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Low Cycle Fatigue (LCF) analysis (last updated 2011-10-05) Kjell Simonsson 1 Aim For high loadings/short lives (with respect to the number of load cycles), fatigue life calculations are generally strain-based. The aim of this presentation is to give a short introduction to strain-based Low Cycle Fatigue (LCF) analysis. Kjell Simonsson 2 Basic observations; Wöhler-diagrams/SN-curves Basquin’s relation Experiments made on smooth test specimens result in stress-life curves (so called SN-curves) of the type shown below (such diagrams are also referred to as Wöhler-diagrams) max min a log a 2 fl Fatigue limit (or Endurance limit), “utmattningsgräns” in Swedish log ' f log fl Nf No. of cycles to failure 2 N f No. of load reversals to failure log 2 N f In the region of long lives, the curve may often be described by Basquin’s relation log a log 'f b log2 N f log a log 'f 2 N f a 'f 2 N f b Kjell Simonsson 3 b Wöhler-diagrams/SN-curves; cont. Basquin’s relation; cont. Basquin’s relation may be rewritten in terms of elastic strains, or total strains, since the stress is low for long lives a 'f 2 N f b log a a log 'f E 'f E 2 N f b or c.f. log a log log 2 N f 'f E b log2 N f Now, it is important to note that even if the experimental curve (blue above) was simply scaled by the Young’s modulus for long lives, this is not the case for short lives, since the presence of plasticity will make the stress-total strain relation non-linear. Kjell Simonsson 4 Wöhler-diagrams/SN-curves; cont. The Coffin-Manson relation In the strain representation introduced on the previous slide, it has been found that the SN-curve for short lives also can be described by a linear relation in the log-log diagram log a log ' f ' f log c.f. log a log 'f c log2 N f or E a 'f 2 N f c log 2 N f The relations above are different versions of the so called CoffinManson relation. Kjell Simonsson 5 Wöhler-diagrams/SN-curves; cont. Putting it all together; Morrow’s relation From a mathematical point of view, it is clear that due to the log-log scale • the contribution to the strain amplitude from Basquin’s relation is negligible compared to the contribution from Coffin and Manson’s relation for short lives • the contribution to the strain amplitude from Coffin and Manson’s relation is negligible compared to the contribution from Basquin’s relation for long lives Furthermore, since both relations lie below the experimental one in the “overlapping” region, a direct addition of the two relations has a potential to reasonably well describe the behavior for all fatigue lives. The so obtained model, found below, is referred to as Morrow’s relation. Kjell Simonsson 6 Wöhler-diagrams/SN-curves; cont. Morrow’s relation; cont. We thus have log a log 'f log ' f a 'f E 2 N f b 'f 2 N f c E log 2 N f Often the two terms in the Morrow relation are identified as the elastic and plastic part of the total strain, but this is only completely true for long lives (with no plastic strains) and approximately true for very short lives (with very small elastic strains compared to the plastic ones). In the “overlapping” region this interpretation is not applicable! 7 Kjell Simonsson Wöhler-diagrams/SN-curves; cont. Morrow’s relation; cont. Strength coefficient The following labeling is common Ductility coefficient log a log 'f log a ' f E 'f E 2 N f b 'f 2 N f c Strength exponent log 2 N f Kjell Simonsson 8 Ductility exponent Wöhler-diagrams/SN-curves; cont. Morrow’s relation; cont. In order to account for mean stress effects, the Morrow relation is modified as exaggeratedly illustrated below for a positive mean stress, where the black curves represents the mean stress corrected relations. log a log 'f log 'f a 'f m E 2 N f b 'f 2 N f c E log 2 N f Kjell Simonsson 9 An example Let us assume that a material is subjected to repeated strain sequences for which we, by a cycle count method (e.g. the Rain Flow Count method), have found the following inherent strain cycles • 1 cycle with a 0.00275 • 2 cycles with a 0.00125 • 2 cycles with a 0.0015 The question is now how many cycles the material can withstand if the influence of the mean stress can be neglected, that the Palmgren-Miner linear damage accumulation hypothesis is applicable and if the material data is given by E 200GPa , 'f 900 MPa , 'f 0.26 , b 0.095 , c 0.47 What we need to do, before being able to consider the damage calculation for one load sequence, is to calculate the number of cycles to failure for each type of load cycle. This is accomplished by using the Morrow equation. Kjell Simonsson 10 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 This is a non-linear equation for Nf1, which needs to be solved numerically. In many pocket-calculators there are features for doing this. However, since one notes that the Right Hand Side (RHS) of the above relation is a decreasing function of Nf1, we may find a solution by a simple trial and error methodology. PLEASE HELP ME! N=10 000 Kjell Simonsson => RHS=? 11 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 Nf1=10 000 => RHS=4.23E-3 larger than the LHS, must increase Nf1 ! Nf1=100 000 => RHS=? Kjell Simonsson 12 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 Nf1=10 000 => RHS=4.23E-3 larger than the LHS, must increase Nf1 ! Nf1=100 000 => RHS=2.25E-3 smaller than the LHS, must decrease Nf1 ! Nf1=50 000 => RHS=? Kjell Simonsson 13 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 Nf1=10 000 => RHS=4.23E-3 larger than the LHS, must increase Nf1 ! Nf1=100 000 => RHS=2.25E-3 smaller than the LHS, must decrease Nf1 ! Nf1=50 000 => RHS=2.67E-3 smaller than the LHS, must decrease Nf1 ! Nf1=40 000 => RHS=? Kjell Simonsson 14 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 Nf1=10 000 => RHS=4.23E-3 larger than the LHS, must increase Nf1 ! Nf1=100 000 => RHS=2.25E-3 smaller than the LHS, must decrease Nf1 ! Nf1=50 000 => RHS=2.67E-3 smaller than the LHS, must decrease Nf1 ! Nf1=40 000 => RHS=2.83E-3 larger than the LHS, must increase Nf1 ! Nf1=45 000 => RHS=? Kjell Simonsson 15 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 Nf1=10 000 => RHS=4.23E-3 larger than the LHS, must increase Nf1 ! Nf1=100 000 => RHS=2.25E-3 smaller than the LHS, must decrease Nf1 ! Nf1=50 000 => RHS=2.67E-3 smaller than the LHS, must decrease Nf1 ! Nf1=40 000 => RHS=2.83E-3 larger than the LHS, must increase Nf1 ! Nf1=45 000 => RHS=2.75E-3 = LHS !!!! Kjell Simonsson 16 An example; cont. For the first load cycle we have 9E8 0.095 0.47 0.00275 0.262 N f 1 2 N f 1 2 E11 By doing the same type of calculations for the other two load cycle types, we get in total Nf1=4.5E4 , Nf2=2.2E6 , Nf3=7.2E5 Palmgren-Miner then gives the following damage for 1 sequence D fs 1 2 2 1 4.5E 4 2.2 E 6 7.2 E 5 38500 Thus, the expected number of load sequences to failure (when D=1) is N f D fs 1 N f Kjell Simonsson 1 38 500 D fs 17 Wöhler-diagrams/SN-curves; cont. It is to be noted that there are other ways to describe the SN-curve and its mean stress sensitivity. However, these are outside the scope of this presentation. Kjell Simonsson 18 Topics still not discussed Topics for the next lecture On the next lecture we are to look more closely into the description of the cyclic plastic behavior of metallic materials, and to introduce approximate methods for estimating the cyclic plastic flow at stress concentrations (in an otherwise elastic structure/body) Kjell Simonsson 19