Stress gradient effect can be taken into consideration through either FKM guideline method or Critical Distance 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 OptiStruct 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. OptiStruct 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 OptiStruct 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.

## Critical Distance Method

To activate Stress Gradient effect using Critical Distance method, the GRD field on FATPARM should be set to GRDCD.

Small stress concentration features or geometries with high stress gradients are less effective in fatigue than larger features or smaller gradients with the same maximum stress. A plate with a small hole, say 0.1mm, will have a much longer fatigue life than one with a large hole of 10mm even though both plates have the same stress concentration factor and maximum stress. In conventional fatigue analysis, the stress gradient effect is taken into account by using an empirical fatigue notch factor, Kf, rather than the stress concentration factor Kt. Since there is no concept of a Kt or nominal stress in a finite element model stress gradient effects are considered directly. All of the holes have the same maximum stress, three times the nominal stress.

Figure 1 that the stresses are independent of size only at the edge of the hole and vary far from the hole. The dashed line in the figure is drawn at 0.5mm. Here the stresses increase as the size of the hole increases. Suppose crack nucleation mechanisms result in a crack with a size of 0.5mm. For the smallest hole, 0.1mm, the stress available for continued growth is only 100 MPa, the nominal stress. The same size crack is subjected to a stress of 275 MPa in the larger hole, nearly equal to the maximum stress.

For nucleation of a crack around a hole of different sizes, it is useful to think about a process zone for crack nucleation. Materials are not continuous and homogeneous on the size scale that crack nucleation mechanisms operate. The grain size of the material is a convenient way to visualize the fatigue process zone. Figure 2 shows the grain size superimposed on the stress distribution from Figure 1. What is the stress in the process zone? A simple first approximation would be to take the stress in the center of the grain. Thus, a stress of 275 MPa would be used to compute the fatigue life of a 10mm hole and a stress of 100 MPa would be used for the 0.1mm hole.

The modern view of fatigue is that when a material is stressed at the fatigue limit a microcrack will form but not grow outside of the process zone. Stress gradient effects are included in the fatigue analysis in a very simple and straightforward manner. In Critical Distance method, stresses and strains at a distance L/2 (Point Method) from the surface are used rather than the surface stresses and strains. For solid elements, the stress and strain at L/2 below surface is an interpolated stress and strain from grid point stresses and strains of an element of interest. In case of 2nd order solid elements, only grid point stresses and strains at corners are used for interpolation.

The critical distance can be expressed in terms of the threshold stress intensity, $\text{Δ}{K}_{TH}$ , and fatigue limit range, $\text{Δ}{\sigma }_{FL}$ , as:(3)
$L=\frac{1}{\pi }{\left(\frac{\text{Δ}{K}_{TH}}{\text{Δ}{\sigma }_{FL}}\right)}^{2}$
The critical distance is a unique material property. If the critical distance of the material in use is known, user can input the critical distance in MATFAT after keyword STSGRD. When you input the critical distance, it is important to define dimension of length in MATFAT as well. Computing the critical distance from the threshold stress intensity, however, is difficult because the threshold stress intensity, particularly for small microcracks, is usually unknown. Fortunately, there is a good direct correlation between the critical distance and fatigue.(4)
$L={\left(7×{10}^{-4}\frac{E}{\text{Δ}{\sigma }_{FL}}\right)}^{1.92}$

If you do not directly input the critical distance, OptiStruct uses Equation 4 to estimate the critical distance in SN fatigue analysis. Fatigue limit $\text{Δ}{\sigma }_{FL}$ is taken after the SN curve adjustment. Dimension of L is mm.

In EN fatigue analysis, the fatigue limit $\text{Δ}{\sigma }_{FL}$ is approximated in the following manner.(5)
$\text{Δ}{\sigma }_{FL}=2E{e}_{nc}$
(6)
${e}_{nc}=\frac{S{\text{'}}_{f}}{E}×{N}_{c}^{b}$
Where,
$S{\text{'}}_{f}$
Fatigue strength coefficient.
${N}_{c}$
Reversal limit of endurance.
$E$
Young’s modulus.

If $\text{Δ}{\sigma }_{FL}$ is 0 or the calculated $L$ is greater than 0.2mm, $L$ will be set to 0.2mm. In case of shell elements, the maximum calculated $L$ is thickness/4.

## Input to Activate Stress Gradient Effect

Choose a method (FKM guideline or Critical Distance) 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.