The behavior of hysteretic, coupled elastic and fluid systems is modeled. The emphasis is on quasistatic equilibrium in response to prescribed chemical potential (mu) protocols and prescribed stress (sigma) protocols. Hysteresis arises in these models either from the presence of hysterons or from the presence of self-trapping internal fields. This latter mechanism is modeled in finite element calculations which serve to illustrate the creation of hysteresis in a range of circumstances that go from conventionally hysteretic systems, a sandstone, to systems like a wood fiber. An essential ingredient in the behavior of these systems, the interaction between the mechanical variables and the fluid variables, is accorded special attention. The proper venue for the exploration of these systems is (mu, sigma) space and appropriate mu protocols, sigma protocols, and combined mu-sigma protocols.