Equilibrium and Compatibility

We can think of an elastic solid as a highly redundant framework — each particle is built-in to its neighbours. For such a framework, we expect to get some equations from considerations of equilibrium, but not as many as there are unknowns. The deficit is made up by compatibility conditions — statements that the deformed components must fit together. These latter conditions will relate the dimensions and hence the strains of the deformed components and in order to express them in terms of the same unknowns as the stresses (forces) we need to make use of the stress-strain relations as applied to each component separately. If we were to approximate the continuous elastic body by a system of interconnected elastic bars, this would be an exact description of the solution procedure. The only difference in treating the continuous medium is that the system of algebraic equations is replaced by partial differential equations describing the same physical or geometrical principles.