True (logarithmic/natural) strain, engineering strain, and true stress for mechanics of materials and tensile testing. All calculations run 100% in your browser.
εe = ΔL / L0 ·
True strain: εt = ln(L/L0) = ln(1+εe) ·
True stress: σt = σe(1+εe)True strain (symbol epsilon_t, also called logarithmic or natural strain) is a deformation measure that uses the instantaneous length as reference, not the initial length. It is defined as epsilon_t = ln(L/L0), where L = current length and L0 = initial length. Unlike engineering strain, which always refers to L0, true strain adds up correctly over sequential deformations.
The formula is epsilon_t = ln(L/L0) = ln(1 + epsilon_e), where epsilon_e = DeltaL/L0 is the engineering strain. Example: a 100 mm bar pulled to 110 mm has epsilon_e = 10/100 = 10%, while epsilon_t = ln(110/100) = ln(1.1) = 9.5310%. The two values are nearly equal at small strains but diverge as strain grows.
Engineering strain epsilon_e = DeltaL/L0 always refers to the initial length L0, while true strain epsilon_t = ln(L/L0) refers to the instantaneous length so each stage’s deformation adds up correctly (the additive property). At large strains they differ markedly: epsilon_e = 50% equals epsilon_t = ln(1.5) = 40.5465%. True strain also handles compression symmetrically: shortening to half length gives epsilon_t = ln(0.5) = −69.3147%.
Use true strain when strains are large (rule of thumb: above 5–10%), for example in tensile-test-to-failure analysis, metal forming simulations (forging, rolling, drawing), and material true stress–strain curves. For small strains such as ordinary elastic design (below 1%), engineering strain suffices: at epsilon_e = 1%, the relative difference of epsilon_t from epsilon_e is only 0.4967%.
Divide the final length by the initial length, then take the natural logarithm: epsilon_t = ln(L/L0). Example: L0 = 250 mm stretched to L = 287.5 mm gives epsilon_t = ln(287.5/250) = ln(1.15) = 13.9762%. Its engineering strain is (287.5−250)/250 = 15%. Enter both lengths in this calculator’s first tab to get both values at once plus their difference.
True strain is negative in compression and has no −100% lower bound like engineering strain. Example: a 100 mm bar compressed to 90 mm has epsilon_e = −10/100 = −10%, while epsilon_t = ln(90/100) = ln(0.9) = −10.5361%. As length approaches zero, epsilon_t tends to negative infinity, which is physically sensible: infinite deformation would be needed to compress a body out of existence.
True stress sigma_t is force divided by the instantaneous cross-section area (which shrinks when pulled), while engineering stress sigma_e uses the initial area A0. Their relation: sigma_t = sigma_e (1 + epsilon_e). Example: sigma_e = 400 MPa at epsilon_e = 25% gives sigma_t = 400 × 1.25 = 500 MPa. The true stress–strain curve keeps rising until fracture; it does not drop like the engineering curve after the necking point.
This formula assumes constant volume during plastic deformation (A0 × L0 = A × L), so A/A0 = 1/(1+epsilon_e). This assumption is reliable for uniform plastic deformation before necking in a tensile test. After necking (localized thinning), deformation is no longer uniform and this simple formula does not apply; a Bridgman correction or direct neck-diameter measurement is needed.
Strain is dimensionless, but is commonly expressed in percent (%), mm/m (millistrain), or micrometers per meter (microstrain). The conversion: 0.1 dimensionless = 10% = 100 mm/m = 100,000 microstrain. This calculator accepts input in all of these units and displays results in the unit of your choice.
Because of the logarithm property: ln(a/b) + ln(b/c) = ln(a/c). True strain from two sequential stages simply adds up to the total strain. Example: a true strain of 9.5310% (100 mm to 110 mm) followed by another 9.5310% true strain (110 mm to 121 mm) totals 19.0620% = ln(1.21). Engineering strain lacks this property: 10% + 10% engineering does not equal 21% engineering.
Yes. True strain is unbounded above: a very large L gives a very large epsilon_t. Example: epsilon_t = 0.5 (50%) equals an engineering strain of e^0.5 − 1 = 64.8721%, meaning a 100 mm bar stretches to 100 × e^0.5 = 164.8721 mm. Engineering strain can never reach −100% (zero length), but negative true strain is likewise unbounded below.
This calculator computes axial true strain from mechanical deformation (tension/compression). The Poisson's ratio calculator computes the lateral-to-axial strain ratio (nu = −epsilon_lateral/epsilon_axial), while the thermal expansion calculator computes strain from temperature change (epsilon = alpha × DeltaT), not from force. For full 2D/3D stress analysis use the principal stress calculator.
Not for final design. This calculator is for education and preliminary estimates using standard mechanics-of-materials theory. Structural components, pressure vessels, and safety-critical parts must follow code standards (e.g., SNI, ASTM, ASME) and be verified by a licensed engineer.
For education and preliminary estimates only — not for certified design. Safety-critical component design must follow code standards (e.g., SNI, ASTM, ASME) and be verified by a licensed engineer.
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