## Viscoelastic Properties

Considering the complex hierarchical organization of tendons and ligaments, surrounding proteins, and ground substance, it is not surprising that they demonstrate nonlinear viscoelastic properties that are both time- and history-dependent. This essentially means that the elongation of the tissue is based not only on the amount of force but on the time and history of force application. A Medline search reveals published literature as early as 1969 on the viscoelastic properties of collagenous tissues (71,72). Time-dependent behavior of connective tissues has classically been examined using creep (21,71,73-84), stress relaxation, or hysteresis experimental methods.

Creep is fundamentally simple to measure, where a constant load is applied to a tissue and the progressive time-dependent elongation is measured. The elongation amount should be measured with reference to markers on the sample rather than the separation of the grips to ensure no slippage occurs. Stress relaxation is measured when the sample is held at a fixed displacement and the corresponding load is measured vs time. Lynch and coworkers (85) recently reported the stress relaxation behavior of ovine tendons relating to fiber orientation and strain rate. These authors examined samples aligned with the transverse to the tendon fiber direction to determine the anisotropic properties of toe-region modulus (E0), linear-region modulus (E), and Poisson's ratio (v). Interestingly, they reported the fiber-aligned linear-region modulus (E1) to be strain-rate-dependent. Poisson's ratio values were not found to be rate-dependent in either the fiber-aligned or transverse direction. They concluded that the lack of strain-rate dependence of transverse properties demonstrates that slow constant strain-rate tests represent elastic properties in the transverse direction. In contrast, when tested along the long axis of the samples, the strain-rate dependence of the modulus in the linear region suggests that incremental stress-relaxation tests are required to determine the tendon's equilibrium elastic properties (85). This study indicates continued advances in our understanding of tendons and ligaments.

Hysteresis is another viscoelastic property that has been considered in the literature. A single cyclic loading and unloading of a tendon or ligament produces two separate curves corresponding to whether a load is applied or withdrawn. These curves form a hysteresis loop on the load-elongation graph. With repeated/cyclical loading of a sample, its properties alter, and the amount of elongation increases with each given load. Increasing cycles cause the loops to become more constant. When a tendon or ligament is exposed to repeated loading, the area of hysteresis for the first few loading cycles is comparatively greater than with later load cycles (Fig. 1). Hence, with repeated loading, the load-elongation curve becomes more constant and is referred to as preconditioning. This effect is important when testing tendons and ligaments and should be calculated into experimental design to prevent inaccurate results.

Quasilinear Viscoelasticity

The quasilinear viscoelasticity (QLV) theory developed by Fung (86) has been used successfully to describe these time- and history-dependent viscoelastic properties for many soft tissues. This theory has also been used for ligaments and tendons (87). Recently, the QLV theory has been further refined to account for a constant strain rate (rather than an infinite strain rate or true step load, which is physically impossible to achieve experimentally) and the subsequent stress relaxation. The QLV theory assumes that the stress-relaxation function of the tissue can be expressed in the form:

where ae(e) is the elastic response, i.e., the maximum stress in response to an instantaneous step input of strain e. G(t) is the reduced relaxation function that represents the time-dependent stress response of the tissue, which is normalized by the stress at the time of step input of strain. If the strain history is considered as a series of infinitesimal step strains (Ae), then the overall stress-relaxation function will be the sum of all individual relaxations. Thus, for a general strain history, the stress at time t, a(t), is given by the strain history and convolution integral over time of G(t):

The lower limit of integration is taken as negative infinity to imply inclusion of all past strain history. In the experimental setting, one assumes that the history begins at t = 0. Once G(t), ae(e), and the strain history are known, the time- and history-dependent stress can be completely described by Eq. 2. For soft tissues whose stress-strain relationship and hysteresis are not over sensitive to strain rates, Fung has proposed the following expression for G(t):

where

is the exponential integral, and C, x1; and t2 are material coefficients.

An exponential approximation has been chosen to describe the elastic stress-strain relationship during a constant strain rate test:

where A and B are material coefficients. It is impossible to experimentally apply an instantaneous strain to the test material and therefore impossible to directly measure ae(e). To better approximate actual experimental conditions, it is necessary to replace the instantaneous step load with a ramp load of constant finite strain rate y to a strain level £ at time t0. The corresponding stress rise during 0 < t < t0 can then be written by combining Equations 3 and 4 as:

eBYT dT

Similarly, the subsequent stress relaxation a(t) from t0 to t can be described as:

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