Fatigue tests of specimens are usually done on components with mirror polished surfaces. However, a component with a larger surface roughness will have a reduced fatigue strength. To account for the effect of this surface roughness to the fatigue strength of a material, a roughness factor K_{R} is used. This surface roughness factor is used to adjust the material S-N curve. This article describes the FKM method for obtaining the surface roughness factor and how to apply this factor to the S-N curve.

## FKM surface roughness factor

The FKM-guideline *Analytical Strength Assessment* defines the roughness factor K_{R} as follows:

$${\displaystyle K_R=1-a_R \cdot log_{10}(R_Z) \cdot log_{10}\left(\frac {2R_m}{R_{m,N,min}}\right)}$$

With:

*R*the surface roughness in µm according to DIN 4768 (see Table 2)_{Z}*R*the tensile strength in MPa_{m}*a*a constant (see Table 1)_{R}*R*the minimum tensile strength in MPa (see Table 1)_{m,N,min}

The roughness factor *K _{R} = 1* for a polished surface. Surface roughnesses higher than that of a polished surface will have values for

*K*.

_{R}< 1Table 1 below gives the values for R_{m,N,min} and for constant a_{R} for different material types.

Material Type | a_{R} |
R_{m,N,min} [MPa] |
---|---|---|

Steel | 0.22 | 400 |

GS | 0.20 | 400 |

GGG | 0.16 | 400 |

GT | 0.12 | 350 |

GG | 0.06 | 100 |

Wrought aluminium alloys | 0.22 | 133 |

Cast aluminium alloys | 0.20 | 133 |

Table 2 below provides some indicative values for the surface roughness R_{Z} of different surface finishes.

Surface Condition | R_{z} [µm] |
---|---|

Polished | 0 |

Ground | 12.5 |

Machined | 100 |

Poor machined | 200 |

Rolled | 200 |

Cast | 200 |

## Example calculation

As an example, let's look at a steel with a tensile strength R_{m} = 600 MPa and with a machined surface.

In Table 1 we find for Steel: a_{R} = 0.22 and R_{m,N,min} = 400 MPa. Table 2 gives us R_{Z} = 100 µm for a machined surface.

Putting these values in the above equation gives us:

$${\displaystyle K_R=1-0.22 \cdot log_{10}(100)\cdot log_{10}\left( \frac{2 \cdot 600}{400}\right)}$$

and we get *K _{R} = 0.79* as result.

## Construction of the S-N curve

Before we get to how the S-N curve is adjusted for the surface roughness factor, let's look at how the S-N curve is constructed (see Figure 1 below - click for larger image).

For the number of cycles N_{f} between 10^{3} and N_{c1} cycles, the S-N curve is defined as:

$${\displaystyle \Delta \sigma(N_f) = SRI1 \cdot N_f^{b_1}}$$

The stress range Δσ is a function of the number of cycles N_{f}. SRI1 is the stress range intercept at 1 cycle and b_{1} is the slope, which is negative. The parameter N_{c1} is called the *Fatigue Transition Point* and is basically the number of cycles by which a kink in the fatigue curve is noticeable. The value of N_{c1} is usually around 10^{6} - 10^{7} cycles.

For N_{f} > N_{c1}, the S-N curve is defined as:

$${\displaystyle \Delta \sigma(N_f) = SRI2 \cdot N_f^{b_2}}$$

SRI2 is the stress range intercept at 1 cycle and b_{2} is the slope, which is negative, for the second part of the S-N curve.

The parameters SRI1, N_{c1}, b_{1} and b_{2} are material parameters derived from the fatigue test data. SRI2 can be derived as follows:

$${\displaystyle SRI2 = SRI1 \cdot (N_{c1})^{b_1-b_2}}$$

## S-N curve adjusted for the surface roughness factor

The surface condition has the greatest effect in the high-cycle regime and becomes progressively smaller towards the low-cycle regime. S-N curves are usually adjusted for the surface roughness by changing the slope b_{1} in the first part of the S-N curve (see figure below) and keeping the fatigue strength at 1000 cycles the same. The slope b_{2} of the second part of the S-N curve remains unchanged.

So, let's find out how the equation of the S-N curve in Figure 2 is modified. For the unadjusted curve (polished surface), the stress range at N_{c1} cycles is:

$${\displaystyle \Delta \sigma(N_{c1}) = 1300 \cdot (10^6)^{-0.0612}=558.14 \text{ MPa}}$$

The stress range Δσ^{'} at N_{c1} cycles for a machined part becomes in our case (with K_{R} = 0.79, as calculated in the beginning):

$${\displaystyle \Delta \sigma^{'}(N_{c1}) = K_R \cdot 558.14 = 440.97 \text{ MPa}}$$

SRI1^{'} and b_{1}^{'} for the first part of the modified curve can be determined from the two equalities below:

$${\displaystyle SRI1 \cdot (10^3)^{b_1} = 851.81 = SRI1^{'} \cdot (10^3)^{b^{'}_1}}$$

$${\displaystyle \Delta \sigma^{'}(N_{c1}) = 440.97 = SRI1^{'} \cdot (10^6)^{b_1^{'}}}$$

When we solve for SRI1^{'} and b_{1}^{'}, we find:

*SRI1 ^{'} = 1645 MPa* and

*b*

_{1}^{'}= -0.0953Since b_{2}^{'} = b_{2}, we can solve below equation for SRI2^{'}:

$${\displaystyle \Delta \sigma^{'}(N_{c1}) = 440.97 = SRI2^{'} \cdot (10^6)^{b_2}}$$

which gives us *SRI2' = 677 MPa*.

The equation of the modified S-N curve between 10^{3} and N_{c1} cycles is then:

$${\displaystyle \Delta \sigma(N_{f}) = 1645 \cdot (N_f)^{-0.0953}}$$

And in the region for N_{f} > N_{c1}:

$${\displaystyle \Delta \sigma(N_{f}) = 677 \cdot (N_f)^{-0.0310}}$$