True stress is the force applied to a material divided by its actual cross-sectional area at that instant. This differs from the more commonly reported engineering stress, which divides force by the original cross-sectional area, ignoring the fact that the material gets thinner as you stretch it. The distinction matters most when metals and other materials undergo large deformations, because the two measurements can diverge significantly.
True Stress vs. Engineering Stress
When engineers pull a metal bar in a tensile test, they record how much force it takes to stretch it. Engineering stress takes that force and divides it by the cross-sectional area the bar started with, before any stretching began. It’s a convenient measurement because you only need to measure the bar once, at the beginning.
True stress uses the same force but divides it by the cross-sectional area the bar has right now, at this moment during the test. As a bar stretches, it gets narrower. That means the actual area carrying the load is smaller than the original area, so the true stress at any given moment is higher than the engineering stress. Early in a test, while deformation is small, the two values are nearly identical. As deformation increases, they split apart.
Why the Difference Matters
On an engineering stress-strain curve, stress rises to a peak called the ultimate tensile strength, then appears to drop before the material breaks. That apparent drop is misleading. The material hasn’t actually gotten weaker. What’s happened is that the bar has started to “neck,” meaning one region narrows dramatically while the rest stays roughly the same. The force needed to keep stretching it may decrease slightly, but the shrinking neck means the real stress on that shrinking section is still climbing.
The true stress-strain curve captures this reality. Instead of dropping after the ultimate tensile strength, true stress continues to rise all the way to fracture. This makes it a more accurate picture of what the material is actually experiencing internally. Engineers working on metal forming, crash simulation, or any process involving large plastic deformation need true stress values because they reflect the material’s real resistance to further deformation.
Converting Between the Two
Before necking begins, you can convert between engineering and true values using a simple relationship. The key assumption is volume conservation: when you stretch a piece of metal plastically, it gets longer and thinner, but its total volume stays the same. This is a foundational principle in plasticity theory, and it holds well for metals undergoing permanent (plastic) deformation, though it’s not perfectly exact for all theoretical frameworks dealing with very large deformations.
With this assumption, if the engineering strain is εe and the engineering stress is σe, then:
- True stress = σe × (1 + εe)
- True strain = ln(1 + εe)
The “ln” is the natural logarithm. True strain is sometimes called logarithmic strain for this reason.
These conversion formulas only work up to the onset of necking. Once necking starts, the strain is no longer uniform along the length of the bar. Different sections are deforming by different amounts, so a single strain value for the whole specimen stops being meaningful. Getting true stress-strain data beyond necking requires direct measurement of the neck’s geometry, which is significantly more involved.
The Strain Hardening Relationship
For many metals, the true stress-strain curve in the plastic region follows a surprisingly clean mathematical pattern known as the Hollomon equation:
σ = Kεn
Here, σ is true stress, ε is true strain, K is a material constant called the strength coefficient (equal to the true stress when true strain equals 1.0), and n is the strain hardening exponent. The exponent n describes how quickly a metal gets stronger as you deform it. A higher n means the material hardens more with deformation, which generally makes it more formable because it resists localized thinning.
For example, annealed copper has a strain hardening exponent of about 0.54, which is relatively high, reflecting copper’s ability to work-harden substantially before failing. Low carbon steels and aluminum alloys have lower n values. This single number, derived from the true stress-strain curve, is one of the most useful parameters in sheet metal forming because it predicts how evenly a material will stretch before it starts to neck.
When Each Measure Is Used
Engineering stress is still widely reported in material data sheets and specifications because it’s straightforward to calculate and perfectly adequate for design work in the elastic range, where deformations are tiny. If you’re selecting a steel beam for a building, engineering stress values give you everything you need.
True stress becomes essential when you’re modeling large plastic deformations. Finite element simulations for car crash testing, forging, deep drawing of sheet metal, and similar processes all require true stress-strain input. Using engineering stress-strain data in those simulations would underestimate the material’s actual strength at high strains and produce inaccurate results. Most commercial simulation software expects material data entered as true stress and true strain for exactly this reason.
In research settings, true stress-strain curves also reveal more about a material’s fundamental behavior. Because they strip away the geometric artifact of a changing cross section, they isolate the material’s intrinsic response to deformation, making it possible to compare materials and develop constitutive models that apply across different loading conditions.