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Biomechanics: viscoelasticity, creep, relaxation in ligament and tendon
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[Rod Lakes]
Ligament: ligament viscoelasticity, ligament creep, ligament relaxation
Interrelation of creep and relaxation: a modeling approach for ligaments, R. Lakes and R. Vanderby, J. Biomechanical Engineering, 121, 612-615, Dec. (1999). (Get pdf)
    Experimental tissue rheology data (Thornton, et al., 1997) show that relaxation proceeds more rapidly (a greater slope on a log log scale) than creep in ligament, a fact not explained by linear viscoelasticity. ligament fiber An interrelation between creep and relaxation is therefore developed for ligament viscoelasticity based on a single-integral nonlinear superposition model. This interrelation differs from the convolution relation obtained by Laplace transforms for linear materials. We demonstrate via continuum concepts of nonlinear viscoelasticity that such a difference in rate between creep and relaxation phenomenologically occurs when the nonlinearity is of a strain-stiffening type, i.e. the stress-strain curve is concave up as observed in ligament. We also show that it is inconsistent to assume a Fung-type constitutive law (Fung, 1972) for both creep and relaxation in ligament viscoelasticity. Using published data of Thornton, et al., the nonlinear interrelation developed herein predicts creep behavior from relaxation data well (R greater than 0.998). Although data are limited and the causal mechanisms associated with viscoelastic tissue behavior are complex, continuum concepts demonstrated here appear capable of interrelating creep and relaxation with fidelity. The image shows ligament structure, after Vanderby.
lig plot     The research is primarily directed toward study of nonlinearly viscoelastic behavior of soft connective tissues, particularly the constitutive behavior representing ligament viscoelasticity. Because of their simplicity, ligaments are chosen as an experimental model. The research is organized as follows. (i) An experimental study is in progress, that defines viscoelastic behavior throughout the range of reversible ligament deformations. This study uses creep and relaxation plus recovery protocols to allow discrimination among various nonlinearities. (ii) A robust viscoelastic constitutive model is developed from experimental data. This model appropriately accounts for nonlinearities and interrelates creep and relaxation. (iii) The reversible deformation limits for levels of applied stress and strain (as a function of load-time) is experimentally identified from the data in (i), thus defining the onset of sub-failure tissue damage for a single overload. A sub-failure criterion for the onset of damage is then formulated. (iv) Creep and relaxation testing are performed on additional specimens to quantify the compromise in mechanical behavior after higher loadings and deformations produce greater levels of tissue damage (from a single overload). (v) The microtrauma and damage associated with irreversible deformations is morphologically characterized using scanning electron microscopy.
    Nonlinear viscoelasticity is phenomenologically observed in all soft connective tissues. It is axiomatic then that any biomechanical study of these tissues must explicitly or implicitly take this behavior into account. Optimal soft tissue repair or replacement must viscoelastically mimic original tissue. Accurate and robust biomechanical models must reproduce this characteristic behavior. It is therefore sine qua non that nonlinear viscoelasticity be well defined in soft connective tissues. Commonly used viscoelastic descriptions have come into question. They appear inadequate to describe the different rates observed in creep and relaxation of ligaments. Since other soft connective tissues share compositional similarities (collagen fiber reinforced materials with relatively high levels of hydration produced by proteoglycans), it is likely that other soft connective tissues have similar constitutive issues that must be addressed in their viscoelastic descriptions. In addition, current descriptions do not allow comparison of studies performed under different experimental modalities and thereby render many previous studies quantitatively inaccessible. This biomechanical deficiency cannot be rectified with existing data, since with most tissues there never has been a study sufficiently complete to allow discrimination among nonlinearities. An experimental program is being conducted (with a ligament model) involving creep and relaxation experiments at different load and strain levels, with the goal of discriminating among nonlinear representations of ligament viscoelasticity and establishing a robust model.
    A particular bonus in performing the above systematic study of nonlinear viscoelasticity throughout the range of reversible deformations is that it also identifies the boundaries of that range, i.e. the onset of tissue damage. This is another fundamental aspect of soft tissue biomechanics that has never been adequately defined. Despite a high clinical incidence, a biomechanical criterion for subfailure damage is largely unknown in soft connective tissues, and sub-failure damage is rarely considered as part of biomechanical studies.
    Ligament viscoelasticity experiment
Ligament tissue viscoelasticity is nonlinear. We study the nonlinear behavior of ligament and determine robust integral formulations to describe the nonlinear behavior.
Provenzano, P., Lakes, R. S., Keenan, T, Vanderby, R. Jr., "Non-linear ligament viscoelasticity", Annals of Biomedical Engineering, 29, 908-914, Nov. (2001)
Abstract
    Ligaments display time dependent behavior, characteristic of a viscoelastic solid, and are non-linear in their stress-strain response. Recent experiments (Thornton et al., 1997) reveal that stress relaxation proceeds faster than more rapidly than creep in medial collateral ligaments, a fact not explained by linear viscoelastic theory but consistent with non-linear theory by Lakes and Vanderby (1999). This study tests the following hypothesis. Non-linear viscoelasticity of ligament requires a description more general than the separable quasi-linear viscoelasticity (QLV) formulation commonly used. The experimental test for this hypothesis involves performing both ligament creep and ligament relaxation studies at various loading levels below the damage threshold. Freshly harvested, rat medial collateral ligaments were used as a model. Results consistently show a non-linear behavior in which the rate of creep is dependent upon stress level and the rate of relaxation is dependent upon strain level. Furthermore, ligament relaxation proceeds faster than ligament creep, consistent with the experimental observations of Thornton et al., (1997). The above results are not consistent with a separable QLV theory. Inclusion of these nonlinearities would require a more general formulation. Get pdf.

    Provenzano, P., Heisey, D., Hayashi, K., Lakes, R. S., and Vanderby, R. Jr., "Subfailure damage in ligament: a structural and cellular evaluation", J. Applied Physiology, 92: 362-371, (2002).
Get pdf. Ligament damage occurs with a different threshold in mechanical tests versus in tests of cellular viability.

    Provenzano, P., Lakes, R. S., Corr, D. T., and Vanderby, R. Jr, "Application of nonlinear viscoelastic models to describe ligament behavior", Biomechanics and Modeling in Mechanobiology, 1: 45-57, (2002).
Get pdf.

    Oza, A., Lakes, R. S. and Vanderby, R., "Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal" Rheologica Acta, 42, 557-568 (2003).
Get pdf.

    Manley E. Jr, Provenzano, P. P., Heisey D., Lakes R., Vanderby R. Jr., "Required test duration for group comparisons in ligament viscoelasticity: a statistical approach", Biorheology, 40(4): 441-50 Apr. (2003). pdf

    Hingorani, R., Provenzano, P., Lakes, R. S., Escarcega, A., and Vanderby, R., Jr. "Experimental evaluation of nonlinear viscoelastic behavior in rabbit ligament", 40th annual meeting, Society of Engineering Science, Ann Arbor, MI, 15 Oct. (2003). Get pdf or image.

    Hingorani, R., Provenzano, P., P., Lakes, R. S., Escarcega, A., Vanderby, R. Jr., "Nonlinear viscoelasticity in rabbit medial collateral ligament", Annals of Biomedical Engineering, 32, 306-312, Feb. (2004). Get pdf.
Results show that within the physiologically relevant region of ligament behavior, the rate of ligament relaxation is strain dependent in the rabbit MCL, with the rate of relaxation decreasing with increasing tissue strain. The rate of ligament creep is stress dependent in the rabbit MCL, with the rate of creep decreasing with increasing stress. Nonlinearities in rates of creep and relaxation cannot be robustly modeled by the QLV formulation (quasi-linear viscoelasticity).

    Jensen, K. T. Dwyer, K. W., Lakes, R. S., and Vanderby, R., Jr., "The rate of viscoelastic recovery is faster than the rate of creep", 50th ORS, paper 0046, March (2004). Get pdf. Ligament creep and recovery do not occur at the same rate. This phenomenon cannot be described by either linear viscoelasticity or quasilinear viscoelasticity (QLV).

    Oza, A., Vanderby, R. and Lakes, R. S., "Generalized solution for predicting relaxation from creep: application to ligament" International Journal of Mechanical Sciences, 48, (6) 662-673 June (2006). Get pdf.
Creep and relaxation are two viscoelastic phenomena that are easily interrelated for a linearly viscoelastic material, but interrelationships are complex for nonlinearly viscoelastic materials. We use a single-integral nonlinear superposition principle to relate creep and relaxation, where the kernel is assumed to be a nonseparable product of strain and time. Herein, we develop time dependence as general power laws with up to four terms for creep compliance and relaxation modulus. Higher-order formulations give better results for ligament in terms of curve fitting and prediction of relaxation from creep. This is illustrated by a comparison between a two- and a three-term formulation on the experimental data of rabbit medial collateral ligaments. Also, an interrelation between several aspects of creep and relaxation is presented for arbitrarily high order, and the nature of high-order interrelation is discussed. The generality of the method makes it suitable to phenomenologically model many complex materials, to predict complex behaviors and to therefore reduce the amount of testing for robust material characterization.

    More recently, the recovery following creep or relaxation was observed to be at a different rate than the creep or relaxation which preceded it. This cannot be predicted or modeled by quasi-linear viscoelasticity (QLV).

    Duenwald, S., Vanderby, R. Lakes, R. S., "Nonlinear Viscoelastic Relaxation and Recovery of Porcine Flexor Tendon," presented at the 45th meeting, Society of Engineering Science, University of Illinois, October (2008). Get pdf

    Duenwald, S. E., Vanderby, Jr., R., Lakes, R. S., "Viscoelastic Relaxation and Recovery of Tendon", Annals of Biomedical Engineering, 37, No. 6, 1131-1140, June (2009). Get pdf. Tendons exhibit complex viscoelastic behaviors during relaxation and recovery. Recovery is critical to predicting behavior in subsequent loading, yet is not well studied. Our goal is to explore time-dependent recovery of these tendons after loading. As a prerequisite, their strain- dependent viscoelastic behaviors during relaxation were also characterized. The porcine digital flexor tendon was used as a model of tendon behavior. Strain-dependent relaxation was observed in tests at 1, 2, 3, 4, 5, and 6% strain. Recovery behavior of the tendon was examined by performing relaxation tests at 6%, then dropping to a low but nonzero strain level. Results show that the rate of relaxation in tendon is indeed a function of strain. Unlike previously reported tests on the medial collateral ligament (MCL), the relaxation rate of tendons increased with increased levels of strain. This strain-dependent relaxation contrasts with quasilinear viscoelasticity (QLV), which predicts equal time dependence across various strains. Also, the tendons did not recover to predicted levels by nonlinear superposition models or QLV, though they did recover partially. This recovery behavior and behavior during subsequent loadings will then become problematic for both quasilinear and nonlinear models to correctly predict.

    Duenwald, S. E., Vanderby, Jr., R., Lakes, R. S., "Constitutive equations for ligament and other soft tissue: evaluation by experiment", Acta Mechanica, 205: 23-33 (2009). get pdf

In cornea from the eye, creep becomes more pronounced as stress increases. The behavior therefore does not follow quasi-linear viscoelasticity (QLV). Boyce, B. L., Jones, R. E., Nguyen, T. D., Grazier, J. M., Stress-controlled viscoelastic tensile response of bovine cornea, J. Biomechanics, 40, 2637-2376, 2007.

Tutorial on interpreting experiments for nonlinear tissue.

We thank the National Science Foundation for support.

[Bone biomechanics] [Wood, bamboo biomechanics] [Osteons] [Structural hierarchy] [Biomechanics class] [Biomaterials book 2nd ed] [Biomaterials book 3rd ed] [Viscoelasticity book, 2009] [Biomaterials] [Salient articles]
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