Rheology Fundamentals

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Rheology Fundamentals

• Lecture Notes 2

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Outline

• Continuum
• Stress
• Strain
• Ideal Solids
• Ideal Liquids
• Non-Newtonian Fluids
• Visco-elastic Behavior

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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

• Stressforce/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 the point on the surface is
given by
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Stress Tensor

• Stress tensor
• Numbers of unknowns
• Sign convention
• Stress in static fluids
• Hydrostatic pressure
• Stress in dynamic fluids
• Pressure and viscous stresses
• Invariant of stress tensor
• Principal stresses

Total stress tensor
Extra stress tensor or Viscous stress tensor
or Dissipative stress tensor
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Viscous Stress Tensor for a Newtonian
Incompressible Fluid
Viscosity
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Deformation

• When you apply an external force to a body,
  you either cause a rigid body movement and/or you
  cause a change in shape. In rheology, we are
  interested in the description of the change in
  shape.
• Change in shape is the change of distances
  between points within a continuum and is a
  geometrical concept.

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Concepts used to Describe Deformation

• Compressible and Incompressible materials
• Lagrange and Euclidian Approaches
• Translation
• Rotation
• Linear and angular deformation
• Rate of deformation
• Velocity
• Acceleration
• Velocity gradient
• Vorticity
• Strain
• Average strain
• Strain tensor
• Rate of change of strain tensor
• Principal strain

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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 SolidsHooks Law 1676
http//www.upscale.utoronto.ca/GeneralInterest/Har
rison/Flash/ClassMechanics/HookesLaw/HookesLaw.htm
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Ideal LiquidsNewtons 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 models

Ludwig Boltzmann
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Voigt (Kelvin-Voigt) Model for Viscoelastic
Behavior
Spring Elastic Modulus Force) ? (Position)
Dashpot Viscosity Force ? (Speed)
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Maxwell Model forViscoelastic Behavior
James Clerk Maxwell 1831-1879
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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, SPE
  Journal, 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.ht
  ml