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Residual Strain is the same thing as permanent strain, as in permanent strain in a beam.
If you refer to a stress vs strain plot, you'll see that this relationship in metals is linear up until the yield point where the metal plastically deforms. After stress slightly decreases due to this yielding, It will continue to increase, along with strain but on a more curve-shaped line.
When stress is removed from the metal, it will slowly decrease back down to 0, but there will be an offset in the strain. The distance from where strain equals zero to this new point along the axis is the the residual strain left in the metal due to plastic deformation.
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