*5.2. Brown–Miller Analysis*

The Brown–Miller algorithm conservative approach for fatigue life prediction, using planes perpendicular to the surface and at 45 degrees to the surface. It uses a critical plane analysis to estimate the fatigue life in reversals to failure, 2*Nf* , by solving the following equation [15] at each node.

$$\frac{\Delta\chi\_{\text{max}}}{2} + \frac{\Delta\varepsilon\_{\text{n}}}{2} = 1.65 \frac{\sigma\_{\text{f}}^{\prime}}{\text{E}} (2\text{N}\_{\text{f}})^{\text{b}} + 1.75 \,\varepsilon\_{\text{f}}^{\prime} (2\text{N}\_{\text{f}})^{\text{c}} \tag{1}$$

With Morrow mean stress correction, Equation (1) is modified to

$$\frac{\Delta\gamma\_{\text{max}}}{2} + \frac{\Delta\varepsilon\_{\text{R}}}{2} = 1.65 \frac{(\sigma\_{\text{f}}^{\prime} - \sigma\_{\text{m}})}{\text{E}} (2\text{N}\_{\text{f}})^{\text{b}} + 1.75 \text{ }\varepsilon\_{\text{f}}^{\prime} (2\text{N}\_{\text{f}})^{\text{c}} \tag{2}$$

where Δγmax 2 is the maximum shear strain amplitude, Δ<sup>ε</sup>n 2 is the strain amplitude normal to the shear stress plane, σm is the mean stress, <sup>σ</sup>f is the fatigue strength coefficient, b is the fatigue strength exponent, εf is the fatigue ductility coefficient, and c is the fatigue ductility exponent.

The critical plane analysis is used to compute the strain tensor at a FE node having three direct and three shear components. The strain tensor is then resolved onto a number of planes, where, at each place the damage associated with the strain is evaluated. The plane resulted with maximum damage is used in strain-life computations. For a Cartesian *x* − *y* − *z* coordinate system, the unique planes can be defined by the orientation the normal of the plane surface makes with respect to the coordinate system [17]. This orientation can be defined by an angle from *x*-axis toward the *y*-axis, and another angle from the *z*-axis toward the *x* − *y* plane [15]. Fe-safe searched for the critical plane having worst damage in 10-degree increments over the 180-degree range of the first angle and 90-degree range of the second angle. The strains are projected to the calculation plane using direction cosines.
