Mean Stress Correction

Generally, S-N curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully-reversed, and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to account for the effect of non-zero mean stresses.

Depending on the material, stress state, environment, and strain amplitude, fatigue life will usually be dominated either by microcrack growth along shear planes or along tensile planes. Critical plane mean stress correction methods incorporate the dominant parameters governing either type of crack growth. Due to the different possible failure modes, shear or tensile dominant, no single mean stress correction method should be expected to correlate test data for all materials in all life regimes. There is no consensus yet as to the best method to use for multiaxial fatigue life estimates. For stress-based mean stress correction method, Goodman and FKM models are available for tensile crack. Findley model is available for shear crack.

Goodman Model

Use the Goodman model to assess damage caused by tensile crack growth at a critical plane.

(1)(1)

$S_{e} = \frac{S_{r}}{(1 - \frac{S_{m}}{S_{u}})}$

Where:

$S_{m}$ is the Mean stress given by $S_{m} = (S_{m a x} + S_{m i n}) / 2$
$S_{r}$ is the Stress Range given by $S_{r} = S_{m a x} - S_{m i n}$
$S_{e}$ is the stress range after mean stress correction (for a stress range $S_{r}$ and mean stress $S_{m}$
$S_{u}$ is the ultimate strength

The Goodman method treats positive mean stress correction in the way that mean stress always accelerates fatigue failure, while it ignores the negative mean stress. This method gives conservative result for compressive mean stress.

A Haigh diagram characterizes different combinations of stress amplitude and mean stress for a given number of cycles to failure.

Findley Model

The Findley criterion is often applied for the case of finite long-life fatigue. The equation for each plane is as follows:(2)(28)

$\frac{Δ τ}{2} + k σ_{n} = τ_{f}^{*} {(N_{f})}^{b}$

Where:

$τ_{f}^{*}$ is computed from the shear fatigue strength coefficient,

$τ_{f}^{'}$ , using: (3)(29)

$τ_{f}^{*} = \sqrt{1 + k^{2}} τ_{f}^{'}$

The correction factor

$\sqrt{1 + k^{2}}$ typically has a set value of about 1.04.

Note:

$τ_{f}^{*}$ must be defined based on amplitude. If

$τ_{f}^{'}$ is not defined by the user, SimSolid calculates it using the following equation:(4)(30)

$\begin{array}{l} τ_{f}^{'} = C f * 0.5 * S R I 1 \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ W h e r e, \\ C f = \frac{2}{1 + \frac{k}{\sqrt{1 + k^{2}}}} \end{array}$

The constant k is determined experimentally by performing fatigue tests involving two or more stress states. For ductile materials, k typically varies between 0.2 and 0.3.

FKM

Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio (R=S_min/S_max) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.

The FKM equations below illustrate the calculation of Corrected Stress Amplitude ( $S_{e}^{A}$ ). The actual value of stress used in the Damage calculations is the Corrected stress range (which is $2 \cdot S_{e}^{A}$ ). These equations apply for SN curves that you input.

Regime 1 (R>1.0): $S_{e}^{A} = S_{a} (1 - M)$

Regime 2 (-∞≤R≤0.0): $S_{e}^{A} = S_{a} + M * S_{m}$

Regime 3 (0.0<R<0.5): $S_{e}^{A} = (1 + M) \frac{S_{a} + (\frac{M}{3}) S_{m}}{1 + \frac{M}{3}}$

Regime 4 (R≥0.5): $S_{e}^{A} = \frac{3 S_{a} {(1 + M)}^{2}}{3 + M}$

Where

$S_{e}^{A}$ is the stress amplitude after mean stress correction (Endurance stress),

$S_{m}$ is the mean stress,

$S_{a}$ is the stress amplitude, and M is the mean stress sensitivity.