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Chapter 2
Elasticity and
Plasticity
Mechanical Testing Machine
Elastic Behavior
Stress–strain curves in an elastic regime. (a) Linear elastic curve , typical for metals, ceramics,
and some polymers. (b) Nonlinear elastic curve, typical for rubber.
Strain Energy
Density
Shear Stress and
Shear Strain
(a) Specimen subjected to shear force.
(b) (b) Strain undergone by small cube in shear region.
(c) (c) Specimen (cylinder) subjected to torsion by a torque T.
Poisson’s 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. Poisson’s ratio, ν, is
the negative ratio of the transverse strain and longitudinal
strain.
Generalized Hooke's Law
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).
Mohr Circle
Pure Shear
Hooke’s Law for Anisotropic Materials
Relations among Elastic Constants for Isotropic Materials
Elastic Compliance and Stiffness Matrixes
Compliance Matrix for a Cubic System
Relationships Among Elastic Constants
Young’s modulus
Shear modulus
Bulk modulus
Poisson’s ratio
Lame΄ constants
11
1E
S
11 12
1
2( )G
S S
11 22 33
11 22 33
1
1( )
3
BK
12
11
S
S
44 11 12
44
12
1 1( )
2C C C G
S
C
Young’s Modulus of Monocrystalline Cu
Young’s Modulus Monocrystalline Zirconia
Young’s Modulus of Monocrystalline Zirconium
Voigt and Reuss Averages for Polycrystals
Voigt average: isostrain
Reuss average: isostress
11 22 32
12 23 13
44 55 66
1 1(3 ' 3 ' ')
5
1' ( )
3
1' ( )
3
1' ( )
3
F G HE
F S S S
G S S S
H S S S
Effect of Porosity on Young’s Modulus
Watchman and Mackenzie:
2
0 1 2(1 )E E f f
1 21.9, 0.9f f
Effect of Microcracks on Young’s Modulus
Effect of Microcracks on Young’s Modulus
3 1
0
[1 1.63 ]E
NaE
3
0
1 1.63E
NaE
Salganik model
O’connell & Budiansky model
Young’s Modulus of Polymers
Young’s Modulus of Polymers as a
Function of Temperature
Viscoelasticity
n n = 0: plastic
n = 1: linear viscous (Newtonian)
n ≠1 : nonlinear
Viscosity and Fluidity
exp( )Q
ART
Viscosity
Fluidity
1
Viscoelasticity
0
0
0 0
0 0
exp[ ( )]
exp[ ( )]
exp (cos sin )
' "
e e i t
i t
E i ie e e
E iE
Viscoelasticity
Tensile storage modulus 0
0
0
0
' cos
" sin
Ee
Ee
Tensile loss modulus
Rubber
Elasticity
2 1
1 1[ ]nKT
11
0
l
l
Stress-Strain Behavior of Biological Materials
(a) Stress–strain 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 stress–strain curve) vs. stress. (Adapted from Y. C. Fung. Biomechanics,
New York: Springer,1993), p. 329.)
Residual Stresses in Arteries
Cartilage
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.)
Mechanical Behavior of Superficial Zone of Cartilage
Stress–strain 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. 171–228.)
Mechanical Testing of DNA
Force vs. Extension for DNA Molecule
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.
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.
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.