Plane harmonic functions in the presence of a surface layer of arbitrary stiffness

Complex variable techniques are employed to characterize two-dimensional solutions $u\left ( {x, y} \right )$ of Laplace’s equation which satisfy the boundary condition ${\left [ {\beta \left ( {{\partial ^2}u/ \\ \partial {y^2}} \right ) + \left ( {\partial u/\partial x} \right )} \right ]_{x = 0}} = 0$, where $\beta$ is referred to as the surface-stiffness parameter. Simple closed-form singular solutions are derived which satisfy this boundary condition and represent source and dislocation singularities. The former is used to synthesize the field generated by a small inclusion of arbitrary shape on which $u = 1$, in the presence of a boundary at $y = 0$ on which $u = 0$. At points not near the inclusion the field has the form of a function of position and surface stiffness multiplied by a strength factor which depends on the size and shape of the inclusion and the surface stiffness. Detailed calculations are presented for two extreme shapes of inclusions—a shallow, wide inclusion on the surface and a deep, narrow inclusion penetrating below the surface—which exhibit the relation between the field near the inclusion and the distant field, and show explicitly the dependence of the strength factor on surface stiffness and inclusion size and shape. The nature and strength of the singularities at the tips of the inclusions are also examined and it is found that a tip singularity at the surface changes character as the surface stiffness varies.