Tensile Method for Measuring the Elastic Modulus of Metal (1)
The simplest form of deformation occurs when a linear or rod-like object is subjected to a tensile force along its length, causing it to elongate. Suppose the wire (or rod) has an original length L and a cross-sectional area S. When a tensile force F is applied within the elastic limit, the wire elongates by ΔL. The ratio F/S represents the force per unit area, known as stress, while the relative elongation ΔL/L is referred to as strain. According to Hooke’s Law, stress and strain are directly proportional within the elastic limit:
(1)
The proportionality constant is given by:
(2)
This constant E is called the elastic modulus, also known as Young’s modulus. The value of E is independent of the shape, size, or length of the object; it solely depends on the material's properties. It is a crucial physical quantity that reflects a material's resistance to deformation. In mechanical design and engineering, understanding this parameter is essential for material selection.
Any solid object undergoes deformation when subjected to external forces. If the force is removed within a certain limit, the object can fully return to its original shape, which is known as elastic deformation.
There are several methods to measure Young’s modulus, including the static method, resonance method, and pulse transmission method. Among these, the resonance and pulse methods offer higher precision. The static method involves applying a load to the object and measuring both the stress and strain, then calculating E using a specific formula. This includes techniques such as the stretching method and the beam bending method.
When a force F is applied to the top of a cubic object while the bottom is fixed (as shown in Figure 1), the object deforms into a slanted parallelepiped. This type of deformation is called shear. After shearing, the displacement from the bottom surface varies (AA’ > BB’), but the relative deformation remains equal, as described in equation (6-3).
In this formula, θ is the shear angle. When θ is small, it can be approximated as tanθ. Experimental results show that the shear angle is proportional to the shear stress within a certain range. Here, S represents the cross-sectional area of the cube parallel to the base. Thus, we have:
(6-4)
The proportionality factor G is referred to as the shear modulus.
Common methods for measuring the shear modulus include the static torsion method and the swing method.
Purpose
1. Understand and apply a method for measuring Young’s modulus.
2. Learn the principle of the optical lever method for measuring small elongations and how to adjust and use the instruments.
3. Acquire knowledge of data processing techniques, particularly the difference method.
Experimental Instruments: Young’s modulus meter, scale reading telescope, optical lever, level, micrometer, vernier caliper (0.02mm accuracy), and 9kg weights.
The detailed setup of the experiment is illustrated in Figure 1. The scale reading telescope consists of a telescope and a scale frame, with adjustable elevation and focus. The Young’s modulus apparatus is a large tripod with two vertical columns. The upper beam allows the wire to be fixed, while the lower part holds the optical lever. The platform can be adjusted vertically along the column. Three leveling screws on the tripod help maintain the stability of the platform.
The optical lever, shown in Figure 2, features a small mirror mounted on a tripod. Two legs are positioned on the same side as the mirror, while the third leg serves as a vertical support, with adjustable length a.
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The simplest form of deformation occurs when a linear or rod-like object is subjected to a tensile force along its length, causing it to elongate. Suppose the wire (or rod) has an original length L and a cross-sectional area S. When a tensile force F is applied within the elastic limit, the wire elongates by ΔL. The ratio F/S represents the force per unit area, known as stress, while the relative elongation ΔL/L is referred to as strain. According to Hooke’s Law, stress and strain are directly proportional within the elastic limit:
(1)
The proportionality constant is given by:
(2)
This constant E is called the elastic modulus, also known as Young’s modulus. The value of E is independent of the shape, size, or length of the object; it solely depends on the material's properties. It is a crucial physical quantity that reflects a material's resistance to deformation. In mechanical design and engineering, understanding this parameter is essential for material selection.
Any solid object undergoes deformation when subjected to external forces. If the force is removed within a certain limit, the object can fully return to its original shape, which is known as elastic deformation.
There are several methods to measure Young’s modulus, including the static method, resonance method, and pulse transmission method. Among these, the resonance and pulse methods offer higher precision. The static method involves applying a load to the object and measuring both the stress and strain, then calculating E using a specific formula. This includes techniques such as the stretching method and the beam bending method.
When a force F is applied to the top of a cubic object while the bottom is fixed (as shown in Figure 1), the object deforms into a slanted parallelepiped. This type of deformation is called shear. After shearing, the displacement from the bottom surface varies (AA’ > BB’), but the relative deformation remains equal, as described in equation (6-3).
In this formula, θ is the shear angle. When θ is small, it can be approximated as tanθ. Experimental results show that the shear angle is proportional to the shear stress within a certain range. Here, S represents the cross-sectional area of the cube parallel to the base. Thus, we have:
(6-4)
The proportionality factor G is referred to as the shear modulus.
Common methods for measuring the shear modulus include the static torsion method and the swing method.
Purpose
1. Understand and apply a method for measuring Young’s modulus.
2. Learn the principle of the optical lever method for measuring small elongations and how to adjust and use the instruments.
3. Acquire knowledge of data processing techniques, particularly the difference method.
Experimental Instruments: Young’s modulus meter, scale reading telescope, optical lever, level, micrometer, vernier caliper (0.02mm accuracy), and 9kg weights.
The detailed setup of the experiment is illustrated in Figure 1. The scale reading telescope consists of a telescope and a scale frame, with adjustable elevation and focus. The Young’s modulus apparatus is a large tripod with two vertical columns. The upper beam allows the wire to be fixed, while the lower part holds the optical lever. The platform can be adjusted vertically along the column. Three leveling screws on the tripod help maintain the stability of the platform.
The optical lever, shown in Figure 2, features a small mirror mounted on a tripod. Two legs are positioned on the same side as the mirror, while the third leg serves as a vertical support, with adjustable length a.
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