Shear Properties

Abstract

Understanding the origins of the multiaxial material properties of soft tissues is crucial for quantifying the tissue material properties that govern the material behavior for accurate predictive modeling of biological systems. The purpose of this study is to quantify the shear material properties of the rabbit patellar tendon. These properties will be examined under different shear strain rates along the fiber direction and perpendicular to the fiber direction. Thin sections of patellar tendon were resected from mature male white New Zealand rabbits and were mounted on a custom-designed testing device, quickly followed by application of predetermined sim-ple shear strains under physiological conditions. The tendons were sequentially sheared to 40% strain at strain rates of 0.1%, 1%, and 10% strain s^-1 using a device driven by a computer-controlled stepper motor. The results suggest that the tendon exhibits direction-dependent viscoelastic shear properties, reflecting the structural anisotropy. These results can be used to generate stress state dependent constitutive models.

Methods

Apparatus

Experiments were conducted using a Mach-1 Micromechanical V500cs test system (Biosyntech, Inc., Laval, Quebec, Canada) equipped with a 10 N load cell connected to the moveable actuator. For measuring thickness, a smooth flat platen, perpendicular to the vertical axis of motion, was connected to the load cell. The shear fixture con-sisted of two parallel aluminum plates, one secured to a vertically stationary base and the other to the Mach-1 load cell, with both plates parallel to the vertical axis of mo-tion. The base was designed to hold liquid so that the test fixture could be submerged in phosphate-buffered saline (PBS). In addition, the entire Mach-1 system resided in a water-jacketed incubator maintained at 37°C.

Specimen Preparation

Tissue was harvested from six skeletally mature white male New Zealand rabbits, which had been euthanized as part of a separate and unrelated research protocol con-ducted with approval of the Institutional Animal Care and Use Committee (IACUC). Hind limbs were removed, wrapped in PBS-soaked gauze, sealed in plastic bags, and frozen at -20°C until the day of testing. On that day, the tissue was thawed at room temperature in PBS, and the patellar tendon was isolated by sharp dissection. The zero position of the Mach-1 was established as the point of contact between the thickness-measuring platen and the base. The intact tendon was then laid flat on the base, widest surface (posterior) down. The actuator was lowered at 100 μm s-1 until the tissue provided 0.118 N (approx. 3.3 kPa) of compressive resistance, at which point the actuator position indicated the specimen thickness. Tissue marking dye (Polysciences, Inc., Warrington, Pa.) was then used to identify the predominant (longitudinal) fiber direction, and square pieces of tendon were obtained using a custom 6 × 6 mm steel punch, the edges of which were aligned parallel and perpendicular to the main fiber direction. Pieces were obtained by punching through the full thickness of the tissue in a direction perpendicular to the wide, anterior surface. Typical sample dimensions were 6 × 6 × 2 mm (length × width × thickness). All portions of tendon throughout the specimen preparation and assembly process leading up to actual testing were kept wet with PBS to prevent loss of tissue fluid. The tendon was kept moist by continuously spraying PBS until the test commenced. Prior to gluing the tissue onto the surface of the paral-lel plates, the tissue areas to be glued were blotted to remove excess PBS.

Testing Protocol

The Mach-1 is a two-axis test system; the base was attached to the machine’s horizontal axis, to which one aluminum plate was rigidly connected. Another plate, parallel to the first, was attached to the load cell on the vertical actuator. The square faces of a specimen were blotted dry and affixed to the plates using cyanoacrylate glue. The use of cyanoacrylate as an adhesive for conducting mechanical tests has shown negligible effects on the material properties of the tissue (Miyazaki and Hayashi, 1999; Yamamoto et al., 1999). The entire surfaces of both sides of the tissue were carefully covered with adhesive to obtain a uniform deformation. The distance between the plates was set to the thickness of the intact tendon and measured under 3 kPa pressure in unconfined compression. The base was then flooded with 37°C PBS, and the sample was allowed to equilibrate for 10 min prior to commencement of testing. Thespecimen was aligned with its predominant fiber direction parallel to the edges of the plates. The plates could be anchored to the machine such that vertical motion of the actuator produced shearing along or transverse to the major fiber axis. The direction of fiber alignment at the start of each test, either parallel or transverse to the shear direc-tion, was randomized. Figure 2 is a schematic of the setup. The distance between the parallel plates was held fixed throughout testing so that vertical motion of the actuator subjected the specimen to simple shear. The applied shear strain was calculated as the vertical displacement divided by the thickness. Each sample was first subjected to 0.025 Hz sinusoidal shear of 20% strain amplitude for 10 cycles, followed by 0.25 Hz sinusoidal shear of 20% strain amplitude for 10 cycles. The rationale behind conducting dynamic tests was first to perform tissue preconditioning by repeated cycling, which has been widely observed to be critical for obtaining accurate viscoelastic results (Weiss et al., 2002). In preconditioning, caution should be taken to ensure that the loading rate does not apply excessive stresses to the tissue, which could prematurely damage the tissue. Secondly, the preconditioning was used to study the frequency-dependent mechanical properties of a patellar tendon by generating shear-strain curves from the load-displacement data from all the cycles (10 cycles at 0.025 Hz and 10 cycles at 0.25 Hz). The goal was to examine the dynamic modulus (G*) and the phase lag, both frequency-dependent mechanical proper-ties. After a 5 min relaxation period under zero applied shear strain, the specimen was then monotonically loaded to 40% shear at 0.1% s-1, 1% s-1, and 10% s-1 with 10 min recovery intervals at zero applied strain between each test. The fixture was then rotated 90° and the same set of tests repeated.In preliminary tests, 40% shear caused no apparent damage to the specimen.

Data Reduction and Analysis

Four material properties were determined from each specimen: the tangent modulus, dynamic modulus, strain energy released per unit volume (area in the hysteresis loop), and phase lag. The tangent modulus was estimated by calculating the slope within the small strain regions of the loading portion of each stress-strain curve. The small strain regions on each curve were linear. The dynamic modulus is the stress strain ratio from dynamic loading, and the phase shift is the change is phase of the dynamic sinusoidal tests with respect to the initial sinusoidal curve.

The shear stress (τ) is given by:

where F is the load; a is the length of the specimen; and b is its width (fig. 1). The shear strain (γ) is given by:

where u is the displacement of the moveable plate with respect to the fixed plate, and c denotes the thickness of the tissue (fig. 2). Viscoelastic parameters were calculated from the oscillatory shear test. The dynamic modulus (G*, Pa) was calculated from the shear stress and strain amplitudes (Aτ and Aγ, respectively):

and

where Aτ and Aγ denote the amplitudes of the cyclic stress-time and strain-time data, respectively, while δσ and δε denote the corresponding phase shifts.

Results

This study determined the strain rate and directional dependency of the material response of rabbit patellar tendon to applied simple shear. Strain rate is defined as the percent of change in strain per unit time.

Although the nature of loading tendons in vivo is primarily tensile, shear properties were studied because of often-associated tissue shearing during its normal physiological functioning. In addition, shearing along the dominant fiber direction characterizes the shear properties of the matrix without the influence of the dominant collagen fibers; likewise, tissue shearing in the transverse direction characterizes the shear properties of the matrix and fibers. Therefore, information regarding the shear properties of the tendon is critical for the development of constitutive models that suitably describe the tendon’s material behavior. Table 1 lists the longitudinal and transverse values for the dynamic modulus, tangent modulus, strain energy dissipation, and phase lag.

The mechanical test results indicated that the mechanical properties of the rabbit patellar tendon were strongly influenced by the strain rate. Figure 3 shows a comparison of forward and reverse loading in longitudinal shear at different applied strain rates. At all strain rates, each of the four properties in the transversely tested samples demonstrated markedly lower magnitudes than those realized in the longitudinal direction. In general, the overall tissue response to shear in either direction (longitudinal, transverse) was nonlinear and viscoelastic (figs. 3 and 4).

Comparison of the strain energy released per unit volume revealed that the data obtained for the lower strain rates were of similar magnitude to ones obtained at higher strain rates in either direction of the study. A conservative level of confidence (p < 0.1) was employed to study the effects of strain rate and direction, due the inherent variability that is often associated in conducting such tests. The data presented strongly suggest a strain rate dependency of the shear properties. Additionally, both tangent modulus and strain energy released per unit volume were observed to be in-sensitive to prominent fiber orientation, because there was of no significant increase in tangent modulus and strain energy released per unit volume with change in fiber orientation. A representative stress-strain curve comparing the material response to simple shear is shown in figure 5.

When comparing material properties based on the dynamic tests (figs. 6 and 7), both the dynamic modulus (G*) and the phase lag were relatively unaffected by the frequency changes or fiber direction. Therefore, we conclude that these dynamic mate-rial properties for simple shear in the longitudinal direction are of similar order to those observed in the transverse direction at these two frequencies. However, both the dynamic modulus and the phase lag in the longitudinal direction were equal to or slightly higher than that of the transversely oriented samples and were consistent with the material response for transversely isotropic fibrous materials.

Citation: Subramanian, S., Elder, S. H., Horstemeyer, M.F., Williams, L.N. "Experimental Investigation of Anisotropic Shear Properties of Rabbit Patellar Tendon," Biological Engineering. 1, 2008: 255-264.