A thermoelastic plate model for shot peen forming metal panels based on effective torque
Conor Rowan
TL;DR
This work develops a thermoelastic Kirchhoff-plate model in which shot peening is represented as spatially varying torques acting through a thermal moment $\tau$. The authors derive the governing bending equations, link $\tau$ to shot peen intensity via a simple calibration using the maximum uniform displacement $|u_{\max}|$ (with $\tau = K I$), and formulate a regularized inverse problem to design realizable peening maps for desired plate contours. Numerical discretization with a Legendre-based, fully-free boundary treatment enables solving for the plate response under nonuniform peening, and experimental validation with quantified uncertainty shows reasonable agreement (about 10% average error) while highlighting variability in the shot peen process. The approach offers a non-invasive calibration path and a practical inverse-design framework, with potential extensions to thermoelastic FE codes for larger, more complex panels and inclusion of distributed weight effects.
Abstract
A common technique used in factories to shape metal panels is shot peen forming. The impacts between the hard steel shot and the softer metal of the panel cause localized plastic deformation used to improve the fatigue properties of the material's surface. The residual stress distribution imparted by impacts also results in bending, which suggests that a torque is associated with it. In this paper, we model shot peen forming as the application of spatially varying torques to a Kirchhoff plate, opting to use the language of thermoelasticity in order to introduce these torque distributions. First, we derive the governing equations for the thermoelastic thin plate model and show that only a torque-type resultant of the temperature distribution shows up in the bending equation. Next, to calibrate from the shot peen operation an empirical effective torque parameter used in the thermoelastic model, a simple and non-invasive test is devised. This test relies only on measuring the maximum displacement of a uniformly shot peened plate as opposed to characterizing the residual stress distribution. After discussing how to handle the unconventional fully-free boundary conditions germane for peened plates, we introduce an approach to solving the inverse problem whereby the peening distribution required to obtain a specified plate contour can be obtained. Given the non-unique relationship between peening distributions and the displacement at discrete points, we explore a regularization of the inverse problem which gives rise to shot peen distributions that match the capabilities of equipment in the factory. In order to validate our proposed model, an experiment with quantified uncertainty is designed and carried out which investigates the agreement between the predictions of the calibrated model and real shot peen forming operations.