Modulus of Elasticity

ByPavithran S. IyerRohan SinhaCMI

• An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i.e., non-permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region.

Modulus of ElasticityYoung’s Modulus

• where λ (lambda) is the elastic modulus; stress is the force causing the deformation divided by the area to which the force is applied; and strain is the ratio of the change caused by the stress to the original state of the object. If stress is measured in pascals, since strain is a unit--less ratio, then the units of λ are pascals as well. An alternative definition is that the elastic modulus is the stress required to cause a sample of the material to double in length. This is not realistic for most materials because the value is far greater than the yield stress of the material or the point where elongation becomes nonlinear, but some may find this definition more intuitive.

Young's modulusYoung's modulus ( (EE) describes tensile elasticity, or the ) describes tensile elasticity, or the tendency of an object to deform along an axis when tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred the ratio of tensile stress to tensile strain. It is often referred to simply as the to simply as the elastic moduluselastic modulus. .

Modulus of ElasticityYoung’s Modulus

When the metal is devoid of force, the molecules are properly aligned.

Modulus of ElasticityYoung’s Modulus

EXTENSION CAUSED DUE TO FORCE, ON AN ELASTIC MATERIAL

THE FIGURE ABOVE SOWS A SECTION OF THE ATOMIC LAYER IN THE ELASTIC MATERIAL

Force

EXTENSION CAUSED DUE TO FORCE, ON AN ELASTIC MATERIAL

THE ABOVE DIAGRAM SHOWS THE ACTION OF FORCE, AT A POINT ON THAT MATERIAL. NOTE: THE ELASTIC NATURE OF THE MATERIAL IS ONLY SEEN WHEN THE FORCES ON DIFFERENT POINTS ON THE MATERIALS ARE DIFFERENT.

EXTENSION CAUSED DUE TO FORCE, ON AN ELASTIC MATERIAL

BUT WHEN FORCE IS APPLIED, THE MOLECULES START SHIFTING AND ON THE WHOLE, THE MATERIAL APPEARS TO BE BENT.THE ELASTIC MATERIAL BENDS, AT THE POINT OF APPLICATION OF FORCE. THIS LENGTH THROUGH WHICH IT BENDS, OR THE HEIGHT OF THE DEPRESSION IS WHAT WE ARE TRYING TO DETERMINE IN THIS EXPERIMENT.

Let’s take the case of a metal strip, where the thickness is very small compared to the length. Let’s take the bending at a distance x from one end and let the shear angle be ‘φ’.

Modulus of ElasticityYoung’s Modulus

• The new depth or bend of the material can be calculated from the Formulations as before as:

Where:

L: Depth

L: Length of the scale between knife edges.

B: Width of the scale

a: Thickness of Scale

Fy: Vertical Force

E: Young’s Modulus of Elasticity

RELATING THE EXTENSION WITH OTHER EXPERIMENTAL PARAMENTERS – A FORMULA

The arrangement of the apparatus can be shown with the help of the following schematic diagram.

The metal strip is kept completely symmetrical with respect to the knife edge so that the torques on either sides balance out. (otherwise they provide a force that counteracts the weight of the mass hung).

The weighs are loaded at the centre since maximum extension occurs there, and errors can be made small. The knife edge on which the mass was hung was ensured to be balanced.

In case of zero error on the dial gauge, the initial reading was taken.

EXPERIMENTAL SET UP AND UNDERSTANDING THE PROCEDURE

•The components are fitted to the correct apparatus.•The knife edges are made to be at the same height using a spirit balance.•The same procedure is repeated for the square-rod holding the dial gauge.•Both the rods (the experimental rod and the rod on which the dial gauge is mounted) are made to be parallel to each other.•The axis of the Dial Gauge is made to be perfectly vertical by rotating the circular rod to which it is attached. If this is not so, then the extension on the rod will not be equal to the reading shown on the dial.•Now, the rod on which measurements are made is kept symmetrically, i.e. the centre of the Strip is at the centre between the knife edges.•Now the mass hanger is kept at the centre and the hanger is attached onto it.•Now the Dial Gauge is brought down onto the knife-edge, so that it is perpendicular to the strips and hence vertical.•The Readings are taken by subtracting the reading before the masses were hung and after the masses were hung.•Now, the necessary variations are made.

BRIEF PROCEDURE AND UNDERSTANDING

Marking the Dial Gauge Readings, we obtain the Shear-Strain Graph as shown (in Black).The ideal Stress-Strain Relation has been shown in green. This is because of the errors in the experimental procedure followed. Since the analog meter showing the extensions was very-very sensitive, any mild disturbances changed the reading of themeter, and the original value could not be reached again. Even while loading the masses a lot of care had to be taken to see that there is no extra push from the loading process. This can cause permanent shift in the extensions.

STRAIN

STRESS

PRECAUTIONS WITH DIAL GAUGE

The following graph shows change in extension observed with the mass attached. As per the Hooke's law, all points must lie in a straight line, but due to the hysteresis observed (the extension dial) while loading and unloading the mass, some random errors had crept in.

OBSERVATIONS – EXTENSION vs. MASS

CALCULATING THE MODULUS OF RIGIDITY FROM DATA

FROM THE ABOVE DATA COLLECTED (IN THE PREVIOUS GRAPH), THE RIGIDITY MODULUS WAS COMPUTED.FOR THE STEEL RODS. ALSO FOR OTHER RODS SUCH AS ALUMINUM AND BRASS ALSO THE SAME WAS COMPUTED.GIVEN BELOW IS A TABLE SHOWING THE COMPUTED VALUE OF THE RIGIDITY MODULUS AND THE CORRESPONDING THEORETICAL VALUE (THAT WAS GIVEN)

MATERIAL EXPERIMENTAL (N/m-2)

THEORITICAL(N/m-2)

STEEL (10X2X500) mm 4.40 X 1010 2.06 X 1010

ALUMINUM 4.22 X 1010 6.70 X 1010

BRASS 5.92 X 1010 9.22 X 1010

Below graph shows the change in Extension for different lengths of the rods used. The effects of hysteresis here appear magnified because the relation is a cubit one. So, any error, in principle will get magnified three times. But one thing that is clear is that it is not a linear relationship. This agrees with the formula that was given earlier.

OBSERVATIONS – EXTENSION VS. LENGTH

OBSERVATIONS – EXTENSION VS. WIDTHThe graph below shows the change in extension with the width of the steel rod used. The relation, as per the formula is inverse cubic. Here we see that there is some considerable error. It can be attributed to systematic error on the dial. Here each time, we need to remove the rod and fix the masses again. Due to this frequent loading and unloading, it must be seen that after each unloading the dial is reset. In case of any hysteresis, the initial reading must be repeated.

OBSERVATIONS – EXTENSIONS AT VARIOUS POINTS OF THE RODIn this section, our intention was to determine the extension caused in the rod, at points further away from the point of suspension of the mass. As expected, we have the greatest extension at the point of suspension of mass, and the extension was seen to decrease linearly with distance from the center.

From this experiment, the relation between various parameters like Force, extension, and the properties of the rod were verified. A major difficulty faced by us during the experiment was that every time the masses were loaded and unloaded from the hanger, there was some hysteresis observed in the readings. Due to this many of the readings did not exactly follow the desired relationships. Also, the nature of the extensions caused all across the rod keeping the point of application of force constant was also observed.

CONCLUSIONS AND INFERENCES