Modelling fibers in growing disks of soft tissues. (July 2015)

• Record Type: Journal Article
• Authors: Wu, Min; Ben Amar, Martine
• Other Names: Destrade Michel guest-editor.; Dorfmann Luis guest-editor.
• Is Part Of: Mathematics and mechanics of solids. Volume 20:Number 6(2015:Jul.)
• Page Start: 663
• Page End: 679
• Publication Date: 2015-07
• Subjects: Bifurcation -- invariants -- neo-Hookean elasticity -- volumetric growth -- fiber; Materials -- Mechanical properties -- Periodicals; Solids -- Periodicals; Materials science -- Mathematics -- Periodicals; 620.11205
• DOI: 10.1177/1081286514550571
• Languages: English
• ISSNs: 1081-2865

Abstract

Despite its simple geometry and pertinence to biological systems, the growth of soft tissue disks has not been studied systematically in finite elasticity. Here with this simple geometry, we perform theoretical studies of constitutive laws concerning fibrous samples in growth or atrophy. Considering the radial growth of an incompressible neo-Hookean disk of matter, reinforced by fibers, we focus on the possible shape bifurcation from the circular to the wavy geometry. By analytical means based on a variational formulation of the theory of elasticity with growth we show that the radial geometry is lost for a critical growth anisotropy coefficient which plays the role of a control bifurcation parameter. Above a threshold which depends on the fiber invariants, the border of the disk becomes undulated with selection of a low wavenumber for low anisotropic coefficient. Radially or circumferentially oriented fibers favor undulations, but not intermediate cross-linked fibers, which inhibit undulations. Our systematic analysis shows the key role of the anisotropic growth coefficient as well as the fiber orientation for the observation of undulated patterns. Such models can explain experimental observations for skin tumors, biofilms and yeast colonies.