Time Dependence and Viscosity

viscosity viscoelasticity time-dependence fluids
Last updated: 2025-01-18

Time Dependence and Viscosity

In the examples of elongating an elastic solid and compressing an ideal gas, stable configurations were assumed before and after deformation. We did not ask how long the deformation lasted. In fact, we tacitly assumed it took place instantaneously. This generally does not happen in biological materials.

The Reality of Time

It is more likely that biological substances deform over considerable length of time for a given set of stresses. In particular, fluids will continue to deform forever when subjected to certain types of stresses.

Rather than asking “how much does it deform?” it becomes more appropriate to ask “how fast does the medium deform?”

Viscosity

The speed of deformation of a given fluid under shear stress is measured by its viscosity.

Simple Definition

Viscosity is a measure of how difficult it is for a fluid to flow.

Mathematical Definition

A constitutive equation for fluid flow is:

σ = η(dε/dt)

where:

  • σ = shear stress
  • η (eta) = viscosity
  • dε/dt = rate of change of shear strain (shear velocity)

Physical Interpretation

For a given stress, the larger the viscosity, the slower the deformation takes place.

Examples of Viscosity

High viscosity (flows slowly):

  • Molasses
  • Honey
  • Cold motor oil
  • Tar

Intermediate viscosity:

  • Warm motor oil
  • Glycerin
  • Liquid soap

Low viscosity (flows easily):

  • Water
  • Gasoline
  • Alcohol
  • Air

Motor Oil Viscosity

The distinction between 30-weight and 10-weight oils for automobile engines is based on viscosity, not actual weight:

  • 30-weight: Higher viscosity (thicker)
  • 10-weight: Lower viscosity (thinner)
  • Cold temperatures increase viscosity
  • Hot temperatures decrease viscosity

Units of Viscosity

SI units: Pascal-second (Pa·s) CGS units: Poise (P) = 0.1 Pa·s

Typical values:

  • Water at 20°C: 0.001 Pa·s
  • Blood: 0.003-0.004 Pa·s
  • Motor oil (SAE 30): 0.2-0.5 Pa·s
  • Honey: 2-10 Pa·s

Viscoelastic Behavior

Human tissues exhibit both elastic and viscous properties—they are viscoelastic.

Figure 2.3: A vivid portrayal of human tissue deformation under impact stress. Heavyweight champion Larry Holmes delivers a punch to Renaldo Snipes.

The Boxer Example

Figure 2.3 vividly illustrates the viscoelastic properties of human tissue during impact stress to a boxer’s head. Facial tissues are grossly deformed due to:

Strong inertial properties:

  • Mass of tissue resists acceleration
  • Face initially doesn’t move with skull

Weak elastic properties:

  • Tissues don’t immediately spring back
  • Temporary storage of deformation energy

Viscous properties:

  • Tissues “flow” slowly back to proper positions
  • Energy dissipated as heat

Three Components

Viscoelastic materials exhibit:

1. Elasticity:

  • Stores energy
  • Determines how complete restoration is
  • Provides restoring force

2. Viscosity:

  • Dissipates energy
  • Determines rate of deformation
  • Resists flow

3. Inertia:

  • Resists acceleration
  • Determines dynamic response
  • Universal property of all materials with mass

The term viscoelastic suffices because inertia is universal. Sometimes called inertio-visco-elastic when all three are emphasized.

Vocal Fold Deformation

Deformations in vocal fold tissues can reach the proportions shown in the boxer illustration, especially during:

  • Violent coughing
  • Throat clearing
  • Loud phonation
  • Impact during glottal closure

Stress Relaxation

If constant strain is applied to viscoelastic tissue, stress gradually decreases over time—called stress relaxation.

Mechanism

Why stress relaxes:

  • Molecular bonds break and reform
  • Internal structure reorganizes
  • Viscous flow redistributes stress
  • Energy dissipates

Mathematical Form

Stress relaxation often follows exponential decay:

σ(t) = σ₀e^(-t/τ)

where:

  • σ(t) = stress at time t
  • σ₀ = initial stress
  • τ (tau) = relaxation time constant
  • e = base of natural logarithm

Vocal Fold Example

When vocal fold tissue is elongated quickly and held at constant length:

  • First second: Rapid drop in stress
  • Subsequent seconds: Continued relaxation in progressively smaller amounts
  • Long term: If held indefinitely, tissue would eventually release all stress and assume new rest length

The stress relaxation is nearly independent of the magnitude of initial strain (at least for strains from 13% to 33%).

Strain Creep

If constant stress (rather than constant strain) is applied to viscoelastic tissue, the tissue continues to increase in length—called strain creep.

Characteristics

Continuous deformation:

  • Material elongates under constant load
  • Rate of elongation decreases with time
  • Never reaches true equilibrium

Plastic deformation:

  • Like taffy or chewing gum
  • Progressive increase in length
  • May not fully recover original shape

Time Scale

Short term (seconds to minutes):

  • Rapid initial deformation
  • Slowing creep rate

Long term (hours to days):

  • Very slow continued deformation
  • Possible permanent changes
  • Tissue remodeling may occur

Measuring Viscosity

Challenge for Biological Tissues

No specific vocal fold viscosity data are presented in this text because:

  • Viscosity varies significantly with temperature
  • Depends on chemical composition of internal and external fluids
  • Affected by blood circulation
  • Difficult to measure accurately in vivo

Variability

Tissue viscosity can vary over many orders of magnitude, whereas elasticity has smaller range for given tissue type.

Factors affecting viscosity:

  • Temperature (higher temperature → lower viscosity)
  • Hydration state
  • Presence of hyaluronic acid and other molecules
  • Edema or inflammation
  • Time of day
  • Hormonal influences

In Vitro Versus In Vivo

There is always question about validity of extrapolating from:

  • In vitro conditions: Where most measurements are taken
  • In vivo conditions: The physiologically relevant state

This is especially true for viscosity, which is highly sensitive to environmental conditions.

Implications for Phonation

Energy Dissipation

Viscosity causes:

  • Conversion of mechanical energy to heat
  • Damping of oscillations
  • Reduction in vibrational amplitude
  • Increased phonation threshold pressure

Vibration Quality

Appropriate viscosity is necessary for:

  • Smooth oscillation
  • Gradual amplitude buildup
  • Prevention of abrupt tissue snapping
  • Distributed stress during collision

Pathology

Too little viscosity (dehydrated tissue):

  • Hard, sharp impacts
  • Increased trauma risk
  • Harsh voice quality

Too much viscosity (edematous tissue):

  • Sluggish vibration
  • High phonation threshold
  • Reduced amplitude
  • Breathy voice quality

Summary

Time dependence is a crucial aspect of biological tissue mechanics:

Viscosity:

  • Measures resistance to flow
  • Determines rate of deformation under stress
  • High viscosity = slow deformation
  • Examples range from water (low) to honey (high)

Viscoelasticity:

  • Combination of elastic and viscous behavior
  • Tissues both store and dissipate energy
  • Allows for deformation while providing restoring forces

Stress Relaxation:

  • Gradual decrease in stress under constant strain
  • Exponential decay with time
  • Molecular bonds break and reform

Strain Creep:

  • Gradual increase in deformation under constant stress
  • Like plastic flow
  • May lead to permanent changes if sustained

Clinical Importance:

  • Affects vibration quality
  • Influences phonation threshold
  • Related to hydration state
  • Changes with pathology

Understanding time-dependent behavior is essential for:

  • Modeling vocal fold vibration accurately
  • Interpreting mechanical measurements
  • Designing therapeutic interventions
  • Explaining voice quality changes

Key Takeaways

  • ✅ Viscosity measures resistance to flow—how quickly materials deform under stress
  • ✅ Biological tissues are viscoelastic, exhibiting both elastic (energy storage) and viscous (energy dissipation) properties
  • ✅ Stress relaxation is gradual decrease in stress when tissue held at constant deformation
  • ✅ Strain creep is gradual increase in deformation when tissue subjected to constant stress
  • ✅ Viscosity can vary over many orders of magnitude, more so than elasticity
  • ✅ Appropriate viscosity is crucial for healthy vocal fold vibration and voice quality

Further Reading

  1. Fung, Y. C. (1981). Biomechanics: Mechanical properties of living tissues. New York: Springer-Verlag.
  2. Alipour-Haghighi, F., & Titze, I. R. (1985). Viscoelastic modeling of canine vocalis muscle in relaxation. Journal of the Acoustical Society of America, 78, 1939-1943.
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