Determination of optimal shot peen forming patterns using the theory of non-Euclidean plates

Abstract

We show how a theoretical framework developed for modelling nonuniform growth can model the shot peen forming process. Shot peen forming consists in bombarding a metal panel with multiple millimeter-sized shot, that induce local bending of the panel. When applied to different areas of the panel, peen forming generates compound curvature profiles starting from a flat state. We present a theoretical approach and its practical realization for simulating peen forming numerically. To achieve this, we represent the panel undergoing peen forming as a bilayer plate, and we apply a geometry-based theory of non-Euclidean plates to describe its reconfiguration. Our programming code based on this approach solves two types of problems: it simulates the effect of a predefined treatment (the forward problem) and it finds the optimal treatment to achieve a predefined target shape (the inverse problem). Both problems admit using multiple peening regimes simultaneously. The algorithm was tested numerically on 200 randomly generated test cases.

Simulation and automation of aluminum panel shot peen forming

“…Representation of peening-induced loads with eigenstrain Following Ref. [13], we virtually divide the plate into two layers of equal thickness to simulate the curving effect of peen forming. We assign in-plane swelling to one layer and in-plane shrinking to the other in the spots where the plate is treated.…”

“…Hence, from the mathematical point of view, the forward problem resolution consists in the determination of a f and b f taking a r,t and a r,b as the input. This is done through numerical minimization of the global elastic energy functional E BL [13]:…”

“…This method can also be reformulated in terms of the theory of non-Euclidean plates [12], which is specifically designed to describe the reconfiguration of plates subjected to eigenstrain. The eigenstrain approach together with the theory of non-Euclidean plates provide a framework for the inverse problem resolution [13].…”

“…Such experiments do not require an automated peen forming system due to the simplicity of the applied pattern. Moreover, the existing literature on the numerical inverse problem resolution in this domain is mostly limited to the description of the developed algorithms [25] or to their numerical validation [11] [13]. Only Faucheux et al have validated their inverse problem resolution strategy experimentally on complex shapes such as a cylinder, a saddle and a wave [21] [23].…”

“…The automated workflow was developed for experimental validation of the numerical forward and inverse problem solvers described in Ref. [13]. The forward problem solver validation involves randomly generated and strongly curved target shapes that have deflection of 10−31 mm.…”