Wrinkling, creasing, and folding in fiber-reinforced soft tissues

Abstract

Many biological tissues develop elaborate folds during growth and development. The onset of this folding is often understood in relation to the creasing and wrinkling of a thin elastic layer that grows whilst attached to a large elastic foundation. In reality, many biological tissues are reinforced by fibres and so are intrinsically anisotropic. However, the correlation between the fiber directions and the pattern formed during growth is not well understood. Here, we consider the stability of a two-layer tissue composed of a thin hyperelastic strip adhered to an elastic half-space in which are embedded elastic fibers. The combined object is subject to a uniform compression and, at a critical value of this compression, buckles out of the plane -it wrinkles. We characterize the wrinkle wavelength at onset as a function of the fiber orientation both computationally and analytically and show that the onset of surface instability can be either promoted or inhibited as the fiber stiffness increases, depending on the fibre angle. However, we find that the structure of the resulting folds is approximately independent of the fiber orientation. We also explore numerically the formation of large creases in fiber-reinforced tissue in the post-buckling regime.

Growth and remodelling of living tissues: perspectives, challenges and opportunities

“…Similar experiments performed on a two-layered brain prototype made of polymeric gels with differential swelling properties reproduce folds similar to the gyri and sulci of a real brain [139]. In this simple two-layer system, it is well appreciated that the pattern adopted by the system depends on a number of important factors such as the relative stiffnesses of the two layers [140], the thickness of the thin layer, the growth of the top layer [141], the curvature of the foundation [142], the adhesion energy between the layers [143,144], the imperfection of the substrate [145], the anisotropic response [146,147], the surface tension and pressure [148] and the nonlinear elastic response of the materials [149]. For small ratios of layer μ l to foundation μ s stiffnesses, m l =m s , 10, as the wrinkling patterns develop, the system localizes this initial deformation and a fold or crease appears as observed in many biological systems.…”

Wrinkling instabilities in soft bilayered systems

“…Various parameters control pattern selection in non-uniformly compressed film-substrate systems. Depending on the stiffness ratio between film and substrate, their mismatch stretch and adhesion energy, creases or wrinkles emerge, which transform into folds, period-doubles, or ridges for higher compression, or delaminate [9][10][11]. Bifurcation analyses can predict the critical conditions for the onset of primary wrinkling.…”

Revisiting the wrinkling of elastic bilayers I: linear analysis

“…Such methods have been used to great success to compute the critical uniaxial compression required to cause buckling of an elastic half-space coated in a thin, stiffer elastic film [6]. Following works have considered variations of the physical setting such as pre-stretching the substrate [7], further compressing the buckled bilayer to induce a second, periodic-doubling bifurcation [8], the limiting behaviour of the system as the stiffness ratio of the layers tends to unity [9], the effect of adding reinforced fibres to the substrate [10] and the replacement of compression with growth as a mechanism to induce buckling [11].…”

“…A last modification we make to the bilayer system is to introduce embedded elastic fibres into the elastic substrate, as considered in [10]. This adds an orientational anisotropy into the system that mimics structures seen in many biological materials.…”