The bending stress in a cracked Kirchhoff plate resting on a Pasternak foundation

The problem of a rectilinear semi-illimitate crack in a Kirchhoff elastic plate laying on a weakly nonlocal elastic two-parameter (Pasternak) foundation and subject to transversal loadings applied on the plate surfaces is studied in the present work. By extending the approach developed by Ang et al. [1] for a plate on a Winkler elastic foundation, the analytical full-field solution is obtained by solving a system of dual integral equations. In particular, the kernel function is factorized by using a procedure based on the Cauchy theorem, similar to the Wiener-Hopf method used in the solution of antiplane fracture problems [2, 3]. Closed-form solutions are thus provided for displacement, bending moment and shear force per unit length along the crack line. The ratioes between the subgrade parameters and flexural rigidity of the plate allow introducing two characteristic lengths, whose influence on the bending stress intensity factor and fracture toughness of the plate is then investigated. The governing equation is closely related to the bending problem of initially curved thin sheets under transversal loading and subjected to a constant normal stress, so that results apply to such situation as well. Accordingly, this is one of the few closed-form solutions available for initially curved cracked thin sheets.