To be in safe side (without breaking); if applied deformation is less than the deformation at UTS then you materials will be deformed. For the so called brittle materials this limit is too small to achieve any practically significant deformation leading to poor workability. Theoretically speaking there is no such limit unless material fails within it's elastic deformation region!!!
In mechanical design the criterion for using the theories of failure by yielding state that the elongation percentage is more than 5%. This may help you. Good Luck
Thangapandian’s question is “…minimum…required…”. For me, it sounds like “what is the minimum percentage of elongation required to obtain the effect of cold working…”.
Some of materials have a plateau in the strain--stress relationship; i.e., linear elastic region, constant stress region then the stress rise up again. The increment of stress after the plateau may be attributed to the work hardening---an effect of the cold working. Therefore, the end of the plateau is the “minimum percentage of elongation required to obtain effect of cold working”. For instance, it is about 2 % of elongation for A5052 annealed material. This is a starting point of interactions between dislocations which are introduced by the plastic deformation.