Why does the yield strength of steel go up if I reduce the thickness of the specimen?

No there is no equation for it, as it is caused by the processing differences of the different sizes and has nothing to do with the size of the material.

• Smaller bars are subject to more work to get them down to the required size as they start out from the same size billet as the larger bars.
• Smaller bars also take less time heat up during heat treatments, so the effects and/or heat treatment are slightly different. (see comment by starrise)

The values in the standards are also not the strength of the material you buy, they are miniumum (and sometimes maximum) values. It can be that a larger diameter bar exceeds the minimum strength of a smaller diameter bar.

The size of the bar can have an effect for materials with a large grain structure when if the bar is only a few grains wide you can have a phenomenon called grain size effect. The other condition I know of where it can happen is with surface treated or coated parts. The same depth of a surface treatment (e.g. carburizing) will have a higher surface to bulk material ratio in smaller bars.

The different mechanical properties for different sheet thicknesses has to do with grain refinement of the material during the rolling process. The following figures show what is happening to the old grain structure.

The new elongated grain are the results of cold working. It is possible through annealing to form new grains (with no bias in direction).

The effects of cold working on Strength, Hardness and Ductility can be seen in the following image