If the contacting bodies are smooth, the gap function of Eq. (1.1) can be expanded as a power series in x and y, for points sufficiently near the origin. Furthermore, since the coordinate system in Fig. 1.2 satisfies the conditions \(g_0(0,0)=0;\;\;\;\frac{\partial g_0}{\partial x}(0,0)=0;\;\;\;\frac{\partial g_0}{\partial y}(0,0)=0,\) the first non-zero terms in this series are the quadratic terms \(g_0(x, y)=Ax^2+By^2+Cxy.\) The problem of Eq. (1.3) is influenced by the gap function only in the contact region \(\mathcal {A}\) which is generally small in non-conformal contact, so the higher order terms in \(g_0\) can often be neglected even when the surfaces are not strictly quadratic. The elastic contact problem for a gap function defined by Eq. (3.2) was first solved by Hertz
(1882) (Journal für die reine und angewandte Mathematik, 92: 156–171, 1882) and the resulting stress and displacement fields are generally referred to as Hertzian contact .

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