True Strain Calculator: Logarithmic Strain, True Stress and Multi-Pass Deformation
True strain, also called logarithmic or natural strain, is the most accurate measure of large plastic deformation because it uses an ever-updating reference length. Enter initial and final lengths, or switch to engineering strain, cross-sectional area reduction, or a multi-pass rolling/drawing schedule to get the true strain instantly. The calculator also converts between engineering and true stress, shows a comparison chart of how the two strain measures diverge, and walks through every step of the math.
Formula
Worked example
A steel bar is stretched from 100 mm to 120 mm. Engineering strain = (120 - 100) / 100 = 0.200. True strain = ln(120 / 100) = ln(1.200) = 0.1823. The difference is 0.0177, about 8.9% of the engineering strain. If the nominal stress is 300 MPa, true stress = 300 x 1.200 = 360 MPa.
What is true strain and why does it differ from engineering strain?
Engineering strain (also called nominal or conventional strain) divides the change in length by the original length: e = (Lf - L0) / L0. This works well for small deformations but breaks down when a specimen stretches or compresses substantially, because the reference length itself changes throughout the process. True strain, also called logarithmic or natural strain, solves this by integrating the incremental strains over the deforming body: de_t = dL / L, which integrates to e_t = ln(Lf / L0). Because it uses a continuously updating reference, true strain is physically meaningful at any deformation level. The two measures differ by less than 0.5% for strains below about 5%, but at an engineering strain of 0.5 the true strain is only 0.405, a difference of nearly 20%. Metal-forming processes routinely operate in this large-strain regime, so the distinction is not academic.
How to calculate true strain from different inputs
- From length measurements: e_t = ln(Lf / L0). Measure the gauge length before and after deformation. Valid for tension and compression; negative for compression (Lf < L0).
- From engineering strain: e_t = ln(1 + e). Useful when a test report gives engineering values and you need to convert. Note that e must be greater than -1 (a specimen cannot reduce to zero length).
- From cross-sectional area: e_t = ln(A0 / Af). Assumes volume constancy (A0 x L0 = Af x Lf), which holds for plastic deformation of metals and most polymers. Note the ratio is inverted relative to the length form: area decreases as length increases in tension.
- Multi-pass (additive property): The most important practical advantage of true strain is that it is additive. If a billet is rolled in three passes with true strains 0.20, 0.15, and 0.10, the total true strain is simply 0.45, equal to what a single-pass operation from the original to final length would give. Engineering strain is not additive.
True stress and the flow-stress curve
When a tensile specimen is pulled, the cross-sectional area decreases, so the actual stress on the material is higher than the engineering stress computed from the original area. True stress corrects for this: sigma_t = sigma_e x (1 + e_e), or equivalently sigma_t = F / A_current. In a flow-stress (true stress vs. true strain) curve, the material response in the plastic regime is commonly described by the Ludwik-Hollomon power law: sigma_t = K x e_t^n, where K is the strength coefficient and n is the strain-hardening exponent. Fitting this model requires true stress and true strain data, which is why FEA software and material databases always use the true quantities.
Volume constancy and the area method
Plastic deformation of metals is essentially incompressible: volume is conserved. This means A0 x L0 = Af x Lf at every instant. The area method (e_t = ln(A0 / Af)) is particularly convenient in compression testing, because measuring diameter is often easier than tracking a changing gauge length. In rolling and wire drawing, the reduction in area is the primary process variable, so strain is routinely expressed via area rather than length. Note that volume constancy is an approximation: it holds for metal plasticity but not for foams, powders, or highly porous materials.
Typical true strain values in metal forming processes
| Process | Typical true strain range | Engineering strain equivalent (approx) |
|---|---|---|
| Wire drawing (single pass) | 0.05 - 0.30 | 5% - 35% |
| Cold rolling (per pass) | 0.05 - 0.20 | 5% - 22% |
| Cold rolling (total, multi-pass) | 0.5 - 3.0 | 65% - 1900% |
| Deep drawing | 0.2 - 0.7 | 22% - 101% |
| Extrusion (moderate) | 1.0 - 2.0 | 172% - 638% |
| Extrusion (severe) | 2.0 - 4.0 | 638% - 5360% |
| Open-die forging | 0.5 - 2.0 | 65% - 638% |
| Tensile test (fracture, ductile steel) | 0.3 - 0.7 | 35% - 101% |
| Superplastic forming | 0.7 - 2.3 | 101% - 900% |
True strain ranges encountered in common industrial deformation operations.
Frequently asked questions
When does true strain matter more than engineering strain?
True strain becomes important above engineering strains of roughly 5-10%. Below that, the two differ by less than 0.5% and either can be used. Above 10% you should use true strain for: FEA material card input, flow-stress fitting, metal-forming force calculations, and any multi-pass deformation where you need to add strains from successive operations. For small-strain structural engineering (buildings, bridges) engineering strain is perfectly adequate.
Why is true strain called logarithmic strain?
Because it is the integral of incremental length changes relative to the current length: e_t = integral of dL/L from L0 to Lf, which evaluates to the natural logarithm ln(Lf/L0). The natural logarithm arises directly from the definition, not as an approximation.
Is true strain always smaller than engineering strain?
Yes, for tensile deformation (Lf > L0), true strain is always smaller in magnitude than engineering strain. This is because ln(1 + e) < e for all positive e. In compression (Lf < L0), both strains are negative and the true strain magnitude is again smaller: |e_t| < |e_e|.
Why is true strain additive but engineering strain is not?
True strain is additive because it is defined with respect to the current (instantaneous) length. When you apply a second deformation, the new reference is the length left by the first operation, and logarithms add: ln(L2/L0) = ln(L1/L0) + ln(L2/L1). Engineering strain uses the original L0 throughout, so the strains from successive passes cannot simply be summed.
Can I use the area formula if volume is not conserved?
No. The relation e_t = ln(A0/Af) requires A0 x L0 = Af x Lf (volume constancy). For elastic strains, volume changes slightly due to Poisson effects, so this formula is only exact in the fully plastic regime. For elastic-plastic deformation you should use the length formula directly, or correct for the elastic volume change using the bulk modulus.
What is a typical true strain in wire drawing?
A single wire-drawing pass typically applies 0.05 to 0.30 of true strain (about 5% to 35% reduction in area). A heavily drawn wire may accumulate 2.0 or more of total true strain across many passes, which is why the additive property is essential for scheduling multi-pass drawing to avoid exceeding the material's ductility.
How do I convert true strain back to engineering strain?
Reverse the formula: e_e = exp(e_t) - 1. For example, a true strain of 0.4055 gives engineering strain = exp(0.4055) - 1 = 1.500 - 1 = 0.500 (50%). This inverse is also available in the "From engineering strain" mode by entering the engineering strain and reading the true strain, or you can swap the calculation direction.