The Effect of Overshooting the Target Strain on Estimating Viscoelastic Properties From Stress Relaxation Experiments

2004 ◽  
Vol 126 (6) ◽  
pp. 844-848 ◽  
Author(s):  
Jonathan A. Gimbel ◽  
Joseph J. Sarver ◽  
Louis J. Soslowsky
Keyword(s):  
Stress Relaxation ◽  
Single Step ◽  
Strain History ◽  
Estimation Of Parameters ◽  
Ramp Rate ◽  
Relaxation Experiment ◽  
Curve Fit ◽  
Target Strain ◽  
Linear Viscoelastic ◽  
Relaxation Experiments

Background: Tendon’s mechanical behaviors have frequently been quantified using the quasi-linear viscoelastic (QLV) model. The QLV parameters are typically estimated by fitting the model to a single-step stress relaxation experiment. Unfortunately, overshoot of the target strain occurs to some degree in most experiments. This has never been formally investigated even though failing to measure, minimize, or compensate for overshoot may cause large errors in the estimation of parameters. Therefore, the objective of this study was to investigate the effect of overshoot on the estimation of QLV parameters. Method of approach: A simulated experiment was first performed to quantify the effect of different amounts of overshoot on the estimated QLV parameters. Experimental data from tendon was then used to determine if the errors associated with overshoot could be reduced when a direct fit is used (i.e., the actual strain history was used in the curve fit). Results: We found that both the elastic and viscous QLV parameters were incorrectly estimated if overshoot was not properly accounted for in the fit. Furthermore, the errors associated with overshoot were partially reduced when overshoot was accounted for using a direct fit. Conclusions: A slow ramp rate is recommended to limit the amount of overshoot and a direct fit is recommended to limit the errors associated with overshoot, although other approaches such as adjusting the control system to limit overshoot could also be utilized.

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.

Correlation of Large Longitudinal Deformations with Different Strain Histories

1967 ◽  
Vol 40 (2) ◽  
pp. 506-516 ◽  
Author(s):  
L. J. Zapas ◽  
T. Craft
Keyword(s):  
Stress Relaxation ◽  
Single Step ◽  
Strain History ◽  
Simple Extension ◽  
Stress Strain ◽  
Fluid Equation ◽  
Elastic Fluid ◽  
Strain Behavior ◽  
Elastomeric Materials ◽  
Single Integral

Abstract In 1963 Bernstein, Kearsley, and Zapas1 presented a theory of an elastic fluid which gave the correct stress-relaxation response for a large variety of elastomeric materials, including vulcanized rubbers. A principle attractiveness of this theory is its relative simplicity; with a single integral in time, it describes the stress-strain behavior for all types of deformation histories. In the case of simple extension, it predicts the behavior in any uniaxial strain history from the results of single step stress-relaxation experiments which cover the same range of extension and time. We designed a series of experiments to check the validity of this theory and found, as is shown in this paper, excellent agreement with experiment in all cases. We are aware that experiments cannot prove a theory. From our results, however, we feel strongly that a single integral expression with a nonlinear integrand such as the BKZ elastic fluid equation is sufficient to describe the stress-strain behavior of elastomeric materials.

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.

A Quasi-Linear, Viscoelastic, Structural Model of the Plantar Soft Tissue With Frequency-Sensitive Damping Properties

2004 ◽  
Vol 126 (6) ◽  
pp. 831-837 ◽  
Author(s):  
William R. Ledoux ◽  
David F. Meaney ◽  
Howard J. Hillstrom
Keyword(s):  
Soft Tissue ◽  
Stress Relaxation ◽  
Structural Properties ◽  
Structural Model ◽  
Elastic Behavior ◽  
Damping Properties ◽  
Significant Difference ◽  
Linear Viscoelastic ◽  
Relaxation Experiments ◽  
Plantar Soft Tissue

Little is known about the structural properties of plantar soft-tissue areas other than the heel; nor is it known whether the structural properties vary depending on location. Furthermore, although the quasi-linear viscoelastic (QLV) theory has been used to model many soft-tissue types, it has not been employed to model the plantar soft tissue. The structural properties of the plantar soft tissue were quantified via stress relaxation experiments at seven regions (subcalcaneal, five submetatarsal, and subhallucal) across eight cadaveric feet. The cadaveric feet were 36.9±17.4 (mean±S.D.) years of age, all free from vascular diseases and orthopedics disorders. All tests were performed at a constant environmental temperature of 35°C. Stress relaxation experiments were performed; different loads were employed for different areas based on normative gait data. A modification of the relaxation spectrum employed within the QLV theory allowed for the inclusion of frequency-sensitive relaxation properties in addition to nonlinear elastic behavior. The tissue demonstrated frequency-dependent damping properties that made the QLV theory ill suited to model the relaxation. There was a significant difference between the elastic structural properties (A) of the subcalcaneal tissue and all other areas p=0.004, and a trend p=0.067 for the fifth submetatarsal to have less viscous damping c1 than the subhallucal, or first, second, or third submetatarsal areas. Thus, the data demonstrate that the structural properties of the foot can vary across regions, but careful consideration must be given to the applied loads and the manner in which the loads were applied.

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.

Methods for Quasi-Linear Viscoelastic Modeling of Soft Tissue: Application to Incremental Stress-Relaxation Experiments

2003 ◽  
Vol 125 (5) ◽  
pp. 754-758 ◽  
Author(s):  
Joseph J. Sarver ◽  
Paul S. Robinson ◽  
Dawn M. Elliott
Keyword(s):  
Soft Tissue ◽  
Stress Relaxation ◽  
Normalization Method ◽  
Time Constants ◽  
Future Studies ◽  
Linear Viscoelastic ◽  
Relaxation Experiments ◽  
Different Strains ◽  
Viscoelastic Modeling ◽  
Strain Dependent

The quasi-linear viscoelastic (QLV) model was applied to incremental stress-relaxation tests and an expression for the stress was derived for each step. This expression was used to compare two methods for normalizing stress data prior to estimating QLV parameters. The first and commonly used normalization method was shown to be strain-dependent. Thus, a second normalization method was proposed and shown to be strain-independent and more sensitive to QLV time constants. These analytical results agreed with representative tendon data. Therefore, this method for normalizing stress data was proposed for future studies of incremental stress-relaxation, or whenever comparing stress-relaxation at different strains.

Stress Relaxation in Combined Torsion-Tension

1970 ◽  
Vol 37 (1) ◽  
pp. 53-60 ◽  
Author(s):  
W. Goldberg ◽  
G. Lianis
Keyword(s):  
Stress Relaxation ◽  
Constitutive Equation ◽  
Constitutive Relation ◽  
Axial Force ◽  
Single Step ◽  
Tension Stress ◽  
Simple Material ◽  
Theoretical Predictions ◽  
Fluid Theory ◽  
Relaxation Experiments

In this paper we briefly examine the form of the isothermal constitutive relation of an isotropic simple material under stress-relaxation conditions. We then compare the predictions of the Signorini form of the Bernstein, Kearsley, Zapas elastic fluid theory and the Lianis constitutive equation for single-step stress-relaxation histories and show that they are identical. Using these theories, we develop theoretical expressions for the torque and axial force in combined torsion-tension stress relaxation. Experiments were performed on samples of an SBR polymer, and the observed response is compared with theoretical predictions.

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