Stress and strain are two of the most important concepts for describing how materials respond to forces. Stress is the internal force per unit area within a material under load ,while strain is the deformation or change in shape of the material that results from the applied force.
However, the relationship between stress and strain goes far beyond theory — it is essential for sound engineering decisions. By comparing them side by side, we can better predict how materials perform, how much they can safely deform, and when they might fail. This article explores their definitions, differences, relationship, and practical applications.
Before we get into the details,you may find this short introductory video on stress and strain helpful:
What Is Stress？
Stress is the internal force per unit area that a material develops to resist an external load. Microscopically, the applied load induces interatomic forces that oppose deformation and “hold” the structure together; this internal resistance is what we measure as stress.
Depending on how the load is applied, stress is classified as:
Tensile stress (σt) and compressive stress (σc): These are normal stresses that act perpendicular to the cross-sectional area.
Shear stress (τ): Caused by tangential forces acting parallel to the cross-sectional area.
Torsional stress (τt): A specific form of shear stress induced by torque or twisting.
Among them, tensile stress is the most fundamental type of stress in engineering design. The calculation formula is:
Where:
σ= stress (Pa or N/m²; sometimes psi)
F = applied force (N)
A = the original cross-sectional area over which the force is applied (m²)
How Stress of Materials Is Measured
Measuring stress directly is not possible, so instead, we must measure either the applied forces or resulting deformations. Below is a concise overview of key measurement techniques:
Method / Technology
Principle
Measurement Device / Tool
Accuracy & Precision
Common Applications
Universal Testing Machine (UTM)
Measures force (F), calculates stress = F/A
UTM with integrated load cell
★★★★★ (High Accuracy)
Fundamental material testing: stress-strain curves, mechanical property evaluation
Strain Gauge
Measures strain (ε), calculates stress via σ = E·ε (assumes linear elasticity)
Strain is a measure of the relative deformation a material undergoes when subjected to an external force. It is expressed as a unitless quantity or as a percentage, representing the change in length (or other dimensions) to the original length (or dimension).
The type of strain corresponds to the stress applied: tensile strain, compressive strain, or shear strain.
The formula for normal strain is:
Where:
ϵ = strain (dimensionless or expressed in %)
ΔL = change in length
L0 = original length
How Strain of Materials Is Measured
Various methods can be used to measure strain. The most commonly used techniques are strain gauges and extensometers. The table below summarizes common methods for measuring strain in materials:
Method
Sensing Principle
Sensor / Transducer
Measurement Scenario
Remarks
Strain Gauge
Resistance change
Foil-type strain gauge
Static or low-frequency strain; commonly used
Widely used in industry; low cost; requires adhesive bonding and wiring connections
Extensometer
Displacement
Clip-on / contact extensometer
Material testing; full-section measurement
High accuracy; unsuitable for dynamic tests or highly localized strain
Non-contact; 2D/3D deformation mapping; expensive system
Piezoelectric Sensor
Piezoelectric effect
Piezoelectric film or crystal
Dynamic strain, pressure, impact, vibration
High frequency response; unsuitable for static strain measurements
Fiber Bragg Grating (FBG)
Optical (Bragg reflection)
FBG optical fiber sensor
Distributed or multiplexed measurement over long distances
Immune to EMI; suitable for aerospace, energy, and smart structures
Laser Doppler Vibrometer (LDV)
Doppler effect
LDV laser probe
Dynamic strain/velocity measurement and surface vibration analysis
Non-contact; high resolution; expensive; sensitive to surface conditions
Key Difference in Stress vs. Strain
Below is a quick table providing a direct overview:
Aspect
Stress
Strain
Formula
σ = F / A
ε = ΔL / L₀
Units
Pa (N/m²), or psi (lbf/in²)
Dimensionless or %
Cause
External force
Deformation caused by stress
Effect
Generates internal forces to counteract external loads; may lead to plastic deformation , fracture, fatigue failure or stress-corrosion cracking if too high
Changes the geometry of the material; recoverable in elastic limit, permanent beyond the yield point
Behavior
The internal force per area that a material must resist. Depending on distribution, it can cause compression, tension, bending or torsion
Describes how much the material deforms under applied stress; can be elastic or plastic
How Stress and Strain Relate to Each Other
Stress causes strain. A stress-strain curve graphs how a material deforms under gradually increasing load by plotting strain (deformation) against applied stress. Let’s review its key points:
1. Elastic Region (Point O–B)
Linear Region(O-A): Stress and strain are perfectly proportional, following Hooke’s law. This linear portion ends at the proportional limit, and its slope is the modulus of elasticity (Young’s modulus), indicating the stiffness of the material. Within this range, deformation is fully elastic: once the load is removed, the material returns to its original shape.
(Nonlinear region A-B): The material still behaves elastically — that is, deformation is fully recoverable, but the relationship becomes nonlinear, meaning Hooke’s law no longer applies. Point B is therefore known as the elastic limit: it represents the maximum force the material can withstand elastically, and the region OB is called the elastic region.
2. Plastic Region (Point B onward)
Yielding(B-C): After point B, the material enters the plastic region and deformation becomes permanent. Point B is also known as the upper yield point, where dislocations suddenly break free from their obstacles, so the required load briefly drops even as the material continues to stretch. The stress then falls to point C—the lower yield point, at which the stress level remains nearly constant while the material keeps elongating in a permanent (plastic) way.
Note that clear “upper → lower” yield plateau (B→C) is most obvious in low-carbon steels. Other alloys often transition more smoothly into plastic deformation without a pronounced stress dip.
Strain Hardening (C–D): After point C, the material work hardens: as dislocations accumulate and interact, the metal’s resistance to further flow increases. Although the section continues to thin and elongate, the increasing resistance to deformation drives the engineering stress higher until it reaches its maximum at point D—the ultimate tensile strength (UTS). This is the highest load the original gauge section can sustain under the test conditions.
Necking and Fracture (D–E): Beyond point D, localized necking begins, causing a rapid reduction in cross-sectional area in one region. In an engineering stress–strain curve, the recorded stress then falls as the material’s load-carrying capacity declines. Eventually the necked region can no longer sustain the load, and the specimen fractures at point E. The strain at E represents the material’s total elongation at failure.
What Is Young's Modulus？
Within the elastic region between O and A, the proportional relationship between stress and strain is defined by Young’s modulus, also known as the modulus of elasticity or tensile modulus. This value quantifies a material’s stiffness via Hooke’s law:
E = stress / strain
Mathematically, that is:
E = σ / ε
where E is Young's modulus with unit Pa or N/m2. The higher the modulus, the less a material deforms under a given stress.
Examples of Stress and Strain in Different Materials
Different materials respond differently under load due to their unique stress–strain behavior. Below are a few examples illustrating this in practice:
Steel in Building Frames
In high-rise buildings, structural steel I-beams and columns carry compressive stresses due to dead and live loads (design stresses are often limited to around 250 MPa). With a Young’s modulus of 200 GPa, the corresponding elastic strain at yield is only 0.125% (ε = σ/E). Beyond the yield point, mild steel strain-hardens and can sustain plastic strains of 10–20% before fracture (ultimate tensile strength 400–550 MPa). In practice, engineers use a safety factor between 1.5 and 2, keeping working stresses below 150 MPa to prevent buckling or permanent deformation.
Aluminum in Aircraft Structures
Aluminum alloys such as 2024-T3 and 7075-T6 experience alternating tensile and compressive stresses up to 300 MPa during takeoff, landing, and turbulence. Their modulus of 70 GPa produces elastic strains of 0.4–0.5%, roughly three times that of steel at the same stress. These alloys offer high ultimate strengths of 500–600 MPa and total elongations of 10–15%. Fatigue life (between 10⁶ and 10⁷ cycles) is managed by monitoring strain amplitudes and crack-growth rates to ensure durability over the service life.
Rubber in Car Tires
Rubber car tires undergo repeated cycles of tension and compression as they rotate and deform against the road surface. Rubber compounds have tensile strengths of 15–25 MPa and low elastic moduli (1–10 MPa), but exhibit reversible strains of 300–600% (some high-performance formulations exceed 1 000%). This large recoverable deformation allows the tire to conform to road irregularities and absorb shocks. Designers also consider viscoelastic hysteresis (energy loss) and fatigue crack growth under millions of load cycles to ensure long-term durability and traction.
Conclusion
Stress-strain curves reveal key mechanical properties—elastic modulus, yield strength, ultimate tensile strength, ductility and toughness—that guide material selection. By analyzing how stress distributes and induces strain, engineers predict deformations and verify that components remain safely in the elastic region, checking against limits like yield or buckling.
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FAQ
1. What is the difference between engineering strain and true strain?
Engineering strain is calculated simply as the change in length divided by the original gauge length, assuming the gauge length remains nearly constant. True strain, by contrast, tracks each tiny length change relative to the specimen’s continually changing length and integrates these incremental strains throughout the deformation process. For small deformations, the two are nearly equal. But as deformation increases, engineering strain underestimates the actual change, whereas true strain provides an exact measure.
2. Is resilience the same as stiffness?
No. Stiffness, quantified by Young’s modulus, is the material’s resistance to elastic deformation (the slope of the stress-strain curve). Resilience is the recoverable energy per unit volume that the material can absorb in that elastic range (the area under the curve up to yield).
