What Are the Dimensions of Strain in Physics?

The concept of strain is central to understanding how materials deform when subjected to stress. In physics, strain characterizes the relative deformation that occurs in a body due to external forces, without any reference to the material’s absolute size. It is a dimensionless quantity used extensively in elasticity and material science.

Definition of Strain

Strain is defined as the ratio of the change in a specific dimension of a body, such as length, area, or volume, to its original value before deformation. This ratio quantifies the extent of deformation and is mathematically expressed for linear strain as $\text{Strain} = \dfrac{\Delta L}{L_0}$, where $\Delta L$ is the change in length and $L_0$ is the original length.

Dimensional Formula of Strain

Since strain represents a ratio of two quantities of the same type, it has no physical dimensions. It is a pure number and considered dimensionless in the context of dimensional analysis.

The dimensional formula of strain is written as $(M^0 L^0 T^0)$. This indicates that strain does not depend on mass, length, or time dimensions.

For instance, linear strain can be detailed as follows:

$\text{Strain} = \dfrac{\text{Change in length}}{\text{Original length}}$

The dimension of both numerator and denominator is $[M^0 L^1 T^0]$. Hence, strain is dimensionless:

$\text{Strain} = \dfrac{[M^0 L^1 T^0]}{[M^0 L^1 T^0]} = [M^0 L^0 T^0]$

Further clarification on dimensional quantities such as mass, length, and time can be found on the Dimensions Of Density page.

Types of Strain

Strain can be categorized based on the nature and direction of deformation induced by applied forces. The primary types are longitudinal strain, volumetric strain, and shear strain. Each type is significant in different physical contexts.

• Longitudinal strain: change in length per original length
• Volumetric strain: change in volume per original volume
• Shear strain: angular deformation due to tangential force

Longitudinal strain occurs when a material is stretched or compressed along one axis. Volumetric strain refers to changes in the entire volume of the material. Shear strain arises from forces applied parallel to a surface.

Unit and Dimensional Analysis

Since strain is a ratio of like physical quantities, it has no units and no dimensions. It is frequently expressed as a simple number or in percentage form for practical calculations.

A detailed analysis of the dimensional formulas for related physical quantities is available on the Dimensions Of Stress and Dimensions Of Force pages.

Strain in Relation to Stress

Strain describes the deformation caused by applied stress. Stress is defined as force per unit area, with its own dimensional formula distinct from strain. The relationship between stress and strain is governed by the mechanical properties of the material, such as Young’s modulus and shear modulus.

The dimensions of stress and strain are fundamentally different even though they are closely related in elasticity theory. More information on the relation and distinctions can be found at Dimensions Of Work.

Application Areas and Examples

Strain is applied in analyzing deformation in rods under tension, changes in the volume of fluids under pressure, and angular shifts in structures due to shear forces. Each example involves a ratio of the change in a physical dimension to its original value.

• Stretching a metal rod produces longitudinal strain
• Compression of a gas leads to volumetric strain
• Twisting a wire generates shear strain

For advanced applications, concepts such as strain energy and strain rate use the dimensionless nature of strain as a basis for further derivation. Related topics include the Dimensions Of Energy and Dimensions Of Volume.