We investigate indentation by a smooth, rigid indenter of a two-dimensional half-space comprised of periodically arranged linear-elastic layers with different constitutive responses. Identifying the half-space’s material parameters as periodic functions in space, we utilize the theory of periodic homogenization to approximate the layered heterogeneous material by a linear-elastic, homogeneous , but anisotropic medium. This approximation becomes exact as the layer thickness becomes infinitesimal. In this way, we reduce the original problem to the indentation of an anisotropic, homogeneous, linear-elastic half-space by a smooth, rigid indenter. The latter is solved analytically by formulating and resolving the corresponding matrix Riemann–Hilbert boundary-value problem in complex analysis. Thus, we obtain an approximate, but analytical solution for the indentation of a layered heterogeneous medium. We then compare this solution with finite element computations of the indentation on the original layered, heterogenous half-space. We conclude that (a) the contact pressure on the layered, heterogenous half-space is well approximated by that obtained through homogenization, and the approximation improves as the layer thickness is decreased, or if the indentation force is increased; (b) the upper bound of the difference between the two contact pressures depends only upon the ratio of the Young’s moduli of the two materials constituting the heterogenous medium and their Poisson’s ratio; and (c) the average variation of the discontinuous von Mises stress in the layered half-space is well approximated by the one found in the homogenized half-space. The approach presented here can be utilized for a diverse array of indentation and contact problems of finely mixed heterogeneous media, and is also amenable to systematic improvements.

中文翻译：

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周期性分层、平面、弹性半空间的压痕

我们研究了由具有不同本构响应的周期性排列的线性弹性层组成的二维半空间的光滑刚性压头的压痕。将半空间的材料参数识别为空间中的周期函数，我们利用周期均质化理论通过线弹性、均质但各向异性的介质来近似层状异质材料。随着层厚度变得无穷小，这种近似变得精确。通过这种方式，我们将原始问题简化为通过光滑刚性压头压入各向异性、均匀、线弹性半空间。后者通过在复分析中制定和求解相应的矩阵 Riemann-Hilbert 边值问题来解析求解。因此，我们得到一个近似值，但是对于分层异质介质的压痕的解析解。然后，我们将此解决方案与原始分层异质半空间上的压痕的有限元计算进行比较。我们得出结论：（a）层状异质半空间上的接触压力与通过均质化获得的接触压力很好地近似，并且随着层厚度的减小或压痕力的增加，近似值会提高；(b) 两种接触压力差的上限仅取决于构成非均质介质的两种材料的杨氏模量与泊松比的比值；(c) 分层半空间中不连续 von Mises 应力的平均变化与在均质化半空间中发现的相似。