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Does surface finish affect steels strength properties? If so, how does strength increase or decrease with the type of surface finish applied?

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Finishing operations on steel will not affect the hardness significantly.

Roughing or forging operations on steel, however, do increase its hardness (by work hardening). Generally, the more a machining process deforms the surface of steel, the harder it becomes.

Here is a research article on, Effects of Milling Condition on the Surface Integrity of Hot Forged Steel

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Shot peening improves the fatigue life , mostly used for steel springs. It does not change the bulk strength much . Shot peening is also used for aluminum. A very smooth ,polished surface can also improve "strength" in VERY highly stressed high strength steels ( this is likely of little use in ordinary applications ; I measured the effect in hydrogen stress cracking coupons stressed nearly to yield.). Generally steels are so tough that surface finish does not have a large affect.

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There are a lot of different types of strength to describe steel (Compressive, tensile, yield, ultimate, fatigue, hardness, toughness, etc). Surface finish affects some of these strength parameters, while some others are independent of surface finish.

Surface finish generally cannot affect the intrinsic property of the underlying steel, which dictates strength parameters that are inherent to the steel material in question. That will include strength parameters such as compressive strength, tensile strength, yield strength, ultimate strength, etc.

Surface finish has significant impacts however to fatigue strength, since different surface finishes will generally have a large impact on how the S-N curve will look like for that particular piece of steel. Generally, smoother surfaces that has been machined or work hardened will have an improved fatigue strength over rougher surfaces.

Hardness, as indicated by @kamran, will generally be impacted by compressive surface finishing techniques applied, making it harder in the process. Stated in @kamran's answer as well, roughening will increase hardness since most roughening methods generally apply some compressive force to steel surfaces. I do want to note that other finishing operations can affect hardness indirectly as well. For example, while milling does not really change hardness at all of the underlying material, you can change the hardness of the surface that you are testing by milling, because hardness of steel do not necessarily stay consistent throughout the entire piece.

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