Viscoelasticity and Biological Tissues

Models for Viscoelastic BehaviorBIOE 3200 Biomechanics

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Represent viscoelastic materials with mechanical analogies (springs and dashpots) Mathematically model viscoelastic behavior using equations that describe stress and strain in spring and dashpot models

Learning objective:

Mechanical analogs allow us to develop viscoelastic constitutive models

Maxwell Model spring and dashpot in seriesModels for Viscoelastic Behavior

Represent viscoelastic materials with mechanical analogies (springs and dashpots) and mathematically refer to http://www.umich.edu/~bme332/ch7consteqviscoelasticity/bme332consteqviscoelasticity.htm- Analogies for viscoelastic materials: Spring (linear elastic material) and Dashpot (Newtonian fluid)Simplest form: Maxwell model (spring and dashpot in series); actually represents a fluid since it relaxes completely to zero stress and undergoes creep indefinitely. 3

Constitutive relationships for Maxwell model of viscoelasticity

The simple constitutive relationship for a spring relates the force (and stress by extension when force is divided by area) to the elongation or displacement (and strain by extension when displacement is normalized by length of the spring):F = E u(s) where Fs is the spring force, E is the elastic modulus of the spring, and u(s) represents the spring displacement.The simple constitutive relationship for a dashpot indicates that the force in the fluid depends on the rate the dashpot is displaced, or equivalently the velocity of the dashpot.Also, the constitutive parameter that relates force (stress) to displacement rate (strain rate) isviscosity, which we denote as. Thus, the constitutive equation for a fluid may be written as [write out F equation with u-dot term], where the dot over the u indicates differentiation with respect to time and the superscript d denotes "dashpot".From geometry, total displacement will be spring plus dashpot displacements:utot = us + ud (elongation strain by extension when displacement is normalized by length of the spring) tot = s + d Also, force on spring = force on dashpot: Fs (t) = Fd(t) (force stress by extension when force is divided by area) s = d Combine s = d with constitutive equations for spring and dashpot, solve for strain rate: s = E s d = d = /E + / (s and d superscripts dropped).

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Constitutive relationships for Maxwell model of viscoelasticity

Calculate response of Maxwell material under fixed strain and fixed stress (assume that strain and stress can be applied instantaneously to a fixed level not possible in reality, but assume for theoretical purposes). Response to an instantaneous strain that is then held constant over time, or stress relaxation behavior (first order differential equation, integrating the strain rate equation with strain rate = 0; constant strain for stress relaxation test)Dividing stress as a function of time by the initial strain, we obtain thestress relaxation functionG(t) for the Maxwell model (exponential decay with time; stress completely relaxes out over time, which is actually characteristic of a viscoelastic fluid rather than a viscoelastic solid)Response to an instantaneous unit step stress held over time, or creep behavior (first order diff eq, integrating strain rate equation with rate of change of stress = 0; constant stress for creep test)Similar to the stress relaxation, we can define acreep functionJ(t) by dividing the strain versus time response by the initial unit step stress. This gives a linear relationship between strain and time clearly not characteristic of a viscoelastic solid, since creep will reach an asymptotic level over time.Although the Maxwell model does not give realistic results for viscoelastic solids, it shows how each mechanical analog contributes to the behavior of the material.

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Voigt Model (aka Kelvin-Voigt Model) spring and dashpot in parallelOther Models for Viscoelastic Behavior

Again, make observations based on the geometry of the Kelvin-Voigt model. First, note that the dashpot will constrain the spring to have the same deformation.Second, note that the total force F in the Voigt model will be equal to the force in the dashpotplusthe force in the spring.Substitute theforce-displacement relationship for the spring, and theforce displacement relationship for the dashpotto give a total force equation with respect to timeBy analogy, the stress-strain differential equation can be defined in terms of E, strain, and strain rate. This equation illustrates an important characteristic of viscoelastic materials: stress in the material depends not only on the strain, but also on thestrain rate.Response of Voigt model to a unit step stress and strainStress relaxation function: gives instantaneous stress relaxation due to the presence of the dashpot (note spike at beginning of stress relaxation curve).

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Solid Linear Viscoelastic Model: spring in series with a Kelvin-Voigt model, or spring in parallel with Maxwell modelModels for Viscoelastic Behavior

Combinations of spring with either Maxwell or Kelvin-Voigt models most closely models viscoelastic behavior of biological tissues. These models account for some instantaneous deformation in creep test, and some permanent deformation in stress relaxation test.7