Plasticity, Yielding & Ultimate Strength

Elastic and Plastic Behaviour

In the elastic range, a material deforms under load and returns to its original shape when the load is removed. Stress is proportional to strain, the relationship is single-valued, and no permanent deformation remains. Once the stress reaches the yield point, the material begins to deform plastically: permanent strain accumulates that is not recovered on unloading. The total strain at any point is the sum of the elastic component (recoverable) and the plastic component (permanent). Beyond yield, the stress–strain relationship is no longer linear, and the material behaviour becomes path-dependent — the current state depends on the entire loading history, not just the current load.

• Elastic strain is recoverable — remove the load and the deformation disappears
• Plastic strain is permanent — it remains after unloading
• Total strain = elastic strain + plastic strain
• Below yield, behaviour is linear (for metals) and path-independent
• Above yield, behaviour is non-linear and path-dependent — the loading history matters

Yield Criteria: When Does Plasticity Begin?

Yielding in a metal is governed by the state of stress, not by any single stress component. A uniaxial tensile specimen yields when the axial stress reaches the yield strength. But in a general three-dimensional stress state, there are six stress components, and the question is which combination of them triggers yielding. This is the role of a yield criterion — a scalar function of the stress tensor that, when it reaches a critical value, indicates the onset of plasticity. Two criteria dominate metal plasticity: von Mises and Tresca.

von Mises equivalent stress:

  σ_vm  =  √[ ½((σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²) ]

  where  σ₁, σ₂, σ₃ = principal stresses

  Yielding occurs when  σ_vm  ≥  σ_yield

  Tresca: yielding occurs when the maximum shear stress reaches a critical value:

  τ_max  =  ½·max(|σ₁−σ₂|, |σ₂−σ₃|, |σ₃−σ₁|)  ≥  τ_yield

Von Mises vs Tresca

The von Mises criterion is based on the distortion energy of the material — it predicts yielding when the energy associated with shape change reaches a critical value. The Tresca criterion is based on the maximum shear stress. For most ductile metals, von Mises agrees better with experimental data, and it has the practical advantage of being a smooth function (the yield surface has no corners), which is convenient for numerical plasticity algorithms. Tresca is simpler to visualise and more conservative — its hexagonal yield surface lies entirely inside the von Mises cylinder — but the corners of the Tresca surface require special handling in computation. In most engineering FEA, von Mises is the default; Tresca is used where a conservative bound is desired or where a particular code requires it.

• Von Mises: distortion-energy criterion; smooth cylindrical yield surface; generally best fit to metal test data
• Tresca: maximum-shear-stress criterion; hexagonal yield surface; conservative relative to von Mises
• The two criteria agree for uniaxial and equi-biaxial states; they differ most under pure shear, where von Mises predicts yield at about 0.577·σ_yield and Tresca at 0.5·σ_yield
• Most FEA solvers default to von Mises for metal plasticity; check which criterion your code or standard requires

Hardening: What Happens After First Yield?

When a metal is strained beyond its initial yield point and then unloaded, the yield stress upon reloading is generally higher than the original yield stress. The material has hardened — it has become stronger as a result of plastic deformation. How the yield surface evolves as plastic strain accumulates is described by a hardening rule, and the choice of rule materially affects the prediction of a structure under cyclic or non-proportional loading.

• Isotropic hardening: the yield surface expands uniformly as plastic strain accumulates; the yield strength in tension and compression increase equally. Simple, appropriate for monotonic loading, but predicts no Bauschinger effect and is wrong for significant load reversal.
• Kinematic hardening: the yield surface translates in stress space without changing size; models the Bauschinger effect where yield in compression after tensile plasticity occurs at a reduced magnitude. Appropriate for cyclic loading and low-cycle fatigue.
• Combined hardening: the yield surface both expands and translates; the most general and the most expensive to calibrate; used where both monotonic and cyclic behaviour must be captured.

Choosing a Hardening Model Honestly

The choice of hardening rule is not a fine detail — it determines whether the analysis captures the right physics for the loading you are analysing. An isotropic hardening model used on a component that sees significant load reversal will predict yield strengths that are too high in the reversed direction, because it cannot represent the Bauschinger effect. A kinematic hardening model used on a component under monotonic loading to large strains will miss the strengthening that isotropic hardening would capture. The honest approach is to match the hardening model to the loading character: monotonic loading permits isotropic; cyclic or reversed loading demands kinematic or combined; and if you do not know which applies, you do not know enough to trust the plasticity result.

Plasticity models need material data that goes beyond the yield point: the hardening curve, the cyclic response, and the parameters that calibrate the chosen hardening rule. A solver with a sophisticated plasticity model fed only with a yield stress and a Young's modulus is not performing a meaningful plastic analysis. Garbage in, garbage out applies to constitutive models as much as to meshing.

Local Yielding, Redistribution and the Plastic Hinge

When a ductile structure is loaded beyond the point at which the most highly stressed location first reaches yield, the behaviour depends on whether the load can find an alternative path. At the yielded location, the stress cannot increase much further (for an elastic–perfectly-plastic material, it cannot increase at all), but the strain can continue to grow. As the yielded region spreads, the stiffness of that region drops, and the load that would otherwise have been carried there is redirected into adjacent material that has not yet yielded. This is load redistribution, and it is the mechanism by which a ductile structure can carry a load far exceeding its first-yield load.

• First yield: the single most highly stressed point reaches the yield criterion
• Redistribution: load sheds from the yielded region into adjacent elastic material
• Plastic hinge: in a beam in bending, the yielded zone spreads through the section until the entire section is plastic and can no longer carry additional moment
• Limit load: the load at which the structure has enough plastic hinges (or enough yielded area) to form a mechanism and collapse
• Ultimate load: the maximum load the structure can carry, which may be the limit load or may be governed by material rupture or instability

YIELDING CAN REDISTRIBUTE LOAD — BUT ONLY IF THE STRUCTURE HAS SOMEWHERE FOR THE LOAD TO GO

The gap between first yield and collapse can be large or it can be zero. It depends on redundancy, load path, and the failure mode of the section. A statically determinate beam in bending reaches its plastic moment capacity at one section and forms a mechanism: one plastic hinge is enough, and the limit load is not much above the first-yield load. A continuous beam over multiple supports requires several plastic hinges to form a mechanism, so the limit load is well above first yield. A thin plate in tension has nowhere for the load to redistribute to — once the entire cross-section has yielded, the section is at its full plastic capacity and the next increment of load causes rupture. The engineer must understand not just that redistribution occurs, but whether the specific structure and load path permit it.

• A statically determinate structure offers no redistribution reserve — one hinge or one yielded section can form a mechanism
• A redundant structure (multiple load paths, continuity, multiple supports) offers redistribution reserve
• A section in uniform tension reaches its full plastic capacity when the entire section yields — there is no further redistribution within the section
• A section in bending redistributes from the extreme fibres inward until the full plastic moment is developed
• If the load path is such that yielding triggers instability (local buckling of the compression flange of a plastic hinge), redistribution is cut short

Redistribution is not a free margin. It requires the structure to accept permanent plastic strain, the load to be maintainable as the structure deforms, and the adjacent material to be able to carry the redirected load without itself yielding or buckling. In a brittle material, in a structure prone to instability, or under a load case that cannot tolerate permanent deformation, redistribution cannot be relied upon. Document the basis on which redistribution is claimed.

Limit Load and Ultimate Load

The limit load is the load at which a perfectly plastic structure forms a mechanism and can no longer sustain additional load. It is a lower bound on the actual ultimate load only if the material has sufficient ductility to allow the redistribution without rupture and the structure does not lose stability first. The ultimate load is the maximum load the structure actually carries, which may coincide with the limit load, may be lower (if instability or rupture intervenes), or may be higher (if strain hardening is significant). In practice, the ultimate load is found by a non-linear analysis that includes both material non-linearity and geometric non-linearity, run until the solver can no longer find equilibrium — the limit point of the load–deflection curve.

Engineering Stress–Strain vs True Stress–Strain

The stress–strain curve produced by a tensile test is usually reported in engineering terms: stress is force divided by the original cross-sectional area, and strain is elongation divided by the original gauge length. This is convenient and standard, but it becomes inaccurate once necking begins. As the specimen deforms, the cross-section reduces, so the true stress (force divided by current area) is higher than the engineering stress. The true strain (logarithmic) also diverges from the engineering strain at large deformations. For analyses that stay in the small-strain regime, engineering data is adequate. For large-strain problems — metal forming, crash, severe plastic deformation — the solver needs true stress–strain data, and the engineer must convert or supply it correctly.

• Engineering stress = F / A₀ (original area); engineering strain = ΔL / L₀ (original length)
• True stress = F / A (current area); true strain = ln(L / L₀) (logarithmic)
• The two coincide at small strains and diverge significantly after necking
• Most structural FEA uses engineering stress–strain data and is valid for strains up to a few per cent
• Large-strain analyses require true stress–strain data — supplying engineering data to a large-strain solver overstates ductility

Ductile Failure and Instability

Even in a ductile material, the path from first yield to ultimate load is not monotonic or guaranteed. As plastic deformation accumulates, the structure may lose stability: a compression flange that has yielded may buckle locally; a tension member that has yielded may neck and rupture; a bent member may form a plastic hinge and then lose lateral support. The interaction of plasticity with stability is one of the reasons that an elastic–perfectly-plastic limit load analysis can be unconservative: it assumes the structure can develop its full plastic capacity without losing stability, which is not always true. A non-linear analysis that includes both geometric and material non-linearity is the only reliable way to capture this interaction.

Stress–Strain Data Requirements

A plasticity analysis is only as good as the material data that feeds it. The minimum data for an elastic–plastic analysis is the yield stress and the hardening curve. For cyclic loading, the cyclic stress–strain curve and the hardening parameters are needed. The data should be at the appropriate temperature, for the appropriate material condition (annealed, work-hardened, heat-treated), and from the appropriate product form (sheet, plate, forging, casting) — because all of these affect the yield and hardening behaviour. Using a generic handbook curve for a material in a condition it was not tested in is a common and frequently unrecognised source of error.

• Yield stress — at the relevant temperature and material condition
• Hardening curve — stress vs plastic strain, beyond yield, up to the strain range expected in the analysis
• Cyclic data — if the loading is cyclic or involves load reversal
• Material condition — annealed, T6, work-hardened, etc.; do not mix conditions
• Product form — sheet, plate, bar, forging, casting; properties differ by form
• Statistical basis — nominal, A-basis, B-basis, or design allowable as required by the methodology

Escalating From First Yield to Limit-Load or Collapse Assessment

When acceptance depends on post-yield reserve, the analysis must move beyond an elastic peak-stress comparison. The relevant question becomes whether plastic redistribution remains stable and whether the structure retains adequate load-carrying capacity before a gross plastic mechanism, excessive deformation, fracture or instability develops. That assessment needs suitable true stress–strain data, an appropriate hardening model, realistic contact and boundary conditions, and a defined criterion for identifying limit load. The dedicated nonlinear and structural-integrity articles provide the canonical methods; this article remains the bridge between elastic static strength and those higher-fidelity assessments.

• Material Non-linearity & Plasticity
• Plastic Collapse & Limit Load Assessment
• Non-linear Buckling & Collapse

Key takeaways

• Elastic strain is recoverable; plastic strain is permanent. Above yield, material behaviour is non-linear and path-dependent.
• Von Mises (distortion energy) is the default yield criterion for ductile metals; Tresca (maximum shear) is simpler and more conservative but less accurate.
• The hardening rule — isotropic, kinematic, or combined — must match the loading character: monotonic loading permits isotropic; cyclic or reversed loading demands kinematic or combined.
• Local yielding does not equal structural collapse. A ductile, redundant structure can redistribute load and carry far more than its first-yield load — but only if the load path and stability permit it.
• The limit load is the mechanism-forming load of a perfectly plastic structure; the ultimate load may be lower if instability or rupture intervenes, and must be found by a non-linear analysis.
• A plasticity analysis is only as good as its material data: yield stress, hardening curve, cyclic response, temperature, material condition and product form must all match the application.