A model for shot peen forming with prestress conditions

Hertzian elastic contact theory and spherical cavity plastic theory were used to derive the stress profiles in the sheet, which can then be considered to be in a uniformly distributed state across the sheet. This was simplified further by ignoring the through thickness stresses and concentrating on the in-plane stresses allowing the analysis to use a 2D representation of the sheet. The stresses were applied by using a temperature profile to generate a thermal strain profile and hence a representative stress profile.

Previous researchers modelled the residual stress from peening as a hot layer but, if prestress conditions are to be included, the model approach must correctly represent the interaction between the peening and prestress. This is achieved by representing maximum peening stress as a cooling of the peened layer whilst restraining the material of the sheet. This causes the peened layer to reach its tensile yield stress in a way that correctly interacts with prestress. The sheet is then returned to ambient temperature and the peened layer is left in the correct state of residual compressive stress. Finally the sheet is released and allowed to deform.

This three step cycle does not allow for the movement of the sheet whilst it is being peened, which can be a significant amount of movement for unconstrained sheets. It is therefore necessary to repeat the three step cycle a number of times. The number of iterations required is established by considering the energy imparted to the sheet by the peens over the duration of the real process. This is used to derive a value for the expected energy imparted to the model, which takes into account energy lost in numerical processing steps required by the model that do not represent a real physical energy loss. The iterations are terminated when the energy reaches the expected value for energy into the model.