Stress Singularities in Swelling Soft Solids

When a swelling soft solid is rigidly constrained on all sides except for a circular opening, it will bulge out to expand as observed during decompressive craniectomy, a surgical procedure used to reduce stresses in swollen brains. While the elastic energy of the solid decreases throughout this process, large stresses develop close to the opening. At the point of contact, the stresses exhibit a singularity similar to the ones found in the classic punch indentation problem. Here, we study the stresses generated by swelling and the evolution of the bulging shape associated with this process. We also consider the possibility of damage triggered by zones of either high shear stresses or high fiber stretches.

Figure 1

Brain bulging following decompressive craniectomy. (a) Left fronto-temporo-parietal craniectomy in human [6]. (b) MRI showing herniation and resulting ischemia (white arrows) after 24 hr following craniectomy on a rat subject to a middle cerebral artery occlusion [7].

Figure 2

Bulging simulation. (a) Cylindrical geometry. (b) Spherical geometry.

Figure 3

(a) Bulge shapes obtained from the exact linear solution (dashed line) compared to the finite-element simulation (solid line). (b) 3D shape of the bulge (linear solution, $h=0.733$). (c) Internal displacements.

Figure 4

Shear stress in the reference configuration. (a) The numerical simulation and (b) the exact solution. (c) Comparison between the exact and approximate solution for the damage drops ($c={\epsilon }_{\max }/h,\nu =0.45,Eh=1$).

Figure 5

Vertical strain in the reference configuration. The maximal value of the stretch is reached on the axis of symmetry. (a) Finite-element simulation. (b) Exact linear solution ($\nu =0.45,Eh=1$).

Figure 6

Swelling in spherical symmetry. The bulging angle as a function of swelling compared to the geometric estimate obtained by overlapping spheres ($\gamma =50°$).

Figure 7

Finite-element simulation for 10% brain swelling. Regions of high shear stress are circled. Inset: transverse section showing radial fiber stretch. Region of high radial stretch is circled.