Theoretical concepts

Stress gradient in fatigue context denote the rate of change of stress with respect to distance. This distance is typically the depth into a component from a stress concentrator or notch. The stress gradient, calculated by the theory of elasticity, is normalized by the maximum stress at the notch, as seen in the figure below:

The stress gradient in reference to the figure is defined as:

and the normalized stress gradient which is used in the formulas is defined as:

Definition of Stress Gradient

Normalized stress gradients for common notch geometries are listed in the figure below:

Normalized Stress Gradients for Representative Notch Types

The normalized stress gradients are used in conjunction with an empirically determined chart that relates the gradient, type of material and offset yield stress of the material, to the stress gradient correction factor (see figure below). The FKM (a German research group of engineers) also has empirical formulae for this relation (Lit.: Rechnerischer Festigkeitsnachweis für Maschienenbauteile aus Stahl Eisenguss- und Aluminiumwerkstoffen 4. erweiterte Ausgabe 2002 p. 108, also available in English Version: Analytical Strength Assessment 5th edition).

Empirical Stress Gradient Correction Factors (Siebel-Stieler 1955)

This is a list of curves implemented within Simcenter 3D and based on the FKM recommendations. The following curves can be chosen (from FKM paper referenced above):

• Stainless steel

• Steel

• Cast steel

• Ductile graphite iron

• Malleable cast iron

• Gray cast iron

• Forgeable aluminum alloy

• Aluminum cast material

The dependency is then calculated from the parameters a,b and the formulas in the table below:

	a	b
Stainless steel	0.4	2400
Steel	0.5	2700
Cast steel	0.25	2000
Ductile graphite iron	0.05	3200
Malleable cast iron	-0.05	3200
Gray cast iron	-0.05	3200
Forgeable aluminum alloy	0.05	850
Aluminum cast material	-0.05	3200

For

The empirical studies provided by, for example, Siebel and Stieler or those leading to the FKM recommendations only take the stress correction at endurance limit into account. If you analyze the effect in more detail, it typically vanishes for large stress cycles. This is often taken into account by a twofold approach:

• Apply the factor n(X) as in the formulae above at endurance limit and

• Apply a n(X)=1 at tensile strength

• Interpolate the factor in between

In the software code both approaches (constant n(X) and n(X) depending on σ) are available.

There are two types of material behavior entries; either you enter the parameters a and b, or you give a list of points (c,n) and a multilinear approximation is used.

This correction factor is applied locally; that is, the stresses are diminished by this factor.

Learn more

The fatigue notch factor

Examples for size effects

Stress gradients

Application of the theory

Macroscopic yielding

Neuber's approach to micro-yielding

Summary for size effects

Surface effects

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Theoretical concepts, Simcenter 3D 2021.1 Series

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