On problems with solution-dependent load

In Continuum Physics, general balance problems between internal fluxes and external sources are considered, mostly on some domain Ω called a material body. Typically, the fluxes are determined by a wanted field – the solution or configuration of the body, possibly augmented by derivatives or other suitable quantities specifying the state of the system. In order to obtain a closed problem formulation, initial and boundary conditions need to be provided as well. In this paper, we discuss some issues appearing in the case that the sources, e.g. forces in a mechanical problem, are not known in advance, but related to the wanted configuration in terms of values and/or derivatives of the solution. An analogy between thermodynamic and mechanical problems is shown.