Micromechanical poroelastic finite element and shear-lag models of tendon predict large strain dependent Poisson's ratios and fluid expulsion under tensile loading

Abstract

As tendons are loaded, they reduce in volume and exude fluid to the surrounding medium. Experimental studies have shown that tendon stretching results in a Poisson's ratio greater than 0.5, with a maximum value at small strains followed by a nonlinear decay. Here we present a computational model that attributes this macroscopic observation to the microscopic mechanism of the load transfer between fibrils under stretch. We develop a finite element model based on the mechanical role of the interfibrillar-linking elements, such as thin fibrils that bridge the aligned fibrils or macromolecules such as glycosaminoglycans (GAGs) in the interfibrillar sliding and verify it with a theoretical shear-lag model. We showed the existence of a previously unappreciated structure-function mechanism whereby the Poisson's ratio in tendon is affected by the strain applied and interfibrillar-linker properties, and together these features predict tendon volume shrinkage under tensile loading. During loading, the interfibrillar-linkers pulled fibrils toward each other and squeezed the matrix, leading to the Poisson's ratio larger than 0.5 and fluid expulsion. In addition, the rotation of the interfibrillar-linkers with respect to the fibrils at large strains caused a reduction in the volume shrinkage and eventual nonlinear decay in Poisson's ratio at large strains. Our model also predicts a fluid flow that has a radial pattern toward the surrounding medium, with the larger fluid velocities in proportion to the interfibrillar sliding.

Keywords: Extracellular matrix; Finite element modeling; Poisson’s ratio; Poroelasticity; Tendon.

Fig. 1

(a) The finite element model consists of staggered fibrils (red rods) interconnected with interfibrillar-linkers (springs) placed in a porous medium (cyan). The coordinate system is placed at the center of the model with x-axis directed along the fibril orientation, and y-axis perpendicular to it. (b) Under tensile loading, interfibrillar-linkers exert a compressive force to the biphasic medium (vertical arrows) and cause the fluid to flow radially outward from the encapsulated matrix. The aspect ratio in the figure is not in scale.

Fig. 2

Predicted Poisson’s ratio for varying interfibrillar-linker elastic moduli.

Fig. 3

(a) The deformed shape of a longitudinal cross-section (at y = 0, Ref to Fig. 1) at 3% strain and K/d = 0.48 MPa. The contour colors show the generated pressure in the tendon. The tendon width at two points x = 0, L/2 and x = L/4, is denoted by d₁ and d₂, respectively. The pressure profile along the centerline of the cross-section (b) is positive, with larger values at fibrils ends where the forces in the cross-linkers are larger. (c) The contraction in tendon width is larger at fibril ends (d₁/a) than the interior point (d₂/a).

Fig. 4

Fluid movement pattern at a tensile strain of 3%, with a interfibrillar-linker elastic modulus K/d = 0.48 MPa. The arrows indicate the fluid velocity field vector. Two transverse cross-sections (A) and (B) at the fibril ends (x = 0, L/2) and interior (x = L/4) points illustrate the fibril location specific fluid velocity magnitudes. The velocity magnitude is larger at cross-section (A), which is located near the fibril ends.

Fig. 5

Fluid velocity at the surface of the tendon at a tensile strain of 3%, with varying interfibrillar-linker elastic moduli. The positive values indicate fluid exudation.

Fig 6

Radius versus spacing of the fibril interfibrillar-linkers for the fixed value of the interfibrillar-linker elastic modulus (K/d).

Fig. 7

The unit cell consists of two neighboring fibrils whose interfibrillar-linkers are spaced by distance, d, and aligned at an angle β to the fibrils. For an element of the matrix (cyan), axial and transverse strains, ε_xx (tensile) and ε_yy (compressive), act on the element.

Fig. 8

With increasing the applied strain, the tensile (ε_xx) and compressive (−ε_yy) strains increase and the Poisson’s ratio (ν_eff) increases rapidly at low strains prior to decreasing at higher strains.

Fig. 9

For the small tensile strain, interfibrillar-linkers exert a large compressive force to the ECM and cause a significant decrease in the volume and fluid exudation, while for the large strains, the force in the interfibrillar-linkers have decreased and the ECM element is under a tensile stretch and volume expansion.