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Shear Behavior Analysis: Modulus of Rigidity in Torsion Testing of Steel, Brass, and Aluminum

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Objectives:

Establishing the correlation between torque (T) and shear stress (τ) with the angle of twist (θ).
Determining the modulus of rigidity (G) of the material.
Identifying the maximum shear stress at both the elastic limit and material failure.

Theory:

General Overview:
- In the elastic range, yielding initiates in the outer fibers, progressing towards the core as twisting continues.
- Plastic deformation extends through the cross-section, accompanied by gradual work hardening.
- Unlike necking, there is no abrupt drop in the curve.
Torsion and Shear Deformation:
- Torsion, a form of shear, is common in machine axles, drive shafts, and twist drills.
- Circular shafts maintain undistorted cross-sections during torsion due to axisymmetry.
Stress, Strain, and Twist Angle:
- A solid circular shaft under torque (T) twists at its free end, causing shear strain represented by the angle of distortion (θ).
- Shear stress (τ) is related to the modulus of rigidity (G) through the equation τ/r = Gθ/L.
Equations and Relationships:
- Torque (T) is related to shear stress and geometry by T/J = τ/r = Gθ/L.
- The simple theory of torsion combines shear stress, torque, and geometry.
- Shear stress in a shaft under pure torsion is given by τ = (Gθ/L)r.
Linear Variation of Shear Stress and Strain:
- Shear stress (τ) and shear strain vary linearly with the radius, reaching maximum values at the outer radius.
- The equation τ = (Gθ/L)r = Gγ relates shear stress, shear strain, and the angle of twist per unit length.

By elucidating these relationships, the theory provides insights into the behavior of materials under torsional deformation, facilitating the experimental objectives.

Specimens and Equipment:

Torsion testing machine – Norwood 50 Nm
Vernier caliper
Torsion specimens: steel, brass, and aluminum

Procedures:

Measure the initial length and gauge length diameter of the specimen.
Mount the specimen between the loading device and torque measurement unit in the hexagon sockets.
Align the specimen by turning the hand-wheel as required.
Adjust the tailstock unit to fully insert the specimen into the hexagon sockets.
Ensure there is no preload on the specimen.
Zero the pointer on the protractor scale at the zero-degree point.
Adjust the digital torque meter to read zero.
Slowly turn the hand-wheel clockwise to load the specimen, turning only for a defined angle increment.
Read the torque value from the digital torque meter, along with the indicated angle of twist.
Repeat steps 8 and 9 until fracture occurs.
Repeat the experiment for other specimens.

JT=LGθ=rτ G=LkJ Where:

T is Torque (Nm)
is Shear stress (Nm−2−2) at radius r
J is Polar moment of area (m4)
θ is Angle of twist (radius, over length L)
G is Shear modulus (MNm−2−2)
L is Length of the bar (m)
r is Radius (m)

The modulus of rigidity (G) can be calculated using the formula G=LkJ, where k is the gradient of the graph depicting torque (T) versus angle of twist (θ).

Comparing the experimental results with the theoretical expectations, we observed torque-angle graphs for different materials.

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Initially, a positive gradient in the elastic region indicated that specimens returned to their original shape when torque was removed.

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As torque increased, the materials entered the plastic region, leading to plastic deformation and eventual fracture.

For Aluminium, the twist angle was proportional to torque due to its ductile nature, but its graph did not exhibit a typical shear stress–strain curve, displaying mainly brittle behavior. Similar trends were observed for Brass and Steel. The shear modulus values obtained experimentally deviated from theoretical values, suggesting imperfections in the materials used.

Discussing the mechanical properties in shear, the shear stress, shear stress proportional limit, ultimate shear stress, and modulus of rigidity were examined. Steel exhibited the highest values, indicating greater brittleness, while Aluminium showed the highest ductility.

Several factors affected experimental results: aged and rusted equipment causing friction, manual application of loads leading to variations, impurities in specimens differing from theoretical purity, irregular torsion force application, parallax errors in measurements, and limitations in time intervals for readings. These factors contributed to discrepancies between experimental and theoretical values.
Furthermore, in the torque-angle graphs, the continuous increase in twist angle even with small increments in torque indicated the materials' deformation behavior. Notably, the deviation from a straight line in the graphs for Aluminium, Brass, and Steel suggested variations in their shear characteristics, with the materials displaying unique behaviors under torsional stress.

Examining the mechanical properties in shear, it was evident that Steel possessed the highest shear stress, shear stress proportional limit, ultimate shear stress, and modulus of rigidity among the tested specimens. This emphasized its inherent brittleness compared to the other materials. Conversely, Aluminium exhibited lower values, highlighting its superior ductility in shear deformation.

Considering the factors influencing experimental outcomes, the impact of old and rusted equipment on friction was particularly significant. The manual application of loads introduced variability, making it challenging to maintain constant torque. Impurities in the specimens and structural imperfections further contributed to discrepancies between theoretical and experimental results.

The control and uniform application of torsion force emerged as a critical factor affecting the experiment. Deviations in the force application, especially due to manual handling, resulted in non-uniform torsion cycles and influenced the final results.

Parallax errors were acknowledged as a source of measurement inaccuracies, particularly in instruments like vernier calipers and protractors. Additionally, the tight time intervals between consecutive readings posed challenges, affecting the accuracy of measurements due to the rigidity of the torsion machine and the need for rapid data acquisition.

In conclusion, while the experimental results provided valuable insights into the shear behavior of Aluminium, Brass, and Steel, discrepancies from theoretical expectations indicated the importance of addressing equipment condition, load application methods, specimen purity, and measurement accuracy in future experiments.
In this experiment, the mechanical characteristics of specimens, including shear stress, shear stress proportional limit, ultimate shear stress, and modulus of rigidity, are determined. When subjected to torque, the specimens undergo torsion until they reach the point of fracture. Additionally, the modulus of rigidity (G) can be calculated by analyzing the shear stress vs. shear strain graphs. The results indicate that steel exhibits the highest modulus of rigidity, followed by brass, while aluminum has the lowest. This suggests that steel is the most brittle, whereas aluminum demonstrates the highest ductility.

The torsion test on materials used in mechanical structures serves to replicate real-world operational conditions. Manufacturers can assess product quality, validate designs, and ensure proper manufacturing techniques by understanding shear stress, shear stress proportional limit, ultimate shear stress, and modulus of rigidity for these materials.

Shear Behavior Analysis: Modulus of Rigidity in Torsion Testing of Steel, Brass, and Aluminum. (2024, Feb 28). Retrieved from https://studymoose.com/document/shear-behavior-analysis-modulus-of-rigidity-in-torsion-testing-of-steel-brass-and-aluminum