A presentation by Group 7

TA in charge: Jagadeesh Babu

For a better design For a good and efficient design, the choice of materials should be

good.

For a better performance of a product The product made with the best quality of materials is likely to be

taken further.

To be a sound Mechanical Engineer The study of materials would enhance the productivity of a

mechanical engineer.

For tackling various problems related to materials. The knowledge base for identifying the problems related to

materials in real life is crucial.

Area of knowledge which deals with the behavior and response of materials to applied forces.

Combination of various disciplines and approaches in understanding response of materials to force.

To study further we need to develop some essential fundamentals of Metallurgy and Mechanics.

Materials selection is necessary in various areas like In-service performance, Deterioration, Economics

METALSThey are characterized by:

1. Good strength and high modulus of elasticity.

2. Relatively high density

3. Good electrical and thermal conductivity

4. Good ductility and reflectivity.

5. They generally have FCC crystal structure.

CERAMICS1. High Modulus.

2. Brittle

3. Their strength in Tension < strength in compression (crushing strength)(~15 times larger)

4. They are usually ionic or covalent bonded materials and can be amorphous or crystalline.

POLYMERS1. Low Moduli (~50 times less)

2. Large elastic deformations.

3. Resistant to various chemicals

4. Low density, conductivity, tensile strength.

5. Excellent surface finish can be obtained.

COMPOSITES1. Consist of more than one material type

2. Designed to display a combination of properties of each component

SEMICONDUCTORS1. Electrical properties between conductors and insulators

2. Electrical properties can be precisely controlled

ELASTOMERS1. Low elastic moduli

2. Generally flammable

3. Poor electrical and thermal properties

4. Easy recovery to their original shape.

The four components of the discipline Material Science and Engineering and their inter-relationship are:

Processing Structure Properties Performance

Which important device helps to study microstructures: TEM (Transmission Electron Microscope)

Consider a bar of cross sectional area A being subjected to equal and opposite forces F pulling at the ends. Stress = F/A

This stress is called the tensile stress because every part of the object is subjected to a tension. The SI unit of stress is the Newton per square meter, which is called the Pascal.

Strain: It is defined as change in dimension per unit original dimension.. Strain has no unit.

Shear Stress and Strain: Now consider a force that is applied tangentially to an object. The ratio of the shearing force to the area A is called the shear stress. If the object is twisted through an angle q, then the strain is tan q.

True stress is the stress determined by the instantaneous load acting on the instantaneous cross-sectional area.

True stress ( ) is related to engineering stress (Assuming material volume remaining constant)

True strain is the rate of instantaneous increase in the instantaneous gauge length.

Elastic Deformation: This type of deformation is reversible. Once the forces are no longer applied, the object returns to its original shape.

Engineers often use Hooke’s law [ ] for calculation in tensile tests. The elastic range ends when the material reaches its yield strength. At this point plastic deformation begins.

Plastic Deformation: This type of deformation is irreversible. However, an object in the plastic deformation range will first have undergone elastic deformation, which is reversible, so the object will return part way to its original shape. Soft thermoplastics have a rather large plastic deformation range as do ductile metals such as copper, silver, and gold.

Metal fatigue: Another deformation mechanism is metal fatigue, which occurs primarily in ductile metals. It was originally thought that a material deformed only within the elastic range returned completely to its original state once the forces were removed. However, faults are introduced at the molecular level with each deformation.

Compressive failure: In long, slender structural elements —such as columns or truss bars — an increase of compressive force F leads to structural failure due to buckling at lower stress than the compressive strength.

Fracture: This type of deformation is also irreversible. A break occurs after the material has reached the end of the elastic, and then plastic, deformation ranges. At this point forces accumulate until they are sufficient to cause a fracture. All materials will eventually fracture, if sufficient forces are applied.

DuctilityDuctile materials are characterized by their ability to yield at normal temperatures.

A stress–strain curve typical of structural steel.

1. Ultimate Strength

2. Yield Strength

3. Rupture

4. Strain hardening region

5. Necking region.

6. A: Apparent stress (F/A0)

B: Actual stress (F/A)

Brittle materialsBrittle materials, which includes cast iron, glass, and stone, are characterized by the fact that rupture occurs without any noticeable prior change in the rate of elongation.

Strain hardening is when a metal is strained beyond the yield point. An increasing stress is required to produce additional plastic deformation and the metal apparently becomes stronger and more difficult to deform.

A material with small grain size will strain harden more rapidly than the same material with a larger grain size.

It can be used to treat material. Prior hardening causes the treated material to have an apparently higher yield stress. Therefore, the metal is strengthened.

A graph of Load v/s Extension of different materials

Some important terms

Specific strength= Strength/Density

Engineering Stress= Load/Initial Area

True Stress=Engineering Stress*(1+Engineering strain)

Engineering Strain= Extension/Initial length

True Strain= ln(1+Engineering strain)

Nanomaterials Porous materials

We would cover the following topics in this section:

1. Atomic Bonding in Solids

2. Energy and Packing

3. Crystal Structures

How do atoms assemble into solid structures? (for now, focus on metals)

How does the density of a material depend on its structure?

When do material properties vary with the sample (i.e., part) orientation?

In the figure the structure of diamond is shown.

The study atomic bonding in materials is essential in proper understanding of the properties and behavior of materials.

The solids are classified into roughly four different types in terms of their bonding

Covalent bonding, which forms network covalent solids (sometimes called simply "covalent solids")

Ionic bonding, which forms ionic solids

Metallic bonding, which forms metallic solids

Weak intermolecular bonding, which forms molecular solids (sometimes anomalously called "covalent solids")

Network covalent solids

Characterized by high strength, modulus, melting point and brittleness.

Ionic solids

Typically they are brittle, crystalline, have intermediate strength and high melting point.

Metallic solids

Can form various alloys, generally ductile and malleable and have high elastic modulus.

Molecular solids

Major intermolecular forces in them are vanderwaal forces. They have low melting point, low density and are bad conductors.

• Non-dense, random packing.

• Dense, ordered packing

Dense, ordered packed structures tend to have lowerenergies.

Crystalline materials- atoms pack in periodic 3D arrays.

Example: metals, many ceramics, some polymers

Non-crystalline materials-atoms have no periodic packing

Occurs for: complex structures and rapid cooling

"Amorphous“=Noncrystalline

A crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid.

A crystal structure describes a highly ordered structure, occurring due to the intrinsic nature of molecules to form symmetric patterns.

A crystal structure can be thought of as an infinitely repeating array of 3D 'boxes', known as unit cells.

A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical transparency.

A crystal structure can be thought of as an infinitely repeating array of 3D 'boxes', known as unit cells. The unit cell is calculated from the simplest possible representation of molecules, known as the asymmetric unit.

The unit cell is a small box containing one or more atoms arranged in 3-dimensions.

The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal.

How can we stack metal atoms to minimize empty space?

Now stack these 2-D layers to make 3-D structures

They tend to be densely packed.

Reasons for their dense packing:

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

b. Metallic bonding is not directional.

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

d. Electron cloud shields cores from each other

They have the simplest crystal structures.

We will examine three such structures...

Atomic packing factor (APF), packing efficiency or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is dimensionless and always less than unity.

For BCC

For HCP

Rare due to low packing density (only Po has this structure).

Close-packed directions are cube edges.

APF for simple cubic structure is 0.52

Atoms touch each other along cube diagonals.

Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

ex: Cr, W, Fe (α), Tantalum, Molybdenum

Coordination number= 8

Atomic Packing Factor of BCC is 0.68

Atoms touch each other along face diagonals.

Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordination #12

Atomic Packing Factor of FCC is 0.74 (maximum achievable APF)

ABCABC... Stacking Sequence

2D Projection

FCC unit cell

ABAB... Stacking Sequence

3D Projection 2D Projection

Coordination # = 12 APF = 0.74

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn c/a = 1.633

• In general, the order of materials in decreasing densities are: Metals, Ceramics, Polymers.

Why?

Metals have

• Closed packing (metallic bonding)

• Often large molecular masses

Ceramics have

• Less dense packing

• Often lighter elements

Polymers have

• Low packing density (often amorphous)

• Lighter elements

In this topic we are covering:

1. Crystals as building blocks

2. Crystallographic directions

3. Miller indices

Atoms may assemble into crystalline or amorphous structures.

We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP).

Common metallic crystal structures are FCC, BCC, and HCP .

Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures.

Crystallographic points, directions and planes are specified in terms of indexing schemes.

Crystallographic directions and planes are related to atomic linear densities and planar densities.

Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).

Materials can be single crystals or polycrystalline.

Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.

X-ray diffraction is used for crystal structure and interplanarspacing determinations.

Some engineering applications require single crystals:

Diamond single Turbine blades

crystals for abrasives

Properties of crystalline materials often related to crystal structure.

Ex: Quartz fractures more easily along some crystal planes than others.

Nb-Hf-W plate with an electron beam weld. Each "grain" is a single crystal. If grains are randomly oriented, overall

component properties are not directional. Grain sizes typ. range from 1 nm to 2 cm(i.e., from a few to millions of atomic layers).

Single Crystals

-Properties vary with direction: anisotropic.

Example: the modulus of elasticity (E) in BCC iron (on right)

Polycrystals

Properties may/may not vary with direction.

If grains are randomly oriented: isotropic.

(E poly iron= 210 GPa)

If grains are textured, anisotropic.

Two or more distinct crystal structures for the same material (allotropy/polymorphism)

Example:

Titanium [α,β-Ti]

Carbon-Diamond, graphite

Point coordinates for unit cell’s center are

a/2, b/2, c/2 ½ ½ ½

Point coordinates for unit cell’s corner are 111

Translation: integer multiple of lattice constants are identical position in another unit cell

The unit cell geometry is completely defined in terms of six parameters: The three edge lengths a, b, and c, and the three inter-axial angles alpha ,beta and gamma

These are termed as the lattice parameters of a crystal structure and there are seven types of crystal systems and the systems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral, monoclinic and triclinic

A crystallographic direction is defined as a line between two points, or a vector. The following steps are utilized in the determination of the three directional indices:

A vector of convenient length is positioned such that it passes through the origin of the coordinate system. Any vector may be translated throughout the crystal lattice without alteration, if parallelism is maintained

The length of the vector projection on each of the three axes is determined, these are measured in terms of the unit cell dimensions a, b, and c.

These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.

The three indices, not separated by commas, are enclosed in square brackets, thus: [uvw]. The u, v, and w integers correspond to the reduced projections along the x, y, and z axes, respectively

Algorithm

Vector repositioned (if necessary) to pass through origin.

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

Adjust to smallest integer values

Enclose in square brackets, no commas

A problem arises for crystals having hexagonal symmetry in that some crystallographic equivalent directions will not have the same set of indices. This is circumvented by utilizing a four-axis, or Miller–Bravais, coordinate system.

The three a1,a2,a3 and axes are all contained within a single plane (called the basal plane) and are at angles 120 to one another. The z axis is perpendicular to this basal plane. Directional indices, which are obtained as described above, will be denoted by four indices, as [uvtw].

Conversion from the three-index system to the four-index system, gives:

u=(2u’-v’)/3

v=(2v’-u’)/3

t=-(u+v)

w=w’

Algorithm

Vector repositioned (if necessary) to pass through origin.

Read off projections in terms of unit cell dimensions ,, , .

Adjust to smallest integer values

Enclose in square brackets, no commas-[utvw]

In all but the hexagonal crystal system, crystallographic planes are specified by three Miller indices as (hkl).Any two planes parallel to each other are equivalent and have identical indices. The procedure employed in determination of the h, k, and l index numbers is as follows:

If the plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation, or a new origin must be established at the corner of another unit cell.

At this point the crystallographic plane either intersects or parallels each of the three axes; the length of the planar intercept for each axis is determined by the parameters a,b and c

The reciprocals of these numbers are taken. A plane that parallels an axis may be considered to have an infinite intercept, and, therefore, a zero index.

An intercept on the negative side of the origin is indicated by a bar or minus sign positioned over the appropriate index. Furthermore, reversing the directions of all indices specifies another plane parallel to, and equidistant from, the origin.

Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.

Algorithm

Read off intercepts of plane with axes in terms of a, b, c

Take reciprocals of intercepts

Reduce to smallest integer values

Enclose in parentheses, no commas i.e., (hkl)

Consider the example of the plane passing through the y=b/2 and parallel to the the x,zaxis The intercepts are

infinity,1/2,infinity And the miller indices

of this will be the reciprocal of the intercepts.

At T < 912 iron has the BCC structure.

In this topic, we are going to cover

Defects (Point, Line)

Classification of dislocations

Movement of dislocations

MOST REAL MATERIALS HAVE ONE OR MORE “ERRORS IN PERFECTION” WITH DIMENSIONS ON THE ORDER OF AN ATOMIC DIAMETER TO MANY LATTICE SITES

Defects can be classified by: 1. According to geometry

(point, line or plane) 2. Dimensions of the defect

In real materials we find: Crystalline Defects or lattice irregularity

We can define this mixture/solution on a weight or “atomic” basis

A similar discussion can apply to “mixtures” of metals – called alloys

Vacancies: Vacant atomic sites in a structure.

Self-Interstitials: -“Extra" atoms positioned between atomic sites.

Vacancydistortion of planes

self‐interstitial

distortion of planes

• The simplest of the point defect is a vacancy, or vacant lattice site.

• All crystalline solids contain vacancies.

• Principles of thermodynamics is used explain the necessity of the existence of vacancies in crystalline solids.

• The presence of vacancies increases the entropy (randomness) of the crystal.

• The equilibrium number of vacancies for a given quantity of material depends on and increases with temperature as follows: (an Arrhenius model)

Two outcomes if impurity (B) added to host (A):

1. Solid solution of B in A (i.e., random dist. of point defects)

2. Solid solution of B in A plus particles of a new phase (usually for a larger amount of B)

Substitutional solid soln.(e.g., Cu in Ni)

Interstitial solid soln.(e.g., C in Fe)

Second phase particle--different composition--often different structure.

Conditions for substitutional solid solution (S.S.)

Hume – Rothery rules1. r (atomic radius) < 15%2. Proximity in periodic table

i.e., similar electronegativities

3. Same crystal structure for pure metals4. Valency equality

All else being equal, a metal will have a greater tendency to dissolve a metal of higher valency than one of lower valency(it provides more electrons to the “cloud”)

Interstitial solid solution applies to carbon in α-iron. The carbon atom is small enough to fit with some strain in the interstice (or opening) among adjacent Fe atoms in this important steel structure.

But the interstitial solubility is quite low since the size mismatch of the site to the radius of a carbon atom is only about 1/4

Random, substitution solid solution can occur in Ionic Crystalline materials as well. Here of NiOin MgO.

The O2− arrangement is unaffected. The substitution occurs among Ni2+ and Mg2+ ions.

A substitution solid solution of Al2O3 in MgO is not as simple as the case of NiO in MgO. The requirement of charge neutrality in the overall compound permits only two Al3+ ions to fill every three Mg2+ vacant sites, leaving oneMg2+

vacancy.

Iron oxide, Fe1−xO with x ≈ 0.05, is an example of a nonstoichiometric compound. Similar to the case of Figure 4.6, both Fe2+ and Fe3+ ions occupy the cation sites, with one Fe2+ vacancy occurring for every two Fe3+ ions present.

Frenkel Defect--a cation is out of place

Shottky Defect--a paired set of cation and anion vacancies

Shottky Defect:

Frenkel Defect

from W.G. Moffatt, G.W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1, Structure, John Wiley and Sons, Inc., p. 78.

Equilibrium concentration of defects kT/QDe~

Are called Dislocations:And:• slip between crystal planes result when dislocations move,• this motion produces permanent (plastic) deformation.

Schematic of Zinc (HCP):

•  before deformation •  after tensile elongation

Adapted from Fig. 7.8, Callister 7e.

slip steps which are the physical evidence of large numbers of dislocations slipping along the close packed plane {0001}

Linear Defects (Dislocations) Are one-dimensional defects around which atoms are

misaligned

Edge dislocation: extra half-plane of atoms inserted in a crystal structure b (the berger’s vector) is (perpendicular) to dislocation

line

Screw dislocation: spiral planar ramp resulting from shear deformation b is (parallel) to dislocation line

Burger’s vector, b: is a measure of lattice distortion and is measured as a distance along the close packed directions in the lattice

Definition of the Burgers vector, b, relative to an edge dislocation.

(a) In the perfect crystal, an m× n atomic step loop closes at the starting point.

(b) In the region of a dislocation, the same loop does not close, and the closure vector (b) represents the magnitude of the structural defect. For the edge dislocation, the Burgers vector is perpendicular to the dislocation line.

Fig. 4.3, Callister 7e.

Screw dislocation: The spiral stacking of crystal planes leads to the Burgers vector being parallel to the dislocation line.

In screw dislocation, the dislocation movement is perpendicular to the applied stress.

A screw dislocation may be thought of as arising from a shear distortion. Its dislocation line passes through the center of an atomic spiral.

Mixed dislocation: This dislocation has both edge and screw character with a single Burgers vector consistent with the pure edge and pure screw regions.

Burgers vector for the aluminum oxide structure.

The large repeat distance in this relatively complex structure causes the Burgers vector to be broken up into two (for O2−) or four (for Al3+) partial dislocations, each representing a smaller slip step.

This complexity is associated with the brittleness of ceramics compared with metals. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)

Dislocations are visible in (T) electron micrographs

Adapted from Fig. 4.6, Callister 7e.

• Structure:  close‐packedplanes & directionsare preferred.

view onto twoclose‐packedplanes.

close‐packed directionsclose‐packed plane (top)

close‐packed plane (bottom)

• Comparison among crystal structures:FCC:  many close‐packed planes/directions;HCP:  only one plane, 3 directions;BCC:  none “super‐close” many “near close”

• Specimens that were tensiletested.

Mg (HCP)

Al (FCC)

One case is a twin boundary (plane) Essentially a reflection of atom positions across the twinning

plane.

Stacking faults For FCC metals an error in ABCABC packing sequence Ex: ABCABABC

Adapted from Fig. 4.9, Callister 7e.

Typical optical micrograph of a grain structure, 100×. The material is a low-carbon steel.

The grain boundaries have been lightly etched with a chemical solution so that they reflect light differently from the polished grains, thereby giving a distinctive contrast. (From Metals Handbook, 8th ed., Vol. 7: Atlas of Microstructures of Industrial Alloys, American Society for Metals, Metals Park, OH, 1972.)

Simple grain-boundary structure. This is termed a tilt boundary because it is formed when two adjacent crystalline grains are tilted relative to each other by a few degrees (θ).

The resulting structure is equivalent to isolated edge dislocations separated by the distance b/θ, where b is the length of the Burgers vector, b. (From W. T. Read, Dislocations in Crystals, McGraw-Hill Book Company, New York, 1953. Reprinted with permission of the McGraw-Hill Book Company.)

The ledge Growth leads to structures with Grain Boundaries.The shape and average size or diameter of the grains for some polycrystalline specimens are large enough to observe with the unaided eye. (Macroscopic examination)

Specimen for the calculation of the grain-size number, G is defined at a magnification of 100×.

This material is a low-carbon steel similar to that shown in Figure 4.18. (From Metals Handbook, 8th ed., Vol. 7: Atlas of Microstructures of Industrial Alloys, American Society for Metals, Metals Park, OH, 1972.)

Two-dimensional schematics give a comparison of (a) a crystalline oxide and (b) a non-crystalline oxide.

The non-crystalline material retains short-range order (the triangularly coordinated building block), but loses long-range order (crystallinity). This illustration was also used to define glass in Chapter 1 (Figure 1.8).

Bernal model of an amorphous metal structure.

The irregular stacking of atoms is represented as a connected set of polyhedra. Each polyhedron is produced by drawing lines between the centers of adjacent atoms.

Such polyhedra are irregular in shape and the stacking is not repetitive.

A chemical impurity such as Na+ is a glass modifier, breaking up the random network and leaving non-bridging oxygen ions.

[From B. E. Warren, J. Am. Ceram. Soc. 24, 256 (1941).]

Schematic illustration of medium-range ordering in a CaO–SiO2 glass. Edge-sharing CaO6octahedra have been identified by neutron-diffraction experiments.

[From P. H. Gaskell et al., Nature 350, 675 (1991).]

Point, Line, Surface and Volumetric defects exist in solids.

The number and type of defects can be varied and controlled T controls vacancy conc. amount of plastic deformation controls # of dislocations Weight of charge materials determine concentration of

substitutional or interstitial point ‘defects’

Defects affect material properties (e.g., grain boundaries control crystal slip).

Defects may be desirable or undesirable e.g., dislocations may be good or bad, depending on whether

plastic deformation is desirable or not. Inclusions can be intention for alloy development

Here we would cover:

1. Slip systems, plane

2. Concept of critical shear stress

3. Grain boundary in polycrystals

4. Mechanisms of Strengthening of Metals

Plastic deformation is permanent, and strength and hardness are measures of a material’s resistance to this deformation.

On a microscopic scale, plastic deformation corresponds to the net movement of large numbers of atoms in response to an applied stress.

During this process, interatomic bonds must be ruptured and then reformed.

There is a preferred plane and in that plane there are specific directions along which dislocation motion occurs.

This plane is called the slip plane; it follows that the direction of movement is called the slip direction. This combination of the slip plane and the slip direction is termed the slip system.

For a particular crystal structure, the slip plane is the plane that has the most dense atomic packing—that is, has the greatest planar density.

Slip plane for fcc

Even though an applied stress may be pure tensile (or compressive), shear components exist at all but parallel or perpendicular alignments to the stress direction. These are termed resolved shear stresses, and their magnitudes depend not only on the applied stress, but also on the orientation of both the slip plane and direction within that plane.

Let Φ represent the angle between the normal to the slip plane and the applied stress direction, and λ the angle between the slip and stress directions; it can then be shown that for the resolved shear stress

τ=σ cosΦ cosλ

Maximum value of t is called the critical resolved shear stress.

Deformation and slip in polycrystalline materials is somewhat more complex.

Because of the random crystallographic orientations of the numerous grains, the direction of slip varies from one grain to another.

Dislocation motion occurs along the slip system that has the most favorableorientation

The dislocation density in a metal increases with deformation or cold work, due to dislocation multiplication or the formation of new dislocations, as noted previously.

Consequently, the average distance of separation between dislocations decreases—the dislocations are positioned closer together.

On the average, dislocation–dislocation strain interactions are repulsive. The motion of a dislocation is hindered by the presence of other dislocations.

As the dislocation density increases, this resistance to dislocation motion by other dislocations becomes more pronounced. Thus, the imposed stress necessary to deform a metal increases with increasing cold work.

Strengthening by Grain Size Reduction

Strengthening by Strain Hardening

Solid-Solution Strengthening

Grain-boundary strengthening (or Hall–Petchstrengthening) is a method of strengthening materials by changing their average crystallite (grain) size.

The size of the grains, or average grain diameter, in a polycrystalline metal influences the mechanical properties.

A fine-grained material (one that has small grains) is harder and stronger than one that is coarse grained, since the former has a greater total grain boundary area to impede dislocation motion. For many materials, the yield strength varies with grain size according to

Hall petch equation

This equation indicates that the yield strength has an inverse square root relation with grain size (d). Theoretically, a material can be made infinitely strong if the grains are made infinitely small.

Variation of Yield Strength with grain size.

Strain Hardening is when a metal is strained beyond the yield point. An increasing stress is required to produce additional plastic deformation and the metal apparently becomes stronger and more difficult to deform.

To go a step further in explaining, if a low-carbon steel is strained passed the yield point, then aged for several days at room temperature, it will have a higher yield stress after the aging. This happens because during the aging carbon or nitrogen atoms diffuse to dislocations, re-anchoring them.

Increasing strength of by strain hardening will also reduce the ductility of the product

For this strengthening mechanism, solute atoms of one element are added to another, resulting in either substitutional or interstitial point defects in the crystal. The solute atoms cause lattice distortions that impede dislocation motion, increasing the yield stress of the material.

Increasing the concentration of the solute atoms will increase the yield strength of a material, but there is a limit to the amount of solute that can be added, and one should look at the phase diagram for the material and the alloy to make sure that a second phase is not created.

Solid solution strengthening increases strength and hardness