AMME1362

Página 1

Introduction to Materials (Chapter 1)

Classification of Materials

Materials can be classified into three basic categories:

Metals

Composed of one or more metallic compounds - Alloys (e.g Iron, Aluminium, Copper) and most often, in small amounts, non-metallic compounds (e.g Carbon, Nitrogen, Oxygen)

Relatively dense in comparison to Ceramics and Polymers

Relatively stiff, strong, and ductile, good conductors (unbound electrons), and lustrous

Widespread use in structural applications

Ceramics

Compounds between metallic and non-metallic elements - oxides, nitrides and carbides (e.g Aluminium Oxide, Silicon Oxide, Silicon Carbide, Silicon Nitride)

Relatively stiff and strong - comparable to metals, hard, good insulators, can be transparent, translucent or opaque

Use in cookware, cutlery and even some automobile engine parts

Polymers

​​​​​​​​​​​​​​Organic compounds that are chemically based on carbon, hydrogen and other non-metallic elements. Includes plastic and rubber materials

Have very large molecular structures, chainlike in nature - backbone of carbon atoms (e.g Polyethylene, Polyvinyl Chloride, Polystyrene)

Typically have low densities, not as stiff or strong as metals and ceramics, extremely ductile and pliable, relatively chemically inert and un-reactive to environments, low conductivity

Drawback includes softening and/or decomposing at modest temperatures

Composites

​​​​​​​Composed of two (or more) individual materials that come from categories above - aims to achieve the best characteristics

Fibreglass = small glass fibres embedded within a polymeric material = relatively stiff, strong and flexible

Processing/Structure/Properties/Performance Correlations

With regard to the design, production, and utilization of materials, there are four elements to consider

Processing

Structure

Properties

Performance

The performance of a material depends on its properties, which in turn are a function of its structure(s);

furthermore, structure(s) is (are) determined by how the material was processed.

Página 2

The Structure of Crystalline Solids (Chapter 3)

• 3.1 – 3.15: Study all parts

• 3.16: Calculations of interplanar spacing not required

• 3.17: Understand the concept

ISSUES TO ADDRESS... • Basic knowledge on crystal structures (for now, focus more on metals) • Structure-density relationship in some crystal systems • Material property-orientation relationship

A crystalline material (metals, many ceramic materials, and certain polymers ) is one in which the atoms are situated in a repeating or periodic array over large atomic distances—that is, long-range order exists, such that upon solidification, the atoms will position themselves in a repetitive three-dimensional pattern, in which each atom is bonded to its nearest neighbour atoms.

Some of the properties of crystalline solids depend on the crystal structure of the material, the manner in which atoms, ions, or molecules are spatially arranged.

Crystal Systems:

Unit Cells:

Small groups of atoms form a repetitive pattern. so it is convenient to subdivide the unit cell structure into small repeat entities

A unit cell is the most basic and least volume consuming repeating structure of any solid

We generally use the unit cell having the highest level of geometrical symmetry and smallest repetitive volume.

Geometry is completely defined in terms of lattice parameters:

the three edge lengths a, b, and c,

the three interaxial angles a, b, and g.

Coordination number: the number of nearest neighbor or touching atoms

Atomic packing factor (APF) = \(\frac{Volume of atoms in a unit cell}{Total unit cell volume}\)

To determine the number of atoms associated with each unit cell:

N = \(\N_i\) + \(\frac{N_f}{2}\) + \(\frac{N_e}{8}\)

Atomic hard sphere model, in which atoms/ions are solid sphere with well-defined diameters, is used to describe crystalline structures. The nearest neighbours touch each other

Metallic Crystal Structures

Tend to be densely packed because

Typically, only one element is present, so all atomic radii are the same.

Metallic bonding is not directional.

Nearest neighbour distances tend to be small in order to lower bond energy.

Electron cloud shields cores from each other

Have the simplest crystal structures.

Types of structures:

Simple Cubic (SB)

Rare due to low packing density (only Polonium has this structure) - Not stable (48% of space is waste)

Close-packed directions are cube edges

Coordination # = 6

APF = 0.52 (1 atom in a unit cell)

Face-Centred Cubic (FCC)

Most popular in metals - eg. Al, Cu, Au, Pb, Ni, Pt, Ag

ABCABC... Stacking Sequence

Atoms touch each other along face diagonals

Coordination # = 12

APF = 0.74 (4 atoms in a unit cell)

Body-Centred Cubic (BCC)

Atoms touch each other along cube diagonals

Coordination # = 8

APF = 0.68 (2 atoms in a unit cell)

Hexagonal Close Packed (HCP)

ABAB... Stacking Sequence

magnesium, titanium, and zinc

Coordination # = 12

APF = 0.74 (6 atoms in a unit cell

c/a = 1.633 (a and c represent, respectively, the short and long unit cell dimensions)

Point Coordinates

Any point can be specified in terms of its coordinates as fractional multiples of the unit cell edge lengths

This is possible using three point coordinate indices: q, r, and s. These indices are fractional multiples of a, b, and c unit cell edge lengths

qa = lattice position referenced to the x axis

rb = lattice position referenced to the y axis

sc = lattice position referenced to the z axis

Crystallographic Direction

A crystallographic direction is defined as a line directed between two points, or a vector.

Algorithm:

1. Vector repositioned (if necessary) to pass through origin.

2. Read off projections in terms of unit cell dimensions a, b, and c

3. Adjust to smallest integer values 4. Enclose in square brackets, no commas - [uvw] - overbar represents a negative index

Página 3

Mechanical Properties (Chapter 6)

Concepts of Stress and Strain

Stress (MPa)

Tensile stress \(\sigma\) is defined as

\(\sigma\) = \(\frac{Ft}{A_0}\)

in which Ft (newtons) is instantaneous load applied perpendicular to the specimen cross - section. And \(A_0\) is the original cross sectional area before any load is applied (\(m^2 \)).

Shear Stress \(\tau\) is defined as

\(\tau\) = \(\frac{Fs}{A_0}\)

in which Fs refers to the load applied parallel to the specimen cross-section (shifts side to side)

Strain (unitless) - the ratio of change in dimension to original dimension of a body when it is deformed

Tensile strain \(\epsilon\) is the response to stress and is defined as

\(\epsilon\) = \(\frac{delta}{l_0}\)

It is the deformation elongation or change in length at some instant, as referenced to the original length. When you squeeze or stretch something you're applying stress. The object strains to accommodate that stress.

Shear strain

\(\gamma\) = \(\frac{deltax}{y}\) = \(\tan\)\(\theta\)

refers to the ratio of the change in dimensions to the original dimension caused due to parallel forces

Lateral Strain

\(\epsilon_L\) = \(\frac{-deltal}{w_0}\)

refers to the contraction in the transverse or lateral direction

Stress-Strain Behaviour

The degree to which a structure deforms or strains depends on the magnitude of an imposed stress. For most metals that are stressed in tension and at relatively low levels, stress and strain are proportional to each other through the relationship

\(\sigma\) = E\(\epsilon\) - (Hooke's Law = stress/strain)

E (GPa) is Young's Modulus/Modulus of Elasticity and is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression - tensile

If an object has a HIGH young's modulus, it requires a LOT of stress to make it collapse.

If an object has a low young's modulus, it requires very little stress to make it collapse

On a stress-strain curve, the linear part is the elastic portion of the curve, and the slope of this is Young's modulus.

Shear stress and strain are proportional to each other through the relationship

\(\tau\) = G\(\gamma\)

where G is the shear modulus, the slope of the linear elastic region of the shear stress–strain curve

For isotropic materials, shear and elastic moduli are related to each other and to Poisson’s ratio according to

E = 2G(1+v)

Elastic Properties of Materials

Elastic Deformation is non-permanent - reversible. As shown in the stress–strain plot application of the load corresponds to moving from the origin up and along the straight line. Upon release of the load, the line is traversed in the opposite direction, back to the origin.

For non linear-behaviour:

tangent modulus is taken as the slope of the stress–strain curve

secant modulus represents the slope of a secant drawn from the origin to some given point of the s-P curve.

Elastic strain is manifested as small changes in the interatomic spacing and the stretching of interatomic bonds.

The magnitude of the modulus of elasticity is a measure of the resistance to separation of adjacent atoms, that is, the interatomic bonding forces.

Furthermore, this modulus is proportional to the slope of the interatomic force–separation curve at the equilibrium spacing:

Time-dependent elastic behaviour is known as an-elasticity, and it is due to time-dependent microscopic and atomistic processes that are attendant to the deformation.

Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression

v = \(\frac{lateral strain}{axial strain}\) = \(\frac{epsilon_L}{epsilon}\)

Plastic Deformation

As the material is deformed beyond the point of elasticity, the stress is no longer proportional to strain (Hooke’s law), and permanent, nonrecoverable. The transition from elastic to plastic is a gradual one for most metals; some curvature results at the onset of plastic deforma- tion, which increases more rapidly with rising stress.

From an atomic perspective, plastic deformation corresponds to the breaking of bonds with original atom neighbors and then the re-forming of bonds with new neighbors as large numbers of atoms or molecules move relative to one another; upon removal of the stress, they do not return to their original positions.

Tensile Properties

Yield Strength (MPa - \(\sigma_y\)

stress level at which plastic deformation begins, or where the phenomenon of yielding occurs.

the initial departure from linearity of the stress–strain curve - the proportional limit

Tensile Strength

After yielding, the stress necessary to continue plastic deformation in metals increases to a maximum and then decreases to the eventual fracture

Maximum stress on engineering stress-strain curve.

Ductility

measure of a material's ability to undergo significant plastic deformation before rupture

expressed quantitatively as either percent elongation or percent reduction in area.

%EL = \(\frac{l_f-l_0}-{l_0}\) x 100 OR %RA = \(\frac{A_f-A0)-{A_0}\) x 100

Toughness

ability of a material to absorb energy and plastically deform before fracturing

Approximate by the area under the stress-strain curve

Resilience

ability of a material to absorb energy when it is deformed elastically, and release that energy upon unloading

Modulus of resilience – the strain energy per unit volume for stress up to the point of yielding

Hardness

measure of a material’s resistance to localized plastic deformation (e.g., a small dent or a scratch)

Large hardness means:

resistance to plastic deformation or cracking in compression.

better wear properties

Hardness tests are more frequent than other mechanical properties because:

•Simple & inexpensive

•Nondestructive – only small indentation

Can be approximately converted to other properties (e.g., tensile strength)

Rockwell:

– Each scale runs to 130 but only useful in range 20-100.

– Minor load 10 kg – Major load 60 (A), 100 (B) & 150 (C) kg

• A = diamond, B = 1/16 in. ball, C = diamond

Hardness specification includes: 1. hardness number 2. scale symbol ex: 80 HRB  80 Rockwell hardness on the B scale

True Stress and Strain

True stress \(\sigma_T\) is defined as the load F divided by the instantaneous cross-sectional area Ai over which deformation is occurring (i.e., the neck, past the tensile point)

\(\sigma_T\) = \(\frac{Ft}{A_i}\)

True strain:

\(\epsilon_T\) = ln\(\frac{l_i}{l_0}\)

If no volume change occurs during deformation—that is, if

A_il_i = A_0l_0

then true and engineering stress and strain are related according to

sigmaT = sigma(1 + epsilon)

epsilonT = ln(1 + epsilon)

Elastic Strain Recovery

Upon release of the load during the course of a stress–strain test, some fraction of the total deformation is recovered as elastic strain. During the unloading cycle, the curve traces a near straight-line path from the point of unloading (point D), and its slope is virtually parallel to the initial elastic portion of the curve (modulus of elasticity). The magnitude of this elastic strain, which is regained during unloading, corresponds to the strain recovery. If the load is reapplied, the curve will traverse essentially the same linear portion in the direction opposite to unloading; yielding will again occur at the unloading stress level where the unloading began. There will also be an elastic strain recovery associated with fracture.