Elastic behaviour, stress–strain and Hooke’s law

Elastic Behaviour of Solids

When an external force is applied to a solid object, it causes a change in its shape or size. This applied force is called a deforming force. The solid, in response, develops an internal restoring force that tries to bring it back to its original shape or size.

The ability of a solid material to regain its original shape or size after the removal of the deforming force is called elasticity. Materials that exhibit this property are called elastic.

If the deforming force is not too large, the solid will return to its original shape. However, if the deforming force exceeds a certain limit, the solid may not regain its original shape completely, even after the removal of the force. This phenomenon is called plastic deformation, and materials that undergo this are called plastic.

The maximum deforming force up to which a solid can regain its original shape is called its elastic limit. Beyond this limit, the material starts to deform permanently.

Stress

When a deforming force is applied to a body, the internal restoring force that comes into play per unit area is called stress. Mathematically, stress is defined as:

$$ \text{Stress} = \frac{\text{Internal Restoring Force}}{\text{Area}} $$

The unit of stress depends on the units of force and area. In the SI system, the unit of force is Newton (N) and the unit of area is square meter (m²). Therefore, the SI unit of stress is N/m², which is also known as Pascal (Pa). In the CGS system, the unit of stress is dyne/cm².

Stress can be of different types, depending on the nature of the deforming force:

• Tensile Stress: When a solid is stretched or compressed along its length by forces acting perpendicular to its cross-sectional area, the stress developed is called tensile stress. If the force tends to increase the length, it's stretching stress; if it tends to decrease the length, it's compressive stress.
• Shear Stress: When forces are applied parallel to the surface of the body, causing it to deform by sliding one layer over another, the stress developed is called shear stress. If F is the tangential force applied and A is the area over which it is applied, then shear stress is given by $ \tau = \frac{F}{A} $.
• Bulk Stress: When a body is subjected to a uniform pressure from all sides (like a solid immersed in a liquid), the stress developed is called bulk stress. It is equal to the magnitude of the applied pressure. $ \sigma_v = \frac{F}{A} = P $, where P is the pressure.

Strain

Strain is defined as the ratio of the change in shape or size of a body to its original shape or size, when subjected to a deforming force. Strain is a dimensionless quantity as it is a ratio of two similar quantities. It is also a unitless quantity.

Strain can also be of different types, corresponding to the types of stress:

• Tensile Strain (or Longitudinal Strain): This is the strain produced due to tensile stress. It is defined as the ratio of the change in length ($ \Delta L $) to the original length ($ L $). $$ \epsilon = \frac{\Delta L}{L} $$
• Shear Strain: This is the strain produced due to shear stress. It is defined as the angle (usually in radians) through which a line perpendicular to the fixed surface and initially vertical, is rotated. If $ \Delta x $ is the displacement of the top surface of area A and L is the length of the body, then shear strain is given by $ \phi = \frac{\Delta x}{L} $.
• Bulk Strain: This is the strain produced due to bulk stress. It is defined as the ratio of the change in volume ($ \Delta V $) to the original volume ($ V $). $$ \epsilon_v = \frac{\Delta V}{V} $$

Hooke’s Law

Robert Hooke, an English scientist, proposed a fundamental law regarding the elastic behavior of materials. Hooke’s Law states that within the elastic limit, the stress developed in a body is directly proportional to the strain produced.

$$ \text{Stress} \propto \text{Strain} $$

This can be written as:

$$ \text{Stress} = E \times \text{Strain} $$

Here, $ E $ is the constant of proportionality and is called the modulus of elasticity of the material. The value of $ E $ depends on the nature of the material.

Hooke's Law is a cornerstone of the study of elasticity and is valid for most materials within their elastic limit. It implies that if you double the strain, you double the stress, provided you stay within the elastic range.

Modulus of Elasticity

The modulus of elasticity is a measure of a material's stiffness or resistance to elastic deformation. It is defined as the ratio of stress to strain within the elastic limit.

There are three main types of moduli of elasticity, corresponding to the three types of stress and strain:

1. Young’s Modulus (Y or E): This is the modulus of elasticity for tensile or compressive stress. It is defined as the ratio of tensile stress to tensile strain. $$ Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta L/L} $$ Young's modulus indicates how much a material will stretch or compress under tensile or compressive load. A higher Young's modulus means the material is stiffer.
2. Shear Modulus (or Modulus of Rigidity, $ \eta $ or G): This is the modulus of elasticity for shear stress. It is defined as the ratio of shear stress to shear strain. $$ \eta = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\phi} $$ Shear modulus measures a material's resistance to shearing deformations.
3. Bulk Modulus (K or B): This is the modulus of elasticity for bulk stress. It is defined as the ratio of bulk stress to bulk strain. Since the volume decreases under pressure, the change in volume is negative. To make the bulk modulus positive, a negative sign is introduced. $$ K = -\frac{\text{Bulk Stress}}{\text{Bulk Strain}} = -\frac{P}{\Delta V/V} = -\frac{PV}{\Delta V} $$ Bulk modulus describes how a substance compresses volumetrically when subjected to uniform pressure. It is particularly relevant for fluids.

Stress-Strain Curve

A stress-strain curve is a graphical representation of the relationship between stress and strain for a material under tension or compression. It provides valuable information about the mechanical properties of the material. Let's consider a typical stress-strain curve for a ductile material like steel.

As we increase the load (and thus the stress) on a specimen, the strain also increases. The curve can be divided into several regions:

1. Elastic Region (OA): In this region, the material obeys Hooke’s Law. Stress is directly proportional to strain. If the load is removed, the material returns to its original shape. Point A is the elastic limit.
2. Plastic Region (AB and beyond): Beyond the elastic limit A, the material starts to deform plastically. If the load is removed, the material will not return to its original shape completely; it will retain some permanent deformation.
3. Yield Point (B): This is the point where the stress remains almost constant, or even slightly decreases, while the strain increases significantly. The material starts to flow. The stress at this point is called the yield strength.
4. Ultimate Tensile Strength (C): This is the maximum stress the material can withstand before it starts to neck (i.e., its cross-sectional area begins to decrease significantly).
5. Fracture Point (D): This is the point where the material breaks. The stress at this point is called the fracture strength.

The region between B and C is where the material undergoes plastic deformation, and the region between C and D is where it experiences necking and eventually fractures.

Ductility and Brittleness

Materials can be classified as ductile or brittle based on their stress-strain behavior.

• Ductile Materials: These materials (like copper, aluminum, mild steel) show significant plastic deformation before they fracture. They can be drawn into wires. Their stress-strain curve has a distinct yield point and an ultimate tensile strength well above the yield strength.
• Brittle Materials: These materials (like glass, ceramics, cast iron) show very little or no plastic deformation before they fracture. They break suddenly with little warning. Their stress-strain curve shows a relatively sharp fracture point, often close to the elastic limit.

Factors Affecting Elasticity

The elastic properties of materials can be influenced by several factors:

• Temperature: Generally, the elasticity of materials decreases with an increase in temperature. At very high temperatures, materials may behave plastically even under small loads.
• Impurities and Alloying: The presence of impurities or the formation of alloys can significantly alter the elastic properties of a metal. For example, adding carbon to iron makes steel much stronger and less ductile.
• Heat Treatment: Processes like annealing, hardening, and tempering can change the internal structure of a material and thus its elastic limit and strength.
• Mechanical Working: Processes like rolling, drawing, and forging can introduce strain hardening, which increases the elastic limit and tensile strength but reduces ductility.

Applications of Elasticity

The principles of elasticity are crucial in the design and engineering of many structures and devices:

• Bridges and Buildings: Engineers must ensure that the materials used can withstand the stresses and strains imposed by loads (like traffic, wind, earthquakes) without exceeding their elastic limits.
• Springs: Springs are designed to deform elastically under load and return to their original shape when the load is removed. Their spring constant is directly related to the material's elastic properties.
• Musical Instruments: The vibration of strings in a guitar or piano, or the bending of reeds in a woodwind instrument, relies on the elastic properties of the materials.
• Medical Devices: Catheters, artificial blood vessels, and prosthetics often require materials with specific elastic properties to function correctly and safely within the body.

Poisson's Ratio

When a material is stretched along one direction, it tends to contract in the perpendicular directions. This phenomenon is described by Poisson's ratio.

Poisson's ratio ($ \sigma $) is defined as the ratio of the lateral strain (strain perpendicular to the applied force) to the longitudinal strain (strain along the direction of the applied force), within the elastic limit.

$$ \sigma = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} $$

The negative sign is included because when a material is stretched (positive longitudinal strain), it contracts (negative lateral strain), and vice versa. For most materials, Poisson's ratio is between 0 and 0.5. For example, rubber has a Poisson's ratio close to 0.5, indicating that its volume remains nearly constant when deformed.

For an isotropic material, there is a relationship between Young's modulus (Y), shear modulus ($ \eta $), and Poisson's ratio ($ \sigma $):

$$ Y = 2\eta(1 + \sigma) $$

And between Young's modulus (Y), bulk modulus (K), and Poisson's ratio ($ \sigma $):

$$ Y = 3K(1 - 2\sigma) $$