Topic 13: Viscoelasticity

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Topic 13 Viscoelasticity

• Soft tissues and cells exhibit several anelastic
  properties
• hysteresis during loading and unloading
• stress relaxation at constant strain
• creep at constant stress
• strain-rate dependence
• i.e., in general, stress in soft tissues depends
  on strain and the history of strain
• These properties can be modeled by the theory of
  viscoelasticity

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Simple Linear Viscoelastic Models
Stress depends on strain and strain-rate Tij
Tij(ekl,Dkl)

• Elastic stress depends on strain (spring)
• Viscous stress depends on strain-rate (dashpot)
• Strains add in series, stresses are equal
• Stresses add in parallel, strains are equal

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Maxwell Fluid Model
Total strain rate spring strain rate dashpot
strain rate
A linear first-order ODE
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Maxwell Model Relaxation Solution
Integrating Eq. 1 for constant applied strain, e0
?
A monoexponential decay Exercise check that the
original ODE (Eq. 1) is satisfied by this solution
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Voigt Solid Model
Total stress spring stress dashpot stress
A linear first-order ordinary differential
equation (ODE) Integrating, for constant applied
stress, T0 ?
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Voigt Model Creep Solution
Initial condition, ?(0) 0
Exercise check that the original ODE (Eq. 1) is
satisfied by this solution
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Asymptotic and Instantaneous Elastic Moduli
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Three-Parameter Models
There are various possible configurations The
Kelvin Standard Linear Solid Model
And the equivalent 3-parameter model constructed
from a Voigt model and an elastic element in
series
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Three-Parameter Models
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Creep Solution
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Relaxation
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3-Parameter Model
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Stress Relaxation
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Harmonic Strain Input
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Linear Viscoelastic ModelsCreep Functions
Voigt Solid
Kelvin Solid
Maxwell Fluid
T
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Linear Viscoelastic ModelsRelaxation Functions
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Generalized Voigt Model
Total strain S(strains in each Voigt model)
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Generalized Maxwell Model
Total stress S(stress in each Maxwell model)
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Quasilinear Viscoelasticity

• Soft tissues exhibit several viscoelastic
  properties
• hysteresis
• stress relaxation
• creep
• strain-rate dependence
• Linear viscoelastic models also display many of
  these properties
• However, soft tissue elasticity is nonlinear
• Quasilinear viscoelasticity combines the
  time-history dependence of linear viscoelasticity
  with nonlinear elasticity (static nonlinearity)

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Linear Viscoelasticity General Property

• In all linear viscoelastic models
• the relaxation function k(t) is proportional to
  e0 and therefore to T0
• the creep function c(t) is proportional to T0 and
  therefore to e0

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Quasilinear Viscoelasticity
K(t,?) T(e)(?).G(t) and c(t,T)
?(e)(T).J(t) i.e. in quasilinear viscoelasticity,
the instantaneous stress may be a nonlinear
function of strain but the strain-dependence and
time-dependence are separable. The reduced
relaxation and creep functions are independent of
strain.
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Reduced Relaxation Function in Tissue
Fig 7.54 page 272 from text book showing mean
normalized relaxation curve for 16 loads from
2-16 g (solid line)
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Reduced Creep Function

• Figure 8.35 in textbook. A typical creep curve,
  plotted as a reduced creep function J(t). Dog
  carotid artery.

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Linear Viscoelasticity Summary of Key Points

• In viscoelastic materials stress depends on
  strain and strain-rate
• They exhibit creep, relaxation and hysteresis
• Viscoelastic models can be derived by combining
  springs with syringes
• 3-parameter linear models (e.g. Kelvin Solid)
  have exponentially decaying creep and relaxation
  functions time constants are the ratio of
  elasticity to damping
• The instantaneous elastic modulus is the
  stressstrain ratio at t0
• The asymptotic elastic modulus is the
  stressstrain ratio as t??

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Quasilinear Viscoelasticity Summary of Key Points

• The stress-strain relation is not unique, it
  depends on the load history.
• The elastic modulus depends on the load history,
  e.g. the instantaneous elastic modulus E0 at t0
  is not, in general, equal to the asymptotic
  elastic modulus E0 at t0.
• The instantaneous elastic response T(e)(t)
  E0e(t).
• Creep, relaxation and recovery are all properties
  of linear viscoelastic models.
• Creep solution can be normalized by the initial
  strain to give the reduced creep function J(t).
  J(0)1.
• Relaxation solution can be normalized by the
  initial stress to give the reduced relaxation
  function G(t). G(0)1.