Stress Gradient Effect

The stress gradient effect is supported for both shells and solid elements. For solid elements, the stress gradient effect is only available with nodal evaluation and for shell elements, the stress gradient effect is only available with elemental evaluation.

The Stress Gradient method is currently supported for Uniaxial and Multiaxial SN, EN with Time series loading. It is not supported for Dang Van FOS, 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 HWTK GUI Toolkit 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, Stress Gradient should be set to FKM Guideline in SN/eN dialog. The equivalent stress σ_eq method to calculate stress gradient should be specified.

The following steps are followed to reduce stresses at the surface to take stress gradient effect into consideration.

Calculate stress gradient of 6 components of a stress tensor, $\frac{Δ σ_{i j} (t)}{Δ z}$ , 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 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, $\frac{Δ σ_{e q} (t)}{Δ z}$ , is calculated in an analytical way at each time step. The equivalent stress can be either von Mises stress or absolute maximum principal stress.
The related stress gradient, $G_{σ}$ is calculated using the following normalization.
$\bar{G} {(t)}_{σ} = \frac{1}{σ_{e q} (t)} \frac{Δ σ_{e q} (t)}{Δ z}$
Calculate the correction factor $n_{σ} (t)$ . Refer to Correction Factor Calculation.
Apply the correction factor $n_{σ}$ to the surface stress tensor to obtain reduced surface stress. Apply the same $n_{σ}$ to corresponding strain tensor to obtain reduced strain tensor when EN fatigue analysis is to be carried out with nonlinear analysis.
$σ'_{i j} (t) = \frac{σ_{i j} (t)}{n_{σ} (t)}$

Correction Factor Calculation

Correction factor calculation is based on relationship between $n_{σ}$ and $G_{σ}$ described in the FKM guidelines.

According to FKM guidelines, the stress correction factor

n_{σ}

is determined by:

$\begin{array}{l} If {\bar{G}}_{σ} \leq 0.1 {mm}^{- 1} \\ n_{σ} = 1 + {\bar{G}}_{σ} \cdot m m \cdot 10^{- (a_{G} - 0.5 + \frac{R_{m}}{b_{G} \cdot MPa})} \end{array}$

$\begin{array}{l} If 0.1 {mm}^{- 1} < {\bar{G}}_{σ} \leq 1 {mm}^{- 1} \\ n_{σ} = 1 + \sqrt{{\bar{G}}_{σ} \cdot mm} \cdot 10^{- (a_{G} + \frac{R_{m}}{b_{G} \cdot MPa})} \end{array}$
$\begin{array}{l} If 1 {mm}^{- 1} < {\bar{G}}_{σ} \leq 100 {mm}^{- 1} \\ n_{σ} = 1 + \sqrt[4]{{\bar{G}}_{σ} \cdot mm} \cdot 10^{- (a_{G} + \frac{R_{m}}{b_{G} \cdot MPa})} \end{array}$

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 $G_{σ}$ is mm. HWTK GUI Toolkit takes care of the unit system for $R_{m}$ and $G_{σ}$ through stress units defined in Material and stress unit and length unit defined in SN/eN dialog. $a_{G}$ and $b_{G}$ values are user input in Material under FKM Stress Gradient tab of My Material. Since the stress gradient has to be calculated in length dimension of mm, define the length units so that HWTK GUI Toolkit can properly locate a point that is 1mm below the surface. If $G_{σ}$ is negative, $n_{σ}$ is set to 1.0. If $G_{σ}$ is greater than 100 mm-1, $n_{σ}$ is set to 1.0 with a warning message.

User-defined Relationship

User-defined relationship between $n_{σ}$ and $G_{σ}$ can be specified through a TABLE (pairs of (xi,yi) = ( $G_{σ}$ , $n_{σ}$ ) in My Materials. If $G_{σ}$ falls outside the range of xi, $G_{σ}$ will be extrapolated. This means that $n_{σ}$ can be lower than 1.0 when $G_{σ}$ is negative depending on how $G_{σ}$ is treated when being negative or greater than 100mm-1. The user-defined relationship takes precedence over the one in FKM guidelines.