Modulus of Rigidity Lab
CALCULATING MODULUS OF RIGIDITY
Modulus of Rigidity
Using the initial length, radius, measurements of torque and angle of twist, it is possible to calculate the Modulus of Rigidity for each specimen. The modulus of rigidity or shear modulus represents the ratio of the shear stress to the shear strain. It is denoted ‘G’.
To calculate two separate values of G for each specimen, it is possible to use two methods.
It is important to determine the polar moment of inertia to be used throughout the experiment:
For the Aluminum sample:
For the Brass sample:
Aluminum:
The first equation utilizes the internal torque (Tint) and angle of twist (Φ) on the fine scale. They are related in the following way with the polar moment of inertia (J) and initial length (L0) of the specimen:
As the initial length and polar moment of inertia remain constant throughout the experiment, G then varies according to the ratio of the torque to the angle of twist on the fine scale. This relationship was shown in FIGURE ……. in the linear elastic region. Therefore, it is now possible to calculate the value of G from the raw data taken in the experiment. A number of G values are found, and of these values, an average is taken: 24088909281 Pa or 24.09 GPa.
It is possible to calculate the shear modulus using a second method/equation. In this method, the ratio of Hooke’s law is used. This involves the ratio of the maximum shear stress to the angle gamma. The maximum shear stress is calculated using the torque, initial radius and the polar moment of inertia.
As the torque varies, so does the maximum shear stress.
The angle gamma is calculated using the initial radius, the angle of twist in radian meters and the initial length of the specimen.
As the angle of twist varies, so does the angle gamma.
As G is the ratio between the shear stress and shear strain:
Due to the numerous values of maximum shear stress and gamma, their relationship is
Modulus of Rigidity
Using the initial length, radius, measurements of torque and angle of twist, it is possible to calculate the Modulus of Rigidity for each specimen. The modulus of rigidity or shear modulus represents the ratio of the shear stress to the shear strain. It is denoted ‘G’.
To calculate two separate values of G for each specimen, it is possible to use two methods.
It is important to determine the polar moment of inertia to be used throughout the experiment:
For the Aluminum sample:
For the Brass sample:
Aluminum:
The first equation utilizes the internal torque (Tint) and angle of twist (Φ) on the fine scale. They are related in the following way with the polar moment of inertia (J) and initial length (L0) of the specimen:
As the initial length and polar moment of inertia remain constant throughout the experiment, G then varies according to the ratio of the torque to the angle of twist on the fine scale. This relationship was shown in FIGURE ……. in the linear elastic region. Therefore, it is now possible to calculate the value of G from the raw data taken in the experiment. A number of G values are found, and of these values, an average is taken: 24088909281 Pa or 24.09 GPa.
It is possible to calculate the shear modulus using a second method/equation. In this method, the ratio of Hooke’s law is used. This involves the ratio of the maximum shear stress to the angle gamma. The maximum shear stress is calculated using the torque, initial radius and the polar moment of inertia.
As the torque varies, so does the maximum shear stress.
The angle gamma is calculated using the initial radius, the angle of twist in radian meters and the initial length of the specimen.
As the angle of twist varies, so does the angle gamma.
As G is the ratio between the shear stress and shear strain:
Due to the numerous values of maximum shear stress and gamma, their relationship is