Our work (1) was based on Green’s solution (2). We thank the discussors for pointing out a different approach taken by Borodich 3 following the work of Galin 4. At the time of publication, we were unaware of the work by Borodich. The usage of our derived solution is straightforward. With modern mathematical software, hypergeometric function will be like a regular elementary function and the final result is easy to be obtain. It can also be used to check analytical expressions for possible misprints.
As it is pointed out in the paper, the power of the “polynomial” can be any non-negative number, such as 0, 2, 1/12, e, π. With this solution, one can use multiple terms to define the punch shape instead of a monomial function of the punch radius.
We appreciate the fact that there exist numerous contributions to this field in the Russian literature, and our understanding of this work is mainly based on the books by Gladwell 5 and Sneddon 6. Johnson 7 also mentioned the solutions by Shtaerman and Galin in his book.