The static Green's functions for 2D and 3D linear elasticity are given in Eq. (5.8) and (5.24) respectively in the book Micromechanics of Defects in Solids by Mura (see the attached photos for these equations).
The Green's function in 3D case, as expected, goes to zero when |x-x'| goes to infinity. However, it is not true when considering the 2D case since there is a growing term ln(|x-x'|). So how should we explain such a difference? Is it physically intuitive to have Green's function keeps increasing as |x-x'|->\infty in 2D case?
The Green's function in the 3D case vanishes at infinity because there is no load there. On the cuntrary, there is a load at infinity in the 2D case, since the forces are uniformly distributed along a line parallel to x3 axis.
Victor J. Аdlucky thanks for your answer!
I also thought about something in that direction after I put this question on ResearchGate. However, what still bothers me is that if it is proper to use this 2D Green's function for solving practical '2D' problems.
For example, we know that for 2D plane-strain problem we consider sufficiently large thickness in the x3 axis such that we ignore the deformation related with x3 direction. But what if now we still only apply a point force (not a line force) at a point in the x1-x2 plane and say, at x3 = 0. Now in this case how should we apply the Green's function technique to solve?
Right now I am thinking of using something like Green's function in half-space. But anyway, to me the 2D's Green's function given by the literature is not very useful when considering a problem described above.
Please feel free to put more comments if you like!
I would recommend an interesting discussion about subject of the question:
https://imechanica.org/node/319
(not answering, just find it interesting and want to join the discussion. I read the iMechenica post and please correct me if my understanding is wrong)
1. Imagine applying a line load in the medium, the 2D displacement field is certainly not growing but should monotonically decreasing as R --> inf.
2. Looks like the Green's function (5.24) was obtained from an integral of something with 1/x (~1/R) so at least its integrand is decreasing as R --> inf.
3. Integral of 1/R gives Log(R), but the Green's function (5.24) would need a infinitely large integral constant to shift the entire log function above the x-axis (if really want to match with the previous thought experiment, i.e., to set the 0-displacement ref at R-->inf), which that integral constant is simply neglected as it is not any beneficial.
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