Strain

Core idea

Overview

Strain represents the physical deformation of a material relative to its original length when subjected to an external force or stress. It is a dimensionless quantity that quantifies how much a body has been stretched or compressed along a specific axis.

When to use: This formula is used to calculate normal (tensile or compressive) strain in materials within their linear-elastic region. It assumes the deformation is uniform throughout the material and that the original length is used as the reference point.

Why it matters: Understanding strain is essential for predicting structural failure and ensuring the safety of engineering designs like bridges and aircraft. It allows engineers to relate deformation to stress, which is fundamental in defining material properties like the Modulus of Elasticity.

Symbols

Variables

$ϵ$ = Strain, $Δ$ L = Extension, L = Original Length

ϵ

Strain

Variable

Δ L

Extension

m

L

Original Length

m

Walkthrough

Derivation

Understanding Direct Strain

Strain is a dimensionless measure of deformation: the change in length relative to the original length.

Deformation is uniform along the specimen length.
Change in length is small compared with the original length (engineering strain).

1

Define the Concept:

Strain $ε$ is extension $Δ L$ divided by original length L.

ε = \frac{Δ L}{L}

Note: Strain has no units. It is sometimes reported as a percentage or in microstrain ( $μ ε$ ).

Result

ε = \frac{Δ L}{L}

Source: Edexcel A-Level Engineering — Engineering Materials

Free formulas

Rearrangements

Solve for $ϵ$

Make e the subject

ϵ = \frac{Δ L}{L}

e is already the subject of the formula.

Difficulty: 1/5

Solve for $Δ L$

Make Delta L the subject

Δ L = ϵ L

To make Extension ( $Δ$ L) the subject of the Strain formula, multiply both sides by Original Length (L) to isolate $Δ$ L.

Difficulty: 2/5

Solve for $L$

Strain: Make L the subject

L = \frac{Δ L}{ϵ}

Rearrange the formula for strain to solve for the original length, L.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin with a slope of 1/L, showing that strain increases at a constant rate as extension increases. For an engineering student, this linear relationship means that doubling the extension will always result in a doubling of the strain, regardless of the starting point. The most important feature is that the slope is determined entirely by the original length, meaning that for a fixed extension, a smaller original length results in a steeper increase in strain.

Graph type: linear

Why it behaves this way

Intuition

Visualize a line segment of material stretching or compressing; strain quantifies how much longer or shorter that segment becomes relative to its initial length.

ϵ

Normal strain, representing the fractional change in length of a material.

It tells you 'how much stretch or squeeze there is per unit of original length.' A value of 0.01 means the material changed length by 1% of its original size.

Δ L

The absolute change in length of the material, measured as the final length minus the original length.

This is the raw amount of elongation (positive) or contraction (negative) experienced by the material.

L

The initial, undeformed length of the material.

This acts as the baseline for comparison, normalizing the deformation so that strain is independent of the object's initial size.

Free study cues

Insight

Canonical usage

Strain is a dimensionless quantity, representing a ratio of change in length to original length. It is typically expressed as a pure number, a ratio (e.g., m/m), or a percentage.

Common confusion

A common mistake is attempting to assign a physical unit (like meters or pascals) to strain, or forgetting that the units of $Δ$ L and L must be consistent for the ratio to be meaningful.

Dimension note

Strain is a ratio of two quantities with the same dimension (length), resulting in a dimensionless quantity. While it can be expressed with 'units' like m/m or in/in to explicitly indicate the ratio, these units

Unit systems

$ϵ$ dimensionless - Strain is inherently dimensionless as it is a ratio of two lengths.

$Δ L$ m - Change in length must be in the same units as the original length for the units to cancel correctly.

$L$ m - Original length must be in the same units as the change in length for the units to cancel correctly.

One free problem

Practice Problem

A steel cable with an original length of 5.0 meters is stretched by 0.025 meters under a heavy load. Calculate the normal strain experienced by the cable.

Original Length5 m

Extension0.025 m

Solve for: eps

Hint: Strain is the ratio of the change in length to the original length.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating stretch in a cable under tension, Strain is used to calculate the eps value from Extension and Original Length. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Study smarter

Tips

Ensure the change in length and the original length use identical units.
Strain is a dimensionless ratio, though it is sometimes expressed as a percentage.
Positive results typically denote elongation (tension), while negative results denote shortening (compression).

Avoid these traps

Common Mistakes

Using total length instead of extension.
Mixing cm and m.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Strain is a dimensionless measure of deformation: the change in length relative to the original length.

This formula is used to calculate normal (tensile or compressive) strain in materials within their linear-elastic region. It assumes the deformation is uniform throughout the material and that the original length is used as the reference point.

Understanding strain is essential for predicting structural failure and ensuring the safety of engineering designs like bridges and aircraft. It allows engineers to relate deformation to stress, which is fundamental in defining material properties like the Modulus of Elasticity.

Using total length instead of extension. Mixing cm and m.

When estimating stretch in a cable under tension, Strain is used to calculate the eps value from Extension and Original Length. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Ensure the change in length and the original length use identical units. Strain is a dimensionless ratio, though it is sometimes expressed as a percentage. Positive results typically denote elongation (tension), while negative results denote shortening (compression).

References

Sources

Beer, F. P., Johnston Jr., E. R., DeWolf, J. T., & Mazurek, D. F. (2015). Mechanics of Materials. McGraw-Hill Education.
Hibbeler, R. C. (2018). Engineering Mechanics: Statics & Dynamics. Pearson.
Wikipedia: Strain (materials science)
Britannica: Strain (physics)
Wikipedia: Strain (mechanics)
Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. Transport Phenomena. John Wiley & Sons.
Halliday, David; Resnick, Robert; Robert. Fundamentals of Physics. John Wiley & Sons.
Beer, F. P., Johnston Jr., E. R., DeWolf, J. T., & Mazurek, D. F. (2020). Mechanics of Materials (8th ed.). McGraw-Hill Education.

Overview

Variables

Derivation

Define the Concept:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Young's Modulus

Frequently Asked Questions

Sources