Making a rope that does not affect its attached objects

I have two GameObjects that I wish to connect visually with a rope that stretches when they are far apart and sags when they are close. However, I don't want these entities to be affected by the rope - in other words, I want the rope to be visual and not physical in the game world.

Currently I have a bunch of GameObjects for links in the rope, connected via HingeJoint2Ds. I tried removing the connections to the two attached objects and instead setting the position of the first and last HingeJoint2Ds to their respective entities in my Update() method, but this seems to have imperfect results - when stretched, links in the rope slide to the center, leaving large gaps on either end.

Any suggestions?