Elastic stability of an internally constrained hyperelastic material

The main part of this research deals with the elastic stability of a thick slab, a half space, and a cylindrical tube made of a Bell constrained material subjected to static loading. Euler\u27s stability criterion is applied to derive equations for the stability analysis. For the thick slab problem, we introduce a potential function to derive a canonical 4th order partial differential equation, and then investigate the solution of this equation for different cases. Further, general formulae for the buckling equations are developed. Based on Biot\u27s idea (1), the half-space problem is studied as an extension of the stability problem of a thick slab. For the cylindrical tube problem, it turns out that the mathematical model is very similar to that of the stability problem studied by Wilkes\u27 (2) for the case of an incompressible material. Hence, the strategy presented in (2) is referred to. However, in contrast to (2), the roots of the characteristic equation are examined for three possible situations in which these roots may be real, pure imaginary, or complex conjugate. The solutions of the system of ordinary differential equations corresponding to these cases are discussed, and the buckling equation is thus derived. The stability problem of a thin-walled tube is also studied. In the remaining part of this research, we use the wave speed criterion to discuss certain restrictions on the coefficients of the aforementioned canonical 4th order paritial differential equation, and the existence of equibiaxial deformations of a Bell constrained material under an all around Caucky stress. It is shown that, for a simple hyperelastic Bell constrained material or a material which obeys Bell\u27s law, equibiaxial deformations under an all around Cauchy stress are inherently unstable, and cannot be sustained in either of these materials