Stress-Strain Relation

Mechanical Properties:

Strength of material depends on the ability to sustain load without deformation and failure.

• Measure of Stress  ⇔  defines the Strength of material.
• Measure of Strain  ⇔  gives magnitude of Deformation.

Applications:

1. Aircraft Manufacturing  ⇔  Aluminium alloys or Carbon-reinforced composites are used (because of light weight, strength and able to withstand cyclic mechanical loading).
2. Honey based composites.
3. Steel used in building (have adequate strength).
4. Bio-compatible Titanium alloy  ⇔  for Bone implantment (have strength and toughness).
5. Scratch Resist coating  ⇔  resist abrasion on Optical lenses.

Modes of Failure:

1. Abrasion  ⇔  Damaging material by means of rubbing.
2. Erosion  ⇔  Removing of surface particles due to the hitting of high velocity particles (which transfer K.E.).
3. Corrosion  ⇔   Oxidation of metal.

Steps for Selection of Material:

Before selecting any material ask yourself following questions:

• Should it be strong, stiff (brittle) or ductile (soft)?
• Will it be subjected to high stress or sudden impact force?
• Micro-structure changes due to temperature?
• Will it bear cyclic nature of stresses (tensile or compressive)?
• What changes occur due to corrosion, erosion or abrasion?
• What are the effects of creep, accidental load or thermal vibrations on material? 
• What are the effects produce during machining and fabrication?

Testing for Finding Mechanical Properties:

These test are conducted using Standard method that generally defines:

• Geometry  ⇔  dimensions, geometry of cross-sections (circle or rectangle).
• Load Rates  ⇔  rate of load increases slowly.
• Boundary Conditions  ⇔  degree of freedom (moment = 0, displacement x,y ≠ 0).

There are different test for finding mechanical properties:

1. Tensile Test  ⇔  Tensile load applied normal to surface to increase its length.
2. Compressive Test  ⇔  Compressive load applied normal to surface to decrease its length.
3. Impact Test  ⇔  Sudden load is applied to material. Impact energy is absorbed by material before failure gives Toughness of material.
4. Torsional Test  ⇔  Apply twisting force.
5. Bending Test  ⇔  Apply bending force.

The standard values are defined by different organizations like American Society for Testing Materials (ASTM), International Standard Organization (ISO). For Example:

• ASTM E8-09  ⇔  Tensile testing for metallic materials.
• ASTM E9-09  ⇔  Compression test for metallic materials.

1. Tensile Testing:

Stress-strain relation defines the load bearing capability without excessive deformation using Standard Specimen which contains

1. Large diameter cross-section  ↠  called Grips (we don't want specimen to fail from grip so increase its diameter thereby decrease in stress), it has threads and rough area called Knurling.
2. Small diameter cross-section  ↠  have large stresses, it has markings (distance between markings are called Gauge Length).

In Tensile testing, Universal Testing Machine (UTM) is used which consist of:

• Movable Upper and Fixed Lower crossheads 
• Jigs or fixtures  ↠  on which we mount tension specimen.

The extension (elongation) is calculated by extensometer. Tensile test consists of gradually loading and noting Load-Extension values until fracture happens. This curve is called as Stress-Strain Diagram.

Tensile failure for Ductile material is Cup & Cone Failure (Necking) and that of brittle is straight failure.

Different tensile/compressive tests for particular material gives almost similar properties but not yields exactly same stress-strain curve because:

1. Material Composition (Non-homogeneous) ↠  different composition during (outer surface reacts with atmosphere which is different from that is in middle).
2. Microscopic Imperfection  ↠  voids, gaps and impurities.
3. Manufacturing Operations  ↠  rolling, forging (if in direction of rolling is different to that of perpendicular to rolling direction).
4. Rate of Loading  ↠  if not follow gives loading results (human error).
5. Temperature  ↠  if temperature changes during experiment.

2. Compression Testing:

Compression testing is just like Tensile testing but compression fixtures are used rather than tensile fixtures. Two geometries are used for Compression testing which are:

1. Block  ↠  having dimensions 0.5 inch x 0.5 inch x 1 inch.
2. Cylinder  ↠  having dimensions diameter = 0.5 inch, length = 1 inch.

Compression failure for Ductile material is Buldging and that of brittle is straight cracks. If compression is applied to:

• Brittle Material (Gray Cast Iron)  ↠  slip of plane at 45-degrees (max. shear).
• Ductile Material (Hot Rolled Steel)  ↠  buldging (not slip).
• Brittle, Ductile Material (Aluminium Alloy)  ↠  Two steps: first is buldging and second is slip of plane at 45 degrees.

3. Impact Testing:

This test is done using two geometries:

1. Charpy  ↠  we lift hammer at height ho and release it which hit specimen and divided into 2 pieces. Dimensions are 10 mm x 10 mm x 55 mm. We get notch in middle at 45 degrees and 2 mm deep.
2. Izod  ↠  lift hammer at height ho and release it which hit specimen and divide it into 2 pieces. Dimensions are 10 mm x 10 mm x 75 mm. We get notch not in middle but at 45 degrees and 2 mm deep.

• Difference between charpy and izod is on what geometry we place material.
• Aluminium shows more impact strength (area under the curve in stress-strain diagram).
• Brittle fracture is a straight fracture.
• Ductile fracture absorb more energy before failure (high toughness).

Stress-Strain Diagram (for Mild Steel):

There are two regions of stress-strain diagram:

1. Elastic Region  ↠  Region in which material regain its initial state without permanent deformation.
2. Plastic Region  ↠  material when unloaded from initial load in plastic region, elastic strain is recovered but plastic strain remains in the material.

Some important points are:

• 0.2 % Offset method is used to find the yield point (where curve starts bending.  
• Permanent Set or Strain Hardened  ↠  When plastic strain remains in material and elastic strain is recovered when loaded in plastic region. 

Different terms are discussed ahead:

• Engineering or Nominal Stress  ↠  ratio of force applied to the original area.
• True Stress  ↠  ratio of force applied to the instantaneous area.
• Young's Modulus (Modulus of Elasticity)  ↠  It is the slope of elastic region.
• Hooke's Law  ↠  stress is directly proportional to the strain within proportional limit.
• Yielding  ↠  Phenomenon in which stress is constant and strain is increasing.
• Strain Hardening  ↠  hardening of material i.e. jam-packing (interlocking) of atoms with one another decreasing its area. 
• Ultimate Stress Point  ↠  maximum load point.
• Necking  ↠  the decrease in the area of material after maximum load point is achieved.
• Modulus of Resilence  ↠  energy stored by material till elastic limit.
• Modulus of Toughness  ↠  energy stored by material before fracture.

Ductile Material:

Steps of failure of ductile material under tensile load are discussed ahead:

1. Application of uni-axial tensile load.
2. Activation of dislocations.
3. Dislocations movement opposite to material flow.
4. Dislocations pile-up.
5. Formation of micro-voids and necking.
6. Coalescence of voids (sum of micro-voids).
7. Formation of crack (sum of coalescence).
8. Ductile Fracture (cup and cone fracture, propagation of sum of cracks).

Dislocations:

Dislocation is defined as, Permanent deformation in metals due to the movement of dislocations.

• In point defects, movement of dislocations is the movement of vacancy or small substitutional atom.
• If plane is absent (edge dislocated line)  ↠  whole plane is dislocated.

Dislocation Interactions

1. Annihilation  ↠  occurs at dislocation barriers where dislocation are immobile with respect to glide.
2. Interlocking  ↠  occurs at dislocation barriers where dislocation are mobile with respect to glide.
3. Glide  ↠  dislocation motion along a crystallographic direction.
4. Pileup  ↠  occur in crystals when a number of similar dislocations gliding in a common slip plane.

Ductility of Material:

Ductility of material is defined as, Material subjected to large strains before it fractures.

Strain Energy:

Energy stored in a body due to its strain deformation is called Strain Energy.

True Stress-Strain Formula:

True or Engineering Stress is found by using the instantaneous area which is given by using Logic of Volume Consistency:

Af = Ao * Lo / Lf 

Ques: Why mild steel has two yield points?

Mild steel (ductile material) has two faces upper and lower or harder and softer. Dislocations on harder faces are locked and require more energy. Whereas for softer ones, it require less energy.

Brittle Material:

Materials that exhibit little or no yielding before failure are called Brittle materials. Since there is no movement of dislocations, therefore it directly breaks.

• If you want to increase strength, compromise on ductility or vice versa.
• Aluminium deform elastically 3 times that in Steel.
• Addition of carbon makes steel stronger.
• Under impact loading, brittle material absorbs less energy than ductile material due to area under the curve.
• Bumper of cars are made of ductile material (usually plastic) to absorb more energy.
• Failure  ↠  material not performing desired function. Failure point for ductile material is Yield point and for brittle point is Fracture point (Max. load point).
• Fracture  ↠  Rupture or breaking into two or more parts.

Fracture of Brittle Material:

Steps of failure of brittle material under tension are discussed ahead:

• Brittle material has little or no plastic deformation.
• Initiation of cracks at small flaws (cracks) which results in stress-concentration.
• Crack propagation rate approaching the velocity of sound in metals which move along crystallographic plane {100} by cleavage.
• Fracture surface is flat and perpendicular to the applied stress.
• Cherron Pattern  ↠  Pattern produced by separate crack proportional at this different levels in the material.

Fractures:

1. Ductile Fracture 
2. Brittle Fracture
3. Inter-granular Fracture  ↠  Fracture on the surface of grains. Crack propagates along the grain boundaries.
4. Trans-granular Fracture  ↠  Fracture within the grains.

Shear Stress-Stain Diagram:

Some important points are:

• Shear Stress  ↠  Application of stress parallel to surface.
• Shear Strain  ↠  It is the angular deformation.
• Modulus of Rigidity (G)  ↠  It is the resistance to shear or deformation.
• Stronger materials have higher shear modulus.

Poisson's Ratio:

• A material property.
• Elongation produced by axial tensile force 'P' (in the direction of force) is accompanied by a contraction in transverse direction.
• Poisson Ratio is defined as the ratio of lateral (or compressive) strain to axial (or tensile) strain.

Consider an element subjected to uni-axial stress.

Equations of Normal and Shear stress-strain component is given by: 

• Lame constant is a mathematical constant.
• Bulk Modulus  ↠  It is the inverse of compressibility (or difficulty in compressing material).

Plane Stress and Plane Strain:

In many practical situations stress component along z-direction is zero and is referred to as Plane Stress.  

• Assumption of plane stress and strain is to make our system simpler.

If strain in one direction is zero then it is referred to as Plane Strain.

• In order to make strain along z-direction zero  ↠  apply load opposite to generated stress.
• Zero strain does not mean zero stress in that direction.
• Plane strain assumption is usually applied on thick cross-section.

Crack Propagation:

Lets take an example of brick. Brick is stronger and the cemented bonding is weaker. So crack propagate through cemented portion.

In order to stop crack propagation, bricks are placed such that when crack propagates it finds harder face (brick). Hence crack stop propagating.

Stress Concentration:

If you sharp pencil such that its area become smaller then after small application of small force, it break due to stress concentration.

If we have a small area, forces break surfaces due to stress concentration. To save it from breaking, we usually blunt surfaces (removing of corners). 

• Aluminium Material that changed the world.
• History of Iron and Steel.

References:

• Material from Class Lectures + Book named Engineering Mechanics of Materials by R.C. Hibbeler (10th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.