Abstract
Compelling evidence that many biological soft tissues display both strain- and time-dependent behavior has led to the development of fully non-linear viscoelastic modeling techniques to represent the tissue’s mechanical response under dynamic conditions. Since the current stress state of a viscoelastic material is dependent on all previous loading events, numerical analyses are complicated by the requirement of computing and storing the stress at each step throughout the load history. This requirement quickly becomes computationally expensive, and in some cases intractable, for finite element models. Therefore, we have developed a strain-dependent numerical integration approach for capturing non-linear viscoelasticity that enables calculation of the current stress from a strain-dependent history state variable stored from the preceding time step only, which improves both fitting efficiency and computational tractability. This methodology was validated based on its ability to recover non-linear viscoelastic coefficients from simulated stress-relaxation (six strain levels) and dynamic cyclic (three frequencies) experimental stress-strain data. The model successfully fit each data set with average errors in recovered coefficients of 0.3% for stress-relaxation fits and 0.1% for cyclic. The results support the use of the presented methodology to develop linear or non-linear viscoelastic models from stress-relaxation or cyclic experimental data of biological soft tissues.
Highlights
Viscoelastic theory describes the time-dependent relationship between stress and strain and is commonly used to describe the mechanical behavior of biological tissues
To increase modeling flexibility and address each limitation above, the present study develops a novel numerical integration technique for fully non-linear viscoelastic modeling
The novel direct fit method presented provides a number of advantages over other non-linear techniques, including that of the comprehensive viscoelastic characterization (CVC) method [13,18,19]
Summary
Viscoelastic theory describes the time-dependent relationship between stress and strain and is commonly used to describe the mechanical behavior of biological tissues. The current stress state is dependent upon all previous loading events. This historydependent behavior complicates numerical analyses of viscoelastic materials because the stress at each step throughout the entire loading history must be computed and stored in order to obtain the current stress. For three-dimensional finite element models, computing and storing the stress tensor at each integration point and time step quickly becomes computationally. A numerical integration method for non-linear viscoelastic modeling intractable. To simplify numerical analyses for linear and quasi-linear viscoelastic materials, a discrete series of exponentials (such as a Prony series) is often used to approximate the continuous time-dependent relaxation spectrum
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