Numerical calculation of the elastic field in a half‐space

In this paper the title problem is solved by discretizing the region of application of the load as a finite union of rectangles with non‐overlapping interiors. The surface traction is discretized by taking it constant over each rectangle. Then the displacement and the displacement gradients at an arbitrary point of the half‐space, due to the discretized load, can be found exactly as a finite concatenation of elementary operations and of arctangents and logarithms. Although this representation is numerically appropriate near the rectangle which generates the (partial) elastic field, it is less suitable far away from it. An asymptotic expression for the elastic field in this region is provided. An algorithm for the calculation of the elastic field is given. The method is applied to two examples, viz. the sub‐surface stresses of the Hertz frictionless problem and the steady state frictional rolling problem, both with a ciruclar contact area.