Brain swelling is a serious condition associated with an accumulation of fluid inside the brain that can be caused by trauma, stroke, infection, or tumors. It increases the pressure inside the skull and reduces blood and oxygen supply. To relieve the intracranial pressure, neurosurgeons remove part of the skull and allow the swollen brain to bulge outward, a procedure known as decompressive craniectomy. Decompressive craniectomy has been preformed for more than a century; yet, its effects on the swollen brain remain poorly understood. Here we characterize the deformation, strain, and stretch in bulging brains using the nonlinear field theories of mechanics. Our study shows that even small swelling volumes of 28 to 56 ml induce maximum principal strains in excess of 30%. For radially outward-pointing axons, we observe maximal normal stretches of 1.3 deep inside the bulge and maximal tangential stretches of 1.3 around the craniectomy edge. While the stretch magnitude varies with opening site and swelling region, our study suggests that the locations of maximum stretch are universally shared amongst all bulging brains. Our model has the potential to inform neurosurgeons and rationalize the shape and position of the skull opening, with the ultimate goal to reduce brain damage and improve the structural and functional outcomes of decompressive craniectomy in trauma patients.