On the reconstruction of residual stresses and strains of a plate after shot peening

The subject of this research is a mathematical description of the shape and the stress-strain state of a steel plate subjected to unilateral shot peening, its experimental verification and application of the results for verification of methods for reconstruction of residual stress and strain fields according to experimental data. Such plate is used in manufacturing as a calibrating sample to determine of shot peening duration required for formation of proper compressive tangential stress in the surface layer of the processed product. The method of calibration is convenient and widely applied in different technologies of surface hardening. In that case the source of the residual stresses is plastic strains in surface layer produced by shot peening. For the statement of the problem a plastic strain tensor field is defined up to an arbitrary function. The shape and the stress-strain state of an elastic plate with the surface layer of plastic strains were calculated numerically. The qualitative behavior of numerical solution allowed us to accept the set of hypotheses to find an analytical solution of the spatial problem of elasticity theory and to weaken the boundary conditions. The exact solution has been found analytically. Within the framework of the plane stress state along the thickness and transverse directions, the result exactly corresponds to the Davidenkov–Birger formula connected the tangential residual stress distribution on depth with the function of deflections. An explicit formula for the dependence of the residual (plastic) deformation on the thickness coordinate is obtained. Sources of errors of the received expressions and methods of their correction are analyzed. An experiment has been carried out on the one-sided shot peening of calibration plate made of hardened 65G steel, for which the layer-by-layer etching of the treated surface and the measurement of the flexure of the plate were made (by Davidenkov method). The profiles of residual stresses and strains were reconstructed numerically with reasonable accuracy using the obtained experimental data. The result is applicable to a wide class of problems for elastic bodies with hardened surface layers. It may serve as a base for experimental research of such problems, help to formulate hypotheses and test them by experiment, help to study relation between physical fields in asymptotic case, help to verify applicability of different ways to account residual stresses in numerical solution. The solution found can be used for verification of stress and displacement fields in different cases of preliminarily stressed shell elements in engineering software for calculation of fatigue endurance of different machine parts with hardened surface layer. It also seems to be a reference for the study of surface-hardened bodies with curved free boundary, to which most of the practically important tasks are reduced.