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Research Papers

Surface Instability of Sheared Soft Tissues

[+] Author and Article Information

School of Electrical, Electronic, and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Ireland

D. A. Prikazchikov

Department of Advanced and Applied Mathematics, The Russian State Open Technical University of Railway Transport, Chasovaya Street, Moscow 22∕2, Russia

G. Saccomandi

Dipartimento di Ingegneria Industriale, Università degli Studi di Perugia, 06125 Perugia, Italy

J Biomech Eng 130(6), 061007 (Oct 10, 2008) (6 pages) doi:10.1115/1.2979869 History: Received March 13, 2008; Revised April 22, 2008; Published October 10, 2008

Abstract

When a block made of an elastomer is subjected to a large shear, its surface remains flat. When a block of biological soft tissue is subjected to a large shear, it is likely that its surface in the plane of shear will buckle (appearance of wrinkles). One factor that distinguishes soft tissues from rubberlike solids is the presence—sometimes visible to the naked eye—of oriented collagen fiber bundles, which are stiffer than the elastin matrix into which they are embedded but are nonetheless flexible and extensible. Here we show that the simplest model of isotropic nonlinear elasticity, namely, the incompressible neo-Hookean model, suffers surface instability in shear only at tremendous amounts of shear, i.e., above 3.09, which corresponds to a $72deg$ angle of shear. Next we incorporate a family of parallel fibers in the model and show that the resulting solid can be either reinforced or strongly weakened with respect to surface instability, depending on the angle between the fibers and the direction of shear and depending on the ratio $E∕μ$ between the stiffness of the fibers and that of the matrix. For this ratio we use values compatible with experimental data on soft tissues. Broadly speaking, we find that the surface becomes rapidly unstable when the shear takes place “against” the fibers and that as $E∕μ$ increases, so does the sector of angles where early instability is expected to occur.

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Figures

Figure 1

Shearing (along the arrows) a block of silicone of approximate size 15×10×1.5cm3 and a block of mammalian skeletal muscle (beef) of approximate size 15×10×3cm3. One does not exhibit surface instability, the other does.

Figure 2

Large plane strain deformation of a unit cube near the surface of a semi-infinite incompressible neo-Hookean solid. When the solid is compressed by 71% (or, equivalently, stretched by 238%), its surface wrinkles. Note that the analysis quantifies neither the amplitude nor the wavelength of the wrinkles.

Figure 3

Large simple shear of a unit square in the surface of a semi-infinite incompressible neo-Hookean solid. When the solid is sheared by an amount K0≃3.09 (figure on the right), its surface wrinkles. The corresponding angle of shear is tan−1K0≃72.0deg, which is physically abnormally large. Then the wrinkles are parallel to the direction of greatest tension, which makes an angle φ0≃16.5deg with the direction of shear (and so, the wrinkles are almost aligned with the sheared faces).

Figure 4

A unit square lying on the surface of a semi-infinite solid reinforced with one family of fibers (thin lines) and subject to a simple shear of amount K=0.5 (the angle of shear is tan−1K≃26.6deg) in the X1 direction. In the reference configuration, the fibers are along the unit vector M at the angle Φ=60deg with the X1-axis. In the current configuration, they are along m. The unit vector n is orthogonal to the wrinkles’ front (when they exist). Finally, the dashed line is aligned with the direction of greatest stretch; it is at an angle ψ≃38deg to the direction of shear.

Figure 5

Variations in the critical amount of shear for surface instability with the angle between the directions of shear and the fibers. The solid is modeled as a neo-Hookean matrix reinforced with one family of fibers (standard reinforcing model); the ratio of the matrix shear modulus to the fiber stiffness is taken in turn as 40.0, 20.0, and 10.0. The three graphs coincide as long as 0<Φ<Φ0, where Φ0=99.0deg, 102.8deg, and 108.1deg, respectively. At Φ≃Φ0, the half-space switches from being very stable (Kcr>3.09) to being easily unstable (Kcr<0.3). The part of the plot corresponding to Kcr>5 is not shown for physical and visual reasons.

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