Elasticity Formula, Definition, Examples

In physics, elasticity refers to a material's innate ability to revert to its original size and shape after being subjected to external forces or deformations. It is a fundamental concept in the field of mechanics that helps us understand how solids respond to applied forces and how they recover their initial state once those forces are removed.

What is Elasticity?

Elasticity refers to the property of a material that allows it to deform under the application of an external force and then return to its original shape and size once the force is removed. Elasticity is a fundamental concept in the study of materials and their mechanical behaviour, providing insights into how solids respond to various forces and stresses.

SI Unit of Elasticity

The SI (International System of Units) unit of elasticity is the pascal (Pa). Elasticity is the capacity of a material to deform under tension and to recover to its initial shape when the force has been removed. The pascal is the unit of pressure and stress in the SI system, and it is defined as one newton per square meter (N/m²).

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What is Elastic Stress?

Elastic stress refers to the internal resistance or force per unit area that a material exhibits when it is subjected to an external force or load. When a material is under elastic deformation, it can temporarily change its shape in response to the applied stress, but it will return to its original shape once the stress is removed.

Types of Elastic Stress

There are three types of stress:

• Longitudinal stress

Longitudinal stress is a type of normal stress that develops within a material when it is subjected to an axial load or force along its length. It's also referred to as axial stress or tensile stress when the material is being stretched.

The formula for calculating longitudinal stress (σ) due to an axial load (F) is:

σ = F / A

• Volume stress or bulk stress

Volume stress, or bulk stress, refers to the stress that occurs within a material due to changes in volume or pressure. It's associated with the deformation of a material in all directions when it is subjected to an isotropic change in pressure.

For a fluid, bulk stress can be related to pressure through the equation:

σ_bulk = -P

• Tangential stress or shear stress

Tangential stress, also known as shear stress, is a type of stress that arises within a material when it experiences deformation by forces acting parallel to its surface. Shear stress leads to the sliding or distortion of adjacent layers of the material along the direction of the applied force.

The formula for calculating shear stress (τ) is:

τ = F / A

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What is Strain?

Strain is a measure of the deformation that a material undergoes when subjected to an external force or load. It quantifies the relative change in shape or size of a material due to applied stress. Strain is a dimensionless quantity expressed as a ratio or a percentage, representing the elongation, compression, or distortion of a material compared to its original dimensions.

Types of Strain

There are three types of Strain:

• Longitudinal strain

Longitudinal strain, also known as axial strain or linear strain, is a type of strain that measures the change in length of a material along a particular direction relative to its original length. It is associated with the deformation of a material under an axial load or force applied along its length.

The formula for calculating longitudinal strain (ε) is:

ε = ΔL / L₀

• Volume strain

Volume strain, also known as volumetric strain or bulk strain, is a type of strain that measures the change in volume of a material relative to its original volume when subjected to external forces or pressure changes. It is particularly relevant in situations where a material's volume changes uniformly due to an isotropic (equal in all directions) force or stress.

The formula for calculating volume strain (ε_v) is derived from the change in volume (ΔV) divided by the original volume (V₀):

ε_v = ΔV / V₀

• Tangential strain

Tangential strain, also known as shear strain or angular strain, measures the change in shape or distortion of a material's cross-sectional elements relative to their original shape due to applied shear stress. It is a type of strain that arises when adjacent layers of a material slide or deform relative to each other along a plane parallel to the applied force.

The formula for calculating shear strain (γ) is typically expressed in terms of the change in angle (Δθ) divided by the initial angle (θ₀) between two lines that experience shear deformation:

γ = Δθ / θ₀

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Elastic Hysteresis

Elastic hysteresis, also known as mechanical hysteresis or elastic energy loss, refers to the phenomenon where a material exhibits energy loss during cyclic loading and unloading, even within its elastic deformation range. Elastic hysteresis can be observed in various materials, including rubber, polymers, and certain types of metals.

Hooke’s Law

Hooke's Law is a fundamental principle in physics and materials science that describes the relationship between the deformation (strain) of a material and the force (stress) applied to it within its elastic deformation range. Named after the English scientist Robert Hooke, who first formulated the law in the 17th century, Hooke's Law is a linear relationship that holds true for many materials under certain conditions.

Mathematically, Hooke's Law is expressed as:

σ = E  ε

Modulus of Elasticity

The modulus of elasticity, often referred to as Young's modulus (E), is a material property that measures the stiffness or rigidity of a material. It quantifies how much a material will deform (strain) in response to an applied force (stress) within its elastic deformation range. Young's modulus is a fundamental parameter used to describe the linear relationship between stress and strain in Hooke's Law for many materials.

Mathematically, Young's modulus is defined as:

E = σ / ε

Types of Elasticity Modulus

There are several types of elasticity moduli, each describing a different aspect of a material's response to stress and deformation. Here are some of the main types of elasticity moduli:

• Young's Modulus (E):

Young's modulus is perhaps the most well-known elasticity modulus. It measures a material's stiffness or resistance to elastic deformation under axial loading (tension or compression). It quantifies the ratio of stress to strain within the linear elastic range.

• Shear Modulus (G):

Shear modulus, also known as the modulus of rigidity, measures a material's resistance to shear deformation. It quantifies how much a material will deform under shear stress. Shear modulus is crucial for understanding materials' behaviour under torsion or shearing forces.

• Bulk Modulus (K):

Bulk modulus measures a material's response to changes in volume under hydrostatic stress (uniform pressure changes). It quantifies how a material compresses or expands under the application of pressure. Bulk modulus is particularly relevant for fluids and materials that experience changes in pressure.

• Poisson's Ratio (ν):

Poisson's ratio describes the relationship between lateral (transverse) and axial (longitudinal) strains when a material is subjected to axial stress. It indicates how much a material contracts laterally when stretched longitudinally (negative Poisson's ratio) or expands laterally (positive Poisson's ratio).

• Stress Concentration Factor (Kt):

This factor describes the effect of a geometric discontinuity, such as a notch or hole, on the stress distribution in a material. It's not exactly an elasticity modulus but is related to stress distribution.

These different elasticity moduli help engineers and scientists characterise and predict material behaviour under various conditions. Each modulus provides insights into a specific aspect of how materials respond to mechanical forces, deformations, and loading scenarios.