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
The FKM-guideline Analytical Strength Assessment defines the roughness factor KR as follows:
KR = 1 - aR ⋅ log10(RZ) ⋅ log10(2 ⋅ Rm / Rm,N,min)
- 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.
|Material Type||aR||Rm,N,min [MPa]|
|Wrought aluminium alloys||0.22||133|
|Cast aluminium alloys||0.20||133|
Table 2 below provides some indicative values for the surface roughness RZ of different surface finishes.
|Surface Condition||Rz [µm]|
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 - click for larger image).
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