Surface Roughness Factor KR

Fatigue test specimens are almost always mirror-polished, so the S‑N data that come out of the laboratory represent the best-case surface condition. A real component with a rougher finish — machined, ground, rolled or as-cast — will have a lower fatigue strength because surface irregularities act as micro-notches that initiate cracks earlier. The surface roughness factor KR accounts for this difference. This article describes the FKM method for calculating KR and shows how to apply it to an S‑N curve, step by step.

FKM Surface Roughness Factor

The FKM guideline Analytical Strength Assessment defines KR as:

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

where:

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

For a polished surface the roughness factor equals unity (KR = 1). Any surface rougher than polished gives KR < 1, meaning a reduction in fatigue strength.

Table 1. Constant aR and minimum tensile strength Rm,N,min for different material types (FKM guideline). GS = cast steel; GGG = nodular cast iron; GT = malleable cast iron; GG = grey cast iron.
Material Type aR Rm,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. Indicative RZ values for different surface finishes (DIN 4768)
Surface Condition RZ [µm]
Polished 0
Ground 12.5
Machined 100
Poor machined 200
Rolled 200
Cast 200

Example Calculation of KR

Consider a steel with a tensile strength Rm = 600 MPa and a machined surface finish.

From Table 1: aR = 0.22 and Rm,N,min = 400 MPa. From Table 2: RZ = 100 µm for a machined surface. Substituting:

$${\displaystyle K_R=1-0.22 \cdot \log_{10}(100)\cdot \log_{10}\!\left( \frac{2 \cdot 600}{400}\right)} = 0.79$$

This means the fatigue endurance limit of the machined part is 79 % of the polished-specimen value — a significant reduction that must be accounted for in any fatigue life assessment.

Construction of the S‑N Curve

Before applying KR, it helps to understand how the S‑N curve is built (see Figure 1). Between 103 and Nc1 cycles the curve is defined as:

$${\displaystyle \Delta \sigma(N_f) = SRI_1 \cdot N_f^{\,b_1}}$$

Here SRI1 is the stress range intercept at 1 cycle and b1 is the slope (negative). Nc1 is the fatigue transition point — the knee in the curve, typically around 106 – 107 cycles.

Beyond Nc1 the curve continues with a shallower slope:

$${\displaystyle \Delta \sigma(N_f) = SRI_2 \cdot N_f^{\,b_2}}$$

Construction of an S-N curve
Figure 1. S‑N curve construction (click for larger image)

SRI1, Nc1, b1 and b2 are material parameters from fatigue testing. SRI2 follows from continuity at the transition point:

$${\displaystyle SRI_2 = SRI_1 \cdot (N_{c1})^{b_1-b_2}}$$

S‑N Curve Adjusted for Surface Roughness

Surface condition has its greatest effect in the high-cycle regime and diminishes towards low-cycle fatigue. The standard approach is to modify the slope b1 of the first segment while keeping the fatigue strength at 103 cycles unchanged. The slope b2 of the second segment remains the same.

S-N curve of a steel with corrected curves for different surface finishes
Figure 2. S‑N curve of a polished steel specimen plotted against corrected curves for different surface finishes. b1 = −0.0612 | b2 = −0.0310 | SRI1 = 1300 MPa | UTS = 600 MPa | Nc1 = 106 cycles

For the unadjusted (polished) curve, the stress range at Nc1 is:

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

After applying KR = 0.79 (machined surface), the stress range at Nc1 becomes:

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

The modified parameters SRI1′ and b1′ are found from the two conditions that must be satisfied simultaneously — the strength at 103 cycles is unchanged, and the strength at Nc1 equals the KR-adjusted value:

$${\displaystyle SRI_1 \cdot (10^3)^{b_1} = 851.81 = SRI_1^{\prime} \cdot (10^3)^{b_1^{\prime}}}$$

$${\displaystyle \Delta \sigma^{\prime}(N_{c1}) = 440.97 = SRI_1^{\prime} \cdot (10^6)^{b_1^{\prime}}}$$

Solving gives:

$$SRI_1^{\prime} = 1645 \; \text{MPa} \quad \text{and} \quad b_1^{\prime} = -0.0953$$

Since b2′ = b2, SRI2′ follows from continuity at the transition point:

$${\displaystyle \Delta \sigma^{\prime}(N_{c1}) = 440.97 = SRI_2^{\prime} \cdot (10^6)^{b_2}}$$

which gives $SRI_{2}^{\prime} = 677\text{ MPa}$.

The modified S‑N curve for the machined steel is therefore:

$${\displaystyle \Delta \sigma(N_{f}) = 1645 \cdot N_f^{\,-0.0953}} \qquad \text{for } 10^3 \leq N_f \leq N_{c1}$$

$${\displaystyle \Delta \sigma(N_{f}) = 677 \cdot N_f^{\,-0.0310}} \qquad \text{for } N_f > N_{c1}$$

Notice that the modified slope is considerably steeper than the original (−0.0953 vs. −0.0612), reflecting the fact that the rough surface penalises the high-cycle regime more than the low-cycle regime. This behaviour is consistent with the physical mechanism: at high strains (low-cycle fatigue) the bulk material yields and surface micro-notches become less relevant, whereas at low strains (high-cycle fatigue) crack initiation from surface defects dominates the total life.

If you want to learn more about fatigue assessment methods including surface corrections, survival probability adjustments and damage accumulation, take a look at our course Introduction to Fatigue Analysis with FEA. For projects that require specialist fatigue engineering, our fatigue analysis team can help.

Frequently asked questions

Common questions about surface roughness and fatigue.

The FKM guideline provides constants for wrought and cast steels, several types of cast iron, and wrought and cast aluminium alloys (see Table 1). For other material families — titanium alloys, copper alloys, polymers — the FKM formula does not directly apply, and material-specific fatigue data or alternative correction methods should be used instead.

In the low-cycle regime the material undergoes significant plastic deformation at every cycle. Under these conditions the bulk plasticity dominates the fatigue behaviour and surface micro-notches have relatively little additional effect. In the high-cycle regime, by contrast, the material remains largely elastic and crack initiation from surface defects controls a large fraction of the total life. The FKM method captures this by pivoting the S‑N curve around the 103-cycle point, leaving the low-cycle strength unchanged while reducing the high-cycle endurance limit by the factor KR.

Yes. Processes that introduce compressive residual stresses into the surface layer — shot peening, roller burnishing, nitriding or laser shock peening — can raise the fatigue strength above the polished-specimen baseline. These effects are not captured by KR alone and require separate treatment factors. The FKM guideline provides additional surface treatment correction factors for common processes.