Theoretical Analysis Of Critical Condition For Surface Instability Of Graded Elastic Cylindrical Structures

For an elastic graded cylinder subjected to an axial compression, when the compressive stress is equal to a critical value, the surface becomes unstable and the wrinkling and creasing will in turn occur. By using the transformation between the Cartesian coordinates and the cylindrical coordinates, the constitutive relations and the equilibrium equations as well as boundary conditions for the increments of the Piola-Kirchhoff stress expressed in Cartesian coordinates, are converted into the corresponding ones in cylindrical coordinates. And then the state space solution of critical condition is developed for the onset of surface instability of a homogeneous cylindrical shell by solving the eigenvalue problem. By dividing the structures into a number of sublayers, and considering continuity of the state vector at the interface of adjacent sublayers, the state space solution of surface instability is obtained for graded elastic cylindrical structures.In this paper, the critical strains and corresponding wave numbers (or wavelength) for the onset of surface instability of a number of examples are presented by the state space method, including:a homogenous cylindrical shell, a soft cylinder covered by a stiff layer and a graded elastic cylinder with its elastic modulus exponentially varying in the radius direction. And then the state space method is validated by comparing the results with the other existing methods. In addition, the influences of geometric parameters and material properties on the critical conditions are also presented by the state space method. For a soft cylinder covered by a bilayer, which is subjected to an axial compression, two different wrinkling modes will be occurred with the increasing of the modulus ratio between the intermediate layer and the substrate layer. And the critical value is obtained by the state space method, which is much more accurate than the ones solved by other methods. Furthermore, a convention finite element analysis is investigated by using the commercial software ABAQUS for understanding this interesting phenomenon well.