Simple finite element algorithm for solving antiplane problems with Gurtin–Murdoch material surfaces

The finite element algorithm is developed to solve antiplane problems involving elastic domains whose boundaries or their parts are coated with thin and relatively stiff layers. These layers are modeled by the vanishing thickness Gurtin–Murdoch material surfaces that could be open or closed, and smooth or non-smooth. The governing equations for the problems are derived using variational arguments. The domains are discretized using triangular finite elements. In general, standard linear elements are used to approximate displacements in the domain. However, to capture the singular behavior of the elastic fields near the tips of the open Gurtin–Murdoch surfaces, a novel blended singular element is devised. Numerical examples are presented to demonstrate the accuracy and robustness of the algorithm developed.