Chapter 2Elasticity 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

Bulk modulus

Poisson’s ratio

Lame΄ constants

11

1E

S

11 12

1

2( )G

S S

11 22 33

11 22 33

11

( )3

BK

12

11

S

S

44 11 1244

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 ' ')

51

' ( )31

' ( )31

' ( )3

F G HE

F S S S

G S S S

H S S S

Effect of Porosity on Young’s Modulus

20 1 2(1 )E E f f Watchman and

Mackenzie: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

0

0

0

0

' cos

" sin

Ee

Ee

Tensile storage modulus

Tensile loss modulus

Rubber Elasticity

2 11 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.