Elastic behaviour, stress strain relationship, Hooke's law

Elastic Behaviour

When a body is subjected to an external force, its dimensions change. This change is called deformation. The external force causing the deformation is called the deforming force. If the deforming force is removed, the body tends to regain its original shape and size. This property of a body to regain its original shape and size is called elasticity.

Materials that exhibit this property are called elastic materials. For example, a rubber band stretches when pulled, and it returns to its original length when the pulling force is removed. Similarly, a steel spring can be compressed or stretched, and it springs back to its original shape.

However, there is a limit to this elastic behavior. If the deforming force is increased beyond a certain limit, the body may not regain its original shape and size completely. This limit is called the elastic limit. Beyond the elastic limit, the material starts to deform permanently.

Materials that do not regain their original shape and size after the removal of the deforming force are called plastic materials. For example, a piece of clay or wax. When a force is applied to clay, it deforms, and it retains its new shape even after the force is removed. This property is called plasticity.

In summary:

• Elasticity: The ability of a body to regain its original shape and size after the removal of the deforming force.
• Plasticity: The tendency of a body to undergo permanent deformation under a deforming force.

Stress

When deforming forces are applied to a body, internal restoring forces are developed within the body. These internal forces act across any internal surface of the body, opposing the deformation. Stress is defined as the internal restoring force per unit area.

Mathematically, stress ($\sigma$) is given by:

$\sigma = \frac{F}{A}$

where $F$ is the internal restoring force and $A$ is the cross-sectional area over which the force acts.

The unit of stress is the same as the unit of pressure, which is Newton per square meter (N/m²) or Pascal (Pa) in the SI system. In the CGS system, it is dyne/cm².

There are three main types of stress, depending on the nature of the deforming force:

1. Tensile Stress: When a body is stretched or compressed by forces acting normally on its surfaces, tensile stress is produced. If a wire is stretched by applying forces at its ends, the stress developed is tensile stress.
2. Compressive Stress: This is a type of tensile stress where the deforming force tends to compress the body.
3. Shear Stress: When deforming forces are applied tangentially to the surface of a body, shear stress is produced. This tends to slide one layer of the body over another. The tangential force per unit area is called shear stress ($\tau$).

   $\tau = \frac{F_t}{A}$

   where $F_t$ is the tangential force and $A$ is the area.
4. Bulk Stress: When a body is subjected to uniform pressure from all sides (like a solid submerged in a liquid), bulk stress is produced. It is the force per unit area acting perpendicularly on the surface. Bulk stress is equal to the applied pressure.

Strain

Strain is a measure of the deformation produced in a body. It is defined as the ratio of the change in dimension to the original dimension. Strain is a dimensionless quantity.

There are three types of strain corresponding to the three types of stress:

1. Tensile Strain: It is the ratio of the change in length ($\Delta L$) to the original length ($L$).

   Tensile Strain = $\frac{\Delta L}{L}$

   This strain is associated with tensile or compressive stress.
2. Shear Strain: When a body is subjected to shear stress, the shape changes but the volume remains nearly constant. Shear strain ($\theta$) is defined as the ratio of the displacement of the top surface ($x$) to the height of the body ($L$).

   Shear Strain = $\theta = \frac{x}{L}$

   For small deformations, $\tan \theta \approx \theta$ (in radians).
3. Bulk Strain: It is the ratio of the change in volume ($\Delta V$) to the original volume ($V$).

   Bulk Strain = $\frac{\Delta V}{V}$

   This strain is associated with bulk stress.

Hooke's Law

Robert Hooke, an English scientist, studied the elastic behavior of materials and formulated a law that describes the relationship between stress and strain for most elastic materials.

Hooke's Law states that within the elastic limit, stress is directly proportional to strain.

Mathematically, this can be written as:

Stress $\propto$ Strain

Stress = $E \times$ Strain

Here, $E$ is the constant of proportionality and is called the modulus of elasticity. The value of $E$ depends on the nature of the material. A material with a high modulus of elasticity is more rigid than a material with a low modulus.

The modulus of elasticity ($E$) has the same units as stress (N/m² or Pa).

Hooke's Law is a fundamental principle in the study of elasticity. It holds true for most materials up to their elastic limit. Beyond this limit, the relationship between stress and strain becomes non-linear.

Stress-Strain Relationship

The relationship between stress and strain can be visualized by plotting a graph between stress (on the y-axis) and strain (on the x-axis) for a material. This graph is called a stress-strain curve. Let's consider the stress-strain curve for a ductile material like steel.

When a metal wire is gradually loaded (stress is increased), the strain also increases. The curve typically shows several important regions:

1. Elastic Region (OA): In this region, stress is directly proportional to strain. The material obeys Hooke's Law. If the load is removed, the wire returns to its original length. The point 'O' is the origin, and 'A' is the elastic limit.
2. Elastic Limit (A): This is the maximum stress that the material can withstand without undergoing permanent deformation. If the stress is removed at any point before 'A', the material returns to its original shape.
3. Yield Point (B): Beyond the elastic limit, if the stress is increased further, the material starts to deform plastically. The yield point is the point where the material begins to deform permanently. After reaching this point, a large amount of strain occurs with little or no increase in stress. The stress at the yield 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 (localize deformation). It represents the maximum stress the material can sustain.
5. Breaking Point (D): This is the point where the material breaks. The stress at the breaking point is called the breaking stress or fracture stress.

The stress-strain curve is crucial for understanding the mechanical properties of materials, such as their strength, stiffness, and ductility.

Modulus of Elasticity

The modulus of elasticity is a measure of a material's resistance to elastic deformation. It is defined as the ratio of stress to strain within the elastic limit. There are three types of moduli of elasticity, corresponding to the three types of stress and strain.

Young's Modulus (Y or E)

Young's modulus is defined as the ratio of tensile stress (or compressive stress) to tensile strain (or compressive strain) within the elastic limit.

Young's Modulus ($Y$) = $\frac{\text{Tensile Stress}}{\text{Tensile Strain}}$

$Y = \frac{F/A}{\Delta L/L} = \frac{FL}{A\Delta L}$

Young's modulus is a measure of the stiffness of a solid material. A higher Young's modulus indicates a stiffer material, meaning it requires more force to produce a given amount of elastic deformation. For example, steel has a much higher Young's modulus than rubber.

Example: Consider a steel wire of length 1 m and cross-sectional area 1 mm². If a load of 1 kg is attached to it, the wire stretches by 0.2 mm. Calculate the Young's modulus of steel.

Given: $L = 1$ m, $A = 1 \text{ mm}^2 = 1 \times 10^{-6} \text{ m}^2$, mass $m = 1$ kg, $g \approx 9.8$ m/s², $\Delta L = 0.2$ mm $= 0.2 \times 10^{-3}$ m.

The force applied is $F = mg = 1 \times 9.8 = 9.8$ N.

$Y = \frac{FL}{A\Delta L} = \frac{(9.8 \text{ N})(1 \text{ m})}{(1 \times 10^{-6} \text{ m}^2)(0.2 \times 10^{-3} \text{ m})}$

$Y = \frac{9.8}{0.2 \times 10^{-9}} = \frac{9.8}{2 \times 10^{-10}} = 4.9 \times 10^{10} \text{ N/m}^2 = 49 \times 10^9 \text{ N/m}^2 = 49$ GPa.

This value is typical for steel.

Bulk Modulus (K)

Bulk modulus is defined as the ratio of bulk stress (pressure) to bulk strain within the elastic limit.

Bulk Modulus ($K$) = $\frac{\text{Bulk Stress}}{\text{Bulk Strain}}$

$K = \frac{-P}{\Delta V/V} = \frac{-PV}{\Delta V}$

The negative sign indicates that an increase in pressure ($P$) leads to a decrease in volume ($\Delta V$ is negative). Bulk modulus relates to the compressibility of a substance. A large bulk modulus means the substance is difficult to compress.

Liquids and solids generally have large bulk moduli, while gases have small bulk moduli. For example, water has a bulk modulus of about $2.2 \times 10^9$ Pa.

The reciprocal of bulk modulus is called compressibility.

Compressibility = $\frac{1}{K}$

Modulus of Rigidity (G or $\eta$)

Modulus of rigidity is defined as the ratio of shear stress to shear strain within the elastic limit.

Modulus of Rigidity ($G$) = $\frac{\text{Shear Stress}}{\text{Shear Strain}}$

$G = \frac{\tau}{\theta}$

The modulus of rigidity is a measure of a material's resistance to shear deformation. It is defined only for solids because fluids (liquids and gases) cannot sustain shear stress and flow instead. Solids generally have a non-zero modulus of rigidity.

For most materials, Young's modulus, bulk modulus, and modulus of rigidity are related. For an isotropic solid, the relationship is:

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

$Y = 2G(1 + \mu)$

where $\mu$ is Poisson's ratio. These relationships are important for advanced studies in elasticity.

Poisson's Ratio ($\mu$)

When a wire is stretched, its length increases, but its diameter decreases. This change in diameter is called lateral strain. Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain, provided the stress is within the elastic limit.

Let $\Delta L$ be the change in length and $L$ be the original length. The longitudinal strain is $\frac{\Delta L}{L}$.

Longitudinal Strain = $\frac{\Delta L}{L}$

Let $\Delta d$ be the change in diameter and $d$ be the original diameter. The lateral strain is $\frac{\Delta d}{d}$. Note that if the wire is stretched (length increases), the diameter decreases, so $\Delta d$ is negative.

Lateral Strain = $\frac{\Delta d}{d}$

Poisson's ratio ($\mu$) is defined as:

$\mu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = -\frac{\Delta d/d}{\Delta L/L}$

The negative sign is included to make Poisson's ratio a positive quantity because lateral strain is opposite in sign to longitudinal strain. For most materials, the value of Poisson's ratio lies between 0 and 0.5.

Example: If a wire is stretched by 0.1% and its diameter decreases by 0.03%, what is its Poisson's ratio?

Longitudinal Strain = 0.1% = 0.001

Lateral Strain = -0.03% = -0.0003 (since diameter decreases)

$\mu = -\frac{-0.0003}{0.001} = \frac{0.0003}{0.001} = 0.3$

A value of $\mu = 0$ means there is no lateral strain when stretched longitudinally. A value of $\mu = 0.5$ means that the volume of the material remains constant during elastic deformation (this is true for incompressible materials like rubber).

Applications of Elasticity

The principles of elasticity are applied in numerous fields:

• Engineering Design: Engineers use the properties of elastic materials (like steel, aluminum) to design bridges, buildings, aircraft, and machinery. They ensure that the stresses developed in these structures under load remain within the elastic limit to prevent failure.
• Springs: Springs are used in vehicles, watches, pens, and many other devices. Their elastic properties allow them to store and release energy.
• Musical Instruments: The vibration of strings in a guitar or violin, which produces sound, is based on their elastic properties.
• Medical Devices: Catheters and other medical instruments often utilize the elastic properties of polymers.
• Tire Manufacturing: The flexibility and resilience of rubber tires are due to the elastic behavior of rubber compounds.

Elastic Fatigue

When a material is subjected to repeated cycles of stress and strain, its elastic properties can degrade over time. This phenomenon is known as elastic fatigue. Even if the stress remains below the elastic limit, repeated loading and unloading can lead to the formation and propagation of micro-cracks, eventually causing the material to fail.

This is why components subjected to cyclic loading, such as airplane wings or engine parts, are designed with a significant safety factor and undergo rigorous testing to ensure their fatigue life.

Temperature Effects on Elasticity

The elastic properties of materials can be affected by temperature. Generally, as temperature increases:

• Young's modulus ($Y$) decreases. The material becomes less stiff.
• Modulus of Rigidity ($G$) decreases.
• Bulk Modulus ($K$) decreases.

This means that materials become more deformable at higher temperatures. For example, metals become softer and more ductile when heated.

Stress Relaxation

Stress relaxation is another time-dependent phenomenon observed in viscoelastic materials. If a material is held at a constant strain, the stress required to maintain that strain gradually decreases over time. This occurs because the internal molecular structure rearranges itself to reduce the internal stresses.

This effect is significant in polymers and other viscoelastic materials and can influence their long-term performance under constant deformation.

Surface Tension and Viscosity (Brief Introduction as related to liquid properties)

While the primary focus is on elasticity, it's worth noting that liquids exhibit other important mechanical properties like surface tension and viscosity.

Surface Tension: This is a property of the surface of a liquid that allows it to resist an external force. It is due to the cohesive forces between liquid molecules. The surface molecules are pulled inwards, creating a net inward force that causes the surface to behave like a stretched elastic membrane.

Viscosity: This is a measure of a fluid's resistance to flow. Viscous fluids (like honey) flow slowly, while less viscous fluids (like water) flow easily. Viscosity arises from the internal friction between layers of the fluid as they move past each other.

These properties are studied under the broader topic of Properties of Liquids and are governed by different physical principles than elasticity, but they are crucial for understanding fluid mechanics.