Rheological Property of Bio-material

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Rheological Property of Bio-material

• Physical Properties of Bio-Materials (IV)

Poching Wu, Ph.D. Department of Bio-Mechatronic
Engineering National Ilan University
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Rheology

• A science devoted to the study of deformation
  and flow.

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Rheological Properties

1. Time-dependent stress and strain behavior
2. Creep
3. Stress relaxation
4. Viscosity

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MacPherson Strut(???????)
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Rheological Equations forMaxwell Model
For the spring
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For the dashpot
The total strain
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Since the same force is carried through the
spring and the dashpot, then
therefore
If the model is subjected to a constant strain,
then
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Trel is the time of relaxation
The solution of the above differential equation is
where e is the base of Naperian logarithm (e
2.72)
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Boundary conditions
at t 0
at t ?
Constants A and C be found to be
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then
for a simple Maxwell model is
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• where ? stress
• ?(t) stress at any time t
• ?d the decay stress
• ?e the stress at equilibrium
• ? strain
• t time
• E stiffness or modulus of the
  spring
• representing the ideal elastic
  body
• E0 instantaneous modulus or modulus
  at zero
• time
• Ee equilibrium modulus or modulus
  after
• infinite time

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• where Ed E0 - Ee decay modulus
• ? viscosity coefficient of the
  liquid in the
• dashpot
• s, v subscripts denoting
  respectively, spring
• and viscous element

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Generalized Maxwell Model
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Stress Relaxation

• Decay of stress with time when material is
  suddenly deformed to a given deformation -
  constant strain.

Relaxation Time The rate of stress decay in
a material subjected to a sudden strain. Its
the time required for the stress in the Maxwell
model, representing stress relaxation behavior,
to decay to 1/e or approximately 37 of its
original value.
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Rheological Equations forKelvin Model
For the spring and dashpot
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If the ratio ?/E Tret, called the time of
retardation
After differentiation
For a constant stress ?0, then
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By integration we obtain
where ?0 is the constant stress ?0 is the
initial strain at t 0.
If the initial strain is zero, then ?0 0 and
yields the following expression for creep under
constant load
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For t 0, ? ?0 0 t ?, ?
?0 /E t Tret, ? (?0 /E)(1 - 1/e)
If the constant load is removed from the strained
body, it will not revert to the un-strained state
after infinite time.
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The Four-Element Burgers Model
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Generalized Kelvin Model
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Creep

• Deformation with time when material is suddenly
  subjected to a dead load - constant stress.

Retardation Time The rate at which the retarded
elastic deformation takes place in a material
creeping under dead load. Its the time required
for the Kelvin model, representing creep
behavior, to deform to (1 - 1/e) or approximately
63 of its total deformation.
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Stress-Strain Behavior

• After a given force (point E), the reaction, as
  given by the Maxwell model, combined elastic and
  viscous types of behavior. The greater the speed
  of testing, the greater is the limiting value of
  the force or stress.

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Stress-Strain Behavior
For a constant rate of strain,
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The homogeneous solution for RE 0 is
The particular solution for the stress ? being
some constant B is
or
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The complete solution can be written as
Initial condition ? 0 at t 0
then
Because
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where E is the slope of the initial portion of
the stress-strain curve.

1. For the initial portion of curve, E is
   independent of R.
2. At higher strain, the slope of the curve depends
   upon the rate of testing R.

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Physical Properties of Bio-Materials Laboratory
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• Title Stress Relaxation Test for Bio-Materials
• Objective To understand the phenomenon of stress
  relaxation, and use a mathematic model to
  describe this phenomenon.
• Instrument and Materials Texture Analyzer, P/100
  Compression Plate, Toast (Please describe the
  materials and instruments brand, model, and
  configuration.)
• Method and Procedure (Please describe the
  testing method and procedure in detail.)
• Result and Discussion (Based on the experimental
  data, to analyze force vs. time result by using
  linear regression Microsoft Excel. Use the
  equation below to calculate the values of A and
  Trel. Assume the stress in the end of this test
  is the residual stress.)
• Conclusion (Summarize based on the experimental
  result and discussion. This is the most
  important part of this lab report.)

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Transformation for Linear Regression

• a intercept
• b slope

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Styrofoam
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Stress Relaxation of Bread
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Styrofoam
Bread
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Physical Properties of Bio-Materials Laboratory
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• Title 3-Point Bending Test for Bio-Materials
• Objective To understand the 3-point bending test
  of bio-material, and operation of texture
  analyzer.
• Instrument and Materials Texture Analyzer,
  HDP/3PB Three Point Bending Rig, Cracker (Please
  describe the materials and instruments brand,
  model, and configuration.)
• Method and Procedure (Please describe the
  testing method and procedure in detail.)
• Result and Discussion(Based on the experimental
  data, to calculate the bending strength, Modulus
  of Elasticity.)
• Conclusion (Summarize based on the experimental
  result and discussion. This is the most
  important part of this lab report.)