Stress gradient effect based on FKM guideline method.

It is supported for both shells and solid elements. For solid elements, the stress gradient effect is only available with grid point stress in fatigue analysis using results of static analysis. For solid elements, SURFSTS field on FATPARM is automatically set to GP when Stress Gradient effect is activated.

The Stress Gradient method is supported for Uniaxial and Multiaxial SN, EN and FOS Fatigue. It is not supported for Weld, Vibration, and Transient Fatigue analyses.

## FKM Guideline Method

In the FKM guideline method, stress gradient effect is considered by increasing fatigue strength by a factor calculated using a rule in FKM guidelines. In HyperLife implementation of FKM guideline method, 6 components of a stress tensor at each time step is reduced by the factor provided by FKM guidelines.

To activate Stress Gradient effect using FKM guideline method, the GRD field on FATPARM should be set to GRDFKM.

The following steps are followed to reduce stresses at the surface to take stress gradient effect into consideration.
1. Calculate stress gradient of 6 components of a stress tensor, $\frac{\text{Δ}{\sigma }_{ij}\left(t\right)}{\text{Δ}z}$ , at each time step after linear combination of stress history. z-direction is an outward surface normal. For a solid element, the gradient is calculated by finite difference between stress at surface and stress at 1mm below the surface. The stress at 1mm below surface is an interpolated stress from grid point stresses of an element of interest. In case of 2nd order solid elements, only grid point stresses at corners are used for interpolation. For shell elements, the gradient is calculated from stresses of both layers and its thickness.
2. Using the stress gradient obtained in Step 1, a gradient of equivalent stress in the surface normal direction, $\frac{\text{Δ}{\sigma }_{eq}\left(t\right)}{\text{Δ}z}$ , is calculated in an analytical way at each time step. The equivalent stress can be either von Mises stress or absolute maximum principal stress.
3. The related stress gradient, ${G}_{\sigma }$ is calculated using the following normalization.(1)
$\overline{G}{\left(t\right)}_{\sigma }=\frac{1}{{\sigma }_{eq}\left(t\right)}\frac{\text{Δ}{\sigma }_{eq}\left(t\right)}{\text{Δ}z}$
4. Calculate the correction factor ${n}_{\sigma }\left(t\right)$ . Refer to Correction Factor Calculation.
5. Apply the correction factor ${n}_{\sigma }$ to the surface stress tensor to obtain reduced surface stress. Apply the same ${n}_{\sigma }$ to corresponding strain tensor to obtain reduced strain tensor when EN fatigue analysis is to be carried out with nonlinear analysis.(2)
$\sigma {\text{'}}_{ij}\left(t\right)=\frac{{\sigma }_{ij}\left(t\right)}{{n}_{\sigma }\left(t\right)}$

## Correction Factor Calculation

Correction factor calculation is based on relationship between ${n}_{\sigma }$ and ${G}_{\sigma }$ described in the FKM guidelines.

According to FKM guidelines, the stress correction factor ${n}_{\sigma }$ is determined by:
Table 1. Example values for Constants ${a}_{G}$ and ${b}_{G}$
Constants Stainless Steel Other steels GS GGG GT GG Wrought Al-Alloys Cast Al- Alloys
${a}_{G}$ 0.40 0.50 0.25 0.05 -0.05 -0.05 0.05 -0.05
${b}_{G}$ 2400 2700 2000 3200 3200 3200 850 3200
Where,
GS
Cast Steel and Heat Treatable cast steel for general purposes.
GGG
Nodular Cast Iron.
GT
Malleable Cast Iron.
GG
Cast Iron with lamellar graphite (grey cast iron).

${R}_{m}$ is UTS in MPa and dimension of ${G}_{\sigma }$ is mm. HyperLife takes care of the unit system for ${R}_{m}$ and ${G}_{\sigma }$ through stress units defined in MATFAT and stress unit and length unit defined in FATPARM. ${a}_{G}$ and ${b}_{G}$ values are user input in MATFAT after keyword STSGRD. Since the stress gradient has to be calculated in length dimension of mm, define the length units so that HyperLife can properly locate a point that is 1mm below the surface. If ${G}_{\sigma }$ is negative, ${n}_{\sigma }$ is set to 1.0. If ${G}_{\sigma }$ is greater than 100 mm-1, ${n}_{\sigma }$ is set to 1.0 with a warning message.

## User-defined Relationship

User-defined relationship between ${n}_{\sigma }$ and ${G}_{\sigma }$ can be specified through TABLES1 Bulk Data. Pairs of (xi,yi) = ( ${G}_{\sigma }$ , ${n}_{\sigma }$ ) can be defined on the TABLES1 entry. A TABLES1 that defines the relationship between ${n}_{\sigma }$ and ${G}_{\sigma }$ should be referenced in MATFAT after keyword STSGRD. If ${G}_{\sigma }$ falls outside the range of xi, extrapolation behavior follows usual TABLES1 behavior. This means that ${n}_{\sigma }$ can be lower than 1.0 when ${G}_{\sigma }$ is negative depending on how ${G}_{\sigma }$ is treated when being negative or greater than 100mm-1. The user-defined relationship takes precedence over the one in FKM guidelines.

## Input to Activate Stress Gradient Effect

Choose a method (FKM guideline) to use on the GRD field after keyword STRESS in FATPARM. If FKM guideline method is chosen, the equivalent stress ${\sigma }_{eq}$ method to calculate stress gradient should be specified on the SCBFKM field in FATPARM. Material properties required for stress gradient effect are to be input after keyword STSGRD in MATFAT.