1. Kjell Simonsson 1 Low Cycle Fatigue (LCF) analysis; cont. (last updated 2011-10-11)

2. Kjell Simonsson 2 Aim For high loadings/short lives (with respect to the number of load cycles), fatigue life calculations are generally strain-based. Since global plastic flow generally is not acceptable, we will focus attention on localized plastic flow at local stress raisers/stress concentrations. Having found the total cyclic strain range occurring there, we may then go on and calculate the expected life by e.g. Morrow’s equation. However, before that, as a requisite, we need to take a look at the typical cyclic plastic behavior of metals.

3. Kjell Simonsson 3 Plastic behavior under monotonic loading The Ramberg-Osgood relation The plastic behavior of metals can for monotonic loadings often be described by the Ramberg-Osgood relation

4. Kjell Simonsson 4 Plastic behavior under cyclic loading The cyclic stress-strain curve When a material is subjected to cyclic loading (inducing plastic flow) its behavior typically gradually change for a number of loading cycles and then stabilize. As an example, for a strain controlled test (Rε=-1) of a cyclically softening material we then schematically get (with the rate of softening strongly exaggerated and with some appropriate units and numbers on the axes)

5. Kjell Simonsson 5 Plastic behavior under cyclic loading; cont. The cyclic stress-strain curve; cont. After making a number of cyclic test, where one for each test finds the stabilized conditions, the so called cyclic stress-strain curve can be set up as schematically illustrated below, where it is to be noted that it does NOT have anything to do with the monotonic loading curve discussed previously, and that it generally does not look exactly as it.

6. Kjell Simonsson 6 Plastic behavior under cyclic loading; cont. The cyclic stress-strain curve; cont. The cyclic stress strain curve can generally be described by a Ramberg- Osgood type of relation (with DIFFERENT parameters compared to the case of monotonic loading)

7. Kjell Simonsson 7 Plastic behavior under cyclic loading; cont. The Masing behaviour At stabilised cyclic conditions we typically find the behavior illustrated below, where the inner curve is the cyclic stress-strain curve, and where the outer curve describes a closed hysteresis loop, given by the relation below

8. Kjell Simonsson 8 Plastic behavior under cyclic loading; cont. The Masing behaviour; cont. The so called Masing behaviour, described by the formulas on the previous slide, can be used to construct closed hysteresis loops of any type and location, as illustrated below.

9. Kjell Simonsson 9 A piece of reality At the Divisions of Solid Mechanics and Engineering Materials some work has been performed on modelling the cyclic plastic behaviour of the nickel-base superalloy IN718. The work has been reported in the following journal article: Gustafsson D., Moverare J.J., Simonsson K. and Sjöström S. (2011), Modelling of the constitutive behaviour of Inconel 718 at intermediate temperatures, Journal of Engineering for Gas Turbines and Power, 133, Issue 9 Plastic behavior under cyclic loading; cont.

10. Kjell Simonsson 10 A piece of reality; cont. Material response 1,6% Strain range, 400°C, R=0 As can be seen in the curves from the material testing, the hardening curve of the first loading is lower than that of the following cycling This is due to a large softening of the material, possibly caused by the formation of planar slip bands As can be seen, limited mean stress relaxation occurs By combining the Ohno-Wang kinematic hardening model with an isotropic softening description, a simple model capable of describing the observed behavior was found Plastic behavior under cyclic loading; cont.

11. Kjell Simonsson 11 A piece of reality; cont. Results 1,6% strain range (good fit with few material parameters) Plastic behavior under cyclic loading; cont.

12. Kjell Simonsson 12 A piece of reality; cont. Results 1,0% strain range (with model optimized for the 1,6% case) Plastic behavior under cyclic loading; cont.

13. Kjell Simonsson 13 Plasticity at notches Stress- and strain concentration factors At plastic flow, the stiffness of the material decreases with respect to the elastic behavior. By this it follows, that we in the case of plastic flow at a notch have Schematically, we have the following situation, where Kt is the “ordinary” stress (and strain) concentration factor for elastic conditions. elastic range plastic range

14. Kjell Simonsson 14 Plasticity at notches; cont. Neuber’s rule Neuber suggested that Leading to or

15. Kjell Simonsson 15 Plasticity at notches; cont. Neuber’s rule; cont. For a specified nominal ( and thus specified maximum elastic) stress or strain, the Neuber rule represents a hyperbola, where the form of it is governed by the RHS of the relation.

16. Kjell Simonsson 16 Plasticity at notches; cont. Neuber’s rule; cont. However, since the material for cyclic conditions also is to obey the cyclic stress-strain behavior, the actual stress and strain state at the notch will be given by the intersection of these curves. Actual stress and strain state

17. Kjell Simonsson 17 Plasticity at notches; cont. Neuber’s rule; cont. Note that the Neuber rule provides a simple way of finding an approximation of the stress and strain state at a notch in which the material flows plastically, without actually carrying out any elasto-plastic calculation

18. Kjell Simonsson 18 Topics still not discussed Topics for the next lecture On the next lecture we are to look at how fatigue analysis can be carried out in an FE-context.