Non-uniqueness and stability of two-family fiber-reinforced incompressible hyper-elastic sheet under equibiaxial loading

The problems on the non-uniqueness and stability of a two-family fiber-reinforced anisotropic incompressible hyper-elastic square sheet under equibiaxial tensile dead loading are examined within the framework of finite elasticity. For a two-family fiber-reinforced square sheet, which is in-plane symmetric and subjected to the in-plane symmetric tension in dead loading on the edges, three symmetrically deformed configurations and six asymmetrically deformed configurations are possible for any values of the loading. Moreover, another four bifurcated asymmetrically deformed configurations are possible for the loading beyond a certain critical value. The stability of all the solutions is discussed in comparison with the energy of the sheet. It is shown that only one of the symmetric solutions is stable when the loading is less than the critical value. However, this symmetric solution will become unstable when the loading is larger than the critical value, while one of the four bifurcated asymmetric solutions will be stable.