Buckling and post-buckling analysis of cracked composite plates via a single-domain Ritz approach

Thin and moderately thick composite multi-layered plates are widely employed in many engineering applications, especially in naval and aerospace structures. These structural components can experience in service the presence of cracks, generated for example by corrosion, fatigue or accidental external causes. Cracks can affect the load carrying capability, buckling and post-buckling behaviour of plates; therefore, their effects need to be investigated and taken into account for fail safe or damage tolerant design. Additionally, attention should be devoted to the interaction of cracks with buckling and post-buckling behaviour, as the energy release rate in post-buckling regimes can be adversely affected and unexpected critical safety issues could manifest. Different approaches have been proposed to model cracked plates and, among others, the Ritz method has been successfully used showing adequate accuracy and computational efficiency. A possible strategy to obtain Ritz solution for the cracked plate problem is based on the decomposition of the domain under consideration into several sub-domains over which standard admissible functions are introduced; continuities of displacements and slopes along interconnecting edges between contiguous sub-domains are then restored by enforcing suitable interface conditions. This strategy provides accurate results also for complex structures, like stiffened panels, but it does not possess the convergence features of the original Ritz method and does not account for the crack tip singular behaviour. To overcome such drawbacks, the original single-domain Ritz formulation has to be applied with special trial functions, which account for the presence of the crack by describing the discontinuity of the solution across the crack and the tip singularity. In the present work, a single-domain Ritz formulation for nonlinear analysis of general quadrilateral multi-layered composite plates with straight cracks is presented, based on the first order shear deformation theory and von Karman assumptions for plate geometrical nonlinearity. The employed trial functions consist of the product of Legendre orthogonal polynomials supplemented with special functions able to describe the discontinuity across the crack and the singularity at the crack tip; boundary functions are used to fulfil the homogeneous essential boundary conditions. The problem governing equations are inferred via the stationarity of an energy function penalized to account for non-homogeneous essential boundary conditions and for the no-interpenetration condition along the crack faces. For post-buckling analysis, the resulting nonlinear system is solved via Newton-Raphson algorithms. Convergence studies and results are presented for buckling and post-buckling of plates with a through-the-thickness crack, highlighting differences in the crack behaviour between pre- and post-buckling regimes, which can noticeably affect the plate residual strength. The performed analyses show the efficiency and potential of the method, which provides accurate results in conjunction with reduced number of degrees of freedom and simplified data preparation, with respect to other techniques