  We study dissipation as a function of sample thickness in solids under global
oscillatory shear applied to the top layer of the sample. Two types of damping
mechanism are considered: Langevin and Dissipative Particle Dynamics (DPD). In
the regime of low driving frequency, and under strain-controlled conditions, we
observe that for Langevin damping, dissipation increases with sample thickness,
while for DPD damping, it decreases. Under force-controlled conditions,
dissipation increases with sample thickness for both damping schemes. These
results can be physically understood by treating the solid as a one-dimensional
harmonic chain in the quasi-static limit, for which explicit equations (scaling
relations) describing dissipation as a function of chain length (sample
thickness) are provided. The consequences of these results, in particular
regarding the choice of damping scheme in computer simulations, are discussed.
