An analysis of compressive stability of elastic solids containing a crack parallel to the surface

An analysis is presented for compressive stability of elastic solids containing a crack parallel to the free surface based on the mathematical theory of elasticity. Basic buckling equations derived from the mathematical theory of elasticity are employed and are reduced to a system of homogeneous Cauchy-type singular integral equations by means of Fourier integral transform. The integral equations are solved numerically by utilizing Gauss-Chebyshev integral formulae. Numerical results for buckling loads are presented for various geometrical parameters and are compared with those obtained from classical theory of beam-plate stability based on the Kirchhoff assumption. The comparison of both results shows that the buckling loads obtained from the classical theory of beam-plate stability are much larger than those obtained from the mathematical theory of elasticity, referring to which the limitations of the classical theory applied to the present buckling problem are discussed. A simple but accurate approximate method for estimating the buckling load is developed by the use of the elastic support coefficient obtained from the present analysis and the Euler formula derived from the classical theory for the case of elastically supported ends. Finally, the numerical results of the buckling wave shapes and the Mode I and II stress factors, which cannot be obtained from the classical theory, are presented.