An Application of the Kane and Mindlin Theory to Crack Problems in Plates of Arbitrary Thickness

Classical plane solutions of the theory of elasticity, which are sometimes more than 100 years old, are still used today and provide a framework for the analysis of many practical problems. But, strictly speaking, these analytical solutions are only applicable to plates with vanishing thickness or infinite thickness, where the stress state could be classified as plane stress or plane strain, respectively. However, the through-the-thickness stresses that exist in a plate of given thickness have a significant impact in a number of practical applications; and these stresses are often inevitably ignored due to the lack of analytical tools. This paper presents new analytical results for crack tip opening displacement (CTOD) for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on the solution for an edge dislocation in infinite plate of arbitrary thickness and an application of the distributed dislocation technique. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional elasto-plastic finite element (FE) results. It is shown that both analytical and numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.