A detailed theory of the thermodynamics of linear viscoelastic fluids in steady shear is presented. This theory is exact and gives simple and easily computed expressions for the change in the internal energy and free energy due to the imposition of steady shear for a general linear viscoelastic fluid. It has strong similarities to the extended irreversible thermodynamics approach, but differs from it in a few crucial ways. An important element of our theory is the division of the viscoelastic work into elastic and viscous parts, and the identification of these components as the reversible and irreversible parts of the work. This leads to a natural definition of reversible heat transfer that can be used in the definition of the steady-state entropy. We present the results of nonequilibrium molecular-dynamics computer simulations and evaluate the thermodynamic properties of a simple model fluid as a function of strain rate, showing how the entropy difference of the fluid between the equilibrium and shearing steady states can be computed. We find that, at low strain rates, the entropy change of a simple atomic fluid at constant temperature is positive and proportional to the square of the strain rate. A helpful geometrical interpretation of the decomposition of viscoelastic work into elastic and viscous components using a hysteresis loop construction is also discussed. (C) 2003 American Institute of Physics.