Analytical mechanical model of grinding-induced warpage in open cylindrical thin-shell mirrors

Thin-shell silicon mirrors are critical components in lightweight X-ray optical systems. However, grinding induces surface compressive stresses and subsurface damage layers that can lead to warpage deformation in these open cylindrical thin-shell structures. The curvature effect of thin-shells results in a unique warping mechanism that has remained unexplored. This study developed a novel analytical mechanical model to predict the warpage behavior, identifying the stress-released effect during the fixture removal as the primary driver. Using Donnell–Mushtari shell theory, the study derived the high-order partial differential equations (PDEs) for governing warpage deflection under bending moments acting on the free boundary. A double finite Fourier integral transform was applied to solve these PDEs and derive exact analytical solutions. The model links warpage directly to shell geometry, material properties, and damage parameters. Ultra-precision grinding experiments on silicon thin-shells with different thicknesses validated the model, showing less than 6% error between predicted and measured deflections. Notably, increasing shell thickness can reduce warpage sensitivity. Grinding parameters that promote ductile removal, such as fine-grit resin-bonded wheels, higher wheel speed, lower feeding rate, and smaller grinding depth, produce shallower subsurface damage layers and help minimize warpage. This theoretical innovation provides an efficient computational approach for warpage prediction, thereby facilitating the high-precision manufacturing of thin-shell structures.