Transcript Slide 1

Chapter 2
Elasticity and Viscoelasticity
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) Strain undergone by small
cube in shear region. (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
E 
G 
Bulk modulus B 
Poisson’s ratio
Lame΄ constants
1
S11
1
2( S11  S12 )
11   22   33
1

1
K
 ( 11   22   33 )
3
S12
 
S11
  C44 
  C12
1
1
(C11  C12 ) 
G
2
S 44
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
1
1

(3F ' 3G ' H ')
E
5
1
F'
( S11  S 22  S32 )
3
1
G' 
( S12  S 23  S13 )
3
1
H '
( S 44  S55  S 66 )
3
Effect of Porosity on Young’s Modulus
Watchman and Mackenzie:
E  E0 (1  f1  f 2 2 )
f1  1.9, f 2  0.9
Effect of Microcracks on Young’s Modulus
Effect of Microcracks on Young’s Modulus
Salganik model
E
 [1  1.63 Na 3 ]1
E0
O’connell & Budiansky model
E
 1  1.63 Na 3
E0
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
Viscosity
  A exp(
Fluidity
1

Q
)
RT
Viscoelasticity
e  e0 exp[i ( t )]
   0 exp[i ( t   )]
0
0

E 

exp i 
(cos   i sin  )
e
e0
 E ' iE "
e0
Viscoelasticity
Tensile storage modulus
E'
0
Tensile loss modulus
E" 
0
e0
e0
cos 
sin 
Rubber Elasticity
  nKT [12  11 ]
1 
l1
l0
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 .