<p>Within the context of linear elasticity, approximate analytical solutions are developed for the energy release rate for axisymmetric planar cracks in elastic thin layers sandwiched between two rigid plates. These solutions are validated by comparing them with finite element solutions, and they are applicable to cracks in constrained thin layers made of compressible, nearly incompressible, or incompressible materials. These analytical solutions provide insights into the effects of geometry and material compressibility on fracture of thin layers. In particular, stability of crack growth is discussed under both displacement and force-controlled loading conditions, summarized in stability maps. Remarkably, it is found that, under force-controlled conditions, stable crack growth is possible in incompressible or nearly incompressible layers, but not in compressible layers. We compare the energy release rates for embedded and interfacial cracks, showing that they differ when the cracks are small but become approximately equal for large cracks. The analytical approach is further extended to non-axisymmetric planar cracks in compressible thin layers. However, a similar extension does not apply for cracks in incompressible or nearly incompressible layers.</p>