The plastic stage of the stress-strain curve captures how a material behaves after yielding, defining yield strength, strain hardening, uniform elongation, necking, and fracture characteristics that underpin structural design and metal forming. This article explains the principles, specimen types, measured parameters, standardized test methods, and industrial applications of plastic-stage tensile testing performed by accredited third-party laboratories in accordance with ISO 6892-1, ASTM E8/E8M, GB/T 228.1, and related standards.
Technical Overview
The stress-strain curve obtained from a uniaxial tensile test is the most widely used record of a material's mechanical response. It consists of an elastic region, in which deformation is fully reversible, and a plastic stage, in which deformation is permanent and remains after unloading. Plastic-stage testing concentrates on the segment of the curve between the onset of yielding and final fracture, encompassing yield phenomena, strain hardening, the attainment of tensile strength, necking, and rupture. The parameters extracted from this region, including yield strength, tensile strength, uniform elongation, total elongation, reduction of area, and the strain hardening exponent, form the basis of design allowables, forming-limit assessments, crash-energy predictions, and incoming-material acceptance criteria across virtually every engineering industry.
The Elastic-to-Plastic Transition
For low-carbon steels that exhibit a pronounced yield phenomenon, the transition is marked by an upper yield point (ReH) followed by a lower yield point (ReL) and a yield plateau, produced by the breakaway of dislocations from interstitial carbon and nitrogen atoms and the propagation of Luders bands along the gauge length. For most other engineering alloys, such as aluminum alloys, stainless steels, and high-strength structural steels, the transition is gradual and no distinct yield point exists. In these materials the onset of plasticity is defined by the proof strength at a specified plastic extension, most commonly Rp0.2 determined by the 0.2 percent offset method, or Rp1.0 for certain elevated-temperature or austenitic applications.
Characteristic Phenomena of the Plastic Stage
Once plastic flow begins, the curve typically displays a sequence of well-defined regimes that together describe the ductility and hardening capacity of the material:
- Yield point elongation (Ae): the horizontal plateau during discontinuous yielding in mild steels, prior to the start of uniform strain hardening.
- Strain hardening: the progressive rise of flow stress caused by dislocation multiplication and interactions, which stabilizes plastic deformation and delays localization.
- Tensile strength (Rm): the maximum engineering stress sustained by the specimen, reached at the end of uniform elongation (Ag).
- Diffuse necking: localized reduction of the cross-section that begins at maximum force in accordance with the Considere criterion, after which engineering stress decreases while deformation concentrates in the neck.
- Fracture: final rupture of the necked region, from which total elongation at fracture (A) and reduction of area (Z) are measured.
Engineering Versus True Stress-Strain Curves
Engineering stress and strain are computed from the original gauge cross-section and length, and they are the basis of all standardized reportable results. True stress and true strain, computed from instantaneous dimensions, continue to rise after the onset of necking and provide a more physically meaningful description of the flow behavior. The plastic region of the true stress-strain curve is frequently approximated by the Hollomon power law, where flow stress equals the product of a strength coefficient K and true plastic strain raised to the exponent n. The strain hardening exponent n typically ranges from about 0.1 to 0.5 for ductile metals and, for Hollomon-type materials, approximately equals the true strain at maximum force, making it a direct indicator of uniform stretchability.
Testing Samples
Plastic-stage testing is applicable to any material that can be loaded in uniaxial tension. Specimen geometry, orientation, and preparation are selected according to product form and the governing standard.
Metallic Sheet and Strip
Flat rectangular specimens are taken from sheet and strip in orientations parallel and transverse to the rolling direction. Proportional specimens with a gauge length of 80 mm or non-proportional specimens with a gauge length of 50 mm are typical, with the original thickness retained. Sheet testing is central to automotive advanced high-strength steel programs, where uniform elongation and the n-value directly govern formability and crash performance.
Machined Round Specimens from Bars, Forgings, and Castings
Round specimens with a proportional gauge length of 5.65 times the square root of the original cross-sectional area, equivalent to five diameters, or 11.3 times for long-gauge specimens, are machined from bars, forgings, castings, and heavy sections. Machined specimens allow testing at mid-thickness or mid-radius locations to characterize bulk properties.
Wire, Rod, and Profile Products
When machining is impractical, wire and small-diameter rod are tested in the full as-delivered section, with results reported as force-extension curves and derived elongation values. Grip arrangements and free lengths are adapted to prevent slip and premature failure in the grips.
Tubes and Pipes
Tubes may be tested as full-section longitudinal segments, as longitudinal strip specimens cut and flattened, or as machined transverse segments depending on diameter and wall thickness. Each approach is documented so that the effect of flattening on yield behavior can be assessed.
Welded Joints and Coated Materials
Transverse cross-weld specimens are used to evaluate the composite strength of welded joints, including the yield and hardening response of the weld metal, heat-affected zone, and base metal. For coated sheet, specimens are tested with the coating as delivered unless the specification requires its removal.
Polymers and Composite Laminates
Although the plastic-stage framework was developed for metals, equivalent tensile characterization of plastics is performed per ASTM D638 and ISO 527, and unnotched laminate tension per ASTM D3039, with yield, drawing, and post-yield necking behavior recorded in a similar curve-based manner.
Testing Items
Plastic-stage testing yields a defined set of reportable quantities, each tied to a specific feature of the force-extension or stress-strain curve:
- Upper yield strength (ReH) and lower yield strength (ReL) for materials exhibiting discontinuous yielding.
- Proof strength, plastic extension (Rp0.2 or Rp1.0), for materials without a distinct yield point.
- Yield point elongation (Ae), characterizing the plateau of discontinuous yielding.
- Tensile strength (Rm), the maximum force divided by the original cross-sectional area.
- Percentage extension at maximum force (Ag and Agt), representing uniform elongation.
- Percentage total elongation at fracture (A, A80mm, or A5.65-S0) measured after reassembling the fractured halves.
- Percentage reduction of area (Z), reflecting the cross-section lost in the neck.
- Strain hardening exponent (n) and strength coefficient (K) per ASTM E646 and ISO 10275.
- Plastic strain ratio (r) per ASTM E517 and ISO 10113, often measured on the same specimens for forming analysis.
- Post-necking elongation and fracture characteristics, used in energy-absorption and toughness screening.
- Digitized engineering and true stress-strain curves supplied as input data for finite-element simulation and crash models.
Accredited third-party laboratories typically report all quantities together with the raw curve, test direction, specimen type, equipment class, and applied strain rates so that results are traceable and comparable.
Testing Methods
Governing Standards
Room-temperature plastic-stage tensile testing is standardized internationally by a harmonized family of methods, and companion standards cover derived hardening parameters:
- ISO 6892-1 and EN ISO 6892-1: Metallic materials, tensile testing at ambient temperature.
- ASTM E8/E8M: Standard test methods for tension testing of metallic materials.
- GB/T 228.1: Chinese national standard for metallic materials tensile testing at room temperature, technically aligned with ISO 6892-1.
- JIS Z 2241: Japanese standard for metallic materials tensile testing at room temperature.
- ASTM E646 and ISO 10275: Determination of the tensile strain hardening exponent (n-value).
- ASTM E517 and ISO 10113: Determination of the plastic strain ratio (r-value).
- ISO 6892-2, ISO 6892-3, and ASTM E21: Elevated-temperature and low-temperature tensile testing when the plastic stage must be characterized away from ambient conditions.
Equipment and Instrumentation
Testing is performed on a closed-loop universal testing machine verified to ISO 7500-1 or ASTM E4, typically of force class 1 or better. Plastic strain is measured with a Class 0.5 or Class 1 extensometer per ISO 9513 or ASTM E83, attached to the gauge length; crosshead displacement alone is never used to determine yield or elongation because it includes machine and grip compliance. Video extensometers and digital image correlation systems permit non-contact measurement through necking and fracture and enable full-field strain mapping of the necked region. Gripping is by wedge, hydraulic, or pneumatic fixtures, and axial alignment is verified per ASTM E1012 to limit bending strain, which can otherwise distort yield-point behavior in plate and bar specimens.
Specimen Preparation
Specimens are machined with controlled surface finish and dimensional tolerances, avoiding heating, grinding burn, or cold work that could alter yield behavior. Decarburized or worked surface layers are removed where required. The original cross-sectional area S0 is measured at multiple positions, and the gauge length L0 is marked with scribing, ink, or a video reference. Specimen orientation relative to the rolling or working direction is recorded, since yield strength and elongation are strongly anisotropic in most rolled products.
Test Procedure and Strain Rate Control
The specimen is mounted, the extensometer is attached, and loading begins under controlled strain rate. ISO 6892-1 Method A applies closed-loop strain-rate control, commonly 0.00025 per second during determination of the yield parameters, followed by higher rates, such as 0.0025 to 0.0167 per second, through strain hardening, maximum force, and fracture. Method B controls stress rate in the elastic range and is retained for legacy specifications. Maintaining the specified rate through the yield region is essential because both ReH and ReL are strain-rate sensitive. The test proceeds to fracture, with the extensometer removed before rupture or a non-contact device retained to capture the complete curve. The fractured halves are then reassembled to measure the final gauge length and minimum necked area.
Data Reduction and Curve Analysis
Yield parameters are determined graphically from the recorded curve: ReH as the first peak before yield propagation, ReL as the lowest stress during plateau yielding, or Rp0.2 as the intersection of the curve with a line offset by 0.2 percent strain. Tensile strength is computed at maximum force, uniform elongation at the same point, and total elongation and reduction of area from post-fracture measurements. For n-value determination, true stress and true plastic strain pairs are calculated over the prescribed strain window between the end of yielding and maximum force, and a logarithmic linear regression of the Hollomon relation yields K and n, with regression quality criteria checked per ISO 10275. When required for simulation, the flow curve is extended beyond the onset of necking using measured neck geometry or inverse identification from digital image correlation data.
Applications
Plastic-stage test data are consumed throughout the product lifecycle, from alloy development to fitness-for-service assessment:
- Automotive engineering: uniform elongation, Agt, and n-values of advanced high-strength steels drive crash-energy absorption models, formability windows, and springback simulation for stamped body and chassis components.
- Aerospace: elongation and reduction-of-area minima are specified for fracture-critical parts, and plastic-stage data support material allowables and supplier qualification.
- Construction and seismic design: uniform elongation at maximum force is a mandatory ductility metric for seismic-grade reinforcement, with requirements such as Agt of at least 9 percent in standards including GB/T 1499.2, ensuring structures can deform plastically without losing load capacity.
- Metal forming: press shops use yield, n, and r values to set draw depths, blank holder forces, and die radii, and to predict thinning and localization in deep drawing and hydroforming.
- Pressure equipment and pipelines: strain-based design of pipelines and assessment of pressure vessels rely on measured uniform elongation and hardening curves to define allowable plastic strains and burst margins.
- Welding qualification: cross-weld plastic-stage testing quantifies softening in heat-affected zones and confirms joint ductility for procedure qualification.
- Additive manufacturing: elongation and reduction of area serve as key acceptance metrics in the qualification of additively manufactured alloys, where anisotropy and porosity strongly affect plastic behavior.
- Failure analysis and quality control: abnormally low uniform elongation or reduction of area reveals embrittlement, strain aging, incorrect heat treatment, or processing defects, supporting root-cause investigation and incoming inspection.
FAQ
What is the plastic stage of a stress-strain curve?
The plastic stage is the portion of the tensile curve beginning at the onset of yielding and extending to fracture, in which deformation is permanent. It includes yield phenomena, strain hardening up to the tensile strength, diffuse necking, and final rupture, and it supplies the ductility and hardening parameters used in design and forming.
How is yield strength determined when a material shows no distinct yield point?
For materials with a continuous transition from elastic to plastic behavior, the proof strength Rp0.2 is used. A line parallel to the elastic slope is offset by 0.2 percent strain, and the stress at its intersection with the stress-strain curve is reported as the yield strength.
What is the difference between uniform elongation and total elongation?
Uniform elongation (Ag or Agt) is the extension reached at maximum force, before deformation localizes in a neck. Total elongation at fracture (A) additionally includes the post-necking extension measured after the specimen is reassembled, so it is always larger and depends on gauge length.
Why does engineering stress decrease after the tensile strength is reached?
After maximum force, deformation localizes in a neck. Because engineering stress is calculated using the original cross-sectional area, the falling load appears as a decline in stress, even though the true stress acting on the shrinking neck cross-section continues to increase until fracture.
What is the strain hardening exponent and why is it important?
The n-value, determined per ASTM E646 or ISO 10275, quantifies how strongly a material hardens as it is deformed plastically. Higher n-values delay strain localization, extend uniform stretchability, and improve distribution of strain in formed parts, making the n-value a primary input for formability and crash simulation.
Which standards govern plastic-stage tensile testing?
The core methods are ISO 6892-1, EN ISO 6892-1, ASTM E8/E8M, GB/T 228.1, and JIS Z 2241, with ASTM E646 and ISO 10275 covering n-value determination and ASTM E517 and ISO 10113 covering the r-value. Elevated- and low-temperature variants are covered by ISO 6892-2, ISO 6892-3, and ASTM E21.
Can the plastic stage be tested at non-ambient temperatures?
Yes. Dedicated standards allow full plastic-stage characterization at elevated and sub-zero temperatures using furnaces or cryogenic chambers, which is essential for pressure components, cryogenic structures, and high-ervice-temperature alloys where yield and hardening behavior change significantly with temperature.
Why must an extensometer be used instead of crosshead displacement?
Crosshead displacement includes machine, load cell, and grip compliance as well as plastic deformation outside the gauge length, which distorts measured strains, especially around yield. A calibrated extensometer, or a non-contact optical system, measures strain directly on the gauge length and is required by the standards for yield and elongation determination.