ) of input parameters
  2. Stochastic fatigue simulation of individual components
  3. Reliability simulation of the entire system

In order to take advantage of Probabilistic Fatigue Simulation, uncertainties in the input and the analysis model must be properly calculated. Two types of input uncertainties are considered:

  1. Reducible uncertainties (or epistemic uncertainties)
  2. Irreducible uncertainties (or aleatoric uncertainties)

Stochastic simulation is performed using a ‘Monte Carlo’ simulation. Also, statistical sampling techniques known as ‘Design of Experiments (DOE)’ are discussed for optimizing the size of the design space matrix. Two broad areas are discussed:

  1. 'Design for Reliability' – exploring the statistical variability of design space
  2. 'Design for Robustness' – exploring the extremities of design space

Reliability analysis of the simulated failures is performed using a Weibull analysis. A case study is presented to demonstrate how reliability analysis is used to:

  1. Optimize the design to achieve the target reliability
  2. Identify potential cost savings by identifying the most influential uncertainties
  3. Validating the simulation model for use in further studies
Keywords: Fatigue, Reliability, Uncertainty Quantification (UQ), Probabilistic Design, Design Of Experiments (DOE)