Stress-Strain Diagram

The behavior of any material under loading can be analyzed using stress strain curve of the material. 

To draw the stress - strain diagram, the testing metal specimen be placed in U.T.M (universal testing machine) and applied an axial load. As the axial load is gradually increased in increments, the total elongation over the gauge length is measured at each increment of the load and this is continued until failure of the specimen takes place. 

Knowing the original cross-sectional area and length of the specimen, the normal stress and the strain can be obtained.The graph of these quantities with the stress along the y-axis and the strain along the x-axis is called the stress-strain diagram.

Engineering stress-strain curve

The curve based on the original cross-section and gauge length is called the engineering stress-strain curve, while the curve based on the instantaneous cross-section area and length is called the true stress-strain curve. Unless stated otherwise, engineering stress-strain is generally used. 

The stress-strain diagram differs in form for various materials. 

Stress-strain diagram of a mild steel as shown in the below figure. OE curve is representing engineering stress strain curve and OE' curve is representing true stress strain curve. 

Proportional Limit (Hooke's Law) 

From the origin O to the point A called proportional limit, the stress-strain curve is a straight line. This linear relation between elongation and the axial force causing was first noticed by Sir Robert Hooke in 1678 and is called Hooke’s Law that within the proportional limit, the stress is directly proportional to strain.  

The constant of proportionality E is called the Modulus of Elasticity or Young’s Modulus and is equal to the slope of the stress-strain diagram from O to A. 

Elastic Limit 

The elastic limit point is the limit beyond which the material will no longer go back to its original shape when the load is removed, or it is the maximum stress that may be developed such that there is no permanent or residual deformation when the load is entirely removed.

Elastic and Plastic Ranges 

The region in stress-strain diagram from O to A is called the elastic range. The region from A to E is called the plastic range. 

Yield Point 

Yield point B is the point at which the material will have an appreciable elongation from B to C or yielding without any increase in load.The stress corresponding to this point is called Yield stress. 

Ultimate Strength

The maximum ordinate in the stress-strain diagram at point D is the ultimate strength or tensile strength. 

Rapture or Fracture Strength 

Rapture strength is the strength of the material at rupture at point E. This is also known as the breaking strength.

Strain Hardening

After a material yields, it begins to experience a high rate of plastic deformation. Once the material yields, it begins to strain harden which increases the strength of the material. In the stress-strain curve, the strength of the material can be seen to increase between the yield point B and the ultimate strength at point D. This increase in strength is the result of strain hardening.

Necking

after the ultimate strength at point D, the increase in strength due to strain hardening is outpaced by the reduction in load-carrying ability due to the decrease in cross sectional area. Between the ultimate strength at point D and the fracture point E,the engineering strength of the material decreases and necking occurs.

Proof stress or Offset Yield stress

Proof stress is the stress that is just sufficient to produce under load, a defined amount of permanent residual strain, which a material can have without appreciable structural damage.

In some materials the location of yield point is difficult to determine. In such a case, the offset yield point (or proof stress) is taken as the stress at which 0.2% plastic deformation occurs as shown in the above figure.

Working Stress or Allowable Stress

Working stress is defined as the actual stress of a material under a given loading. The maximum safe stress that a material can carry is termed as the Allowable stress.

The allowable stress should be limited to values not exceeding the proportional limit. However, since proportional limit is difficult to determine accurately, the allowable stress is taken as either the yield point or ultimate strength divided by a factor of safety. 

Factor of Safety 

The ratio of the strength (ultimate or yield strength) to allowable strength is called the factor of safety.

For Ductile materials,
Factor of safety = Ultimate Stress / Allowable stress
For Brittle materials,
Factor of safety = Yield Stress / Allowable stress