Viscoelasticity - 2 BME 615 “It’s a poor sort of memory that only works backwards” - Lewis Carroll, Alice through the Looking Glass

Dirac  function Limiting case of Gauss distribution Area under curve is always 1 Animation from Wikipedia

Step (Heaviside) function -  function t t Note: Area under the curve is 1! t t

        time  t) Boltzmann Integrals – ,t) t time  t) t     Then as    (t-   )    (t-   )    (t-   )    (t-   )

        time  t) Boltzmann Integrals – ,t) t time  t) t     Then as    (t-   )    (t-   )   (t-   )     (t-   )

Boltzmann Integrals Also called superposition integrals, or heretitary integrals, or convolution integrals with the following notation and useful properties for closed form solutions

Cosine wave function using complex math Figure from Fung “Biomechanics” 2 nd ed.

Cosine wave function using complex math Illustration of a cosine wave's fundamental relationship to angular changes around a circle. Wikipedia

Dynamic Mechanical Analysis DMA is the measurement of mechanical response of a material as it is deformed under periodic stress or strain. DMA measures these viscoelastic properties Modulus & strength (elastic properties) Viscosity (strain rate dependence) Damping characteristics Stress relaxation & relaxation modulus Creep Used because all the variables can be changed and controlled

Harmonic strain loadings Note: relaxation-like drop off of peak strain in early cycles! Typical test protocols apply ~10 loading cycles before taking data. This is to report a repeatable, “steady state” behavior. Is this reasonable? What does this throw away? Flexor tendon behavior, data from Jaclyn Kondratko   Cycle 1 Cycle 2 SS cycles repeat Note that stress in any given cycle is not sinusoidally shaped. Why? Is this an affine transformation? This method of reporting an artificial, steady state behavior as elastic behavior is called Pseudoelasticity

Harmonic strain loadings Note how stress precedes strain! Why does this occur??

Harmonic Testing Note: stress precedes strain by phase angle  For a linear material in steady state, a peak applied strain   produces a peak stress  0 Storage modulus Loss modulus Complex dynamic modulus

Harmonic (fully reversed) response Diagram for a linearly viscoelastic material. The role of the phase angle or loss angle  (delta) is illustrated. The loss angle , or the loss tangent tan , is a fundamental measure of damping in a linear material.

3D Viscoelasticity For a Hookean material/tissue For a viscoelastic material/tissue This constitutive equation can accommodate any level of anisotropy with each component having independent elastic and time dependent properties. Consider, for example, a time dependent Lame formulation. Note  in above equations is the Kronecker delta for bookkeeping!

Linear system behavior Figure from Fung “Biomechanics” 2 nd ed. Idealized behavior for SLS looks like this. Why does it bend over instead of being linear?? t    Forcing function Resulting stress This stress curve influenced by relaxation from early increments of loading

Isochronal behavior (provides plot of  vs  relaxation artifact) ()()  (t)(t) t         First, perform a series of experiments with different step loads (strains here) At a fixed time, plot stress vs strain          t0t0 t1t1      t1t1

 Relaxation & Creep Functions Up until now, we solved DEQs from lumped linear models for: 1) Stress [  (t) = k(t)] when the applied strain is a unit step function ε(t) =1(t) is a relaxation function. 2) Strain [  (  )  = c(t)] when the applied stress is a unit step function  (t)=1(t) is a creep function. This method is the hard way, especially if we have test system. Further, these linear models do not admit the nonlinear behavior seen in biological soft tissues. Alternatively, we could test a specimen with a step load and curve fit the resulting behavior.  t 1

Linear damping behavior Figure from Fung “Biomechanics” 2 nd ed.   Idealized damped behavior for SLS looks like this. Why does it bend over instead of being linear?? t 

Rate-dependent SLS behavior Figure from Fung “Biomechanics” 2 nd ed.   slow fast

SLD vs experimentally observed behavior   slow fast Problems with SLS (for modeling soft tissues) Not strain-stiffening Not uniform hysteresis over wide range of load frequencies Need a better theory! What we have with SLS model   slow fast What we observe from soft biological tissues Most common approach is QLV

Quasi-linear viscoelasticity (QLV) Relaxation function k(t) is the viscoelastic stress response to a unit step strain For a linear 3 parameter (Kelvin) solid (see pdf for details): where: and bracketed term is linear time-dependent response. Then, with Boltzmann integral we can evaluate the stress for any strain history is linear elastic stiffness (strain-dependent response) How can we add non-linearity?

QLV – Reduced Relaxation Function Consider a tissue that is non-linear in elastic stiffness but linear in its time dependent response If elastic stiffness can be separated from its time-dependent response in integral, we have a quasi-linear material, i.e. Assume phenomenological function for elastic behavior, e.g., then is the slope of stress-strain curve or and is the “reduced relaxation function” (labeled as G(t) in refs)

QLV – in Boltzmann Integral We again use a Boltzmann superposition integral to calculate stress for any strain history Reduced relaxation Nonlinear stiffness Now have non-linear elastic behavior, but still need more uniform damping

QLV – Spectral Damping Typically, QLV models use “spectral damping” which is the summation of many Kelvin (3 parameter solids) in series. The reduced relaxation function is: and are material properties associated with damping. where c, is associated with a low frequency where the spectral damping begins to assume a constant value and is associated with a high frequency at which the damping diminishes. The constant c is associated with the magnitude of the damping. is the exponential integral defined as: where z is a dummy variable that disappears after integration. Note that when a singularity exits in the integral at.

Spectral Damping Figure from Fung “Biomechanics” 2 nd ed. In the limit, spectral damping is combined behavior of many linear dampers (from SLS model in series) that together can produce a fairly uniform hysteresis (energy loss) over a wide range of physiologic loading frequencies

Spectral damping Figure from Fung “Biomechanics” 2 nd ed. Since the stiffness curve starts from zero and goes to an asymptotic value, what linear lumped parameter model was likely used??

QLV – Applications QLV – despite the complexity of spectral damping – has been used frequently in biomechanics. QLV has been used to describe viscoelastic behavior in Skin Arteries Veins Ligaments Tendons Lung Pericardium Relaxed muscle Cells Etc. QLV captures some, but not all nonlinearities and complexities in the time-dependent behaviors of biological tissues.