Formula & Calculator
True Stress
Corrects engineering stress for the actual instantaneous cross-sectional area of a specimen as it necks down during tensile testing.
Interpretation
σ_true = σ_eng × (1 + ε_eng). True stress accounts for current area. More accurate for large plastic deformation. Used in forming and fracture analysis.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| sigma_true | True stress | MPa |
| sigma_eng | Engineering stress | MPa |
| epsilon_eng | Engineering strain (dimensionless) |
What it means
True stress is the stress calculated using the actual (instantaneous) cross‑sectional area during deformation, as opposed to engineering stress which uses the original area. For uniaxial tension, the relationship is σ_true = σ_eng (1 + ε_eng), assuming constant volume. True stress more accurately represents the actual state of stress, especially after necking begins. It is essential for modelling large plastic deformation processes like rolling, forging, and extrusion. In stress‑strain curves, true stress continues to rise even after the engineering stress maximum (UTS) because of strain hardening. The true stress‑strain curve is used in finite element simulations and in fracture mechanics. Understanding true stress is important for process engineers and designers of high‑strain applications to predict material behaviour accurately.
Worked example
True Stress – Two Examples
Real‑World| Parameter | Value |
|---|---|
| σ_eng | 300 MPa |
| ε_eng | 0.1 |
| Parameter | Value |
|---|---|
| σ_eng | 500 MPa |
| ε_eng | 0.3 |
Common mistakes
- True stress: Based on the instantaneous area (actual cross‑section during loading) – not original area.
- Engineering stress σ_eng: Force / original area.
- Engineering strain ε_eng: (L−L₀)/L₀.
- Formula: σ_true = σ_eng · (1 + ε_eng) – valid up to necking (for uniform deformation).
- Necking: After necking, the stress distribution is no longer uniform; use a correction factor.
Applications
True stress is the load divided by the instantaneous cross‑sectional area, accounting for the reduction in area during deformation. It provides a more accurate representation of the material's stress state, especially in the plastic region. Engineers use true stress‑strain data for advanced material modelling, such as in finite element simulations of forming processes (forging, extrusion, sheet metal forming) and in the design of components that undergo large deformation. By using true stress, they can better predict failure and optimise manufacturing processes. It is essential for understanding material behaviour beyond the necking point, guiding process design and product performance.
- Finite element modelling of large‑deformation processes
- Design of forming operations (rolling, forging, extrusion)
- Material characterisation for plasticity and ductile fracture
- Prediction of forming limits and springback
- Failure analysis in ductile materials
Frequently Asked Questions
True stress is the load divided by the instantaneous (current) cross‑sectional area: σ_true = F / A_instantaneous. It accounts for the reduction in area during plastic deformation.
Engineering stress uses the original area (A₀), which is constant. True stress uses the actual area (A), which decreases with deformation. True stress is always higher than engineering stress after necking.
True stress provides a more accurate description of the material's actual state of stress during large deformations, such as in forming operations, and is used in models of plastic flow and fracture.
Before necking, the deformation is uniform, so σ_true = σ_eng · (1 + ε_eng). This is derived from the conservation of volume.
Using the conversion formula (σ_true = σ_eng(1+ε_eng)) after necking has begun, because the deformation is no longer uniform. After necking, true stress must be measured directly from the instantaneous area.
By simultaneously recording the load and the reduction in cross‑sectional area. This can be done using video extensometry or by measuring the diameter after fracture and assuming constant volume.
It is the load at fracture divided by the final cross‑sectional area. It is often much higher than the engineering ultimate tensile strength.
The true stress‑true strain curve is the fundamental material response in the plastic region. It is used to derive constitutive laws (e.g., power‑law hardening: σ_true = K·ε_true^n).
Engineering stress is used in design and standard specifications because it is based on original dimensions. True stress is used for scientific understanding and in advanced plasticity models.
- It is harder to measure than engineering stress.
- Assumes uniform deformation (constant volume) for the conversion.
- Does not account for triaxiality effects at the crack tip.