Plastic strain is the permanent deformation left in a material after loading, and you calculate it from a stress-strain curve by subtracting the elastic (recoverable) strain from the total strain at any point beyond yielding. The procedure is straightforward once you understand what each part of the curve represents, but several real-world complications can trip you up, from materials that yield abruptly to curves that shift depending on temperature or loading speed.
Why Total Strain Splits Into Two Parts
Every stress-strain curve records two kinds of deformation happening simultaneously once the material starts to yield. Elastic strain is the portion that springs back when you remove the load. Plastic strain is the portion that stays. At any point on the curve past yielding, the total strain you read off the horizontal axis is the sum of these two components. The foundational assumption behind nearly all elastic-plastic material models is this additive split: total strain rate equals elastic strain rate plus plastic strain rate.
This additive decomposition is not just a textbook convenience. It is the working principle embedded in finite-element codes and testing-lab procedures alike.1LAMÉ Manual. Elastic-Plastic Model In practice, “calculate the plastic strain” almost always means: figure out how much elastic strain exists at that stress level and subtract it from the total.
Step by Step at Any Point on the Curve
Suppose you have a stress-strain curve from a standard tensile test and you want the plastic strain at some stress level beyond yield. Here is the procedure:
- Pick your point: Choose the point on the curve where you want to know the plastic strain. Read the stress (vertical axis) and the total strain (horizontal axis) at that point.
- Draw the elastic unloading line: From that point, draw a straight line with a slope equal to the material’s elastic modulus (Young’s modulus, E). This line should be parallel to the initial linear portion of the curve.
- Find where it hits zero stress: Where that line intersects the strain axis (zero stress) is your plastic strain. The horizontal distance from the origin to that intersection is the permanent deformation the specimen would retain if you unloaded it at that point.
- Or just calculate it: Plastic strain equals total strain minus elastic strain, where elastic strain at that stress level is simply stress divided by E.
The graphical method and the arithmetic method give the same answer. The graphical version is useful when you are working from a printed or plotted curve and do not have a clean data file. The arithmetic version is what you would use in a spreadsheet or script when processing raw test data.
One thing to watch: this works cleanly only in the region of the curve where the initial elastic modulus still governs unloading. For most metals loaded monotonically at room temperature, that assumption holds well. In situations involving very large strains, damage accumulation, or repeated cycling, the effective unloading stiffness can shift, and the simple subtraction becomes an approximation rather than an exact answer.
The Offset Method for Finding Yield Strength
Before you can talk about plastic strain at arbitrary points, you usually need to identify where plastic deformation begins, and that is where the offset method comes in. Most metals do not have a single crisp point where elastic behavior ends and plastic behavior begins. The transition is gradual, and the stress-strain curve bends smoothly rather than snapping to a new slope.
The standard convention for metals with this gradual transition is to define the yield strength at 0.2% offset strain (0.002 in absolute terms). You draw a line from the 0.2% strain mark on the horizontal axis, parallel to the initial elastic slope, and where it intersects the curve is the 0.2% offset yield strength. At that intersection, by definition, the plastic strain is 0.002. This is an engineering convention, not a physical threshold: the material has accumulated a small but measurable amount of permanent deformation, and the community has agreed that this amount is enough to call it “yielded.”
Some applications use tighter offset definitions. Research on copper single crystals, for instance, found that at a conventional 0.01% offset, the specimen had accumulated almost no genuine permanent strain: the nonlinearity observed at small offsets actually came from higher-order elastic effects, not true plasticity. Permanent strain of 0.01% did not actually appear until the offset reached about 0.05% and total strain reached roughly 1%.2CrossRef API. Finite Strain and the 0.01 Percent Offset Yield Strength This is a useful reminder that the offset method defines a convenient reference point, not a fundamental material property. The number you get depends on which offset you choose, and at very small offsets, you may not even be measuring real plastic strain.
Materials With a Sharp Yield Point
Not every material yields gradually. Low-carbon steels and some other alloys exhibit a sharp yield point, where the stress-strain curve reaches a peak (the upper yield point), drops suddenly, and then plateaus at a lower stress (the lower yield point) while the material stretches at roughly constant load. During this plateau, localized bands of plastic deformation, called Lüders bands, form and propagate through the specimen.
Research on ferrite-pearlite steels has documented this process in detail using full-field strain measurement. Plastic deformation first concentrates at a stress riser, then a distinct plastic band nucleates at the upper yield point and sweeps across the specimen width. At the upper yield point, a significant concentration of plastic strain is already present at the band’s initiation site, even though the rest of the specimen is still elastic.3PubMed Central. Yield-Point Phenomenon and Plastic Bands in Ferrite–Pearlite Steels
For calculating plastic strain from these curves, the sharp yield point creates a practical wrinkle. During the Lüders plateau, the strain is not uniform along the gauge length. One part of the specimen may be at several percent plastic strain while an adjacent region is still elastic. The average strain reported by the extensometer is a mixture of yielded and unyielded zones. Once the Lüders bands have swept through the entire gauge length, uniform deformation resumes and the stress-strain curve begins to rise again (work hardening). From that point onward, the standard subtraction method works normally. But within the plateau itself, the “plastic strain” you calculate from the average curve is somewhat artificial: it represents an average of wildly different local conditions.
If your material shows a sharp yield point and you need to define the yield stress, the usual convention is to report the lower yield point rather than the upper one. The upper yield point is sensitive to specimen geometry, alignment, and loading rate, so it is less reproducible. The lower yield point better represents the stress at which plastic deformation propagates through the bulk.
Plastic Strain During Cyclic or Interrupted Loading
In real engineering scenarios, loads are not always monotonically increasing. Components may be loaded, partially unloaded, and reloaded. In fatigue testing, specimens cycle through thousands or millions of load reversals. Each time you unload and reload, you can measure the actual plastic strain directly rather than inferring it from the monotonic curve.
Cyclic loading-unloading-reloading tensile tests are a standard experimental technique for tracking how plastic strain evolves. When you unload from a point on the stress-strain curve, the specimen does not retrace the original curve. It follows a nearly linear path (with a slope close to E) back toward zero stress. The gap between where you started loading and where the unloading line hits zero is the plastic strain at that point. By repeating this at several load levels during a single test, you build a detailed picture of how plastic strain accumulates.
These cyclic tests also reveal something subtler. The unloading path is not perfectly linear. There is a small amount of recoverable strain that lags behind, called inelastic back strain. In metallic laminates (layered composites of different metals), this back strain depends strongly on how different the constituent layers are in their yield strengths and stiffnesses, and on how many interfaces exist in the laminate.4Metals. Quantifying Co-Deformation Effects in Metallic Laminates by Loading–Unloading–Reloading Tensile Tests For single-material specimens under modest plastic strains, the effect is small and the simple elastic subtraction works well. But if you are working with composite or laminated materials, be aware that the “elastic” unloading is not as clean as it looks, and the plastic strain you calculate may include a small error from this nonlinear recovery.
Engineering Strain vs. True Strain
The stress-strain curve you see depends on whether the data are plotted as engineering quantities or true quantities. The difference matters a great deal when plastic strains get large.
Engineering strain is the change in length divided by the original length. It is what most basic tensile tests report. True strain is the natural logarithm of the ratio of current length to original length. At small strains (a few percent or less), the two are nearly identical. Once plastic strain pushes past about 10%, the gap widens. At 20% engineering strain, true strain is about 18%. At 50% engineering strain, true strain is about 40%. For large-deformation problems like metal forming, crash simulation, or forging, true strain is the more physically meaningful quantity because it accounts for the continuously changing gauge length.
The conversion is simple: true strain equals the natural logarithm of (1 plus engineering strain). And true stress equals engineering stress multiplied by (1 plus engineering strain), assuming volume stays roughly constant during plastic deformation, which is a reasonable assumption for most metals.
When you read a plastic strain value from a stress-strain curve, check which type of curve you are working with. The subtraction method (total strain minus stress over E) works the same way in both systems, but you need to use the correct modulus. In the true stress-true strain system, E is the same modulus (it is measured at small strains where the two systems coincide), so the arithmetic is unchanged. The difference shows up in the values of strain themselves, not in the calculation procedure.
When the Curve Reaches Its Peak
On an engineering stress-strain curve, the stress reaches a maximum and then drops. That peak corresponds to the onset of necking, where deformation localizes into a narrow band and the specimen’s cross-section starts to shrink rapidly in one region. The Considère criterion describes this point: necking begins when the rate of work hardening can no longer compensate for the reduction in cross-sectional area.
After the onset of necking, the engineering stress-strain curve is no longer a reliable source of plastic strain for the material as a whole. The strain reported by the extensometer is an average over the gauge length, but the actual plastic strain in the neck is much higher than elsewhere. If you need plastic strain values past the peak, you have two options: convert the data to true stress and true strain (which pushes the useful range further but still eventually becomes inaccurate once the neck develops a complex triaxial stress state), or use local measurement techniques that capture the strain field directly in the neck region.
Full-Field Strain Measurement With Digital Tools
Traditional extensometers give you a single average strain value over a fixed gauge length. Digital image correlation (DIC) goes further by tracking the deformation of a speckle pattern painted on the specimen surface, producing a full map of strain across the entire field of view.
DIC has been validated against conventional extensometry and strain gauges for tensile testing. Comparisons on aluminum and other materials have shown that DIC produces values closely matching those from contact-based methods for quantities like Young’s modulus and tensile properties, with DIC sometimes reading slightly higher than classical extensometry.5Elsevier / Materials Today: Proceedings. Experimental investigation of the tensile test using digital image correlation (DIC) method
Where DIC really earns its keep for plastic strain measurement is in the situations described above: necking, Lüders band propagation, and any test where strain is not uniform. Instead of calculating a single plastic strain from a single average strain value, you get a map showing exactly how much plastic strain has accumulated at every point on the specimen surface. For materials that neck severely or form localized bands, this is the only way to get meaningful local plastic strain data.
If you are setting up DIC for plastic strain measurement, the main practical considerations are surface preparation (the speckle pattern needs to be fine enough for your field of view), camera resolution and frame rate (you need enough frames to capture the deformation as it happens), and calibration of the imaging system. The subtraction method still applies to each point in the DIC field: local plastic strain equals local total strain minus local stress divided by E, though extracting local stress from DIC data alone requires some additional modeling or assumptions about the constitutive behavior.
How Temperature and Loading Speed Shift the Curve
A stress-strain curve measured at room temperature and a slow loading rate can look quite different from one measured at high temperature or under dynamic impact. The flow stress of most metals increases at higher strain rates and decreases at higher temperatures. This means the plastic strain at a given total strain level is also affected, because the yield point and the hardening behavior both change.
The Johnson-Cook model is one of the most widely used descriptions for how flow stress depends on plastic strain, strain rate, and temperature simultaneously. It expresses the flow stress as a product of three terms: a strain-hardening term that depends on the equivalent plastic strain, a strain-rate term that scales logarithmically with the ratio of current strain rate to a reference rate, and a thermal-softening term that scales with how close the temperature is to melting.6Journal of Computational Design and Engineering. Identification of plastic constitutive Johnson–Cook model parameters by optimization-based inverse method The parameters of this model are fitted from experimental stress-strain curves at various temperatures and strain rates.
For your purposes as someone reading a stress-strain curve, the practical takeaway is this: if the curve you are working from was generated under specific temperature and rate conditions, the plastic strain values you extract are valid only for those conditions. You cannot take a room-temperature, quasi-static curve and use it to calculate plastic strain in a crash event or a high-temperature forging operation without accounting for how the flow stress changes. If your application involves conditions different from those of the test, you need a constitutive model like Johnson-Cook (or one of its many descendants) to map between conditions.
Plastic Strain Ratios in Sheet Metal
In sheet metal forming, a single plastic strain value is often not enough. You need to know how the plastic strain distributes across different directions. When you pull a sheet metal coupon in tension, it gets thinner and narrower. The ratio of the width strain to the thickness strain, both measured in the plastic regime, is called the plastic strain ratio or Lankford coefficient (commonly written as “r”).
This ratio matters because it tells you how the sheet resists thinning. A material with a high r-value distributes more of its plastic strain into width reduction rather than thickness reduction, which means it is more resistant to thinning during deep-drawing operations and other forming processes. Determination of the plastic strain ratio from texture data using crystal plasticity methods is one of the most common applications of computational materials modeling in the sheet metal industry.7IOP Conference Series: Materials Science and Engineering. Revisiting plastic strain ratio determination in aluminium using crystal plasticity
To measure r experimentally, you run a tensile test on a specimen cut from the sheet, load it into the plastic range (typically to about 15-20% engineering strain, though this varies by standard), and then measure the final width and thickness of the specimen. From those measurements, you compute the plastic width strain and the plastic thickness strain, subtract the elastic contributions, and take the ratio. Because most sheet metals are anisotropic, r varies with the angle at which you cut the specimen relative to the sheet’s rolling direction. Testing at 0, 45, and 90 degrees to the rolling direction gives you three r-values, and an average of these (weighted by the standard formula) gives the normal anisotropy, which is the single number most often used to characterize a sheet’s formability.
This is an area where the simple “subtract elastic from total” approach is necessary but not sufficient. You are decomposing strain in multiple directions simultaneously, and the anisotropy of the material means the yield surface is not symmetric. If you are doing this calculation from raw test data rather than relying on automated software, keep careful track of which direction each strain component corresponds to, and make sure you are subtracting the correct elastic strain for each direction using the appropriate directional modulus if the material is significantly anisotropic.
Common Mistakes That Produce Wrong Numbers
A few errors come up repeatedly when people extract plastic strain from curves. Being aware of them can save you from reporting a number that looks right but is not.
- Using the wrong modulus: If you use a textbook value for E rather than the slope actually measured from the linear portion of your curve, your elastic subtraction will be off. Real specimens sometimes have an apparent modulus that differs from the handbook value because of grip slippage, extensometer misalignment, or compliance in the load frame. Always measure E from your own data.
- Reading past the neck: After the onset of necking on an engineering stress-strain curve, the average strain no longer represents what is happening at any particular point in the specimen. Plastic strains extracted from this region are averaged values that underestimate the true local plastic strain in the neck and overestimate it everywhere else.
- Ignoring the Lüders plateau: If your material has a sharp yield point, strain within the yield-point elongation region is not uniform. Any plastic strain value you calculate from the average curve during this stage is a spatial average, not a local measurement.
- Confusing engineering and true quantities: At large strains, mistaking engineering strain for true strain (or vice versa) produces errors that grow with deformation. A 50% engineering strain corresponds to about 40% true strain. Mixing the two in a single calculation will give a meaningless result.
- Assuming unloading is perfectly linear: For most monotonic tensile tests on simple metals, this is fine. For cyclic loading, composites, or materials that have accumulated damage, the unloading path has some curvature. Using a straight-line approximation when the unloading is nonlinear will slightly overestimate the plastic strain.
If you are processing data programmatically, a good sanity check is to verify that your calculated plastic strain is zero (or very close to zero) at stress levels well within the elastic range. If it is not, something is wrong with your modulus value or your strain offset. Similarly, the plastic strain should never decrease with increasing total strain on a monotonic loading curve. If it does, you have a data-processing error somewhere, likely related to noise in the raw data or an incorrect alignment of the stress and strain channels.