Formula & Calculator
True Strain
Defines strain using the natural log of the length ratio, giving an additive strain measure valid for large deformations.
Interpretation
ε_true = ln(1 + ε_eng). True strain is the natural logarithm of the ratio of lengths. Additive for large strains. Used in plasticity and forming analysis.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| epsilon_true | True strain (dimensionless) | |
| epsilon_eng | Engineering strain (dimensionless) |
What it means
True strain (or logarithmic strain) is defined as the integral of the infinitesimal strain increments, giving ε_true = ln(L/L₀) = ln(1 + ε_eng). It is more fundamental than engineering strain because it is additive: the total true strain after a sequence of steps is the sum of the true strains of each step. This makes it useful for large deformations (plasticity). True strain is essential for representing the flow curve in metal forming, and for constitutive models like power‑law hardening. It also appears in the definition of true stress and is used in finite element analysis. Understanding true strain is crucial for engineers dealing with plasticity, forging, and any process involving significant shape change.
Worked example
True Strain – Two Examples
Real‑World| Parameter | Value |
|---|---|
| ε_eng | 0.2 |
| Parameter | Value |
|---|---|
| ε_eng | 0.5 |
Common mistakes
- True strain: Based on instantaneous length: ε_true = ln(L/L₀) = ln(1 + ε_eng).
- Natural logarithm: Use ln (base e) – not log₁₀.
- Additive: True strains can be added (while engineering strains cannot).
- Range: For small strains, ε_true ≈ ε_eng; for large strains, they differ significantly.
- Sign: Positive for tension, negative for compression.
Applications
True strain (ε_true = ln(1 + ε_eng)) is the logarithmic measure of deformation, providing an additive strain measure for large deformations. It is used in plasticity analysis, forming simulations, and material modelling. Engineers use true strain to describe the deformation history of materials, to calibrate constitutive models, and to predict failure in processes like wire drawing and deep drawing. Unlike engineering strain, true strain accurately represents the cumulative deformation when incremental strains are added. This is essential for accurate simulation of manufacturing processes and for understanding the work hardening behaviour of materials at large strains.
- Finite element simulations of metal forming and plastic deformation
- Material model calibration (e.g., Johnson‑Cook, Hollomon)
- Process design for extrusion, drawing, and rolling
- Analysis of strain hardening and ductility
- Prediction of fracture during forming operations
Frequently Asked Questions
True strain (or logarithmic strain) is defined as the natural logarithm of the ratio of the current length to the original length: ε_true = ln(L / L₀). It is also equal to the integral of dL/L over the deformation history.
Engineering strain is ε_eng = (L – L₀)/L₀. True strain is ε_true = ln(1 + ε_eng). For small strains, they are nearly equal; for large strains, true strain is smaller than engineering strain.
True strains are additive, meaning that the total true strain for sequential deformations is the sum of the individual true strains. This property is important for modelling large deformations and forming processes.
The conversion is ε_true = ln(1 + ε_eng). This holds up to the onset of necking.
Using true strain in calculations that require engineering strain (e.g., for stiffness) without converting. Also, applying the conversion after necking is not valid because deformation is no longer uniform.
True strain represents the cumulative deformation per unit of current length. It reflects the actual material deformation more accurately than engineering strain.
By continuously measuring the gauge length (e.g., with an extensometer) and using the logarithmic definition. The true strain can also be derived from the reduction in area.
The true stress‑true strain curve is used to define the material's flow behaviour. In the plastic region, many materials follow the power law: σ_true = K · ε_true^n.
- Additivity.
- Better representation of large deformations.
- Strain rate independence for some materials.
- Used in advanced constitutive models.
- Harder to interpret for engineers not familiar with logarithms.
- Requires continuous length measurement.
- Not directly used in standard design codes (which use engineering strain).