Upper bound analysis of a shape-dependent criterion for closing central rectangular defects during hot rolling

A triangular velocity field for central defect closure is proposed in this paper. With the proposed velocity field, the minimum upper bound power is calculated. Then, the stress state coefficient as a function of defect thickness δ and aspect ratio η is obtained by the upper bound theorem. By applying the limit condition of the stress state coefficient with respect to defect size and letting the defect size be zero, the analytical solution of critical shape factor depending on aspect ratio solely is derived from a quadratic equation. Ultimately, a shape-dependent criterion for closing rectangular defects during hot rolling is established by relating the derived critical shape factor to the actual one. It is shown that as the aspect ratio increases, the critical defect size decreases and the applied energy required for closing central defects increases. The increases in relative reduction and roller radius or the decrease in initial plate thickness are in favor of defect closure. Validation of the present result with available simulation result and a current rolling schedule shows that the present analytical criterion matches well with the simulation result and can be used for optimizing rolling parameters to close central defects. Experimental research in laboratory shows that a defect can be closed well if the actual shape factor reaches or exceeds the corresponding critical shape factor.