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Rheology Fundamentals
Lecture Notes 2
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Continuum Approach
• What is a continuum?
• What is continuum approach?
• Eulerian method of description• Lagrangian method of description
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Stress
• Stress=force/unit area
• Tensile
• Compressive• Shear
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Stress
•Body and surface forces
• Simple and isolated stresses
• Tensile
• Compression• Shear
• Stress vector
Shear and normal components
Fluid element
The stress vector at thepoint on the surface isgiven by
0
limS
f
S
f
S
y
x
z
n̂
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Stress Tensor
• Stress tensor Numbers of unknowns
Sign convention
•Stress in static fluidsHydrostatic pressure
• Stress in dynamic fluidsPressure and viscous stresses
• Invariant of stress tensor
• Principal stresses
ˆ (3.58) f dSn
Total stress tensor
(3.83) pI
Extra stress tensor or
Viscous stress tensor or
Dissipative stress tensor
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Viscous Stress Tensor for a
Newtonian Incompressible Fluid
( ) jiij
j i
vv
x x
Viscosity
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Deformation
When you apply an external force to abody, you either cause a rigid bodymovement and/or you cause a change in
shape. In rheology, we are interested inthe description of the change in shape.
Change in shape is the change of distances between points within acontinuum and is a geometrical concept.
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Strain
• Strain is the absolute amount of distortion
which occurs, expressed as a fraction of
an original dimension of the unstressed
sample. Strain is dimensionless.
• Tensile strain
• Compressive strain
• Shear strain
• Rate of strain
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Relation Between Force and
Deformation
• Relation for ideal solids
• Relation for ideal fluids
• Relation for viscoelastic materials
• Constitutive equations or rheological
equations of state
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Important Deformations in
Rheology
• Shear
•Elongation
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Ideal Solids
Hook’s Law 1676
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMe
chanics/HookesLaw/HookesLaw.html
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Ideal Liquids
Newton’s Law The resistance of which arises from
the lack of slipperiness originating
In a fluid, other things being equal, is
proportional to the velocity by which
the parts o f the fluid are being separated
from each other. Isaac N. Newton (1687)From Rheology Principles, Measurements, and Applications
By C. W. Macosko, 1994, VCH Publishers, page 65
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Visco-elastic Behavior
• Elasticoviscous fluids
• Viscoelastic solids
• Creep• Relaxation
• Material modelsLudwig Boltzmann
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Voigt (Kelvin-Voigt) Model forViscoelastic Behavior
SpringElastic Modulus
Force)
(Position)
DashpotViscosity
Force (Speed)
Woldemar Voigt(1850-1919)
Lord Kelvin(1824-1907)
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Maxwell Model for
Viscoelastic Behavior
James Clerk Maxwell1831-1879
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Non-Newtonian Behavior
• Bingham body
• Yield stress
• Psudoplastic behavior
• Dilatancy• St. Venant body
• Apparent viscosity
• The power law or Ostwald model (Ostwald de Walle)
• Time-dependent behavior • Thixotropic
• Rheotropic or anti-thixotropic
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References
• Chapter 1 of your text book
• Rheology Study of Flow by George Smoluk, SPEJournal, Vol 27, 1971.
• Chapter 5 of Introduction to Fluid Mechanics by Fox,McDonald, and Pritchard, 6th edition, Wiley
• Rheology Study of Flow, G.R. Smoluk, SPE Journal, Vol.
27, December 1971, pages 19-30
• http://www.rheologyschool.com/rheology_glossary.html
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