Full field solution for a rectilinear crack in an infinite Kirchhoff plate supported by a Pasternak elastic foundation.

The failure of cracked ceramic components is governed by the stresses in the neighbouring of the crack tip, which is described by the stress intensity factor (SIF). Despite the availability of several handbooks for SIFs, very few full-field solutions are available for cracked plates resting on an elastic foundation. This lack of results is problematic, since this situation often occur in practice (e.g. roadways, pavements, floorings, etc.). Furthermore, when some results are available, they never involve the foundation's mechanical properties alone. For instance, the problem of a finite crack in an infinite Kirchhoff plate supported by a Winkler foundation is considered in [1] and it is reduced to a singular integral equation. However, since two length scales exist in the problem (the crack length and the foundation relative Winkler modulus), the SIF may be related to some dimensionless ratio of them and not directly to the foundation's mechanical property. In actual facts, this outcome stems from the Winkler approximation to the foundation and not from the physical feature of the problem. The full-field solution for a semi-infinite rectilinear crack in an infinite Kirchhoff plate resting on a Winkler foundation is found in [2]. Since this is a self-similar problem, no characteristic length scale exists. Application of the above results to road and airport pavements is given in [3]. As a result, the influence of the pavement foundation on the SIF cannot be properly assessed. Several papers address crack problems in plate theory and a literature review of the stress field at the crack tip in thin plates and shells is given in [4] along with comparison with the available experimental results. The present work deals with the elastostatic problem of a semi-infinite rectilinear crack in an infinite Kirchhoff plate resting on a two-parameter elastic foundation under very general loading conditions. The foundation, also termed Pasternak-type, is weakly non-local, as it accommodates for coupling among the independent springs of a purely local model (i.e. the Winkler model). The same model governs the problem of a Kirchhoff plate equi-biaxially loaded in its mid-plane. The Pasternak foundation accounts for two length scales such that the whole problem is governed by a parameter η expressing the soil to plate relative stiffness. The discussion was addressed in a previous paper [5] but therein limited to the range 0 < η < 1, where the limiting case as η → 0 recovers the non-local Winkler model. In the present work the analysis has been extended to the full range of values for the paremeter η. The problem is formulated in terms of a pair of dual integral equations solved through the Wiener–Hopf technique. Numerical results are given for the full field bending and shear stress field within the plate, the corresponding SIFs are obtained and some conclusions drawn.