BME 315 BiomechanicsMaterial Properties of Tendon: introduction

Adapted by R. Lakes from D. Thelen and C. Decker, U. Wisconsin, ’09

I. Introduction

In this lab, we will investigate the material properties of a tendon. Tendons and ligaments arevery similar in structure and composition. They are both composed of collagen and elastinfibers, with a structural hierarchy (Fig. 1). Structurally, tendons connect muscle to bone, andligaments connect bone to bone. An example of a tendon would be the many tendons of thelower leg muscles (Fig. 2a) which act to move the foot. An example of a ligament would be oneof the many ligaments that hold together the bones of your shoulder (Fig. 2b), such as theacromioclavicular ligament and the corocoacromial ligament (named after the locations on thebones that they connect). The material composition and properties of tendons and ligamentsreflect the differences in loading that they will commonly experience. As tendons will transmitsignificant forces from the muscle into movement of the bone very regularly, they are a slightlystronger tissue than ligaments, and are composed of more collagen fibers (roughly 85% versus70% [1]).

Figure 1: Hierarchical structure of tendon and ligament [2].

Tendons and ligaments experience primarily tensile loading in vivo. Hence in this lab, we willbe performing a tensile loading test in which we make continuous measurements of the load anddeflection until failure of the specimen. The measured loads and deflections will be normalizedby initial cross-sectional area and length, respectively. These normalized quantities are referredto as normal stress and strain. Typical engineering materials (e.g. steel and aluminum) usuallyexhibit linear, elastic, homogeneous, and isotropic properties, which is reflected in a linearstress-strain curve for loads below the elastic limit (Fig. 3). In contrast, biological materialsoften exhibit non-linear, inelastic, non-homogenous and anisotropic behavior (Fig. 4).

BME 315 BiomechanicsMaterial Properties of Tendon

Figure 2: Lower leg muscles and tendon attachments [3] (a) and ligaments of the shoulder [4] (b).

The non-linear behavior of tendon can be understood by inspection of the material substructureand hierarchy shown in Figure 1, and the inset images in Figure 4. In the tendon, the fibrils arecomposed of many crimped collagen fibers, and many fibrils are adhered to each other. Whenthe tissue is stretched, the fibers uncrimp and straighten (Fig. 4), which contributes to the toe-region of the curve. As load is increased, continuously increasing amounts of fibers arerecruited. Finally, in the linear portion of the stress – strain curve, all fibers have been recruitedand are straight. A linearly increasing stress versus strain response is then exhibited. At largerstrains, the tendon will begin to undergo micro-damage, leading to the eventual macro-damageand failure.

Figure 3: Typical stress – strain curve of a metal.

BME 315 BiomechanicsMaterial Properties of Tendon

Figure 4: Typical stress – strain curve of tendon or ligament connective tissue.

Biological materials also exhibit viscoelastic behavior. The viscoelastic behavior is manifest inthe time dependent aspect of the material response; in damping of vibrations; and in theattenuation of waves including waves used in clinical diagnostic ultrasound. The viscoelasticbehavior of biological materials is due various interactions of collagen with the proteins, water,and ground substance when it is loaded. As this is a rate dependent phenomenon, a material willrespond differently if loaded quickly as opposed to loading more slowly. The slope of the stress– strain curve will increase with an increasing strain rate (Fig. 5), and the apparent elasticmodulus, the constant of proportionality relating the strain to the stress, will increaseaccordingly. If a viscoelastic material is stretched to a constant deformation, it will slowly relax,with the stress in the material decreasing. A relaxation curve is a plot of the viscoelastic stressversus time response (Fig. 6).

Figure 5: Effect of changing strain rate on stress – strain curve for a viscoelastic material [5].

BME 315 BiomechanicsMaterial Properties of Tendon

Figure 6: Stress relaxation of a viscoelastic material under a constant deformation.A formal relaxation curve displays modulus vs. time. The zero of the time scale is taken halfway through the rise time sothat the flat portion to the left (which is not relaxation) is not shown. It is common to use a logarithmic time scale that of

course, does not show the zero.II. Theoretical background

Stress, strain, and Young’s modulus:

In order to obtain property information about a material that is independent of the geometry,measurements of stress and strain are calculated from force and deflection. The specimengeometry shown in Figure 7 will be referenced in these calculations.

Figure 7: Tension test configuration of tendon specimen.

We will be using engineering stress, which is the load normalized by the test specimen initialcross-sectional area:

(1)

BME 315 BiomechanicsMaterial Properties of Tendon

Engineering strain is defined as the deflection, normalized by the unloaded length:

or (2)

where and are the final and initial lengths, respectively. In our testing, is the gage lengthof the specimen at which the tendon initially becomes taught and generates a tensile load (Fig.8). Young’s Modulus, E, for a 1D loading can then be calculated from the linear region of thestress – strain curve (Fig. 4) by Hooke’s law:

(3)

Viscoelastic behavior:

As described previously, the viscoelastic behavior is manifest in the time dependent aspect of thematerial response. There are various models that are used to describe viscoelastic behavior ofmaterials. Some models utilize combinations of springs and dashpots to describe the observedbehavior of a material. These are popular in elementary introductions to the subject because onecan visualize springs and viscous dampers and the models lead to simple differential equationsthat can be readily solved. The force in a spring is linear with the displacement u, with theconstant of proportionality relating the two being the spring constant, k (Fig. 8, Eq. 4). The forcein a dashpot is dependent on the rate of displacement, , and proportional to the damping

coefficient Greek eta (Fig. 9, Eq. 5). Mechanically, a dashpot is a device which providesdamping usually due to the displacement of a viscous fluid within it, and it is often used torepresent this quality in other systems.

Figure 8: Spring model.

(4)

Figure 9: Dashpot model.

eta (5)

BME 315 BiomechanicsMaterial Properties of Tendon

The three most common models combining these elements are the Maxwell, Voight, and Kelvinstandard linear solid model (Fig. 10). Deriving the equations modeling these systems is beyondthe scope of this course. However we will look at the time dependent behavior of the tendon andhow it changes with strain rate and under the application of constant strain.

Figure 10: Three mechanical models of viscoelasticity. a) Voight, b) Maxwell, and c) Kelvinstandard linear solid.

Under constant strain, the tendon exhibits stress relaxation. This means that the stress willdecrease (relax) with time. The predicted stress relaxation response of these three models isshown in Fig. 11 below. The parameters associated with the initial and final force as well as thedeformation are shown for their respective models in Fig. 11. The exponential decay of forcepredicted by models (a) and (c) gives rise to a time constant, not shown.

Figure 11: a) Maxwell, b) Voight, and c) standard linear solid model of relaxation behavior for deformation u [5]. Aformal relaxation curve displays modulus vs. time. The zero of the time scale is taken halfway through the rise time so

that the flat portion to the left (which is not relaxation) is not shown.

Similarly, in creep, which is the time-dependent strain response to step stress, the response ofmodel (c) is a single exponential.

BME 315 BiomechanicsMaterial Properties of Tendon

Is the behavior in fact exponential? Many decaying curves superficially resemble exponentials.For example, a power law in time, in which A and n are constants, is as follows.

F(t) = A t-n (6)Power laws are also used to model time-depenent materials. Even with a curve fit, it may be achallenge to distinguish different models if the time window is narrow.

To distinguish models, use a wide window in which the ratio of the shortest time to the longesttime is as large as possible. The shortest time available is limited by how fast the deformation isapplied by the test instrument. The time from zero to full deformation is called the rise time.For zero of the time scale, use the time halfway through the rise time. Record the first data pointfor a time a multiple of three of the rise time.

To distinguish models it is customary to plot the time dependence on a logarithmic scale in time.An exponential function shows up as a sigmoid shape occupying about a factor of ten on the logtime scale. If the scale is logarithmic on both axes, a power law shows up as a straight line. Thusthe exponential and power law models can be easily distinguished.

A formal relaxation curve displays modulus vs. time. The zero of the time scale is taken halfwaythrough the rise time. Zero time of course does not appear explicitly in the logarithmic scale.