In this paper, based on a diffusion and mechanical coupled continuum model, the two-dimensional chemo-mechanical coupled problem in the polar coordinates is studied under plane strain assumption. The transient analytical expressions for concentration and stresses are obtained using a displacement potential function and Airy stress function. An axisymmetric problem is considered to verify the correctness of these expressions. After that, a numerical example of a long enough and traction-free cylinder with variant concentration distribution on its cylindrical surface by an angle is given, and the results show that the shear stress on the cross section is generated during the process of diffusion and the axial stress increased with the increment of concentration. The in-plane stress for this traction-free cylinder will vanish at a steady-state when the concentration was linearly distributed.