Chapter 2 Elasticity and Viscoelasticity

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Chapter 2Elasticity and Viscoelasticity
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Mechanical Testing Machine
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Elastic Behavior
Stressstrain curves in an elastic regime. (a)
Linear elastic curve , typical for metals,
ceramics, and some polymers. (b) Nonlinear
elastic curve, typical for rubber.
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Strain Energy Density
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Shear Stress and Shear Strain
(a) Specimen subjected to shear force. (b) Strain
undergone by small cube in shear region. (c)
Specimen (cylinder) subjected to torsion by a
torque T.
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Poissons Ratio
(a) Unit cube being extended in direction Ox3.
(b) Unit cube subjected to tridimensional stress
only stresses on the three exposed faces of the
cube are shown. Poissons ratio, ?, is the
negative ratio of the transverse strain and
longitudinal strain.
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Generalized Hooke's Law
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Mohr Circle
(a) Biaxial (or bidimensional) state of stress.
(b) Mohr circle construction, general
orientation (c) Mohr circle and construction,
principal stresses and maximum shear
stresses (Method I).
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Mohr Circle
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Pure Shear
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Hookes Law for Anisotropic Materials
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Relations among Elastic Constants for Isotropic
Materials
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Elastic Compliance and Stiffness Matrixes
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Compliance Matrix for a Cubic System
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Relationships Among Elastic Constants
Youngs modulus
Shear modulus
Bulk modulus
Poissons ratio
Lame? constants
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Youngs Modulus of Monocrystalline Cu
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Youngs Modulus Monocrystalline Zirconia
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Youngs Modulus of Monocrystalline Zirconium
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Voigt and Reuss Averages for Polycrystals
Voigt average isostrain
Reuss average isostress
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Effect of Porosity on Youngs Modulus
Watchman and Mackenzie
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Effect of Microcracks on Youngs Modulus
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Effect of Microcracks on Youngs Modulus
Salganik model
Oconnell Budiansky model
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Youngs Modulus of Polymers
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Youngs Modulus of Polymers as a Function of
Temperature
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Viscoelasticity
n 0 plastic n 1 linear viscous
(Newtonian) n ?1 nonlinear
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Viscosity and Fluidity
Viscosity
Fluidity
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Viscoelasticity
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Viscoelasticity
Tensile storage modulus
Tensile loss modulus
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Rubber Elasticity
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Stress-Strain Behavior of Biological Materials
(a) Stressstrain response of human vena cava
circles-loading squares-unloading. (Adapted from
Y. C. Fung, Biomechanics (New York Springer,
1993),p. 366.) (b) Representation of mechanical
response in terms of tangent modulus (slope of
stressstrain curve) vs. stress. (Adapted from Y.
C. Fung. Biomechanics, New York Springer,1993),
p. 329.)
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Residual Stresses in Arteries
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Cartilage
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Mesostructure of Cartilage
(a) Mesostructure of cartilage (consisting of
four zones) showing differences in structure as a
function of distance from surface the bone
attachment is at bottom. (From G. L. Lucas, F. W.
Cooke, and E. A. Friis, A Primer on Biomechanics
(New York Springer, 1999), p. 273.) (b)
Cross-section of human cartilage showing regions
drawn schematically in (a). (Courtesy of K. D.
Jadin and R. I. Sah.)
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Mechanical Behavior of Superficial Zone of
Cartilage
Stressstrain curve for samples from the
superficial zone of articular cartilage. Samples
were cut parallel and perpendicular to collagen
fiber orientation. (From G. E. Kempson,
Mechanical Properties of Articular Cartilage. In
Adult Articular Cartilage, ed. M. A. R. Freeman
(London Sir Isaac Pitman and Sons Ltd., 1973),
pp. 171228.)
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Mechanical Testing of DNA
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Force vs. Extension for DNA Molecule
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Stresses in a Thin Film
Effect of stresses in a thin film on bending of
substrate (a) tensile stresses in thin film (b)
compressive stresses in thin film.
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Elastic Constant and Bonding
Two atoms with an imaginary spring between them
(a) equilibrium position (b) stretched
configuration under tensile force (c) compressed
configuration under compressive force.
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Attraction and Repulsion between Two Atoms
(a) Interaction energies (attractive and
repulsive terms) as a function of separation
(b) Force between two atoms as a function of
separation slope decreases as separation
increases.