Notch analysis

The damage parameters are useful once the elastic-plastic cyclic stress-strain behavior at likely fatigue crack initiation sites have been determined. However, in many instances, only the elastic stresses and strains are initially known when designing a component. These elastic stresses or pseudo stresses may have been determined by elastic finite element analysis or by stress concentration tables from handbooks. Using these elastic stresses often lead to predict a much to short fatigue life.

In these cases, it is useful to use approximate formulas that relate the elastic stress-strain behavior at the stress concentration, or notch, to the elastic-plastic behavior of the material at the notch. Some well-known approximate notch analysis formulas are Neuber's rule, Glinka's formula (also known as the Equivalent Strain Energy Density or ESED approach), and the Seeger-Beste approach.

Neuber's rule

Neuber's rule states that the product of the stress and strain computed from the loads computed by theory of elasticity is equal to the product of the elastic-plastic stress and strain, or

where the superscript e denotes stresses and strains calculated by theory of elasticity, also known as pseudo stresses and strains. Schematically, Neuber's rule can be shown in figure Neuber's Rule below. The product of pseudo-stress and pseudo- strain are represented by the shaded rectangle on the axes on the left side of the figure, and the product of the elastic-plastic stress and strain are represented by the rectangle on the left side of the figure. The Ramberg-Osgood relation, together with the relation from equal areas of the rectangles, provide enough information to obtain the elastic-plastic stress and strain.

Neuber's Rule

The ESED approach

The equivalent strain energy density relation, also known as Glinka;s formula states that the strain energy density at a notch of an elastic-plastic body will have the same strain energy density at a notch of an elastic body, at the same load level. Schematically, this is represented in the figure ESED Approach below. The area under the pseudo-stress pseudo-strain curve (the strain energy density) is indicated in the figure on the left, and the point on the elastic-plastic stress-strain curve with an equal strain energy density is shown in the figure on the right. The Ramberg-Osgood equation, together with this relationship can be used to determine the notch stresses and strains.

The ESED Approach

Limit load plasticity

Neuber's Rule and the ESED relations in the forms described previously assume nominally elastic behavior without regard to the global geometry and loading.

In the figure below, the problem is sketched. Whereas for pure tension the whole structure yields after the yielding starts locally. In this case there is no positive effect of microscopic yielding.

For the case of bending the state of global yielding is reached much later.

The Limit Load Ratio

In actual structures there is a limiting value of load that the structure can sustain. The limit load ratio gives the ratio between the limiting load and the load when yielding starts.

The shape of the load-notch strain curve or the pseudo stress- notch strain curve can be influenced by the value of the limit load ratio. In many cases, the load-notch strain curve has less slope (flattens out) as the limit load is approached. This behavior is more pronounced with an increased amount of plasticity at the notch, although this influence can extend to load levels having much smaller amounts of plastic deformation.

For calculating the limit load ratio, the extreme case of the fully plastic limit load for a structure made of an ideal perfectly-plastic material is computed. The axial and shear stress-strain curves of an elastic perfectly-plastic material are shown in the figure Elastic Perfectly Plastic Material below. The yield stress in tension and torsion are σp and τp, respectively.

For certain geometries, the fully plastic limit load can be determined in closed form. For example, consider the notched round shaft of figure Shaft Tension and Torsion below, with nominal diameter, d, subjected to tension and torsion.

Elastic Perfectly Plastic Material

Shaft Tension and Torsion

For only torsional loading of the structure, the stress distribution will initially be as in the figure Elastic Perfectly Plastic Material (a) with the material is still elastic. The largest torque that can be applied to the shaft as it first yields (TF) can be found by the elastic shear stress formula:

The torque at which the cross-section of the shaft becomes fully plastic (see the figure Elastic Perfectly Plastic Material (b)) can be determined by equilibrium. A small increment of the force at a radial distance, r, from the center of the shaft is given by

dF = τpdA

and the total resultant torque can be determined by

The ratio of the fully plastic limit load to the load at the onset of yielding is

In a similar manner, the ratio of the fully plastic limit load to the load at the onset of yielding can be determined for other loading conditions (refer to the figure The Limit Load Ratio above for the cases of bending and tension). For complex geometries, an elastic-plastic finite element analysis may be employed, or judgment may be used to determine this ratio.

Universal Neuber approximation

The Neuber approximation discussed previously can be modified to account for the limit load behavior. In terms of the applied load-notch strain curve and the stress-strain curve in the figure Load Strain Curve below, Neuber's rule can be written as

when

L > LF

where

In these relations, the local stress is σ, and the local strain is ε. The modified nominal stress is computed from the elastic stress (L · c) divided by the limit load ratio, Kp. The modified nominal strain is the strain corresponding to the modified nominal stress, S*, as seen in the figure.

Load Strain Curve

Typical values for the ratio are in the range 1 < Kp < 5. If there is no influence of the plastic limit load, a value of Kp = 30 is often used.

Seeger-Beste approximation

The Seeger-Beste formula for approximating local stress-strain behavior at notches also makes use of the idea of the plastic limit load ratio. The formula was developed based on fracture mechanics considerations of plate with cracks. It is typically less conservative at low loads when compared to Neuber's rule.

where

when

L > LF

where

as before.

Comparison of the different load notch strain approximation formulas

The method you want to use will depend on the special circumstances for your analysis. However, there are some hints on the global behavior of the different approximation formulae.

For applied loads that do not lead to local plasticity there is no difference between the approximations. On the other hand, for large loads the predictions of the individual formulae do differ.

For a given value of Kp and a given material, the Neuber formula gives the highest local strains and therefore the highest damage for a given load history. The ESED approach predicts much lower local strains and therefore gives much less conservative results. The results of the Seeger-Beste approximation lie between the two other methods. This is sketched in the example results in the figures below. Here the load-strain relations and matching single amplitude life-curves for the three approaches are given for Kp = 30, that is, no influence of the limit load ratio.

Load-Strain Relation for the Different Load-Notch Approximations

Corresponding Single Amplitude Life-Curves

Comparison of the influence of different limit load ratios

The main influence of the limit load ratio applied to the load-notch strain approximation formulas is restricted to the low cycle fatigue regime, that is, the parts of a component where the local plasticity is to be considered.

Therefore if you want to analyze a complete structure you need not care about regions with moderate stress contributions when choosing a value for the limit load ratio.

The value of 2.5 that is predefined in some of the Load Notch Strain Relation durability simulation object sets that come with Specialist Durability may therefore be used to get a first idea about the damage distribution of a structure. After you identified the important spots you may construct element sets and assign accurate values of the limit load ratio for those regions individually. Values larger than 10 do not change the overall behavior anymore.

Load Notch Strain Relations for Different Values of the Limit Load Ratio

Results for Different Values of the Limit Load Ratio

In the figures it may be seen that for the high cycle fatigue regime there is no influence of different limit load ratios.

Keep in mind that if you want to compare results to calculations that ignore the effect of the limit load ratio you should choose a value of 30 to get similar results.

Elastic-plastic calculation—field option

In addition to these methods to approximate the load-notch strain curve, you can also specify a field containing discrete points on the load-notch strain curve. By using this option, experimental results or calculations by elastic-plastic finite element analysis can be directly used for the notch stress-strain calculations.

The syntax of this file is explained in detail in the following lines:

The first column contains the elastic stress, e~σ. This elastic stress is always proportional to the external load, even if the material has yielded. The second column contains the unitless total notch strain. The first point should be zero stress and zero strain.

Tab characters should not be used in the file. Comments in the file are marked with #.

Make sure that the material properties are consistent with those used to obtain this load-notch strain curve. For example, if the curve was obtained using non-linear finite element analysis, the finite element analysis material properties should be taken from the same material to be used in the fatigue analysis. It is important that the channel names are chosen exactly as in the example below. Be sure that the entries are ordered such that both columns are increasing.

BEGINCHANNELNAME = ['LOAD','STRAIN']COLUMNTYPE = ['R4','R4']END0.000000e+00 0.000000e+009.259259e-01 1.450555e-051.851852e+00 2.942633e-052.777778e+00 4.466388e-055.555555e+00 8.653605e-051.018519e+01 1.576316e-041.481481e+01 2.275840e-041.944444e+01 2.988200e-042.222222e+01 3.419045e-042.407407e+01 3.705882e-04

Parameter	Meaning	Additional Input
Neuber	Neuber approximation including limit load ratio correction	Limit load ratio (Kp)
Seeger/Beste	Seeger-Beste approximation including limit load ratio correction	Limit load ratio (Kp)
ESED	Equivalent strain energy density including limit load ratio correction	Limit load ratio (Kp)
Field	Read from field	Field name

To specify these parameters in Specialist Durability, use the Simulation Objects command to create a load notch strain relation simulation object.

Learn more

Local stress-strain behavior

Constant amplitude life curves

Endurance limit and static failure

Determining material properties

Mean stress and damage parameters

Axial versus torsion tests

The strain-life analysis in Specialist Durability

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