AkCalculators

Stress and strain — definitions, elastic & plastic behaviour, and engineering practice

Stress and strain describe how materials respond to external loads. Stress quantifies internal forces within a body, while strain quantifies deformation. Together they form the stress–strain curve — a central tool in materials science and mechanical engineering.

Engineering (nominal) stress and strain

Engineering stress (also called nominal stress) is defined as the applied axial force divided by the original cross-sectional area:

σ = F / A₀

Engineering strain (ε) is the ratio of the change in length to the original length:

ε = ΔL / L₀

These simple definitions assume the original dimensions for normalization (A₀ and L₀). They are convenient for reporting and basic design calculations, and are the basis of the classical engineering stress–strain curve.

Elastic region and Hooke's law

At small deformations most engineering materials exhibit a linear relation between stress and strain: Hooke’s law. The slope of this linear portion is Young’s modulus (E):

E = σ / ε

Young's modulus is a material property that measures stiffness (e.g., ~210 GPa for steel, ~69 GPa for aluminium). Within the elastic limit the deformation is reversible — remove the load and the part returns to its original shape. The elastic region is the domain where the calculator’s Elastic tab computes E using measured σ and ε.

Yield, plastic deformation and the 0.2% offset method

Beyond the elastic limit materials begin to yield — they deform plastically and part of the deformation becomes permanent. For some materials (e.g., low-carbon steels) the yield point may be distinct; for others it is gradual. Engineers commonly define a practical yield strength using the 0.2% offset method:

1. Draw a line parallel to the initial elastic slope but offset along the strain axis by ε_offset = 0.002 for a 0.2% offset.
2. The intersection of this offset line with the engineering stress–strain curve defines the offset yield strength σ_y (0.2%).

This approach standardises yield definition for materials lacking a sharp yield point and is widely used in specifications and standards.

True stress and true strain

Engineering stress and strain use original dimensions. True stress and true strain account for instantaneous geometry and therefore describe the material under actual conditions during large plastic deformations.

σ_true = F / A_inst

ε_true = ln(L / L₀)

There are approximations to convert engineering values to true values for moderate strains. A common approximation for true stress is:

σ_true ≈ σ_eng × (1 + ε_eng)

This is useful for tensile testing prior to necking. True stress is essential for material modelling, plasticity analysis and predicting failure under large deformations.

Interpreting a stress–strain curve

A typical monotonic tensile stress–strain curve begins with a linear elastic rise (slope = E), reaches yield, then shows strain hardening (stress increases with further plastic strain) until maximum (ultimate tensile strength, UTS), and then softening to failure (necking and fracture). Key points include:

• Elastic limit — end of linear behaviour.
• Yield strength — onset of significant plastic deformation (commonly defined by 0.2% offset).
• Ultimate tensile strength (UTS) — maximum engineering stress.
• Fracture — final failure point.

Practical calculations and pitfalls

When using stress and strain calculators note these common issues:

• Units: Ensure consistent units for force, area and length. Convert mm² to m² or psi to Pa as needed.
• Mass vs force: When given mass (kg), convert to force using F = m·g (g ≈ 9.80665 m/s²) before computing stress.
• Engineering vs true: Use engineering values for simple comparisons; use true values for large plastic strains and for constitutive modelling.
• Offset selection: The 0.2% offset is standard for many metals; other offsets (e.g., 0.1%, 0.5%) may be used depending on materials and standards.

Worked example — Elastic calculation

Given a tensile specimen with original cross-sectional area A₀ = 100 mm² (0.0001 m²), original length L₀ = 50 mm, subjected to an axial force F = 50 kN, and measured elongation ΔL = 0.6 mm:

1. Convert area: 100 mm² = 100×10⁻⁶ m² = 1e-4 m².
2. Stress: σ = F / A₀ = 50,000 N / 1e-4 m² = 500×10⁶ Pa = 500 MPa.
3. Engineering strain: ε = ΔL / L₀ = 0.6 mm / 50 mm = 0.012.
4. Young's modulus (approx): E = σ / ε = 500 MPa / 0.012 ≈ 41.67 GPa.

Interpretation: The computed E is lower than typical steel values — likely the specimen is not purely elastic or measurement error exists; always check test conditions and data validity.

Worked example — 0.2% offset yield

Using the same engineering curve, draw a line with slope E passing through ε = 0.002; find intersection with the curve (graphical). Numerically you can estimate by shifting strain by 0.002 and solving for stress at that shifted strain on the curve data.

Application notes

Stress–strain calculations are used in material selection, structural design, failure analysis, and quality control. For certification and critical design use instrumented tensile tests, traceable calibration and proper specimen preparation. This calculator is for quick estimation, teaching and initial design checks, not certification.

Understanding the difference between engineering and true quantities and recognising the limits of linear elasticity is essential for reliable mechanical design. Use this tool to perform rapid calculations, check units, and prepare CSV exports for lab notes.