The mechanism by which cells measure the dimension of the organ in which they are embedded, and slow down their growth when the final size is reached, is a long-standing problem of developmental biology. The role of mechanics in this feedback is considered important. Morphoelasticity is a standard continuum framework for modeling growing elastic tissues. However, in this theory, in the absence of additional variables, the feedback between growth and mechanical stress leads to either a collapse or unbounded growth of the tissue, but usually prohibits reaching a finite asymptotic size ('size control'). In this article, we modify this classical setting to include an energetic cost associated with growth, leading to the physical effect of size control. The present model simultaneously provides a qualitatively correct residual stress profile and has a naturally emerging necrotic core, all of which have previously been experimentally established in multicellular spheroids. This is achieved through a local feedback mechanism derived from a thermodynamical framework. The model delivers testable predictions for experimental systems and could be a step towards the understanding of the role of mechanics in the multifaceted question of how growing organs attain their final size.