I would like to solve a problem in which the system has no fully relaxed configuration in which there are no stresses. As a concrete example, consider a beam that is to be pulled into a ring and that ends up connected with some other elastic material. What is the best way to implement this? In the relaxed configuration for the beam itself, the connecting material would be very stressed. If the left and right end of the beam end up with their faces fully in contact, the problem could be setup with displacement boundary conditions (see top of figure). However, it does not seem feasible to use such boundary conditions for other types of connections (i.e., some part of the long edges of the left and right sides of the beams being connected; see bottom of figure). It seems that the mesh itself needs to be connected.
Is there a standard way to setup the mesh in situations where there is no configuration in which everything is unstressed?
Update: I found that in ABAQUS, what I want to do is use what they term “tie constraints.” I found that someone asked if this was possible in FENICS, but as of 2014 such a thing had not been implemented.