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Mechanical forces influence the development and behavior of biological tissues. In many situations, these forces are exerted or resisted by elastic compliant structures such as the own-tissue cellular matrix or other surrounding tissues. This kind of tissue-elastic body interactions are also at the core of many state-of-the-art in situ force measurement techniques employed in biophysics. This creates the need to model tissue interaction with the surrounding elastic bodies that exert these forces, raising the question of which are the minimal ingredients needed to describe such interactions. We conduct experiments in which migrating cell monolayers push on carbon fibers as a model problem. Although the migrating tissue is able to bend the fiber for some time, it eventually recoils before coming to a stop. This stop occurs when cells have performed a fixed mechanical work on the fiber, regardless of its stiffness. Based on these observations, we develop a minimal active-fluid model that reproduces the experiments and predicts quantitatively relevant features of the system. This minimal model points out the essential ingredients needed to describe tissue-elastic solid interactions: an effective inertia and viscous stresses.