**Objective**

-To measure strains on a mild steel specimen loaded in tension (shown below) within elastic limit upto 3000N and obtain the stress strain plot (shown in fig below) for the specimen

-Verify the linear stress-strain relation, and find the slope of stress-strain graph and hence the Young's modulus in a Universal Testing Machine. NOTE: Significant well-correlated readings are obtained only beyond loads of 600-700N, as true material response is obtained only then.

**Introduction**

When stress and strain (refer "Introduction to MoS") are plotted, a curve as shown in figure 1 is obtained. The slope of the Stress strain curve in elastic region is called Young's Modulus.

__The Stress-strain diagram__

The stress strain relationship of any material is of primary importance as it gives a good idea of the mechanical behaviour of the material in real life conditions. This is generally accomplished using the tension-compression tests. When calculating the nominal or engineering stress, we assume that the stress is constant over the entire cross section of the specimen’s central portion along the gage length. Thus,

Where, *A*_{0} is the area of cross-section and *P *is the applied load.

Similarly, the nominal or engineering strain (ϵ) is found using the strain gage reading or by diving the change in gage length (ΔL) by the original gage length of the specimen. After this, the values of stress and corresponding ϵ are plotted graphically with strain values on the x-axis. This diagram is known as the stress-strain diagram.

In this experiment, we will be performing tensile test on the specimen to determine its Young’s modulus.

In the stress strain curve, from the origin to a certain limit, the change in strain is proportional to the stress which produces a linear straight line in the initial portion of the stress strain curve. This region is said to obey Hooke’s law. (see figure 1) For glass, at room temperature, this holds true for the entire region. For material such as concrete, there is hardly any region where it exists. For mild steel, it is true up to some point. The point up to which the material obeys Hooke’s law is termed as the yield point. For mild steel, this point can be clearly observed and a yield plateau follows it where strain happens for little or no stress. Then the region of strain hardening follows till the point of ultimate tensile strength which is practically the highest value of stress the specimen can endure. Necking follows this where the specimen displays significant lateral strain.

*Figure 1: Stress strain curve*

Symbolically, Hooke’s law can be expressed as,

Where the constant of proportionality is Young’s Modulus or Modulus of Elasticity represented as E. Since strain is dimensionless, E has the same units of stress which are newtons per square meter (Pa, pascals - SI units) or pounds per square inch (US standard).

Physically, E represents the stiffness of the material. It can be obtained from the stress strain graph by measuring the slope of the initial linear portion. For all steels, the value of E ranges from 200 to 210 GPa. This is generally defined for isotropic materials where the properties remain constant when considered across the entire cross section of the specimen in all directions. For anisotropic materials such as wood, the properties vary a lot and this is not true. For linear elastic material, the loading and unloading happens across the same path when tested up to the elastic point. Beyond this, a phenomenon called hysteresis (figure 2) comes into play where the return loop is offset by some amount. When solids are put into tension or compressed, there is an overall deformation in the transverse as well as lateral dimensions. That is, if the solid is compressed, the transverse dimension reduces but the lateral dimension increases. Similarly, if the solid is put into tension, the lateral dimension decreases and transverse dimension increases.

*Figure 2: Hysteresis*

__The Tension test__

*Fig 3: Specimen after(left) and before(right) Tension test*

*Fig 4: fig(a): general curve of materials; fig(b): mild steel; fig(c): rubber*

The image shown above is of a mild steel specimen before and after tension test. The scale is in centimetres. Fig 4 (a) is a general stress strain curve, fig (b) is for mild steel and fig (c) is for rubber.

The tension test is carried out using a Universal Testing Machine (UTM) which can test the specimen by putting it in tension or compression. The American Society for Testing and Materials (ASTM) has published standard guidelines for performing Tension tests on sample specimens that will help determine the application limits of materials. ASTM E8 lays out tension testing methods to determine the tensile strength, yield strength, yield point elongation, elongation and reduction area of various metal samples.

While conducting a tension test, the portion of the specimen with uniform cross section is marked out as gage length. The change in gage length is recorded as a function of the applied load on the specimen. To measure this change in gage length, we can use an extensometer or a wire strain gage. The ratio of increase in length (ΔL) to the original length (L) is termed as the strain. These electric strain gages are generally expendable in nature. These tiny strain gages are glued on to the surface of the specimen in consideration. As the specimen elongates or contracts on application of the load, concurrently the strain gage is also affected and this change in length of the thin wires alter the resistance, which can be measured to get the magnitude of the change in the physical dimension of the specimen.

Normal strain is the strain that is normally associated with normal stress. Since strain is a ratio of length by length, it is a dimensionless quantity. But still the value of strain is measured as microstrain (μm/m), mm/mm, in/in or m/m. Generally, the quantity of strain (ϵ) is in the order of magnitude 0.1%. In our experiment, we make use of two strain gages, one placed vertically on one side of the specimen which measures the vertical elongation and one on the other side which measures the horizontal strain. Both the strain gages are placed at the centre of the gage length.

*Fig 7: Young's modulus test specimen with strain gages attached on to it. The strain values are obtained from the vertical strain gages for this experiment*

__Specimen details__

Material : Mild steel

Specimen test section width: 22 mm

Specimen gage length: 99 mm

Specimen thickness: 3.4 mm

The load applied by the UTM is measured using a Load Cell which is a force transducer. The transducer when subjected to an external load deforms a strain gage within. The deformation is reflected as a change in the electrical resistance. Depending on the number of strain gages used, the load cell is categorised as Quarter Bridge (one strain gage), half bridge (two strain gages) or Wheatstone bridge (four strain gages). The output is generally measured in millivolts which can be amplified using instrumentation. The same principle is used in a piezoelectric load cell where a piezoelectric material is used. Here, the deformation on the material produces a voltage output. Further, various types of other load cells are also in use such as capacitive load cell, hydraulic load cell and pneumatic load cell.

*Fig 8: The 50kN capacity electrical Universal testing Machine used in Amrita Virtual Lab*