Till now we have learned that solids have fixed shapes and dimensions i.e. they are rigid. You must have observed that if a weight is hung from the end of a vertically hung steel spring, it gets stretched. When the weight is removed, it goes back to its original shape and size. This shows that steel spring has some elastic behavior!
Stretching of Springs
In this document, we will be studying the Elasticity of solids in detail.
The property of the body to regain its original configuration (length, volume, or shape) when the deforming forces are removed, is called elasticity.
Spring-ball model for the illustration of elastic behavior of solids
1. Deforming force
A force that when applied changes the normal position of the molecules thus resulting in the change of configuration of the body.
2. Elastic body
A body that regains its original configuration immediately and completely after the removal of deforming force from it is called a perfectly elastic body. Quartz and phosphorus bronze are examples of nearly perfectly elastic bodies.
Elastic Materials
3. Plastic body
The inability of a body to return to its original size and shape even on removal of the deforming force is called plasticity and such a body is called a plastic body.
Plastic Materials
Stress is defined as the ratio of the internal force F, produced when the substance is deformed, to the area A over which this force acts.
In equilibrium, this force is equal in magnitude to the externally applied force. In other words,
The SI Unit of stress is newton per square meter (Nm-2).In CGS units, stress is measured in dyne cm-2. The dimensional formula of stress is ML-1T-2
Tensile and compressive stress
2. Tangential stress: When the elastic restoring force or deforming force acts parallel to the surface area, the stress is called tangential stress or shear stress.
Tangential Stress
It is defined as the ratio of the change in size or shape to the original size or shape. It has no dimensions, it is just a number.
1. Longitudinal strain: If the deforming force produces a change in length alone, the strain produced in the body is called longitudinal strain or tensile strain.
It is given as:
Longitudinal strain
2. Volumetric strain: If the deforming force produces a change in volume alone, the strain produced in the body is called volumetric strain. It is given as:
Volumetric strain3. Shear strain: The angle tilt caused in the body due to tangential stress expressed is called shear strain. It is given as:
Shear strain
The maximum stress to which the body can regain its original status on the removal of the deforming force is called the elastic limit.
Hooke’s law states that, within elastic limits, the ratio of stress to the corresponding strain produced is a constant.
This constant is called the modulus of elasticity. Thus,
Stress-strain curves are useful to understand the tensile strength of a given material. The given figure shows a stress-strain curve of a given metal.
Stress-strain curve
Young's modulus (Y) quantifies the elasticity of a material in terms of its ability to withstand changes in length under longitudinal stress. It is defined as the ratio of longitudinal stress to longitudinal strain within the elastic limit, which is the range where the material returns to its original shape after the deforming force is removed.
Where:
Unit: Newton per square meter (N/m2)
Dimensions:
The bulk modulus (K or B) measures a material's resistance to uniform compression. It is the ratio of volume stress to volume strain, applicable when the deforming force is applied equally from all directions, causing a change in volume.
The negative sign indicates that an increase in pressure results in a decrease in volume.
Unit: Newton per square meter (N/m2)
Compressibility: The reciprocal of the bulk modulus
The modulus of rigidity (η), also known as the shear modulus, measures a material's resistance to shear deformation. It is defined as the ratio of shearing stress to shearing strain within the elastic limit.
Where:
Poisson's ratio (σ) is a measure of the transverse strain (contraction or expansion) relative to the longitudinal strain when a material is stretched or compressed.
Where:
When a wire is stretched, the work done is stored as elastic potential energy. For a wire of original length L0 stretched by a distance ΔL:
Increasing temperature generally reduces the elastic properties of materials. This means the elastic constants, such as Young's modulus, decrease, leading to increased plasticity (permanent deformation). However, some materials, like INVAR steel, maintain consistent elastic properties over a range of temperatures, making them valuable for applications requiring dimensional stability.
Impurities can alter the elastic properties of a material. Typically, adding impurities increases the Young's modulus because the impurities can enhance the intermolecular forces within the material, making it more resistant to deformation under an applied force. This increased resistance results in a stiffer material with higher elasticity.
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1. What is Hooke's Law? |
2. What is Young's Modulus of Elasticity? |
3. What is the significance of the Stress-Strain Curve in the study of elastic behavior of solids? |
4. How is Poisson's Ratio related to the elastic behavior of solids? |
5. How is the Bulk Modulus of Elasticity different from Young's Modulus of Elasticity? |
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