**Objective**

-To measure the lateral strain and longitudinal strain of a mild steel specimen under tension in the linear elastic range of loading upto 1500N.

-To calculate its ratio, by obtaining plot of longitudinal and lateral strains between any two points, hence obtaining the Poisson's ratio.

**Introduction**

When a component is being subjected to tension in one direction (elongated), it gets compressed in its perpendicular direction. For example, when a cylinder is pulled along its axis, its length increases, but its diameter decreases. The ratio of this lateral to longitudinal strain is defined as Poisson's ratio.

__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.

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

Poisson's ratio, named after Siméon Poisson, also known as the coefficient of expansion on the transverse axial, is the negative ratio of transverse to axial strain. 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 (nu) is a measure of this effect. The Poisson ratio is the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes.

*Figure 1: Poissons ratio explained*

Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. This is a common observation when a rubber band is stretched, when it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion, and will have the same value as above. In certain rare cases, a material will actually shrink in the transverse direction when compressed (or expand when stretched) which will yield a negative value of the Poisson ratio.

The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than −1.0 nor greater than 0.5 due to the requirement that Young's modulus, theshear modulus and bulk modulus have positive values. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (beforeyield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume. Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0: showing very little lateral expansion when compressed. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these auxetic materials are stretched in one direction, they become thicker in perpendicular direction. Some anisotropic materials have one or more Poisson ratios above 0.5 in some directions.

*Figure 2: Poisson's ratio*

__The Tension test__

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

The image shown above is of a mild steel specimen before and after tension test. The scale is in centimetres.

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 6: Poisson's ratio test specimen with strain gages attached on to it. The strain values are obtained from both the vertical and horizontal strain gages for this experiment. The strain values from horizontal strain gage gives lateral strain readings and strain values from vertical strain gages give longitudinal strain readings*

__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.