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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

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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

(a) Specimen subjected to shear force. (b) Strain undergone by small cube in shearregion. (c) Specimen (cylinder)subjected to torsion by a torque T.


Poisson’s Ratio

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


Mohr Circle

(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).


Mohr Circle


Pure Shear


Anisotropic Effects


Elastic Modulus


Elastic Compliance and Stiffness Matrix


Anisotropy of Cubic System

Zener’s anisotropy ratio

Young’s modulus

Shear modulus

Bulk modulus

Poisson’s ratio

Lame constants


Young’s Modulus and Shear Modulus of Monocrystalline Cu


Young’s Modulus and Shear Modulus Monocrystalline Zirconia


Tridimensional Polar Plot for Zirconium


Polycrystal

Voigt average: assume strain is same everywhere

Reuss average: assume stress is same everywhere


Porosity on Young’s Modulus

Watchman and Mackenzie:


Microcracks vs. Young’s Modulus


Microcracks vs. Young’s Modulus (cont’d)

1973: Salganik model

1974: O’connel & Budiansky model


Elastic Properties of Polymers


Viscoelasticity

n=0: plastic

n=1: linear viscous (Newtonian)

n: power law

Viscosity coefficient

Fluidity:


Viscoelasticity (cont’d)

Tensile storage modulus

Tensile loss modulus


Rubber Elasticity

From thermodynamics, one can derive:

Extension ratio:


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.


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