International Conference on Mechanics and Energy
December 18-20, 2017, Sousse, TUNISIA ICME'2017
Application of A Basquin Model On Fatigue Tests of a Fragile Materials
S. Achouri,S.Tlili ,O.Ghelloudj, D.Berdjane.
Research Center in Industrial Technologies CRTI , P.O.Box 64, Cheraga 16014 Algiers, Algeria s.achouri@crti.dz
B. Redjel
Civil Engineering Laboratory, Badji Mokhtar University, 23000 ,PO Box 12, Annaba, Algeria.bredjel@yahoo.fr
Abstract:Cyclic tests of repeated fatigue loads were carried out in three-point bending on a fragile material. The tests were conducted to burden with R = 0 (ratio of minimum stress to maximum stress), a charging frequency set to a minimum of 75 cycles / min, 1,25Hz and a sinusoidal signal. The specimens used were cut on medium thickness molded plaques 4 mm, dimensions of 80 mm length and 15 mm width according to the recommendations of the standard EN ISO 14125. These specimens were subjected to various loading levels versus the maximum charge of static failure in three-point bending, either: 90%, 80%, 85%, 70%, 60%, 55%, 45%, 40% for each load level, a minimum of three test specimens was tested. The curve traced Wohler is distinguished by a wide dispersion in the lifetimes between the specimens subjected to the same level of loading and tested under the same conditions of cycling and was modeled by a straight line. This dispersion is a consequence of the heterogeneity of the studied fragile materiel. Indeed, the characteristics of specimens such as reinforcement’s rate, distribution, density and shape defects and static strength are not comparable from one specimen to another. This phenomenon of dispersion that the life of forecasts studied fragile materials can be estimated with a high probability by the curve of Wöhler.
However, the route of the latter giving the middle part a good and acceptable performance can still be used as a comparison corresponding to variations in compositions, test frequency, cycling parameters …. etc . The deterioration of the fragile material takes place in the early fatigue loading cycles and gradually increases on the surface and within the volume to the final fracture. The state of fatigue damage is characterized mainly by a combination of density and orientation of microcracks. The stages of evolution of the damage in the case of cyclic loading are the same as thoseencountered in static loading but chronology and different magnitudes.
Keywords: fatigue, damage, frequency, static, Wöhler curve.
Graphical Abstract
*Corresponding author: Sihem achouri E-mail: souma_sihem@yahoo.fr.
s.achouri@crti.dz
Achouri et al. / ICME’2017, December 18-20, 2017
1. Introduction
Several studies show that in the case of fatigue of fragile materials, for a given stress level σi, a dispersion value of the number of cycles to failure is observed.
The problem of dispersal of results is the consequence of the structural heterogeneity of the defects since the elaboration of material [1]. B. Soh Fotsing et al. [2]
have been reported that since these defects present in the material are difficult to measure, it is advisable to use as a random variable, for a given stress level, the number of cycles to failure to explain the existence of defects by a probabilistic approach. N. Ngarmaïm et al.
[3] have formulate a probabilistic model of a new expression of the SN curve in fatigue based on the concept of "weakest link" of Weibull introducing a new parameter Nc, the number characteristic of cycles corresponding to the failure probability equal to 1.
His confrontation with the terms of the curve most used SN, particular those Basquin, the Wöhler and the Stromeyer, the fatigue tests and martensite steels P22O 100C6 data provided errors of about 5% maximum for models Basquin,Wöhler and that proposed.
2. Results and Discussion
In this study, the minimal stress σmin of repeated sollicitation has been maintained constant and equal to zero for all the tests. Only the maximal stress σmax and consequently the amplitude of loading was varied (figure 1) During a cyclic loading, the behavior describes a hysteresis loop (σ-ε). It corresponds to a discrepancy between the evolution of the stress and the evolution of the deformation as a function of time The surface of these loops, which can be quantitatively measured for the energy consumed, increases with each cycle until the total sample break. This measurement can also be an effective means of quantifying the damage during cycling.
Fig. 1 Distribution of the experimental points of the strength amplitude according to the failure number of cycles
All the results of the experimental tests are reported on figure 2 which shows the distribution of the experimental points of the strength amplitude according to the failure number of cycles.
Fig. 2 Curve of Wöhler.
As it appears on the curve the results of materials lifetimes between specimens subjected to the same loading level and tried in the same conditions of repeated sollicitation are characterised by a significant scatter. This scatter is the consequence of the heterogeneity of the studied laminated from fragile materials. Indeed, the characteristics of specimens such as the dimensions, the position, the shape, the orientation, the distribution, the density and the shape of the defects, the rate of the reinforcements, which are conditioned by the manufacturing process as well as the static mechanical resistance are not comparable and differ from a test tube to the other one and from a pick- up position to another one in the plates.
The random character of the presence of the defects in the material and of their size is also a consequence of defects initiatings which lead to a higher order dispersion of lifetimes in fatigue.
The results of the fatigue tests are described according to the linear equation of Wöhler:
σfat = A - B. log NR
where σfat and NR represent the applied maximal stress and the fracture number of cycles respectively.
According to Mandell et al. [4-5] A and B are constants of the tested material. A represents the static fracture strength of the sample.
The calculation software performed by linear regression allowed to identify the values of both coefficients A and B of the curve of endurance. The linear regression performed on the experimental points of the figure 2 is given by the following equation of Wôhler:
σfat = 70 - 7 log NR.
The comparison betwen the value of the coefficient A and that of σst shows a 30 % gap in favour of the first
Achouri et al. / ICME’2017, December 18-20, 2017
one. This gap which remains significant can be partially explained by the large gap between the speeds of loading as well as by the large dispersion of the results of the fatigue test and the consequences of the material inhomogeneities.
The value of the coefficient of linear correlation of the curve of Wöhler is slightly low translating the significant dispersion of material lifetimes. This one is largely bound to the heterogeneous nature of the studied fragile material as well as to the nature of the specimens which rarely have comparable characteristics.
The normalized Wöhler’s curve is given by the following equation:
σfat / A = 1 - B/A. log NR.
This shape of this representation illustrated by figure 3 allows to reveal a rate of degradation quantified by the absolute value of the slope (B/A) of the equation of Wöhler and which is so constant by decade of cycles.
The slope of the normalized straight line is 9,5 %.
This constant degradation rate by decade of cycles is in accordance with those reported on fragile materials in the literature and which are generally esteemed in numerous works at 10 % [6].
Fig. 3 Normalized curve of Wöhler
Thus, in order to account for an asymptotic branch for the large number of cycles, Basquin proposed [7]:
σmax = σfat (2NR)x.
σmax and NR respectively represent the maximum stress of the cycle considered and the number of cycles at break. σfat and x are constants of the material and represent respectively the coefficient of fatigue stress and the Basquin exponent.
Fig.4 Basquin exponent
3. Conclusions
-The fatigue lifetime of these fragile materials results show a significant scatter.
-The fatigue damage in these materials is characterized by complex mechanisms such as, microcracking, delamination. This behavior makes fatigue life forecasts very difficult.
-The general trend of the experimental results on a wide range of number of cycles in the field of the limited endurance shows a convergence of the models of Hwang and Hanf, Wöhler and Basquin around the experimental points
-The straight line of Wöhler remains an interesting approach for its simplicity and offers of the middle part an often acceptable representation in view of the significant dispersif aspect of the material lifetime results.
References
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[2] B. Soh Fotsing, E. maronne, N. Nadjitonon J.-L. Robert.
Intégration d’une démarche fiabiliste dans l’exploitation des critères de fatigue multiaxiaux. 23ème Journée de Printemps de la Commission de Fatigue, du 25 au 26 mai 2004.
Achouri et al. / ICME’2017, December 18-20, 2017
[3] N. Ngarmaïm, B. Tikri, B. Bassa, N. Kimtangar, F. Pennec &
J-L. Robert « A New Expression of the Curve S-N in Fatigue based on the Concept of the "Weakest Link" of Weibull » Global Journal of Researches in Engineering : A Mechanical and Mechanics Engineering Volume 14 Issue 3 Version 1.0 2014, pp 21- 26.
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[6] Krawczak P., « Essais des plastiques renforcés », Techniques de l’Ingénieur, traité plastiques et fragile materials, AM5 405, p.8, 26 pages, Doc. 1-10.
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