An Improved Method to Analyze the Stress Relaxation of Ligaments Following a Finite Ramp Time Based on the Quasi-Linear Viscoelastic Theory

2004 ◽  
Vol 126 (1) ◽  
pp. 92-97 ◽  
Author(s):  
Steven D. Abramowitch ◽  
Savio L.-Y. Woo
Keyword(s):  
Stress Relaxation ◽  
Soft Tissues ◽  
Viscoelastic Behavior ◽  
Cyclic Stress ◽  
Elastic Response ◽  
Curve Fit ◽  
Order Of Magnitude ◽  
Linear Viscoelastic ◽  
Linear Viscoelastic Theory ◽  
Experimental Values

The quasi-linear viscoelastic (QLV) theory proposed by Fung (1972) has been frequently used to model the nonlinear time- and history-dependent viscoelastic behavior of many soft tissues. It is common to use five constants to describe the instantaneous elastic response (constants A and B) and reduced relaxation function (constants C, τ1, and τ2) on experiments with finite ramp times followed by stress relaxation to equilibrium. However, a limitation is that the theory is based on a step change in strain which is not possible to perform experimentally. Accounting for this limitation may result in regression algorithms that converge poorly and yield nonunique solutions with highly variable constants, especially for long ramp times (Kwan et al. 1993). The goal of the present study was to introduce an improved approach to obtain the constants for QLV theory that converges to a unique solution with minimal variability. Six goat femur-medial collateral ligament-tibia complexes were subjected to a uniaxial tension test (ramp time of 18.4 s) followed by one hour of stress relaxation. The convoluted QLV constitutive equation was simultaneously curve-fit to the ramping and relaxation portions of the data r2>0.99. Confidence intervals of the constants were generated from a bootstrapping analysis and revealed that constants were distributed within 1% of their median values. For validation, the determined constants were used to predict peak stresses from a separate cyclic stress relaxation test with averaged errors across all specimens measuring less than 6.3±6.0% of the experimental values. For comparison, an analysis that assumed an instantaneous ramp time was also performed and the constants obtained for the two approaches were compared. Significant differences were observed for constants B, C, τ1, and τ2, with τ1 differing by an order of magnitude. By taking into account the ramping phase of the experiment, the approach allows for viscoelastic properties to be determined independent of the strain rate applied. Thus, the results obtained from different laboratories and from different tissues may be compared.

The Time and History-Dependent Viscoelastic Properties of the Canine Medial Collateral Ligament

1981 ◽  
Vol 103 (4) ◽  
pp. 293-298 ◽  
Author(s):  
S. L.-Y. Woo ◽  
M. A. Gomez ◽  
W. H. Akeson
Keyword(s):  
Medial Collateral Ligament ◽  
Viscoelastic Properties ◽  
Cyclic Stress ◽  
Elastic Response ◽  
Collateral Ligament ◽  
Stress Strain ◽  
Linear Viscoelastic ◽  
Viscoelastic Theory ◽  
Linear Viscoelastic Theory

The viscoelastic properties of the canine medial collateral ligament (MCL) were investigated. Stress-strain relationships at different strain rates, long-term stress relaxation and cyclic stress-strain curves of the MCL were obtained experimentally using a bone-MCL-bone preparation. The experimental data were used in conjunction with the quasi-linear viscoelastic theory as proposed by Fung [15] to characterize the reduced relaxation function, G(t) and elastic response σe (ε) of this tissue. It was found that the quasi-linear viscoelastic theory can adequately describe the time and history-dependent rheological properties of the canine medial collateral ligament.

An Experimental and Analytical Evaluation of the Nonlinear Viscoelastic Properties of the Healing Medial Collateral Ligament in a Goat Model

2003 ◽  
Author(s):  
S. D. Abramowitch ◽  
T. D. Clineff ◽  
R. E. Debski ◽  
S. L.-Y. Woo
Keyword(s):  
Stress Relaxation ◽  
Medial Collateral Ligament ◽  
Viscoelastic Properties ◽  
Soft Tissues ◽  
Viscoelastic Behavior ◽  
Collateral Ligament ◽  
Relaxation Experiment ◽  
Nonlinear Viscoelastic ◽  
Linear Viscoelastic Behavior ◽  
Linear Viscoelastic

The medial collateral ligament (MCL) is one of the most frequently injured ligaments in the knee. Although it can heal spontaneously after rupture, laboratory studies have shown that the mechanical properties of the healing MCL remain inferior to normal for up to two years after injury (1). Additionally, the healing MCL has been shown to display increased amounts of stress relaxation and creep (2). In order to more completely describe the viscoelastic properties of healing ligaments, we propose to use the Quasi-Linear Viscoelastic (QLV) theory formulated by Fung (1972). This theory has been used to successfully describe the viscoelastic properties of many soft-tissues (3). Recently, our research center has developed an improved approach to determine the constants describing the QLV theory based on data collected from a stress relaxation experiment that utilizes a slow strain rate during loading. This approach allows for experimental errors that commonly result from fast strain rates to be avoided (ex. overshoot) (4). Therefore, the objective of this study were to use this new approach to determine the constants describing the quasi-linear viscoelastic behavior of the healing goat MCL at 12 weeks after injury.

scholarly journals THE EFFECT OF PREGNANCY AND POSTPARTUM RECOVERY ON THE VISCOELASTIC BEHAVIOR OF THE RAT CERVIX

2012 ◽  
Vol 12 (01) ◽  
pp. 1250009 ◽  
Author(s):  
WILLIAM R. BARONE ◽  
ANDREW J. FEOLA ◽  
PAMELA A. MOALLI ◽  
STEVEN D. ABRAMOWITCH
Keyword(s):  
Viscoelastic Behavior ◽  
Collagen Content ◽  
Elastic Response ◽  
Pregnancy And Postpartum ◽  
Normal Functional ◽  
Linear Viscoelastic ◽  
Viscoelastic Theory ◽  
Matrix Organization ◽  
Linear Viscoelastic Theory ◽  
Postpartum Recovery

The objective of this study was to elucidate the normal functional adaptations of the cervix in pregnancy. Utilizing a Long-Evans rodent model, the cervix was divided into distal and proximal portions for virgin, mid-pregnant, and four weeks postpartum animals. The quasi-linear viscoelastic theory describes the elastic and viscous behavior of the cervix. A hydroxyproline assay was used to measure collagen content. The nonlinearity of the elastic response significantly increased throughout the entire cervix during pregnancy when compared to virgin samples (p < 0.05) and was similar to virgin samples postpartum. All viscous behavior, except for the short-term relaxation of the proximal cervix, significantly differed for pregnant specimens (p < 0.05) and remained similar to pregnant samples postpartum. Collagen content was found to increase by mid-pregnancy only in the proximal cervix when compared to virgin. Distal and proximal portions, however, were found to differ in collagen content at all time points (p < 0.05). This study finds that the cervix becomes elastically stiffer with increasing strain and exhibits increased viscous behavior during pregnancy, with incomplete recovery postpartum. These alterations allow for quick dissipation of loads, and are likely related to altered matrix organization and porosity reported by others.

The Effects of Refreezing on the Tensile Properties of the Medial Collateral Ligament-Bone Complex: A Rabbit Model

2003 ◽  
Author(s):  
Daniel K. Moon ◽  
Mary T. Gabriel ◽  
Steven D. Abramowitch ◽  
Yoshiyuki Takakura ◽  
Savio L.-Y. Woo
Keyword(s):  
Stress Relaxation ◽  
Tensile Properties ◽  
Medial Collateral Ligament ◽  
Soft Tissues ◽  
Rabbit Model ◽  
Frozen Storage ◽  
Cyclic Stress ◽  
Specimen Preparation ◽  
Freezing And Thawing ◽  
Collateral Ligament

Biomechanical tests of soft tissues, especially those from human cadavers, are generally done after a period of postmortem storage by freezing. In some instances, specimen preparation and testing can be complex and can take place over several days, thus necessitating an additional period of frozen storage before biomechanical evaluation is completed. Studies have been done in the past, which investigated the effects of postmortem freezing on the tensile properties of ligament-bone complexes (Viidik and Lewin 1966; Noyes and Grood 1976; Dorlot 1980; Barad 1982; Nikolaou 1986; Woo, Orlando et al. 1986). It has been shown in our laboratory that careful postmortem freezing for up to three months did not significantly change the mechanical properties of the ligament midsubstance and the cyclic stress relaxation behavior and the structural properties of the ligament-bone complex, but the area of hysteresis was significantly reduced in the stored specimens for the first few cycles of cyclic stress relaxation (Woo, Orlando et al. 1986). Hence, it is important to ensure that an additional freezing and thawing cycle will not further change the tensile properties of these tissues. Therefore, the objective of this study was to evaluate the effects of freezing and thawing twice at 20°C on the structural and viscoelastic properties of femur-medial collateral ligament-tibia complex (FMTC) in a rabbit model.

Comparison of a Novel Nonlinear Viscoelastic Finite Ramp Time Correction Method to a Heaviside Step Assumption

2011 ◽  
Author(s):  
Kevin L. Troyer ◽  
Christian M. Puttlitz
Keyword(s):  
Stress Relaxation ◽  
Soft Tissues ◽  
Viscoelastic Behavior ◽  
Relaxation Modulus ◽  
Correction Method ◽  
Nonlinear Viscoelastic ◽  
Instantaneous Strain ◽  
Relaxation Experiments ◽  
Time Correction

Connective soft tissues exhibit time-dependent, or viscoelastic, behavior. In order to characterize this behavior, stress relaxation experiments can be performed to determine the tissue’s relaxation modulus. Theoretically, the relaxation modulus describes the stress relaxation behavior of the tissue in response to an instantaneous (step) application of strain. However, a step increase in strain is experimentally impossible and a pure ramp load is intractable due to the inertial limitations of the testing device. Even small deviations from an instantaneous strain application may cause significant errors in the determination of the tissue’s relaxation modulus.

A Nonlinear Anisotropic Viscoelastic Model for the Tensile Behavior of the Corneal Stroma

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
T. D. Nguyen ◽  
R. E. Jones ◽  
B. L. Boyce
Keyword(s):  
Soft Tissues ◽  
Viscoelastic Model ◽  
Corneal Stroma ◽  
Viscoelastic Behavior ◽  
Cyclic Stress ◽  
Uniaxial Tensile ◽  
Nonlinear Creep ◽  
Stress Strain ◽  
Highly Nonlinear ◽  
Tensile Strip

Tensile strip experiments of bovine corneas have shown that the tissue exhibits a nonlinear rate-dependent stress-strain response and a highly nonlinear creep response that depends on the applied hold stress. In this paper, we present a constitutive model for the finite deformation, anisotropic, nonlinear viscoelastic behavior of the corneal stroma. The model formulates the elastic and viscous response of the stroma as the average of the elastic and viscous response of the individual lamellae weighted by a probability density function of the preferred in-plane lamellar orientations. The result is a microstructure-based model that incorporates the viscoelastic properties of the matrix and lamellae and the lamellar architecture in the response of the stroma. In addition, the model includes a fully nonlinear description of the viscoelastic response of the lamellar(fiber) level. This is in contrast to previous microstructure-based models of fibrous soft tissues, which relied on quasilinear viscoelastic formulations of the fiber viscoelasticity. Simulations of recent tensile strip experiments show that the model is able to predict, well within the bounds of experimental error and natural variations, the cyclic stress-strain behavior and nonlinear creep behavior observed in uniaxial tensile experiments of excised strips of bovine cornea.

scholarly journals Stress relaxation of porcine tendon under simulated biological environment: experiment and modeling

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Sylwia D. Łagan ◽  
Aneta Liber-Kneć
Keyword(s):  
Stress Relaxation ◽  
Viscoelastic Model ◽  
Model Fitting ◽  
Viscoelastic Behavior ◽  
Strain Range ◽  
Viscoelastic Response ◽  
Physiological Strain ◽  
Linear Viscoelastic ◽  
Biological Environment ◽  
Environment Experiment

Purpose: The aim of the study was to investigate the viscoelastic response in the low and high physiological strain with the use of experimental and modeling approach. Methods: Viscoelastic response in the low, transition and high physiologic strain (3, 6 and 9%) with consideration of simulated biological environment (0.9% saline solution, 37 °C) was measured in relaxation tests. Preconditioning of tendons was considered in the testing protocol and the applied range of load was obtained from tensile testing. The quasi-linear viscoelasticity theory was used to fit experimental data to obtain constants (moduli and times of relaxation), which can be used for description of the viscoelastic behavior of tendons. The exponential non-linear elastic representation of the stress response in ramp strain was also estimated. Results: Differences between stress relaxation process can be seen between tendons stretched to the physiological strain range (3%) and exceeding this range (6 and 9%). The strains of 6% and 9% showed a similar stress relaxation trend displaying relatively rapid relaxation for the first 70 seconds, whereas the lowest strain of 3% displayed relatively slow relaxation. Conclusions: Results of the model fitting showed that the quasi-linear viscoelastic model gives the best fit in the range of low physiological strain level.

Quasi-Linear Viscoelastic Theory is Insufficient to Comprehensively Describe the Time-Dependent Behavior of Human Cervical Spine Ligaments

2010 ◽  
Author(s):  
Kevin L. Troyer ◽  
Christian M. Puttlitz
Keyword(s):  
Cervical Spine ◽  
Stress Relaxation ◽  
Strain Range ◽  
Relaxation Curve ◽  
Physiological Strain ◽  
Dependent Behavior ◽  
Linear Viscoelastic ◽  
Viscoelastic Theory ◽  
Relaxation Experiments ◽  
Linear Viscoelastic Theory

Stress relaxation experiments were conducted on cervical spine ligaments at multiple strain magnitudes to determine the validity and applicability of the quasi-linear viscoelastic (QLV) theory to model their dynamic behavior. The results indicate that the shape of the stress relaxation curve is dependent upon the magnitude of the applied strain. Thus, a more general, nonlinear formulation is required to model these ligaments within the physiological strain range.

Characterization of Viscoelastic Behavior of Stereoregular Poly(Isoprenes) by Relaxation Experiments

1974 ◽  
Vol 47 (1) ◽  
pp. 1-18
Author(s):  
L. Szilagyi ◽  
T. Riccò ◽  
F. Danusso
Keyword(s):  
Viscoelastic Behavior ◽  
Supermolecular Structure ◽  
Relaxation Modulus ◽  
Mechanical Relaxation ◽  
Recent Theory ◽  
Mean Values ◽  
Linear Viscoelastic ◽  
Viscoelastic Theory ◽  
Linear Viscoelastic Theory ◽  
Relationship Of

Abstract The mechanical relaxation of twelve samples of unvulcanized cis poly-(isoprene)s, including both natural and synthetic polymers, was studied over a range of temperatures. Master curves of relaxation modulus obtained from these data were used to derive relaxation spectra according to linear viscoelastic theory. A recent theory was used to calculate mean values of quantities related to the supermolecular structure which occurs spontaneously in these materials and is responsible for their viscoelastic properties. This structure is schematized in a model consisting of a system of macromolecules which interact with each other by elastic forces and frictions corresponding to points of entanglement between chains. The analysis leads to the determination, for each sample, of the number of entanglements per molecule, the physical network density, the value of relaxation parameters, and the relationship of each of these quantities to molecular weight.

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