When I look at the yield of one material then you find more than one yield for different thickness.
why the yield changes with changing the thickness. is there an equation to prove this ?
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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.
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).