Chapter 2: Elasticity and Plasticity. Tensile Strength Testing Machine. Elasticity. Stress–strain curves in an elastic regime. (a) Typical curve for metals and ceramics. (b) Typical curve for rubber. Strain Energy Density. Shear Stress and Strain.
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Chapter 2:Elasticity and Plasticity
Tensile Strength Testing Machine
Stress–strain curves in an elastic regime. (a) Typical curve for metals and ceramics. (b) Typical curve for rubber.
Strain Energy Density
Shear Stress and Strain
(a) Specimen subjected to shear force. (b) Strain undergone by small cube in shearregion. (c) Specimen (cylinder)subjected to torsion by a torque T.
In an isotropic material, ε11 is equal to ε22.
(a) Unit cube being extended in direction Ox3. (b) Unit
cube in body subjected to tridimensional stress; only stresses
on the three exposed faces of the cube are shown.
More Complex State of Stresses
(a) Biaxial (or bidimensional) state of stress.
(b) Mohr circle and construction of general orientation 0X1 X2
(c) Mohr circle and construction of principal stresses and
maximum shear stresses (Method I).
Elastic Compliance and Stiffness Matrix
Zener’s anisotropy ratio
Tridimensional Polar Plot for Zirconium
Voigt average: assume strain is same everywhere
Reuss average: assume stress is same everywhere
Watchman and Mackenzie:
1973: Salganik model
1974: O’connel & Budiansky model
n=1: linear viscous (Newtonian)
n: power law
Tensile storage modulus
Tensile loss modulus
From thermodynamics, one can derive:
Elastic Properties 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.)
Blood Vessels Dimensions
Residual Stresses in Arteries
Cartilage Fiber Network
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. Shah.)
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 Properties of a DNA
Stretching Force vs. Relative Extension for a DNA Molecule
Stresses Acting on a Thin Film
Effect of stresses acting on 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; notice decrease in slope as separation increases.