True Strain Calculator

Enter a specimen's original and final length to get true (logarithmic) strain ε = ln(L₁/L₀), engineering strain, percent elongation, and the stretch ratio.

Quick Facts

True strain formula
ε = ln(L₁ / L₀)
Also called logarithmic or natural strain; integrates dL/L over the deformation.
Relation to engineering strain
ε = ln(1 + e)
e = (L₁ − L₀)/L₀ is the engineering (nominal) strain most tensile-test reports use.
Constant-volume relation
ε = ln(A₀ / A₁)
Holds for plastic deformation, where A₀L₀ = A₁L₁ because volume is conserved.
Sign convention
ε > 0 tension, ε < 0 compression
True strain is negative whenever the specimen ends up shorter (L₁ < L₀).

Your Results

Calculated
True Strain (ε)
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ε = ln(L₁ / L₀), dimensionless
Engineering Strain (e)
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e = (L₁ − L₀) / L₀
Percent Elongation
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Engineering strain expressed as a percentage
Stretch Ratio (λ)
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λ = L₁ / L₀ (extension ratio); ε = ln(λ)

Ready

Enter the initial and final length, then press Calculate.

True Strain Formula and Method

True strain (also called logarithmic strain or natural strain) measures deformation relative to the specimen's instantaneous length at every stage of stretching or compression, rather than only its original length. It is defined as ε = ln(L₁ / L₀), where L₀ is the original (gauge) length and L₁ is the final length. This calculator takes those two lengths and returns the true strain, the more familiar engineering strain, the percent elongation, and the stretch (extension) ratio.

How true strain is derived

Engineering strain compares total elongation to a single fixed reference: e = (L₁ − L₀) / L₀. True strain instead adds up strain increments dL/L as the length changes continuously from L₀ to L₁: ε = ∫ dL/L = ln(L₁/L₀). Because each increment is measured against the current length rather than the original one, true strain and engineering strain are related by ε = ln(1 + e) — they agree closely for small deformations (roughly e < 5%) but diverge as deformation grows, which is why true strain is the standard choice for large-strain plasticity, metal forming, and true stress-strain curves.

True strain vs. engineering strain

The key practical advantage of true strain is that it is additive across sequential deformation steps: stretching a bar in two stages gives a total true strain equal to the sum of the two individual true strains, while engineering strains from separate stages cannot simply be added together. True strain is also symmetric in tension and compression — doubling a length gives ε = ln(2) ≈ 0.693, and halving it gives ε = ln(0.5) ≈ −0.693 — whereas engineering strain would show +100% for doubling but only −50% for halving.

Assumptions and practical notes

The length-based formula assumes uniform, uniaxial elongation over the measured gauge length; in a tensile test this holds up to the onset of necking, after which strain localizes and gauge-length measurements no longer represent the whole specimen. The equivalent area-based formula, ε = ln(A₀/A₁), assumes constant volume during deformation, which is a good approximation for plastic flow in metals but not for elastic deformation. Always keep the two lengths (or areas) in the same unit before dividing — true strain, engineering strain, and the stretch ratio are all dimensionless, so the unit itself cancels out.

Frequently Asked Questions

What is the difference between true strain and engineering strain?
Engineering strain e = (L₁ − L₀) / L₀ measures elongation relative to the fixed original length. True strain (also called logarithmic or natural strain) is ε = ln(L₁ / L₀), which measures elongation relative to the instantaneous length at every point during deformation. They are related by ε = ln(1 + e), and the two nearly match for small strains (under about 5%) but diverge noticeably at large plastic deformations.
Why does true strain use a natural logarithm?
True strain is defined as the integral of incremental strain dL/L from the original length L₀ to the final length L₁, which evaluates to ln(L₁/L₀). Because it is built from infinitesimal steps, true strain is additive: the true strain of two sequential deformations equals the sum of their individual true strains, which is not true for engineering strain.
Can true strain be negative?
Yes. True strain is negative whenever the final length is shorter than the original length (compression), since ln(L₁/L₀) is negative for L₁ < L₀. Notably, stretching a bar to double its length gives ε = ln(2) ≈ 0.693, and compressing it to half its length gives ε = ln(0.5) ≈ −0.693 — the same magnitude with opposite sign, a symmetry engineering strain does not have.
How does true strain relate to cross-sectional area?
For plastic deformation, volume is approximately conserved, so A₀L₀ = A₁L₁. Substituting into the length-based definition gives ε = ln(A₀ / A₁), which is useful when area or diameter is easier to measure than length, such as in compression tests or wire drawing.

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