Plastic strain is the permanent deformation left in a material after you remove the load, and you calculate it by subtracting the elastic (recoverable) strain from the total strain. The core formula is simple: plastic strain equals total strain minus elastic strain. But applying that formula depends on what data you have, whether you’re working from a stress-strain curve, an FEA simulation, or a tensile test result.
The Basic Formula
Total strain in a material has two components: elastic strain (which springs back when the load is removed) and plastic strain (which stays permanently). The relationship is additive:
ε_plastic = ε_total − ε_elastic
Elastic strain follows Hooke’s law, so you can always calculate it if you know the stress and the material’s elastic modulus (Young’s modulus):
ε_elastic = σ / E
where σ is the current stress and E is the elastic modulus. Combining these two expressions gives you the practical working formula:
ε_plastic = ε_total − (σ / E)
This works for uniaxial loading (pulling or compressing in one direction). If you stretch a steel bar with a modulus of 200 GPa to a total strain of 0.015 and the stress at that point is 400 MPa, the elastic strain is 400/200,000 = 0.002, and the plastic strain is 0.015 − 0.002 = 0.013, or 1.3%.
Reading Plastic Strain From a Stress-Strain Curve
On a stress-strain curve, plastic strain is the horizontal distance between the unloading path and the origin. When you load a material past its yield point and then release the load, it doesn’t return to zero strain. Instead, it unloads along a line parallel to the original elastic slope. The strain remaining when stress reaches zero is the residual plastic strain.
Mathematically, if you unloaded from a strain of ε_ul at a stress of σ_ul, the plastic strain is:
ε_plastic = ε_ul − σ_ul / E
This is the same subtraction formula, just applied at the specific point where unloading begins. You’re essentially sliding back down the elastic line from your current position and seeing where you land on the strain axis.
Finding Where Plastic Strain Begins
Before you can calculate plastic strain, you need to know when it starts. Most materials don’t have a sharp, obvious yield point. The standard engineering approach is the 0.2% offset method: you draw a line parallel to the elastic portion of the stress-strain curve, but shifted 0.002 (0.2%) to the right along the strain axis. Where that offset line intersects the stress-strain curve is the yield strength.
The logic is straightforward. Industry practice uses strains of 0.1% and 0.3% to define the linear (elastic) slope of the curve, then offsets a parallel line by 0.2% strain. Any strain beyond that intersection point has a plastic component. The 0.2% value is a convention, not a physical law. Some applications use 0.05% or 0.1% offsets when tighter tolerances matter, and some polymers or additively manufactured materials need different approaches entirely.
For materials that do show a distinct yield point (like mild steel with its obvious “knee” in the curve), you can read yield directly from the data without the offset construction.
Engineering Strain vs. True Strain
The formulas above use engineering strain, which is based on the original length of the specimen. For small deformations (a few percent), engineering and true strain are practically identical. For large plastic deformations, they diverge, and true strain becomes more physically meaningful.
True strain is defined as:
ε_true = ln(1 + ε_engineering)
where ln is the natural logarithm. At 10% engineering strain, true strain is ln(1.10) = 0.0953, a small but real difference. At 50% engineering strain, true strain is only 0.405, a significant gap. If you’re working in finite element software or analyzing metal forming operations, you’ll almost always need true plastic strain. To get it, convert your total true strain and subtract the elastic component calculated from true stress and the modulus.
Plastic Strain Under Multiaxial Loading
Real components rarely experience simple tension or compression in one direction. Bolts, pressure vessels, and crash structures see stress from multiple directions simultaneously. In these cases, you need an equivalent plastic strain that collapses the full 3D strain state into a single number you can compare against material data.
For materials that follow the von Mises yield criterion (most metals), the equivalent plastic strain increment combines the individual plastic strain components from all three axes and the shear planes into one value. The calculation involves squaring each directional plastic strain increment, combining them with weighting factors, and taking the square root. In practice, you rarely compute this by hand. Finite element software (Abaqus, ANSYS, LS-DYNA) tracks equivalent plastic strain automatically at every integration point. It’s typically labeled PEEQ or EQPS in your results.
The important thing to understand is that equivalent plastic strain is cumulative. It only increases, never decreases, even if the loading direction reverses. It represents how much total permanent deformation the material has experienced regardless of direction.
Strain Hardening Models for the Plastic Region
Once a material yields, stress typically continues to rise with further plastic strain, a behavior called strain hardening. Several empirical equations describe this relationship, and choosing the right one matters if you’re predicting plastic strain at a given stress level (or vice versa).
The most widely used is the Hollomon equation:
σ = K × ε_p^n
where σ is true stress, ε_p is true plastic strain, K is the strength coefficient, and n is the strain-hardening exponent. The exponent n typically ranges from 0.1 to 0.5 for metals. A higher n means the material hardens more gradually and can spread deformation over a larger area before necking. You can rearrange this to solve for plastic strain at a known stress: ε_p = (σ/K)^(1/n).
The Swift equation adds a small initial strain offset: σ = K_s × (ε_p + ε_0)^n_s. This handles materials that don’t start hardening from exactly zero plastic strain. The Voce equation takes a different approach, modeling stress as an exponential saturation curve: σ = σ_0 − Aσ_0 × exp(−β × ε_p). Voce works well for materials whose hardening rate tapers off at large strains, which is common in aluminum alloys. Most FEA software lets you input data in any of these forms, or you can provide a tabular stress-strain curve directly.
Practical Steps for Common Scenarios
From Tensile Test Data
If you have raw load-displacement data from a tensile test, convert to engineering stress and strain first (stress = force/original area, strain = change in length/original length). Identify the elastic modulus from the initial linear region. Then, for any point on the curve past yield, plastic strain equals the total strain at that point minus the stress at that point divided by the modulus.
For FEA Material Input
Most finite element codes require true plastic strain paired with true stress. Start with your engineering data, convert both stress and strain to true values, then subtract the elastic strain at each point. Your table should start at zero plastic strain (the yield point) and increase from there. A common mistake is feeding in total strain instead of plastic strain, which shifts the entire hardening curve and produces incorrect results.
From a Known Load on a Structure
If you know the stress at a particular location exceeds yield, and you know the material’s hardening behavior (from a Hollomon fit or a stress-strain table), you can read or calculate the corresponding plastic strain directly. For a quick estimate without hardening data, assume the material is elastic-perfectly-plastic (no hardening after yield), and all strain beyond σ_yield/E is plastic.