Micromechanical models of helical superstructures in ligament and tendon fibers predict large Poisson's ratios

Abstract

Experimental measurements of the Poisson's ratio in tendon and ligament tissue greatly exceed the isotropic limit of 0.5. This is indicative of volume loss during tensile loading. The microstructural origin of the large Poisson's ratios is unknown. It was hypothesized that a helical organization of fibrils within a fiber would result in a large Poisson's ratio in ligaments and tendons, and that this helical organization would be compatible with the crimped nature of these tissues, thus modeling their classic nonlinear stress-strain behavior. Micromechanical finite element models were constructed to represent crimped fibers with a super-helical organization, composed of fibrils embedded within a matrix material. A homogenization procedure was performed to determine both the effective Poisson's ratio and the Poisson function. The results showed that helical fibril organization within a crimped fiber was capable of simultaneously predicting large Poisson's ratios and the nonlinear stress-strain behavior seen experimentally. Parametric studies revealed that the predicted Poisson's ratio was strongly dependent on the helical pitch, crimp angle and the material coefficients. The results indicated that, for physiologically relevant parameters, the models were capable of predicting the large Poisson's ratios seen experimentally. It was concluded that helical organization within a crimped fiber can produce both the characteristic nonlinear stress-strain behavior and large Poisson's ratios, while fiber crimp alone could only account for the nonlinear stress-strain behavior.

Figure 1

Schematic of tendon and ligament microstructure, adapted from Kastelic et al.

Figure 2

Separate models were constructed with 7, 19, 37, 61 and 91 discrete fibrils. Model C, which had 37 fibrils, was considered to be the base model and was used for most simulations.

Figure 3

A - Untransformed model; B - Planar crimp model; C - Helically transformed model; D - Helically transformed model combined with planar crimp. The top models show the full mesh while bottom models show just the fibrils with the matrix material removed.

Figure 4

A - A sinusoidally crimped fiber was defined by its diameter D, crimp angle θ_crimp, crimp amplitude A and crimp period λ. B - The helical pitch angle was defined as the angle between the vertical (z) axis and the fibrils. All models shared the same coordinate system. The fiber axis was aligned with the z axis and the x-y plane was transverse to the fiber.

Figure 5

A - Plot of the stress vs. strain for models with θ_crimp =10, 15 and 20 degrees. B - Plot of the stress vs. strain for crimped models (θ_crimp=15 degrees) with a helical pitch of θ_pitch = 0, 12, 16 and 23 degrees. C - The Poisson function plotted vs. strain for models with crimp angles of θ_crimp = 0, 10 and 20 degrees.

Figure 6

A - The effective Poisson’s ratio plotted as a function of mean helical pitch for models with a modulus ratio of M_r=10³. B - The effective Poisson’s ratio as a function of mean helical pitch for models with a modulus ratio of M_r =10⁴. The top curve in both plots corresponds to a helical crimp of 0°, with successive curves featuring an increase in crimp angle by 5° until an angle of 35° is reached at the bottom.

Figure 7

A - The percent of the maximum Poisson’s ratio plotted as a function of the total number of fibrils. The mean pitch for all models was 15° and the modulus ratio was 10⁴. B - Sensitivity of the effective Poisson’s ratio to changes in the modulus ratio. Mean pitch = 23° and compressibility ratio C_r= 1. (C) Sensitivity of the effective Poisson’s ratio to changes in the compressibility ratio. Mean pitch = 23° and modulus ratio M_r =10⁴.