The axisymmetric three-body, double-unilateral contact problem for an elastic sphere is treated by analytical techniques. It is assumed that the elastic sphere is put on the surface of an elastic layer of finite thickness and, afterwards, is indented at the upper pole of the sphere with a rigid spherical punch. Friction is neglected at both contact interfaces. A first-order asymptotic solution for the displacement–force relation, which generalizes the corresponding Hertzian formula, is derived in explicit form in terms of dimensionless asymptotic constants that account for the finite sizes of the elastic sphere and layer. Under the assumption that the sphere/substrate contact diameter is less than the substrate thickness, the presented results can cover a wide range of parameter combinations, which may be used for the purpose of benchmark assessments in the finite element analysis. The constructed asymptotic model is remarkably simple and elegant, and yet can be applied to various contact problems of practical importance (including a very timely problem of indentation of spherical viruses). The underlying theoretical framework is versatile and can be further extended for analysis of multiple contacts involving elastic spheres and elastic layers.

通过分析技术处理弹性球体的轴对称三体双单边接触问题。假定将弹性球放置在厚度有限的弹性层的表面上，然后再用刚性球形冲头在球的上极上压入。两个接触面的摩擦均被忽略。位移-力关系的一阶渐近解采用了无量纲渐近常数的显式形式，该解推广了对应的赫兹公式，该无因次渐近常数说明了弹性球体和层的有限大小。在假设球体/基材接触直径小于基材厚度的情况下，给出的结果可以涵盖各种参数组合，可用于有限元分析中基准评估的目的。所构建的渐近模型非常简单，优雅，并且可以应用于各种具有实际重要性的接触问题（包括非常及时的球形病毒压痕问题）。基本的理论框架是通用的，可以进一步扩展以分析涉及弹性球体和弹性层的多个接触。