# Stress Gradient Effect

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.

- Calculate stress gradient of 6 components of a stress tensor,
$\frac{\text{\Delta}{\sigma}_{ij}\left(t\right)}{\text{\Delta}z}$
, 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 2
^{nd}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. - Using the stress gradient obtained in Step 1, a gradient of equivalent stress in the surface normal direction, $\frac{\text{\Delta}{\sigma}_{eq}\left(t\right)}{\text{\Delta}z}$ , is calculated in an analytical way at each time step. The equivalent stress can be either von Mises stress or absolute maximum principal stress.
- The related stress gradient,
${G}_{\sigma}$
is calculated using the following
normalization.
(1) $$\overline{G}{\left(t\right)}_{\sigma}=\frac{1}{{\sigma}_{eq}\left(t\right)}\frac{\text{\Delta}{\sigma}_{eq}\left(t\right)}{\text{\Delta}z}$$ - Calculate the correction factor ${n}_{\sigma}\left(t\right)$ . Refer to Correction Factor Calculation.
- 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.

- $\begin{array}{l}\text{If}{\overline{G}}_{\sigma}\le 0.1{\text{mm}}^{-1}\\ {n}_{\sigma}=1+{\overline{G}}_{\sigma}\cdot mm\cdot {10}^{-\left({a}_{G}-0.5+\frac{{R}_{m}}{{b}_{G}\cdot \text{MPa}}\right)}\end{array}$

- $\begin{array}{l}\text{If}0.1{\text{mm}}^{-1}{\overline{G}}_{\sigma}\le 1{\text{mm}}^{-1}\\ {n}_{\sigma}=1+\sqrt{{\overline{G}}_{\sigma}\cdot \text{mm}}\cdot {10}^{-\left({a}_{G}+\frac{{R}_{m}}{{b}_{G}\cdot \text{MPa}}\right)}\end{array}$
- $\begin{array}{l}\text{If}1{\text{mm}}^{-1}{\overline{G}}_{\sigma}\le 100{\text{mm}}^{-1}\\ {n}_{\sigma}=1+\sqrt[4]{{\overline{G}}_{\sigma}\cdot \text{mm}}\cdot {10}^{-\left({a}_{G}+\frac{{R}_{m}}{{b}_{G}\cdot \text{MPa}}\right)}\end{array}$

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 |

- 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.

`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.

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.

`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.

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

- $S{\text{'}}_{f}$
- Fatigue strength coefficient.
- ${N}_{c}$
- Reversal limit of endurance.
- $E$
- Young’s modulus.

If $\text{\Delta}{\sigma}_{FL}$ 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.