0
Research Papers

On the Computation of Stress in Affine Versus Nonaffine Fibril Kinematics Within Planar Collagen Network Models

[+] Author and Article Information
Thomas J. Pence, Ryan J. Monroe

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824

Neil T. Wright

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824pence@egr.msu.edu

J Biomech Eng 130(4), 041009 (Jun 03, 2008) (9 pages) doi:10.1115/1.2917432 History: Received April 16, 2007; Revised December 06, 2007; Published June 03, 2008

Some recent analyses modeled the response of collagenous tissues, such as epicardium, using a hypothetical network consisting of interconnected springlike fibers. The fibers in the network were organized such that internal nodes served as the connection point between three such collagen springs. The results for assumed affine and nonaffine deformations are contrasted after a homogeneous deformation at the boundary. Affine deformation provides a stiffer mechanical response than nonaffine deformation. In contrast to nonaffine deformation, affine deformation determines the displacement of internal nodes without imposing detailed force balance, thereby complicating the simplest intuitive notion of stress, one based on free body cuts, at the single node scale. The standard notion of stress may then be recovered via average field theory computations based on large micromesh realizations. An alternative and by all indications complementary viewpoint for the determination of stress in these collagen fiber networks is discussed here, one in which stress is defined using elastic energy storage, a notion which is intuitive at the single node scale. It replaces the average field theory computations by an averaging technique over randomly oriented isolated simple elements. The analytical operations do not require large micromesh realizations, but the tedious nature of the mathematical manipulation is clearly aided by symbolic algebra calculation. For the example case of linear elastic deformation, this results in material stiffnesses that relate the infinitesimal strain and stress. The result that the affine case is stiffer than the nonaffine case is recovered, as would be expected. The energy framework also lends itself to the natural inclusion of changes in mechanical response due to the chemical, electrical, or thermal environment.

FIGURES IN THIS ARTICLE
<>
Copyright © 2008 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Panel (a) is an illustration of an arbitrary element in the reference configuration. It is composed of springs a, b, and c connected at Point G, all of which are enclosed in a smooth contour, in this case, a circle. Panel (b) shows the deformed element for what might represent a uniaxial loading of the global body. The deformed locations of the boundary nodes a,b,c follow from λx and λy, but the deformed location of the internal node g depends on the kinetic relation.

Grahic Jump Location
Figure 2

Geometry of a highly symmetric base element (panel (a)) and for the element rotated by ϕ≠0 (panel (b))

Grahic Jump Location
Figure 3

The stiffness kernels for the highly symmetric base element (depicted in Fig. 2) as a function of the rotation ϕ. The curves for the affine and force balance kinetic relations are so labeled. The vertical scale is in units of k∕πR, where k is the spring stiffness and R is the element radius.

Grahic Jump Location
Figure 4

The geometry of the three different types of base elements. In I, spring b is nonorthogonal to springs a and c. In II, spring b is orthogonal to springs a and c, but shifted so that the intersection G moves by R−La. In III, spring b is either lengthened or shortened and collinear springs a and c are displaced from the center by R−Lb.

Grahic Jump Location
Figure 5

The isotropic stiffness for the multielement realization with Base Element II in the case of small strains. The curves for the affine kinetic relation and the force balance kinetic relation are labeled. The solid curves are for the C¯xx and the dashed curves are for the C¯xy. The vertical scale is in units of k∕πR. Locations on the vertical axis correspond to La=R and so recover the results for the base element in Fig. 2.

Grahic Jump Location
Figure 6

The isotropic stiffness for the multielement realization with Base Element III in the case of small strains. The curves for the affine kinetic relation and the force balance kinetic relation are labeled. The solid curves are for the C¯xx and the dashed curves are for the C¯xy. The vertical scale is again in units of k∕πR. Locations on the vertical axis correspond to Lb=R, and so recover the results for the base element in Fig. 2.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In