How this instrument works
Strain records how much a thing has stretched relative to its own size, and two honest ways exist to do that dividing. An engineering measure refers every increment of stretch back to the original length L₀ and never changes its denominator. A logarithmic measure refers each increment to whatever length a specimen happens to have at that moment, dε = dL ⁄ L, and integrating from L₀ up to L yields a natural logarithm, ln(L ⁄ L₀). Neither approximates the other; they answer different questions, and their answers separate as deformation grows.
Paul Ludwik proposed this logarithmic form in 1909 while studying metals pushed far past yield, calling it natural strain. Heinrich Hencky rebuilt it on continuum-mechanics foundations in 1928, which is why much of the literature says Hencky strain instead. Its appeal is arithmetic: deform a bar in stages and logarithmic values from each stage simply add, because a logarithm turns multiplied length ratios into a sum. Engineering values refuse to behave that way. Any rolling mill, wire-drawing bench or forging press tracking cumulative reduction across several passes therefore books its work logarithmically — halving a thickness is 0.693 however the schedule gets split.
Both measures assume deformation spread evenly along the gauge length, which is precisely what fails when a tensile specimen necks. Past ultimate tensile stress, elongation collects into a short band, and any figure computed across the full gauge understates what that band endures; testers then switch to an area form, ln(A₀ ⁄ A), reading the minimum diameter, which agrees with its length counterpart only while plastic flow conserves volume. Logarithms also fix the domain: final length must stay positive, and squeezing toward zero thickness drives results to negative infinity — a sensible warning, since no finite amount of work flattens material into nothing.
- Enter Original length — your undeformed gauge span L₀, in mm, cm or m. It must be whatever span your extensometer or gauge marks actually straddle, not the whole coupon.
- Enter Final length: that same span after deformation. For a compression test it is the smaller figure, which this sheet handles without complaint.
- Read True (logarithmic) strain, an integrated measure ln(L ⁄ L₀), carried to nine decimals.
- Compare it with Engineering strain shown alongside. Below about 0.02 both agree to within one percent of a reading; past that point their gap widens quickly.
Worked example — a 100 mm gauge pulled to 110 mm
A tensile coupon carries gauge marks 100 mm apart, and pulling continues until those marks sit 110 mm apart. Put 0.1 m into Original length and 0.11 m into Final length. Engineering strain returns exactly 0.1 — ten percent on the nose, and what any datasheet would quote. True (logarithmic) strain returns ln(0.11 ⁄ 0.1) = ln(1.1) = 0.0953101798043 — some 4.7 percent below its engineering counterpart, and already too big a discrepancy to wave away in forming work.
Pull that coupon again, 110 mm out to 121 mm, and additivity shows its hand. Across the whole history an engineering value of 0.21 appears, not 0.1 + 0.1 as intuition offers. Its logarithmic counterpart is 0.0953101798043 taken twice, 0.190620359609, which is exactly ln(1.21). Stage strains add; stage ratios multiply. That single fact is why a forging engineer quoting a reduction of 1.2 can be understood without also being told how many passes produced it.
Questions
Why does true strain differ from engineering strain at all?
Because one denominator moves and one does not. Engineering strain divides every scrap of stretch by a fixed starting length; its logarithmic cousin divides each increment by whatever length exists at that instant, and integrating gives ln(L ⁄ L₀). While stretches stay small, running length barely differs from starting length, so both nearly coincide — at one percent they part by roughly half a percent of a reading. At ten percent that gap reaches 4.7 percent, at fifty percent about nineteen percent, and beyond that they diverge without limit.
What units does true strain use?
None. It is metres divided by metres, a pure number, which the SI Brochure describes as having unit one. So 0.0953 may equally be written 9.53 percent, or 95 310 microstrain (µε) in laboratory shorthand, where one microstrain is 10⁻⁶. Keep percent and raw ratio firmly apart when feeding results into other formulas — passing 10 where 0.1 was wanted is a classic hundred-fold blunder, and nothing in arithmetic will flag it.
When should I use true strain rather than engineering strain?
Once deformation passes a few percent, or whenever contributions from several stages have to be added. Metal forming works almost exclusively in logarithmic terms: rolling, drawing, extrusion and forging routinely reach values of 0.5 to 3, where an engineering figure stops being useful bookkeeping. Elastic design, structural codes and published elongation-at-break numbers stay engineering, because working deformations are tiny and that convention is universal. Always say which one you mean; a bare figure is ambiguous.
Does compression give a negative value?
Yes, and symmetrically so, which is this measure's tidiest property. Squash a specimen to half its height and ln(0.5) = −0.693 comes back; stretch it to double and ln(2) = +0.693. Equal magnitude, opposite sign. An engineering counterpart returns −0.5 and +1.0 for those same two deformations, an asymmetry that turns awkward whenever compression tests get compared against tension tests. Set Final length below Original length here and that negative value appears directly.
Does this still hold once necking begins?
No. Both formulas assume elongation shared evenly along the gauge. After a tensile specimen passes its ultimate stress, deformation localises into a narrow band, and anything computed from overall gauge length understates conditions inside that band by a wide margin. Testers move to an area form, ln(A₀ ⁄ A), measured from the minimum diameter, which stays valid because plastic flow conserves volume almost exactly. Elongation quoted on a datasheet is a gauge-length average and carries that same caveat.
How does this relate to Young's modulus and stress?
Young's modulus is defined on the straight elastic portion of a stress–strain curve, where deformation rarely exceeds 0.002 for metals and both strain measures agree to better than a tenth of a percent — so E = σ ⁄ ε is untroubled by which one you pick. This distinction only earns its keep out in the plastic region, and there it usually travels alongside true stress, force divided by the instantaneous shrunken area rather than the original one. Pairing true stress against true strain is what lets a flow curve extrapolate sensibly.