Buckling condensation in constrained growth
- Dervaux, Julien and Ben Amar, MartineJOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 59
, 538-560 (2011)
Abstract : The multiple complexities inherent to living objects have motivated the
search for abiotic substitutes, able to mimic some of their relevant
physical properties. Hydrogels provide a highly monitorable counterpart
and have thus found many applications in medicine and bioengineering.
Recently, it has been recognized that their ability to swell could be
used to unravel some of the universal physical processes at work during
biological growth. However, it is yet unknown how the microscopic
distinctions between swelling and biological growth affect macroscopic
changes (shape, stresses) induced by volume variations. To answer this
question, we focus on a clinically motivated example of growth. Some
solid tumors such as melanoma or glioblastoma undergo a shape transition
during their evolution. This bifurcation appears when growth is confined
at the periphery of the tumor and is concomitant with the transition
from the avascular to the vascular stage of the tumor evolution. To
model this phenomenon, we consider in this paper the deformation of an
elastic ring enclosing a core of different stiffness. When the volume of
the outer ring increases, the system develops a periodic instability. We
consider two possible descriptions of the volume variation process:
either by imposing a homogeneous volumetric strain (biological growth)
or through migration of solvent molecules inside a solid network
(swelling). For thin rings, both theories are in qualitative agreement.
When the interior is soft, we predict the emergence of a large
wavelength buckling. Upon increasing the stiffness of the inner disc,
the wavelength of the instability decreases until a condensation of the
buckles occurs at the free boundary. This short wavelength pattern is
independent of the stiffness of the disc and is only limited by the
presence of surface tension. For thicker rings, two scenarios emerge.
When a volumetric strain is prescribed, compressive stresses accumulate
in the vicinity of the core and the deformation localizes itself at the
boundary between the disc and the ring. On the other hand, swelling
being an instance of stress-modulated growth, elastic stretches near the
core saturate and the instability occurs primarily at the free boundary.
Besides its implications for the mechanical stability of avascular
tumors, this work provides important results concerning layered tissues
growth and the role of hydrogels as biological tissues substitutes. (C)
2010 Elsevier Ltd. All rights reserved.