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 KR 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 KR

The FKM-guideline Analytical Strength Assessment defines the roughness factor KR as follows:

KR = 1 – aR ⋅ log10(RZ) ⋅ log10(2 ⋅ Rm / Rm,N,min)

With:

• RZ the surface roughness in µm according to DIN 4768 (see Table 2)
• Rm the tensile strength in MPa
• aR a constant (see Table 1)
• Rm,N,min the minimum tensile strength in MPa (see Table 1)

The roughness factor KR = 1 for a polished surface. Surface roughnesses higher than that of a polished surface will have values for KR < 1.

Table 1 below gives the values for Rm,N,min and for constant aR for different material types.

Table 2 below provides some indicative values for the surface roughness RZ of different surface finishes.

## Example calculation

As an example, let’s look at a steel with a tensile strength Rm = 600 MPa and with a machined surface.

In Table 1 we find for Steel: aR = 0.22 and Rm,N,min = 400 MPa. Table 2 gives us RZ = 100 µm for a machined surface.

Putting these values in the above equation gives us:

KR = 1 – 0.22 ⋅ log10(100) ⋅ log10(2 ⋅ 600 / 400)

and we get KR = 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).

For the number of cycles Nf between 103 and Nc1 cycles, the S-N curve is defined as:

Δσ(Nf) = SRI1 ⋅Nf b1

The stress range Δσ is a function of the number of cycles Nf. SRI1 is the stress range intercept at 1 cycle and b1 is the slope, which is negative. The parameter Nc1 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 Nc1 is usually around 106 – 107 cycles.

For Nf > Nc1, the S-N curve is defined as:

Δσ(Nf) = SRI2 ⋅Nf b2

SRI2 is the stress range intercept at 1 cycle and b2 is the slope, which is negative, for the second part of the S-N curve.

The parameters SRI1, Nc1, b1 and b2 are material parameters derived from the fatigue test data. SRI2 can be derived as follows:

SRI2 = SRI1 ⋅ (Nc1)b1 – b2

## 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 b1 in the first part of the S-N curve (see figure below) and keeping the fatigue strength at 1000 cycles the same. The slope b2 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 Nc1 cycles is:

Δσ(Nc1) = 1300 ⋅ (106)-0.0612 = 558.14 MPa

The stress range Δσ at Nc1 cycles for a machined part becomes in our case (with KR = 0.79, as calculated in the beginning):

Δσ(Nc1) = KR ⋅ 558.14 = 440.97 MPa

SRI1 and b1 for the first part of the modified curve can be determined from the two equalities below:

SRI1 ⋅ (103)b1 = 851.81 = SRI1 ⋅ (103)b1

Δσ(Nc1) = 440.97 = SRI1 ⋅ (106)b1

When we solve for SRI1 and b1, we find:

SRI1 = 1645 MPa and b1 = -0.0953

Since b2 = b2, we can solve below equation for SRI2:

Δσ(Nc1) = 440.97 = SRI2 ⋅ (106)b2

which gives us SRI2′ = 677 MPa.

The equation of the modified S-N curve between 103 and Nc1 cycles is then:

Δσ(Nf) = 1645 ⋅ (Nf)-0.0953

And in the region for Nf > Nc1:

Δσ(Nf) = 677 ⋅ (Nf)-0.0310

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