A hybrid analytical–numerical solution for 3D notched plates

• A hybrid analytic and numerical method was developed to analyze 3D notched plates.
• The method separates the 3D equilibrium equation into primary and shadow parts.
• A least-squares method is used to find the factors of these primary and shadow modes.
• The proposed method can accurately simulate the singularities near a sharp V-notch.
• We can get a whole displacement field with analytic singular effect at the notch tip.

This study used a hybrid analytical and numerical method to analyze three-dimensional (3D) elastic bodies with sharp-V notches. The proposed method separates the 3D equilibrium equation into primary and shadow parts, where the solution of the primary part is the analytical solution under the generalized plane-strain theory, and the shadow part is solved numerically using a weak form based on the finite element theory. A least-squares method is then used to find the multiplication factors of these primary and shadow modes using 3D finite element results. Numerical simulations indicate that the proposed method can accurately simulate the singularities near a sharp V-notch. The major advantage of this method is that a 3D whole displacement field with the singular effect based on the theoretical solution near the notch can be obtained for anisotropic materials under arbitrary loads.