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We investigate the thick brane model in Palatini $f(\mathcal{R})$ gravity.The brane is generated by a real scalar field with a scalar potential. We solvethe system analytically and obtain a series of thick brane solutions for the$f(\mathcal{R})=\mathcal{R}+\alpha \mathcal{R}^2$-brane model. It is shown thattensor perturbations of the metric are stable for$df({\mathcal{R}})/d{\mathcal{R}}>0$. For nonconstant curvature solutions, thegraviton zero mode can be localized on the brane, which indicates that thefour-dimensional gravity can be recovered on the brane. Mass spectrum ofgraviton KK modes and their corrections to the Newtonian potential are alsodiscussed.

Tensor perturbations of Palatini $f(\mathcal{R})$-branes