The Boussinesq problem, namely the indentation of a flat-ended cylindrical punch into a viscoelastic half-space is studied. We assume a linear viscoelastic model wherein the linearized strain is expressed as a function of the stress. However, this expression is not invertible, which makes the problem very interesting. Based on the Papkovich–Neuber representation in potential theory and using the Fourier–Bessel transform for axisymmetric bodies, an analytical solution of the resulting time-dependent integral equation is constructed. Consequently, distribution of the displacement and the stress fields in the half space with respect to time is obtained in the closed form.

研究了Boussinesq问题，即将平端圆柱冲头压入粘弹性半空间。我们假设线性粘弹性模型，其中线性化应变表示为应力的函数。但是，此表达式不是不可逆的，这使问题变得非常有趣。基于势理论中的Papkovich-Neuber表示并针对轴对称物体使用傅里叶-贝塞尔变换，构造了所得的时变积分方程的解析解。因此，以封闭的形式获得了相对于时间的半空间中的位移和应力场的分布。