Stress And Strain StressStress is defined as "force per area".

Direct Stress or Normal StressStress normal to the plane is usually denoted "normal stress" and can be expressed as

σ = Fn / A         (1)whereσ = normal stress ((Pa) N/m2, psi)Fn = normal component force (N, lbf)A = area (m2, in2)

Shear StressStress parallel to the plane is usually denoted "shear stress" and can be expressed as

τ = Fp / A         (2)whereτ = shear stress ((Pa) N/m2, psi)Fp = parallel component force (N, lbf)A = area (m2, in2)

StrainStrain is defined as "deformation of a solid due to stress" and can be expressed as

ε = dl / lo = σ / E         (3)where

dl = change of length (m, in)lo = initial length (m, in)ε = unitless measure of engineering strainE = Young's modulus (Modulus of Elasticity) (Pa, psi)

Hooke's Law -  Modulus of Elasticity (Young's Modulus or Tensile Modulus)Most metals have deformations that are proportional with the imposed loads over a range of loads. Stress is proportional to load and strain is proportional to deformation expressed by the Hooke's law like

E = stress / strain = (Fn / A) / (dl / lo)         (4)whereE = Young's modulus (N/m2) (lb/in2, psi)

Modulus of Elasticity or Young's Modulus are commonly used for metals and metal alloys and expressed in terms 106 lbf/in2, N/m2 or Pa. Tensile modulus are often used for plastics and expressed in terms 105 lbf/in2 or  GPa

Please note: The above article is taken from www.engineeringtoolbox.com.

Creep Behavior of Materials When a metal or alloy is under a constant load or stress, it may undergo progressive plastic deformation over a period of time, even though applied stress is

less than the yield strength at that temperaure. This time dependent strain is called creep (above definition is taken from AMIE study material) More information is taken from Wikipdeia and shown below.

In materials science, creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses. It occurs as a result of long term exposure to high levels of stress that are below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods, and near melting point. Creep always increases with temperature.

The rate of this deformation is a function of the material properties, exposure time, exposure temperature and the applied structural load. Depending on the magnitude of the applied stress and its duration, the deformation may become so large that a component can no longer perform its function — for example creep of a turbine blade will cause the blade to contact the casing, resulting in the failure of the blade. Creep is usually of concern to engineers and metallurgists when evaluating components that operate under high stresses or high temperatures. Creep is a deformation mechanism that may or may not constitute a failure mode. Moderate creep in concrete is sometimes welcomed because it relieves tensile stresses that might otherwise lead to cracking.

Stages of Creep

In the initial stage, or primary creep, the strain rate is relatively high, but slows with increasing strain. This is due to work hardening. The strain rate eventually reaches a minimum and becomes near constant. This is due to the balance between work hardening and annealing (thermal softening). This stage is known as secondary or steady-state creep. This stage is the most understood. The characterized "creep strain rate" typically refers to the rate in this secondary stage. Stress dependence of this rate depends on the creep mechanism. In tertiary creep, the strain rate exponentially increases with strain because of necking phenomena.

GENERAL CREEP EQUATION

where is the creep strain, C is a constant dependent on the material and the particular creep mechanism, mb are exponents dependent on the creep mechanism, Q is the activation energy of the creep mechanism, σ is the applied stress, d is the grain size of the material, k is Boltzmann's constant, and T is the absolute temperature. and

Creep in materials must be taken into consideration before designing machine components which work in

high temperature / high stress environments. Other components in which creep is important design consideration include Bulb filaments, Crown Glass, Metal Paper clips, ...

Notes on Various Fractures Hello Everyone,

Have a blessed day. Hoping your preparation is going cool unlike mine. I just thought I will share brief notes on various fractures to give a brief overview.Brittle Fracture     A fracture which takes place by rapid propagation of crack with a negligible deformation. In amorphous materials, the fracture is completely brittle. In crystalline materials, it occurs after small deformation.

Ductile Fracture:     A fracture which takes place by a slow propagation of crack with appreciable plastic deformation. This type of fracture comes into play in materials which don't work harden much. 

Creep Fracture     A fracture which takes place due to excessive creeping of materials, under steady load. Creep is exhibited in iron, nickel, copper and alloys at higher temperature.  Creep resistance may be increased by addition of certain elements such as cobalt, nickel , manganese, tungsten, ...

Fatigue Fracture:

    A fracture that occurs when a material is subjected to cyclic loading. If the loads are above a certain threshold, microscopic cracks will begin to form at the surface. Eventually a crack will reach a critical size, and the structure will suddenly fracture.

Brittle Fracture In brittle fracture, no apparent plastic deformation takes place before fracture. In brittle crystalline materials, fracture can occur by cleavage as the result of tensile stress acting normal to crystallographic planes with low bonding (cleavage planes). In amorphous solids, by contrast, the lack of a crystalline structure results in a conchoidal fracture, with cracks proceeding normal to the applied tension.

The theoretical strength of a crystalline material is (roughly)

where: -E is the Young's modulus of the material,

γ is the surface energy, and

ro is the equilibrium distance between atomic centers.

On the other hand, a crack introduces a stress concentration modeled by

(For sharp cracks)

where: -σapplied is the loading stress,

a is half the length of the crack, and

ρ is the radius of curvature at the crack tip.Putting these two equations together, we get

Looking closely, we can see that sharp cracks (small ρ) and large defects (large a) both lower the fracture strength of the material.

Recently, scientists have discovered supersonic fracture, the phenomenon of crack motion faster than the speed of sound in a material. This phenomenon was recently also verified by experiment of fracture in rubber-like materials.

The above info is taken from www.ubstech.com.

Ductile Fracture

In ductile fracture, extensive plastic deformation takes place before fracture. The terms rupture or ductile rupture describe the ultimate failure of tough ductile materials loaded in tension. Rather than cracking, the material "pulls apart," generally leaving a rough surface. In this case there is slow propagation and an absorption of a large amount energy before fracture.

Many ductile metals, especially materials with high purity, can sustain very large deformation of 50–100% or more strain before fracture under favorable loading condition and environmental condition. The strain at which the fracture happens is controlled by the purity of the materials. At room temperature, pure iron can undergo deformation up to 100% strain before breaking, while cast iron or high-carbon steels can barely sustain 3% of strain.

Because ductile rupture involves a high degree of plastic deformation, the fracture behavior of a propagating crack as modeled above changes fundamentally. Some of the energy from stress concentrations at the crack tips is dissipated by plastic deformation before the crack actually propagates.

The basic steps are: void formation, void coalescence (also known as crack formation), crack propagation, and failure, often resulting in a cup-and-cone shaped failure surface.

The steps are clearly shown in the figure given here. The above info is taken from http://www.websters-online-dictionary.org/. Please do refer to them for further info.

Brittle Fracture VS Ductile Fracture Brittle Fracture:

o Caused due to high impact blows on the material

o Plastic deformation is zero or very very less

o Once crack is formed,  the crack is unstable in nature and propagates very rapidly.

o Crack propagates nearly perpendicular to the direction of the applied stress

o Crack often propagates by cleavage - breaking of atomic bonds along specific crystallographic planes (cleavage planes).

Ductile Fracture:

o Caused due to tensile forces acting on the material

o Necking can be observed i.e. excessive plastic deformation takes place

o Crack is stable i.e. once crack is formed, it resists propagation unless further stress is applied

o Micro-voids are formed fist, then by shear forces the crack propagates and results in fracture

Good material about Fractures is available here & here

Griffith's Theroy: Mechanism of Brittle Fracture

      It has been observed that the stress required for a material, at which it fractures, is only a small fraction of

cohesive strength. This discrepancy led Griffith to suggest that the low observed strengths were due to presence of micro-cracks, which act as the points of stress concentration. 

       According to the Griffith's criterion. the crack will propagate under the effect of a constant applied stress if an incremental increase in length produces no change in total energy of the systems. Mathematically the above criterion is explained as 

A proper explanation of the above theory is given as below by Wikipedia:

Fracture mechanics was developed during World War I by English aeronautical engineer, A. A. Griffith, to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts:

The stress needed to fracture bulk glass is around 100 MPa (15,000 psi).

The theoretical stress needed for breaking atomic bonds is approximately 10,000 MPa (1,500,000 psi).

        A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could

not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.

      To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length (a) and the stress at fracture (σf) was nearly constant, which is expressed by the equation:

An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed.

      The growth of a crack requires the creation of two new surfaces and hence an increase in the surface energy. Griffith found an expression for the constant C in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was:

Compute the potential energy stored in a perfect specimen under an uni-axial tensile load.

Fix the boundary so that the applied load does no work and then introduce a crack into the specimen.

The crack relaxes the stress and hence reduces the elastic energy near the crack faces. On the other hand, the crack increases the total surface energy of the specimen.

Compute the change in the free energy (surface energy − elastic energy) as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases by increasing the crack length, i.e. by causing fracture. Using this procedure, Griffith found that

where E is the Young's modulus of the material and γ is the surface energy density of the material. Assuming Eγ = 1 J/m2 gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass. = 62 GPa

Stress Strain Curve, Mild Steel Hello Everyone,

Those who are familiar with concept of Stress - Strain curve, please do continue with this post to understand about upper yeild strenght and lower yield strenght for mild steel. Those who are not familiar please take time to go through this article

From the image given here taken from etomica.org it is clear that materials like mild steel have two yield strenghs. The first called upper yield strenght and the second called lower yield strength.

Once the stress reaches the upper yield strength, the internal relaxation comes into play and the strain can be observed even at lower amount of stress. The stain is bound to osciallte between both the limits. The lower yield strength is about half the tensile strength of the material.  The explaination can be summarized as follows   "At elastic limit, sudden yield happens & fall-off of load takes place. Hence material continues to defrom at lower load until material hardening sets in"

Answers.com says the reason for such behavior is Low carbon steels suffer from yield-point runout where the material has two yield points. The first yield

point (or upper yield point) is higher than the second and the yield drops dramatically after the upper yield point. If a low carbon steel is only stressed to some point between the upper and lower yield point then the surface may develop Lüder bands.

True Stress Vs Engg. Stress Engineering stress assumes that the area a force is acting upon remains constant, true stress takes into account the variation in the cross sectional area as a result of the stress induced deformation (strain) of a material.

For example a steel bar in tension once it's yield point or stress is reached will start to "neck". Necking is the localised concentration of strain in a small region of the material, causing a reduction in cross sectional area at this point.

To calculate the engineering stress in the above case, the applied load is divided by the original cross sectional area, however the true stress would be equal to the load divided by the new deformed cross sectional area. Therefore true stress is likely to be significantly higher than engineering stress. Note that while the material is deforming elastically before thwe yield point is reached there will be some difference between true and enginnering stress (as the material is changing shape) but it will be much smaller than the difference after the yield point is reached.

A rock core in a uniaxial compression test will typically expand radially under loading. Therefore in this case, the engineering stress (based on the original diameter) will be larger than the true stress within the material.

Tensile Toughness Toughness:    Energy observed by material prior to fracturing is called toughness. It depends on both strength and ductility of the material in question. A pic from www.etomica.org is given below to show the relationship the three entities in question viz tensile toughness, ductility and strength.

 

From the figure, it can be concluded that tensile toughness is the are under the stress - strain curve. It is high if a material has high amount of strength and ductility. Materials with low ductility of low strength don't posses ample tensile toughness. 

The word toughness is usually used for tensile toughness.  In tesile toughness, the strain rate is relatively slow. There is another type of toughness called as impact toughness. Please do read about it here to understand the difference.

This post is made from the study material provided by IEI and from the www.etomica.org

Tensile Test, Part One Tensile Test:    Tensile test is a simple test, wherein the specimen in question is subjected to uni-axial load (pulled apart) till failure. This test is used to plot the stress - strain curve there by coming to conclusion about

Yield point

Elasticity limit

Point of  Proportionality

and lot more factors including, true breaking stress, fracture point load, ...

 

   A sample of specimen is taken, and is pulled apart in apparatus known as Universal testing machine. The length and cross section area of sample are decided as per our needs. Nomenclature of the specimen is shown in figure.

    Once the equipment is set up, the load on the specimen is gradually increased noting down the stress and strain levels till the point of rupture (i.e. fracture). A typical curve for ductile materials is shown here:

In figure, the points to be noted include: 

Proportionality zone .. i.e the zone where HOOK's law is valid

The region where elasticity is exhibited

The zone where material yields and plastic deformation happens

The ultimate tensile strength & the uniform elongation of specimen till then

The fracture point and the local necking which happens before fracture

PS:  Some info has been taken from Wikipedia and from http://invsee.asu.edu/.

Tensile Test, Part Two Please read the previous part of this article here

The typical stress strain curve for ductile material is given here: 

The points to note from this picture are:

Elastic Limit  The material is elastic till this point in the curve. The stress strain ratio is constant when the curve is linear within this zone. The material deforms in this zone but regains original shape and size once the load is removed. The working stress is always much below the Elastic Limit

Yield Point:   Plastic deformation happens at this point. The deformation is permanent in nature and the original shape and size are not restored once the load is removed.

Creep:   A small amount of creep may come into play due to sudden elongation of material. This effect of creep is not shown in this picture. Creep usually appears for negligible time and is not taken into account.

Ultimate Strength:    Stress is necessary to obtain stain from the yield point onwards. Ultimate tensile strength (UTS), is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract. This is the highest point in the curve

Fracture Point:

    Once the Ultimate stress is crossed, the material starts necking (i.e. non-uniform reduction in area of cross section in specimen). The material then breaks apart.

Elasticity and Plasticity Elasticity:       External loads tend to deform materials from their original shape and size. Elasticity is the ability of a material to return to its original shape and size after removing the load applied. Elastic deformation (change of shape or size) lasts only as long as a deforming force is applied to the object, and disappears once the force is removed.This is so because the atoms in the metal change their position due to external stress but can't take new positions because the change of position is too small (or in acceptable range)

     Steels and other such materials are elastic over good range.

Plasticity:     External loads tend to deform materials from their original shape and size. Plasticity is the ability of a material to retain the deformation even after external load is removed.In plastic deformation, the atoms in the material due to external force are displaced and take up new positions. They cannot come back to their natural positions once force is removed.

     This is very desirable property in machine tools. Lead has good plasticity at room temperature. Cast Iron has no plasticity even at very high temperature.    

PS: Some of the info here is taken from science.jrank.org

What are Mechanical Properties of Materials?? Hello Everyone,

From what I studied, I can come to conclusion that:

External loads are always applied on Materials during their service. The properties which describe the re-action of a material to those external loads are all classified as Mechanical Properties of materials.

It is important to ascertain the mechanical properties of material with standard laboratory tests in which the loads on the materials in real environment  are applied. This lets us determine behavior of materials and ensure we choose the right materials

 The important Mechanical Properties include:

Ductility

Hardness

Elasticity and Plasticity

Poission's ratio

Creep

Stress and Strain W.R.T Tensile stress, Shear Stress

Poisson's Ratio Poisson's ratio (ν), is the ratio, when a sample

object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).

When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio ν (nu) is a measure of the Poisson effect

In the above picture, the stress is acting in X axis, but change in object is evident in Y and Z axes also. Poisson's effect is all about this change and Poisson's ratio is a measure of this effect and is given by 

For ideal material, the ratio is 0.5. But in general it ranges from 0.25 to 0.40

Relation between E, G and Poisson's Ratio The definite relationship between Young's

modulus, Shear modulus & Poissons ratio  is asked many a times in our old question papers though for two marks only.

So I thought I will put up the answer here:

Let young's modulus = E, Shear modulus = G, Bulk Modulus = K and

poisson's ratio = v E = 3K(1-2v) E = 2G(1+v)

Mechanical Properties of Materials Hello Everyone,  Well I am a little tensed. Exam dates have been released. My preparation till now amounts to almost nothing and the pressure is huge. So started serious study from today morning itself.

  So stated with chapter called as Mechanical Properties of Materials. So the important topics to be studied here include:

Tensile strength

Poission's Ratio

Yield strength

Elastic and viscoelastic properties

Creep, stress relaxation and impact.

Fracture behaviour.

Ductile fracture

Griffith theory

effect of heat treatment and temperature on properties of metals.

Fatigue Fracture         Fatigue fracture is a fracture that occurs

when a material is subjected to cyclic loading and unloading. If the loads are above a certain threshold, microscopic cracks will begin to form at the surface. Eventually a crack will reach a critical size, and the structure will suddenly fracture.

        Rotating shafts, connecting rods, aircraft wings and leaf springs are some examples of structural and machine components that are subjected to millions of cycles of alternating stresses during service. Majority of fractures in such components is due to fatigue.

         Fatigue fracture occurs by crack propagation. The crack usually initiates at the surface of the specimen and propagates slowly at first into the interiors. At some critical stage, crack propagation becomes rapid culminating in fracture.

        The fatigue behavior can be understood from results of fatigue test, which are presented in from of S-N curves.  Samples of material are subjected to alternating stresses of different levels. The number of cycles of stress reversals N required to cause fracture is plotted against the applied stress level S. Some materials such as mild steel show a clearly defined fatigue limit. If the applied stress is below the fatigue limit, (aka Endurance Limit) the material will withstand any number of stress reversals. If materials don't show clearly defined limit, the fatigue limit is defined as stress that would cause failure after a specified number of stress reversals.

The above info is taken from Material Science and Engineering by Raghavan and the picture shown here is taken from http://www.fea-optimization.com/. Please do refer to them for more info. 

Charpy and Izod Tests

I was telling about Impact hardness earlier on this blog. The Charpy and Izod tests are useful in determining the Impact hardness of the materials.

The materials with high impact hardness are ductile in nature and the materials with low impact hardness are brittle in nature.

Charpy Test:

The specimen used is 55mmx10mmx10mm in size with a V notch (making 45 degrees) as shown in figure. The specimen is held horizontally and a hammer repeatedly strikes the specimen till the specimen fails

A hammer attached to pendulum strikes the specimen when released from a height. The pendulum swings back after striking the specimen. The angle from which the pendulum is released and the angle to which the pendulum raises back after breaking the specimen are noted down and used in calculation of Impact hardness.

A picture of apparatus used is shown in the figure. The formula for calculating the energy required for breaking the specimen is given by formula     Enegry = WR(cosα  - cosβ)

with W => weight of pendulum and hammer        R => distance between center of gravity of pendulum to its striking edge         α  => Initial angle from which pendulum is released          β => The angle to which the pendulum rose after breaking the specimen

Izod Test:

Here the specimen is of size 75mmX10mmX10mm with a V notch (making 45 degrees). The specimen is held vertically and the test is performed. The entire test is similar to Charpy Test.

The pictures in the post are taken from Vhttp://www.soawe.com/time/?tag=What-is-a-charpy-test

Impact hardness   We have been speaking about tensile toughness. Tensile toughness can be defined as the resistance offered by material to plastic deformation i.e. the ability to resit indentation and penetration or abrasion. Here, the load is applied slowly and the strain rate is quite slow too.

But in real life materials are also subjected to sudden blows. The resistance offered by materials to such blows (or impacts) can be called as impact toughness.

Hard, strong materials with good tensile toughness also falter under sudden impacts and exhibit brittle nature and undergo brittle fracture. The brittleness of materials and the reliability of materials under impacts can be studied using Charpy test and Izod test.

The tests are described in the further sections of this blog. Please do take time to go through the same.

Hardness tests of materials

Hardness of a material refers to the resitance the material offers to permanent plastic deformation when an external force is applied. Wikipedia defines it as the measure of how resistant solid matter is to various kinds of permanent shape change when a force is applied. 

In view of syllabus of material science, Impact hardness tests are important and necessary. Impact hardness refers to resitance offered by material when the force applied is impact in nature i.e. for short period with high magnitude. 

The four important tests covered in the syllabus are

Rockwell Hardness test 

Knoop Hardness test

 Vickets hardness test

Brinell's Hardness test

The hardness tests are performed since 

They are easy, simple 

The set up is in-expensive The test doesn't damage the entire specimen.

Usually small specimen is sufficient

Other physical properties can be told from this tests

Resilience of Material Hello Everyone,

   Every wondered why objects like spring give back energy when they uncoil?? Well one of the reasons fro this behavior is resilience of material with which spring is manufactured. 

   Resilience of material is the ability of it to absorb energy when deformed elastically due to applied stress and return the energy back when unloaded. 

    Modulus of Resilience is the measure of this property and as per the wikipedia,  Modulus of Resilience can be calculated using the following

formula: , where σy is yield stress, E is Young's modulus, and is strain.

Determination of yield Strength Hello Everyone,

  In the previous articles, I told what is yield strength is? Now how to determine it is always a problem. Many a ductile materials get deformed (elastic and plastic). But the boundaries of deformation cannot be strictly defined due to hell lot of reasons. 

   So the Americans devised a plan to find out the yield strength. They define the same as the stress

at which a predetermined amount of permanent deformation occurs. To find yield strength, the predetermined amount of permanent strain is set along the strain axis of the graph, to the right of the origin (zero). It is indicated in Figure as Point (D).

  A straight line is drawn through Point (D) at the same

slope as the initial portion of the stress-strain curve. The point of intersection of the new line and the stress-strain curve is projected to the stress axis. The stress value, in pounds per square inch, is the yield strength. It is indicated in Figure 5 as Point 3. This method of plotting is done for the purpose of subtracting the elastic strain from the total strain, leaving the predetermined "permanent offset" as a remainder. When yield strength is reported, the amount of offset used in the determination should be stated. For example, "Yield Strength (at 0.2% offset) = 51,200 psi."

 Notes for the above article is taken from www.engineersedge.com.

Rockwell Hardness Test     The Rockwell test determines the hardness by

measuring the depth of penetration of an indenter under a large load (60Kgf - 200Kgf) compared to the penetration made by a preload (10Kgf). There are different scales, which are denoted by a single letter, that use different loads or indenters. The result, which is a dimensionless number, is noted by HRX where X is the scale letter.

      The determination of the Rockwell hardness of a material involves the application of a minor load followed by a major load, and then noting the depth of penetration, vis a vis, hardness value directly from a dial, in which a harder material gives a higher number. The chief advantage of Rockwell hardness is its ability to display hardness values directly, thus obviating tedious calculations involved in other hardness measurement techniques. 

This method is widely used in Industry as the reading is available easily & quickly. 

Brinell Hardness Test        The Brinell hardness test method consists of

indenting the test material with a 10 mm diameter hardened steel or carbide ball subjected to a load of 3000 kg. 

     The objective of harness test is define the hardness number which represents an arbitrary quantity used to provide a relative idea of material properties. The hardness number derived in this test is called Brinell harness number and is designated as BHN

      For softer materials the load can be reduced to 1500 kg or 500 kg to avoid excessive indentation. The full load is normally applied for 10 to 15 seconds in the case of iron and steel and for at least 30 seconds in the case of other metals. The diameter of the indentation left in the test material is measured with a low powered microscope. 

      The Brinell harness number is calculated by dividing the load applied by the surface area of the indentation. The formula is shown in the picture shown below.

Where F = Force applied in kgF          D = diameter of indenter       

     This method is not used in industry since it is quite slow, deforms the specimen excessively and requires setup to calculate the depth of the indentaion..

    The above information has been taken from www.gordonengland.co.uk. Please do refer to them for more info.

Knoop Hardness Test       The Knoop hardness test  is a microhardness

test - a test for mechanical hardness used particularly for very brittle materials or thin sheets, where only a small indentation may be made for testing purposes

    A pyramidal diamond point is pressed into the polished surface of the test material with a known force, for a specified dwell time, and the resulting indentation is measured using a microscope. The geometry of this indenter is an extended pyramid with the length to width ratio being 7:1 and respective face angles are 172 degrees for the long edge and 130 degrees for the short edge. The depth of the indentation can be approximated as 1/30 of the long dimension. 

The Knoop hardness HK or KHN is then given by the formula:

where: L = length of indentation along its long axis

Cp = correction factor related to the shape of the indenter, ideally 0.070279

P = load

        The advantages of the test are that only a very small sample of material is required, and that it is valid for a wide range of test forces. The main disadvantages are the difficulty of using a

microscope to measure the indentation (with an accuracy of 0.5 micrometre), and the time needed to prepare the sample and apply the indenter.

Status Chapter 02, Defects in solids Well Hello Everyone,

Hope your preparation for AMIE is going on at good pace. I studied a little about Crystal Defects Earlier and am posting notes here. The articles I posted here related to this chapter include:

Question about Degrees of Freedom Cool PPT on Crystal Defects

No. Of Atoms in Zinc Unit Cell

Old Questions. Material Science. Chap. 02.

Atomic Packing Factor

Diffusion In Solids

Line defects and Surface defects

Point Defects in Crystal

Status Chap 02

Simple, Body Centered & Face Centered Cubic System..

Crystal Systems. Bravias Lattices.

Miller Index

Prerequisites for understanding defects in crysta...

Chap. 02. Defects in Crystals.

Chap. 02. Defects in Crystals.    Horrible chapter as per me. Yeah, you read it well, Its horrible I say. I have read the syllabus @ study material provided. It speaks of all types of defects including point defects, line defects, surface defects and volume defects. Also speaks of lot of other stuff like further classifications in the defects and such.

    I am of opinion that one should first read the internals of crystal and crystal structures, bravias crystals, Atomic Packing Factor, Space lattice stuff and such. APF is important and seems to have appeared in the exams a lot of times. I think the book Material Science by RS Khurmi & RS Sedha is really cool for this purpose. But geez its such a pain to study the same. Hard stuff regarding Simple lattice, Body Centered Lattice , FCC, .. volumes, Avagordo's numbers volumes and masses of atoms and all such stuff.

     Added to that I still couldn't start the actual topic of defects still.Should see but its absorbing to study the same. I will keep you posted about my developments. 

Point Defects in Crystals One more hard day of studying. So I was going through point defects in crystals. Before reading about point defects please understand what crystals are and why you need to study their internal structures and how defects effect the structure sensitive properties of the material

This stuff is really interesting in itself. You gotta study a lot from vacancy, interstitial defects, constitutional impurity, Schotty defect, Frenkel defect and all such. 

Also study what enthalpy of formation is? How to calculate the same? The effects of defects on the bonds between atoms, on the stress and strain and elasticity ....

The material science & Engineer by Raghavan &  Material Science by RS Khurmi & RS Sedha are good books for the same. I am following the later

The basics of point defects in crystals can be summarzied as follows:

Vacancy – missing atom at a certain crystal lattice position; 

Interstitial impurity atom – extra impurity atom in an interstitial position

Self-interstitial atom – extra atom in an interstitial position; 

Substitution impurity atom – impurity atom, substituting an atom in crystal lattice; 

 Frenkel defect – extra self-interstitial atom, responsible for the vacancy nearby.

The above summary is taken from www.substech.com. Please refer to them for further info. 

A lot more stuff has to be studied regarding effects of defects in the crystal, the calculation of equilibrium concentration of vacancies, drawing miller indices for given plane and for given crystal direction and all such.... Geez will this ever end???

Prerequisites for understanding defects in crystals     Hello EveryOne, So about crystal defects, I am not sure if I can complete the chapter by this weekend or not. I had to go through a lot of old question papers to clearly understand what all must be studied to complete the portion of this chapter. All I can tell you is, the notes from study material is not sufficient at all. 

    The prerequisites  for studying the material in the study material are as follows as per me:

1. What are crystalline structures? What is lattice? What is unit cell?

2. Study about the 7 crystal systems. Study about the 14 Bravias lattices

3. Understand the concept of Miller indices. Find miller indice for a given plane. Sketch a plane when miller indices are given.

4. Understand about crystal directions. Find crystal directions when miller indices are given.

5. Find out the relation between radius of atom and parameter of lattice in a Bravias lattice. Find out the no. of atoms in a given lattice. Find out the no. of atoms in sq. mm. of cube

6. Find out Atomic Packing Factor. Know how to calculate the density of cube using Avagordo's Number.

    Yeah, I must say I made some progress in this chapter anyway. I know a little about everything I listed above and practiced some problems on the same.

     A note about questions from these topics in old papers. The questions are given here. Do try to solve them.

1. Calculate volume of FCC unit cell in terms of atomic radius. (W-05)

2. Show that Atomic Packing Factor of FCC unit cell is greater than the Atomic Packing Factor of BCC unit cell. (W-05)

3. Calculate the Atomic Packing Factor of a FCC unit cell. (W-06)

4. If lattice parameter of alpha iron is 286 pm, find out the atomic radius (W-06).

    There is a lot more to study including types of defects and the effects of these defects on the structure intensive properties of crystalline materials .... Everything happens at a pace. All we can do is, improve the pace. Thats what I intend to do this time.

Miller Index Miller indices are a notation system in crystallography for planes and directions in crystal (Bravais) lattices.In particular, a family of lattice planes is determined by three integers ℓ, m, and n, the Miller indices. They are written (hkl), and each index denotes a plane orthogonal to a direction (h, k, l) in the basis of the reciprocal lattice vectors. By convention, negative integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. Miller index 100 represents a plane orthogonal to direction ℓ; index 010 represents a plane orthogonal to direction m, and index 001 represents a plane orthogonal to n.There are also several related notations

the notation {ℓmn} denotes the set of all planes that are equivalent to (ℓmn) by the symmetry of the lattice.

In the context of crystal directions (not planes), the corresponding notations are:

[ℓmn], with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice; and

similarly, the notation 〈hkl〉 denotes the set of all directions that are equivalent to [ℓmn] by symmetry.

Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller. The method was also historically known as the Millerian system, and the indices as Millerian, although this is now rare.The precise meaning of this notation depends upon a choice of lattice vectors for the crystal, as described below. Usually, three primitive lattice vectors are used. However, for cubic crystal systems, the cubic lattice vectors are used even when they are not primitive (e.g., as in body-centered and face-centered crystals).

The above information is taken from Wikipedia. Please refer to Wikipedia for better information.

Simple, Body Centered & Face Centered Cubic Systems The three Bravais lattices which form cubic crystal systems are

Simple cubic (P)

Body-centered cubic (I)

Face-centered cubic (F)

The simple cubic system (P) consists of one lattice point on each corner of the cube. Each atom at the lattice points is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (1⁄8 × 8).The body-centered cubic system (I) has one lattice point in the center of the unit cell in addition to the

eight corner points. It has a net total of 2 lattice points per unit cell (1⁄8 × 8 + 1).The face-centered cubic system (F) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 atoms per unit cell (1⁄8 × 8 from the corners plus 1⁄2× 6 from the faces).Attempting to create a C-centered cubic crystal system (i.e., putting an extra lattice point in the center of each horizontal face) would result in a simple tetragonal Bravais lattice.

Line defects and Surface defects     Yeah, I was reading about these topics only from last Saturday. Please head to my advice, this is the wrong way of spending your weekend. So basically there is a lot to study, but everything is quite easy to understand. Video tutorials can be of great help in this issue, but alas, I couldn't surf for the same till now. I shall update the same here at the earliest.

What all needs to be covered:

1. What are line defects, what are dislocations? ( both are same basically )

2. Types of line defects viz screw dislocation & edge dislocation.

3. How to find out the burgers vector for a given defect. What is burgers circuit, how to find out the direction of burgers vector. What is Burgers Vector?

4. What are various types of surface defects? 

5. Study about grain boundary and interface, tilt boundary, twinning, stacking faults & such.

Note:

Twinning is important concept as the same is asked many many times in the old question papers.

For simple definitions of defects and classification please refer to material science by RS Khurmi & RS Sedha 

For brief and clear cut explanation please refer to the  study material provided by AMIE

Resources:

A good pdf which helps to visualize the dislocations is here

A good image to help visualize point defects is here

Another point defect explanation here

Diffusion In Solids I was studying about diffusion in solids. The simplest definition of diffusion is "movement of atoms in solids under thermal energy and a gradient" is called diffusion. Where the gradient can be concentration or Electric / Magnetic field ...

This is relatively simple topic and can be easily understood. But the topic is a little important as

questions regarding time taken for carburization are repeatedly asked in the question papers.

The important topics to study include:

Diffusion mechanisms (vacency, interstitial ..)

Rate of diffusion in steady state and non steady state (Ficks first law and second law)

Kirkendall effect

Perfect resource for this topic is study material by IEI. But the book by Raghavan is also very very good. The book is available here

Atomic Packing Factor          In crystallography, atomic packing factor (APF) or packing fraction is the fraction of volume in a crystal structure that is occupied by atoms. It is dimensionless and always less than unity. For practical purposes, the APF of a crystal structure is determined by assuming that atoms are rigid spheres. The radius of the spheres is taken to be the maximal value such that the atoms do not overlap. For one-component crystals (those that contain only one type of atom), the APF is represented mathematically by

where Natoms is the number of atoms in the unit cell, Vatom is the volume of an atom, and Vunit cell is the volume occupied by the unit cell. It can be proven mathematically that for one-component structures, the most dense arrangement of atoms has an APF of

about 0.74. In reality, this number can be higher due to specific intermolecular factors. For multiple-component structures, the APF can exceed 0.74.

WORKED OUT EXAMPLE

Body-centered cubic crystal structure

BCC structure       The primitive unit cell for the body-centered cubic (BCC) crystal structure contains nine atoms: one on each corner of the cube and one atom in the center. Because the volume of each corner atom is shared between adjacent cells, each BCC cell contains two atoms.Each corner atom touches the center atom. A line that is drawn from one corner of the cube through the center and to the other corner passes through 4r, where r is the radius of an atom. By geometry, the length of the diagonal is a√3. Therefore, the length of each side of the BCC structure can be related to the radius of the atom by

     Knowing this and the formula for the volume of a sphere((4 / 3)pi r3), it becomes possible to calculate the APF as follows:

  The above information is taken from Wikipedia. Please do refer to the same for further info.

No. Of Atoms in Zinc Unit Cell I have been going through the question papers again. I found a rather interesting question viz "Calculate the no. of atoms in zinc unit cell?"

First some basics about Calculation of Number of Particles per Unit Cell of a Cubic Crystal System.Keeping the following points in mind we can calculate the number of atoms in a unit cell.

An atom at the corner is shared by eight unit cells. Hence an atom at the corner contributes 1/8 to the unit cell

An atom at the face is a shared by two unit cells

Contribution of each atom on the face is 1/2 to the unit cell.

An atom within the body of the unit cell is shared by no other unit cell

Contribution of each atom within the body is 1 to the unit cell.

An atom present on the edge is shared by four unit cells

Contribution of each atom on the edge is 1/4 to the unit cell.By applying these rules, we can calculate the number of atoms in the different cubic unit cells of monoatomic substances.

(The above concepts are taken from tutorvista.com. Please refer to them from more info)

Now I intend to answer this question in this post as I find it little cool. Zinc is having HCP structure as per Material Science and Engineering by Raghavan. A cool image showing the HCP structure is here:

    It is clear from the image that 3 atoms are inside the body of the HCP unit structure. So the count of atoms is 3 as of now.  

    It is clear from the image that the hexagonal faces have one atom each. But each such atom is shared by two HCP units equally. Hence the count of atoms comes to "4" here.

   Now, the atoms at the twelve points ( hexagonal edges on top and bottom ) make another two atoms. Hence count is 6 totally. 

    A simple explaination of the same given at answers.com is given below. Please do refer to the same.

   

    There are 6 atoms in the hcp unit cell. The hex shape has six atoms at the points that are direct translations of each other making 1 atom for the top hex and one atom for the bottom hex. That's 2. The atom in the center of the top and center of the bottom are translations giving 1 more. That's 3. Then there are 3 atoms in the middle region of each cell bringing the total to 6

Chapter 03, Phase Diagrams Hello Everyone,

     After a week or so spent lethargically, I am back to study. So I intend to study about "Phase diagrams". I

was going through the Syllabus the other day and felt that this chapter is cool. What do we need to cover in this chapter??? As per me the following are must, please do let me know if I am missing something here

Phases & micro-structure's basic concepts 

Solubility and limitations Mono-component and binary systems

Non-equilibrium system

Phase diagram and. application in crystalline and non-crystalline solids.

 Lever rule and applications

Effects of phases on Mechanical properties 

    Hello Everyone again, After starting the real study ( i.e. the books and such ) the above given points are not ample as per me. So I am updating a list of topics that should be studied for clearing all questions which may arise from this topic:

Introduction to phases, phase diagrams & need for studying the same

Mono-component systems, Gibbs phase rule & applications

Binary phase diagrams, triple points, invariant points

Isomorphic systems, applications of phase diagrams (lever rule)

Non - equilibrium solidification of alloys, problems related to cooling

Alloy Systems (Binary Eutectoid, Hypo-Eutectoid plain carbon steels, Hyper-Eutectiod plain carbon steels )

Non - crystalline solids and phases in them

Basics of Phase diagrams       Phase diagrams are one of the most important

sources of information concerning the behavior of elements, compounds and solutions. They provide us with the knowledge of phase composition and phase stability as a function of temperature (T), pressure (P) and composition (C). Furthermore, they permit us to study and control important processes such as phase separation, solidification, sintering, purification, growth and doping of single crystals for technological and other applications. Although phase diagrams provide information about systems at equilibrium, they can also assist in predicting phase relations, compositional changes and structures in systems not at equilibrium.

     The phase rule, also known as the Gibbs phase rule, relates the number of components and the number of degrees of freedom in a system at equilibrium by the formula                                       F = C – P + 2

where F equals the number of degrees of freedom or the number of independent variables, C equals

the number of components in a system in equilibrium and P equals the number of phases. The digit 2 stands for the two variables, temperature and pressure.

    The number of degrees of freedom (F) of a system is the number of variables that may be changed independently without causing the appearance of a new phase or disappearance of an existing phase.

    Please note that the value of F cannot be less than 0. So the maximum number of phases can be found out using the Gibbs Formula with taking thermodynamics into consideration.

    The point at which F = 0 , is called invariant point. The point at which the three phases can co-exist is called triple point. 

Triple Points & Gibbs Rule Hello Everyone,

    I intend to write small note on triple point which I came across while studying for Phase Diagrams chapter of Material Science in section A of AMIE.

    In thermodynamics, the triple point of a substance is the temperature and pressure at which three phases (for example, gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium. For example, the triple point of mercury occurs at a temperature of −38.8344 °C and a pressure of 0.2 mPa.

In addition to the triple point between solid, liquid, and gas, there can be triple points involving more than one solid phase, for substances with multiple polymorphs. Helium-4 is a special case that presents a triple point involving two different fluid phases (see lambda point). In general, for a system with p possible phases, there are

triple points.

        The triple point of water is used to define the kelvin, the SI base unit of thermodynamic temperature. The number given for the temperature of the triple point of water is an exact definition rather than a measured quantity. The triple points of several substances are used to define points in the ITS-90 international temperature scale, ranging from the triple point of hydrogen (13.8033 K) to the triple point of water (273.16 K).

Tie Line Tie Line: An imaginary horizontal line (isotherm) spanning

a two-phase region of an equilibrium phase diagram, terminating at the nearest phase boundaries on either side.

 Tie lines are important when using phase diagrams to predict the constitution of two-phase materials.

Eutectic reaction Eutectic reaction:

A three-phase reaction in which, upon cooling, a liquid transforms to give two solid phases. 

e.g:                                 L +

Eutectic Systems        A eutectic system is a mixture of chemical compounds or elements that has a single chemical composition that solidifies at a lower temperature than any other composition. This composition is known as

the eutectic composition and the temperature is known as the eutectic temperature. On a phase diagram the intersection of the eutectic temperature and the eutectic composition gives the eutectic point. Not all binary alloys have a eutectic point; for example, in the silver-gold system the melt temperature (liquidus) and freeze temperature (solidus) both increase monotonically as the mix changes from pure silver to pure gold.

The eutectic reaction is defined as follows:

        This type of reaction is an invariant reaction, because it is in thermal equilibrium; another way to define this is the Gibbs free energy equals zero. Tangibly, this means the liquid and two solid solutions all coexist at the same time and are in chemical equilibrium. There is also a thermal arrest for the duration of the reaction.

        The resulting solid macrostructure from a eutectic reaction depends on a few factors. The most important factor is how the two solid solutions

nucleate and grow. The most common structure is a lamellar structure, but other possible structures include rodlike, globular, and acicula